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We introduce a generative model that can at test-time sample any arbitrary subset of video frames conditioned on any other subset and present an architecture adapted for this purpose. Doing so allows us to efficiently compare and optimize a variety of schedules for the order in which frames in a long video are sampled and use selective sparse and long-range conditioning on previously sampled frames. We demonstrate improved video modeling over prior work on a number of datasets and sample temporally coherent videos over 25 minutes in length. We additionally release a new video modeling dataset and semantically meaningful metrics based on videos generated in the CARLA autonomous driving simulator. + +# 1 Introduction + +Generative modeling of photo-realistic videos is at the frontier of what is possible with deep learning on currently-available hardware. Although related work has demonstrated modeling of short photorealistic videos (e.g. 30 frames [36], 48 frames [6] or 64 frames [16]), generating longer videos that are both coherent and photo-realistic remains an open challenge. A major difficulty is scaling: photorealistic image generative models [4, 8] are already close to the memory and processing limits of modern hardware. A long video is at very least a concatenation of many photorealistic frames, implying resource requirements, long-range coherence notwithstanding, that scale with frame count. + +Attempting to model such long-range coherence makes the problem harder still, especially because in general every frame can have statistical dependencies on other frames arbitrarily far back in the video. Unfortunately fixed-lag autoregressive models impose unrealistic conditional independence assumptions (the next frame being independent of frames further back in time than the autoregressive lag is problematic for generating videos with long-range coherence). And while deep generative models based on recurrent neural networks (RNN) theoretically impose no such conditional independence assumptions, in practice they must be trained over short sequences [12, 26] or with truncated gradients [31]. Despite this, some RNN-based video generative models have demonstrated longer-range coherence, albeit without yet achieving convincing photorealistic video generation [26, 3, 7, 20, 2]. + +In this work we embrace the fact that finite architectures will always impose conditional independences. The question we ask is: given an explicit limit $K$ on the number of video frames we can jointly model, how can we best allocate these frames to generate a video of length $N > K ?$ One option is to use the previously-described autoregressive model but, if $K = N / 4$ , we could instead follow Ho et al. [16] by training two models: one which first samples every 4th frame in the video, and another which (in multiple stages) infills the remaining frames conditioned on those. To enable efficient exploration of the space of such sampling schemes, we propose a flexible architecture based on the denoising diffusion probabilistic model (DDPM) framework. This can sample any subset of video frames conditioned on observed values of any other subset of video frames. It therefore lets us explore a wide variety of previously untested sampling schemes while being easily repurposed for different generation tasks such as unconditional generation, video completion, and generation of videos of different lengths. Since our model can be flexibly applied to sample any frames given any others we call it a Flexible Diffusion Model, or FDM. + +![](images/d3896c398df1e076a51acfb5819f046231ccb319e84ddbaf5049df13dc706653.jpg) +Figure 1: A long video (25 minutes, or approximately 15 000 frames) generated by FDM for each of CARLA Town01 and MineRL, conditioned on 500 and 250 prior frames respectively. We show blocks of frames from three points within each video, starting from the final observed frame on the left. Blocks are marked with the time elapsed since the last observation and frames within them are one second apart. We observe no degradation in sample quality even after $> 1 5 0 0 0$ frames. + +Contributions (1) At the highest level, we claim to have concurrently developed one of the first denoising diffusion probabilistic model (DDPM)-based video generative models [16, 40]. To do so we augment a previously-used DDPM image architecture [15, 22] with a temporal attention mechanism including a novel relative (frame) position encoding network. (2) The principal contribution of this paper, regardless, is a “meta-learning” training objective that encourages learning of a video generative model that can (a) be flexibly conditioned on any number of frames (up to computational resource constraints) at any time in the past and future and (b) be flexibly marginalized (to achieve this within computational resource constraints). (3) We demonstrate that our model can be used to efficiently explore the space of resource constrained video generation schemes, leading to improvements over prior work on several long-range video modeling tasks. (4) Finally, we release a new autonomous driving video dataset along with a new video generative model performance metric that captures semantics more directly than the visual quality and comparison metrics currently in widespread use. + +# 2 Sampling long videos + +Our goal in this paper is to sample coherent photo-realistic videos v with thousands of frames (see Fig. 1). To sample an arbitrarily long video with a generative model that can sample or condition on only a small number of frames at once, we must use a sequential procedure. The simplest example of this is an autoregressive scheme, an example of which is shown in Fig. 2a for a video completion task. In this example it takes seven stages to sample a complete video, in that we must run the generative model’s sampling procedure seven times. At each stage three frames are sampled conditioned on the immediately preceding four frames. This scheme is appealing for its simplicity but imposes a strong assumption that, given the set of four frames that are conditioned on at a particular stage, all frames that come afterwards are conditionally independent of all frames that came before. This restriction can be partially ameliorated with the sampling scheme shown in Fig. 2b where, in the first three stages, every second frame is sampled and then, in the remaining four stages, the remaining frames are infilled. One way to implement this would be to train two different models operating at the two different temporal resolutions. In the language of Ho et al. [16], who use a similar approach, sampling would be carried out in the first three stages by a “frameskip-2” model and, in the remaining stages, by a “frameskip-1” model. Both this approach and the autoregressive approach are examples of what we call sampling schemes. More generally, we characterize a sampling scheme as a sequence of tuples $[ ( \mathcal { X } _ { s } , \mathcal { Y } _ { s } ) ] _ { s = 1 } ^ { \bar { S } }$ , each containing a vector $\mathcal { X } _ { s }$ of indices of frames to sample and a vector $\mathcal { { D } } _ { s }$ of indices of frames to condition on for stages $s = 1 , \ldots , S$ . + +Algorithm 1 Sample a video v given a sampling scheme $[ ( \mathcal { X } _ { s } , \mathcal { Y } _ { s } ) ] _ { s = 1 } ^ { S }$ . For unconditional generation, the input v can be a tensor of zeros. For conditional generation, the observed input frames should contain their observed values. + +![](images/754ba6c3882c936adc15285b6eb3db96a76d55df22201eb208337df69c833391.jpg) + +![](images/600712d51e1e42029bc1b4c4cd2cbf9b83301ebb8fae3df91ba6c5a7c1851508.jpg) +Figure 2: Sampling schemes to complete a video of length $N = 3 0$ conditioned on the first 10 frames, with access to at most $K = 7$ frames at a time. Each stage $s$ of the sampling procedure is represented by one row in the figure, going from top to bottom. Within each subfigure, one column represents one frame of the video, from frame one on the left to frame 30 on the right. At each stage, the values of frames marked in blue are sampled conditioned on the (observed or previously sampled) values of frames marked in red; frames marked in gray are ignored; and frames marked in white are yet to be sampled. For every sampling scheme, all video frames have been sampled after the final row. + +Algorithm 1 lays out how such a sampling scheme is used to sample a video. If the underlying generative model is trained specifically to model sequences of consecutive frames, or sequences of regularly-spaced frames, then the design space for sampling schemes compatible with these models is severely constrained. In this paper we take a different approach. We design and train a generative model to sample any arbitrarily-chosen subset of video frames conditioned on any other subset and train it using an entirely novel distribution of such tasks. In short, our model is trained to generate frames for any choice of $\mathcal { X }$ and $\mathcal { V }$ . The only constraint we impose on our sampling schemes is therefore a computational consideration that $| \mathcal { X } _ { s } | + | \mathcal { V } _ { s } | \le K$ for all $s$ but, to generate meaningful videos, any valid sampling scheme must also satisfy two more constraints: (1) all frames are sampled at at least one stage and (2) frames are never conditioned upon before they are sampled. + +Such a flexible generative model allows us to explore and use sampling schemes like those in Fig. 2c and Fig. 2d. We find in our experiments that the best video sampling scheme is dataset dependent. Accordingly, we have developed methodology to optimize such sampling schemes in a dataset dependent way, leading to improved video quality as measured by the Fréchet Video Distance [33] among other metrics. We now review conditional DDPMs (Section 3), before discussing the FDM’s architecture, the specific task distribution used to train it, and the choice and optimization of sampling schemes in Section 4. + +# 3 A review of conditional denoising diffusion probabilistic models + +Denoising diffusion probabilistic models, or DDPMs [28, 15, 22, 30], are a class of generative model for data $\mathbf { x }$ , which throughout this paper will take the form of a 4-dimensional tensor representing multiple video frames. We will describe the conditional extension [32], in which the modeled $\mathbf { x }$ is conditioned on observations $\mathbf { y }$ . DDPMs simulate a diffusion process which transforms $\mathbf { x }$ to noise, and generate data by learning the probabilistic inverse of the diffusion process. The diffusion process happens over timesteps $0 , \ldots , T$ such that $\mathbf { x } _ { 0 } = \mathbf { x }$ is data without noise, $\mathbf { x } _ { 1 }$ has a very small amount of noise added, and so on until $\mathbf { x } _ { T }$ is almost independent of $\mathbf { x } _ { \mathrm { 0 } }$ and approximates a random sample from a unit Gaussian. In the diffusion process we consider, the distribution over $\mathbf { x } _ { t }$ depends only on $\mathbf { x } _ { t - 1 }$ : + +$$ +\begin{array} { r } { q ( \mathbf x _ { t } | \mathbf x _ { t - 1 } ) = \mathcal N ( \mathbf x _ { t } ; \sqrt { \alpha _ { t } } \mathbf x _ { t - 1 } , ( 1 - \alpha _ { t } ) \mathbf I ) . } \end{array} +$$ + +Hyperparameters $\alpha _ { 1 } , \ldots , \alpha _ { T }$ are chosen to all be close to but slightly less than 1 so that the amount of noise added at each step is small. The combination of this diffusion process and a data distribution + +![](images/3fc8bc3620910017f4fb64feeca7720f8dbd1c2dc54a5bb68290e6287af2087d.jpg) +Figure 3: Left: Our DDPM iteratively transforms Gaussian noise $\mathbf { x } _ { T }$ to video frames $\mathbf { x } _ { \mathrm { 0 } }$ (shown with blue borders), conditioning on observed frames $\mathbf { y }$ (red borders) at every step. Right: The U-net architecture used within each DDPM step. It computes $\epsilon _ { \theta } ( \mathbf x _ { t } , \mathbf y , t )$ , with which the Gaussian transition $p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } )$ is parameterized. + +$q ( \mathbf { x } _ { 0 } , \mathbf { y } )$ (recalling that $\mathbf { x } _ { 0 } = \mathbf { x }$ ) defines the joint distribution + +$$ +q ( \mathbf x _ { 0 : T } , \mathbf y ) = q ( \mathbf x _ { 0 } , \mathbf y ) \prod _ { t = 1 } ^ { T } q ( \mathbf x _ { t } | \mathbf x _ { t - 1 } ) . +$$ + +DDPMs work by “inverting” the diffusion process: given values of $\mathbf { x } _ { t }$ and $\mathbf { y }$ a neural network is used to parameterize $p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } )$ , an approximation of $q ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } )$ . This neural network lets us draw samples of $\mathbf { x } _ { \mathrm { 0 } }$ by first sampling $\mathbf { x } _ { T }$ from a unit Gaussian (recall that the diffusion process was chosen so that $q ( \mathbf { x } _ { T } )$ is well approximated by a unit Gaussian), and then iteratively sampling $\mathbf { x } _ { t - 1 } \sim p _ { \theta } ( \cdot | \mathbf { x } _ { t } , \mathbf { y } )$ for $t = T , T - 1 , \dots , 1$ . The joint distribution of sampled $\mathbf { x } _ { \mathrm { 0 : } T }$ given $\mathbf { y }$ is + +$$ +p _ { \theta } ( \mathbf { x } _ { 0 : T } | \mathbf { y } ) = p ( \mathbf { x } _ { T } ) \prod _ { t = 1 } ^ { T } p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } ) +$$ + +where $p ( \mathbf { x } _ { T } )$ is a unit Gaussian that does not depend on $\theta$ . Training the conditional DDPM therefore involves fitting $p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } )$ to approximate $q ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } )$ for all choices of $t$ , $\mathbf { x } _ { t }$ , and $\mathbf { y }$ . + +Several observations have been made in recent years which simplify the learning of $p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } )$ . Sohl-Dickstein et al. [28] showed that when $\alpha _ { t }$ is close to 1, $p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } )$ is approximately Gaussian [28]. Furthermore, Ho et al. [15] showed that this Gaussian’s variance can be modeled well with a non-learned function of $t$ , and that a good estimate of the Gaussian’s mean can be obtained from a “denoising model” as follows. Given data $\mathbf { x } _ { \mathrm { 0 } }$ and unit Gaussian noise $\epsilon$ , the denoising model (in the form of a neural network) is fed “noisy” data $\mathbf { x } _ { t } : = \sqrt { \tilde { \alpha } _ { t } } \mathbf { x } _ { 0 } + \sqrt { 1 - \tilde { \alpha } _ { t } } \epsilon$ and trained to recover $\epsilon$ via a mean squared error loss. The parameters $\begin{array} { r } { \tilde { \alpha } _ { t } : = \prod _ { i = 1 } ^ { t } \alpha _ { i } } \end{array}$ are chosen to ensure that the marginal distribution of $\mathbf { x } _ { t }$ given $\mathbf { x } _ { \mathrm { 0 } }$ is $q ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } )$ as derived from Eq. (1). Given a weighting function $\lambda ( t )$ , the denoising loss is + +$$ +\mathcal { L } ( \theta ) = \mathbb { E } _ { q ( \mathbf { x } _ { 0 } , \mathbf { y } , \epsilon ) } \left[ \sum _ { t = 1 } ^ { T } \lambda ( t ) | | \epsilon - \epsilon _ { \theta } ( \mathbf { x } _ { t } , \mathbf { y } , t ) | | _ { 2 } ^ { 2 } \right] \quad \mathrm { w i t h } \quad \mathbf { x } _ { t } = \sqrt { \tilde { \alpha } _ { t } } \mathbf { x } _ { 0 } + \sqrt { 1 - \tilde { \alpha } _ { t } } \epsilon . +$$ + +The mean of $p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } , \mathbf { y } )$ is obtained from the denoising model’s output $\epsilon _ { \theta } ( \mathbf x _ { t } , \mathbf y , t )$ as $\begin{array} { r } { \frac { 1 } { \alpha _ { t } } \mathbf { x } _ { t } - \frac { 1 - \alpha _ { t } } { \sqrt { 1 - \tilde { \alpha } _ { t } } } \epsilon _ { \theta } \big ( \mathbf { x } _ { t } , \mathbf { y } , t \big ) } \end{array}$ . If the weighting function $\lambda ( t )$ is chosen appropriately, optimising Eq. (4) is equivalent to optimising a lower-bound on the data likelihood under $p _ { \theta }$ . In practice, simply setting $\lambda ( t ) : = 1$ for all $t$ can produce more visually compelling results in the image domain [15]. + +In our proposed method, as in Tashiro et al. [32], the shapes of $\mathbf { x } _ { \mathrm { 0 } }$ and $\mathbf { y }$ sampled from $q ( \cdot )$ vary. This is because we want to train a model which can flexibly adapt to e.g. varying numbers of observed frames. To map Eq. (4) to this scenario, note that both $\mathbf { x } _ { \mathrm { 0 } }$ and $\mathbf { y }$ implicitly contain information about which frames in the video they represent (via the index vectors $\mathcal { X }$ and $\mathcal { V }$ introduced in the previous section). This information is used inside the neural network $\epsilon _ { \theta } ( \mathbf x _ { t } , \mathbf y , t )$ so that interactions between frames can be conditioned on the distance between them (as described in the following section) and also to ensure that the sampled noise vector $\epsilon$ has the same shape as $\mathbf { x } _ { \mathrm { 0 } }$ . + +$: \mathcal { X } : = \{ \} ; \mathcal { Y } : = \{ \}$ +2: while True do +3: $n _ { \mathrm { g r o u p } } \sim$ UniformDiscrete $( 1 , K )$ +4: $s _ { \mathrm { g r o u p } } \sim \mathrm { L o g U n i f o r m } ( 1 , ( N - 1 ) / n _ { \mathrm { g r o u p } } )$ +5: $x _ { \mathrm { g r o u p } } \sim \mathrm { U n i f o r m } ( 0 , N - ( n _ { \mathrm { g r o u p } } - 1 ) \cdot s _ { \mathrm { g r o u p } } )$ +6: ogroup ∼ Bernoulli(0.5) +7: $\tilde { \mathcal { G } } : = \{ \lfloor x _ { \mathrm { g r o u p } } + s _ { \mathrm { g r o u p } } \cdot i \rfloor | i \in \{ 0 , \ldots , n _ { \mathrm { g r o u p } } - 1 \} \} \backslash \mathcal { X } \backslash \mathcal { Y }$ +8: if $| \mathcal { X } | + | \mathcal { Y } | + | \mathcal { G } | > K$ then +9: return set2vector $( \mathcal { X } )$ , set2vector $( { \mathcal { V } } )$ +10: else if $| \mathcal { X } | = 0$ or $o _ { \mathrm { g r o u p } } = 0$ then +11: $\mathcal { X } : = \mathcal { X } \cup \mathcal { G }$ +12: else +13: $y : = \mathcal { y } \cup \mathcal { G }$ + +![](images/5026656df62c5d1435c78fbe9221db88072c7b53140f572331883e0731269e40.jpg) +Figure 4: Left: Samples from $u ( \mathcal { X } , \mathcal { Y } )$ with video length $N = 3 0$ and limit $K = 1 0$ on the number of sampled indices. Each row shows one sample and columns map to frames, with frame 1 on the left and frame $N$ on the right. Blue and red denote latent and observed frames respectively. All other frames are ignored and shown as white. Right: Pseudocode for drawing these samples. The while loop iterates over a series of regularly-spaced groups of latent variables. Each group is parameterized by: the number of indices in it, $n _ { \mathrm { g r o u p } }$ ; the spacing between indices in it, $s _ { \mathrm { g r o u p } }$ ; the position of the first frame in it, $x _ { \mathrm { g r o u p } }$ , and an indicator variable for whether this group is observed, $O _ { \mathrm { { g r o u p } } }$ (which is ignored on line 10 if $\mathcal { X }$ is empty to ensure that the returned value of $\mathcal { X }$ is never empty). These quantities are sampled in a continuous space and then discretized to make a set of integer coordinates on line 7. The process repeats until a group is sampled which, if added to $\mathcal { X }$ or $\mathcal { V }$ , will cause the number of frames to exceed $K$ . That group is then discarded and $\mathcal { X }$ and $\mathcal { V }$ are returned as vectors. The FDM’s training objective forces it to work well for any $( \mathcal { X } , \mathcal { Y } )$ pair from this broad distribution. + +# 4 Training procedure and architecture + +Training task distribution Different choices of latent and observed indices $\mathcal { X }$ and $\mathcal { V }$ can be regarded as defining different conditional generation tasks. In this sense, we aim to learn a model which can work well on any task (i.e. any choice of $\mathcal { X }$ and $\mathcal { V }$ ) and so we randomly sample these vectors of indices during training. We do so with the distribution $u ( \mathcal { X } , \mathcal { Y } )$ described in Fig. 4. This provides a broad distribution covering many plausible test-time use cases while still providing sufficient structure to improve learning (see ablation in Section 6 and more details in Appendix C). To cope with constrained computational resources, the distribution is designed such that $\left| \mathcal { X } \right| + \left| \mathcal { V } \right|$ is upper-bounded by some pre-specified $K$ . Sampling from $q ( \mathbf { x } _ { 0 } , \mathbf { y } )$ in Eq. (4) is then accomplished by randomly selecting both a full training video $\mathbf { v }$ and indices $\mathcal { X } , \mathcal { Y } \sim u ( \cdot , \cdot )$ . We then extract the specified frames $\mathbf { x } = \mathbf { v } [ \mathcal { X } ]$ and $\mathbf { y } = \mathbf { v } [ \mathcal { V } ]$ (where we use $\mathbf { v } [ \mathcal { X } ]$ to denote the concatenation of all frames in $\mathbf { v }$ with indices in $\mathcal { X }$ and and $\mathbf { v } [ \mathcal { V } ]$ similarly). + +Architecture DDPM image models [15, 22] typically use a U-net architecture [24]. Its distinguishing feature is a series of spatial downsampling layers followed by a series of upsampling layers, and these are interspersed with convolutional res-net blocks [14] and spatial attention layers. Since we require an architecture which operates on 4-D video tensors rather than 3-D image tensors we add an extra frame dimension to its input, output and hidden state, resulting in the architecture shown on the right of Fig. 3. We create the input to this architecture as a concatenation $\mathbf x _ { t } \oplus \mathbf y$ , adding an extra input channel which is all ones for observed frames and all zeros for latent frames. For RGB video, the input shape is therefore ( $K$ , image height, image width, 4). Since the output should have the same shape as $\mathbf { x } _ { t }$ we only return outputs corresponding to the latent frames, giving output shape $\vert \mathcal { X } \vert$ , image height, image width, 3). We run all layers from the original model (including convolution, resizing, group normalization, and spatial attention) independently for each of the $K$ frames. To allow communication between the frames, we add a temporal attention layer after each spatial attention layer, described in more detail in the appendix. The spatial attention layer allows each spatial location to attend to all other spatial locations within the same frame, while the temporal attention layer allows each spatial location to attend to the same spatial location across all other frames. This combination of a temporal attention layer with a spatial attention layer is sometimes referred to as factorized attention [32, 16]. We found that, when using this architecture in conjunction with our meta-learning approach, performance could be improved by using a novel form of relative position encoding [27, 38]. This is included in our released source code but we leave its exposition to the supplementary material. + +Training batch padding Although the size $| \mathcal { X } \oplus \mathcal { Y } |$ of index vectors sampled from our training distribution is bounded above by $K$ , it can vary. To fit examples with various sizes of index vectors into the same batch, one option would be to pad them all to length $K$ with zeros and use masks so that the zeros cannot affect the loss. This, however, would waste computation on processing tensors of zeros. We instead use this computation to obtain a lower-variance loss estimate by processing additional data with “training batch padding”. This means that, for training examples where $| x \oplus y | < K$ , we concatenate frames uniformly sampled from a second video to increase the length along the frame-dimension to $K$ . Masks are applied to the temporal attention mechanisms so that frames from different videos cannot attend to eachother and the output for each is the same as that achieved by processing the videos in different batches. + +Sampling schemes Before describing the sampling schemes we explore experimentally, we emphasize that the relative performance of each is dataset-dependent and there is no single best choice. A central benefit of FDM is that it can be used at test-time with different sampling schemes without retraining. Our simplest sampling scheme, Autoreg, samples ten consecutives frames at each stage conditioned on the previous ten frames. Long-range is similar to Autoreg but conditions on only the five most recent frames as well as five of the original 36 observed frames. Hierarchy-2 uses a multi-level sampling procedure. In the first level, ten evenly spaced frames spanning the non-observed portion of the video are sampled (conditioned on ten observed frames). In the second level, groups of consecutive frames are sampled conditioned on the closest past and future frames until all frames have been sampled. Hierarchy-3 adds an intermediate stage where several groups of variables with an intermediate spacing between them are sampled. We include adaptive hierarchy-2, abbreviated Ad. hierarchy-2, as a demonstration of a sampling scheme only possible with a model like FDM. It samples the same frames at each stage as Hierarchy-2 but selects which frames to condition on adaptively at test-time with a heuristic aimed at collecting the maximally diverse set of frames, as measured by the pairwise LPIPS distance [41] between them. + +Optimizing sampling schemes An appealing alternative to the heuristic sampling schemes described in the previous paragraph would be to find a sampling scheme that is, in some sense, optimal for a given model and video generation/completion task. While it is unclear how to tractably choose which frames should be sampled at each stage, we suggest that the frames to condition on at each stage can be chosen by greedily optimizing the diffusion model loss which, as mentioned in Section 3, is closely related to the data log-likelihood. Given a fixed sequence of frames to sample at each stage $[ \mathcal { X } _ { s } ] _ { s = 1 } ^ { S }$ we select $\mathcal { \partial } _ { s }$ for each $s$ to minimize Eq. (4). This is estimated using a set of 100 training videos and by iterating over 10 evenly-spaced values of $t$ (which reduced variance relative to random sampling of $t$ ). See the appendix for further details. We create two optimized sampling schemes: one with the same latent indices as Autoreg, and one with the same latent indices as Hierarchy-2. We call the corresponding optimized schemes Opt. autoreg and Opt. hierarchy-2. + +# 5 CARLA Town01 Dataset + +In addition to our methodological contributions, we propose a new video-modeling dataset and benchmark which provides an interpretable measure of video completion quality. The dataset consists of videos of a car driving with a first-person view, produced using the CARLA autonomous driving simulator [9]. All 408 training and 100 test videos (of length 1000 frames and resolution $1 2 8 \times 1 2 8 )$ ) are produced within a single small town, CARLA’s Town01. As such, when a sufficiently expressive video model is trained on this dataset it memorizes the layout of the town and videos sampled from the model will be recognisable as corresponding to routes travelled within the town. We train a regression model in the form of a neural network which maps with high accuracy from any single rendered frame to $( x , y )$ coordinates representing the car’s position. Doing so allows us to plot the routes corresponding to sampled videos (see left of Fig. 5) and compute semantically-meaningful yet quantitative measures of the validity of these routes. Specifically, we compute histograms of speeds, where each speed is estimated by measuring the distance between the regressed locations for frames spaced ten apart (1 second at the dataset’s frame rate). Sampled videos occasionally “jump” between disparate locations in the town, resulting in unrealistically large estimated speeds. To measure the frequency of these events for each method, we compute the percentage of our point-speed estimates that exceed a threshold of $1 0 \mathrm { m / s }$ (the dataset was generated with a maximum simulated speed of $\mathrm { 3 m / s }$ ). We report this metric as the outlier percentage (OP). After filtering out these outliers, we compute the Wasserstein distance (WD) between the resulting empirical distribution and that of the original dataset, giving a measure of how well generated videos match the speed of videos in the dataset. We release the CARLA Town 01 dataset along with code and our trained regression model to allow future comparisons.2 + +![](images/91d03113bf431ff2ce8593f9dcf8c30de45d678fc436461bb3511e1f415c4d07.jpg) +Figure 5: Left: Map of the town featured in the CARLA Town01 dataset. We visualize two video completions by FDM by showing coordinates output by our regressor (discussed in Section 5) for each frame. Those corresponding to the initial 36 observed frames are shown in red and those for the 964 sampled frames are shown in blue. Right: For each completion, we show one of the initially observed frames followed by four of the sampled frames (at positions chosen to show the progression with respect to visible landmarks and marked by black dots on the map). The town’s landmarks are usually sampled with high-fidelity, which is key to allowing the regressor to produce a coherent trajectory on the left. However there are sometimes failures: a blue square near the top-right of the map shows where the video model “jumped” to a wrong location for a single frame. + +# 6 Experiments + +We perform our main comparisons on the video completion task. In keeping with Saxena et al. [26], we condition on the first 36 frames of each video and sample the remainder. We present results on three datasets: GQN-Mazes [10], in which videos are 300 frames long; MineRL Navigate [13, 26] (which we will from now on refer to as simply MineRL), in which videos are 500 frames long; and the CARLA Town01 dataset we release, for which videos are 1000 frames long. We train FDM in all cases with the maximum number of represented frames $K = 2 0$ . We host non-cherry-picked video samples (both conditional and unconditional) from FDM and all baselines online3. + +Comparison of sampling schemes The relative performance of different sampling schemes varies significantly between datasets as shown in Table 1. We report Fréchet Video Distances (FVDs) [33], a measure of how similar sampled completions are to the test set, on all datasets. In addition on GQN-Mazes we we report the accuracy metric [26], which classifies videos based on which rooms are visited and measures how often a completion is given the same class as the corresponding test video. For CARLA Town01 we report the previously described percentage outliers (PO) and Wasserstein distance (WD) metrics. + +![](images/20181a2ed912845712556e621993f86259d7ed3f838de77c7ab7cd3878c59243.jpg) +Figure 6: Speed distributions measured from sampled and ground-truth dataset videos. + +We can broadly consider the aforementioned sampling schemes as either being in the “autoregressive” family (Autoreg and Long-range) or in the “hierarchical” + +Table 1: Evaluation on video completion with various modes of our method along with several baselines from the literature. Error bars denote the standard error computed with 5 random seeds. Higher is better for the accuracy metric [26] and lower is better for all other metrics shown. + +
ModelSampling schemeGQN-MazesMineRLCARLA Town01
FVD AccuracyFVDFVDWDOP
CWVAE [26]CWVAE837±882.6± 0.51573±511610.66644.4
TATS [11]TATS163 ± 2.677.0± 0.8807±143291.64842.4
VDM [16]VDM66.7 ± 1.577.8± 0.5271±8.81690.50116.9
FDM (ours)Autoreg86.4± 5.269.6 ± 1.3281±102220.5790.51
Long-range64.5 ± 1.977.0 ± 1.4267 ±4.02130.6530.47
Hierarchy-253.1 ± 1.182.8± 0.7275±7.71200.3183.28
Hierarchy-353.7 ± 1.983.8 ± 1.1311± 6.81490.3634.53
Ad. hierarchy-255.0 ± 1.483.2 ± 1.3316 ±8.91170.3113.44
+ +family (the remainder). Those in the hierarchical family achieve significantly better FVDs [33] on GQN-Mazes. Our samples in the appendix suggest that this is related to the autoregressive methods “forgetting” the colors of walls after looking away from them for a short time. In contrast, for MineRL the autoregressive methods tend to achieve the best FVDs. This may relate to the fact that trajectories in MineRL tend to travel in straight lines through procedurally-generated “worlds”[13, 26], limiting the number of long-range dependencies. Finally on CARLA Town01 we notice qualitatively different behaviours from our autoregressive and hierarchical sampling schemes. The hierarchical sampling schemes have a tendency to occasionally lose coherence and “jump” to different locations in the town. This is reflected by higher outlier percentages (OP) in Table 1. On the other hand the autoregressive schemes often stay stationary for unrealistically long times at traffic lights. This is reflected in the histogram of speeds in Fig. 6, which has a larger peak around zero than the ground truth. The high variance of the sampling scheme’s relative performance over different datasets points to a strength of our method, which need only be trained once and then used to explore a variety of sampling schemes. Furthermore, we point out that the best FVDs in Table 1 on all datasets were obtained using sampling schemes that could not be implemented using models trained in prior work, or over evenly spaced frames. + +Comparison with baselines The related work most relevant to ours is the concurrent work of Ho et al. [16], who model 64-frame videos using two trained DDPMs. The first is a “frameskip-4” model trained to generate every fourth frame and the second is a “frameskip-1” model trained on sequences of nine consecutive frames and used to “fill in” the gaps between frames generated in the first stage. To compare against this approach, which we denote VDM, we train both a “frameskip-4” and a “frameskip-1” model with architectures identical to our own.4 Since VDM requires two trained DDPMs, we train it for more GPU-hours than FDM despite the fact that FDM is meta-learning over a far broader task distribution. We also compare against TATS [11], which embeds videos into a discrete latent space before modelling them with a transformers, and the clockwork VAE (CWVAE) [26], a VAE-based model specifically designed to maintain long-range dependencies within video. + +Both the diffusion-based methods, FDM and VDM, achieve significantly higher FVD scores than TATS and CWVAE. This may point toward the utility of diffusion models in general for modeling images and video. Table 1 also makes clear the main benefit of FDM over VDM: although there is no sampling scheme for FDM which always outperforms VDM, there is at least one sampling scheme that outperforms it on each dataset. This speaks to the utility of learning a flexible model like FDM that allows different sampling schemes to be experimented with after training. + +Optimized sampling schemes As mentioned in Section 4, another advantage of FDM is that it makes possible a model- and dataset-specific optimization procedure to determine on which frames to condition. Table 2 shows the results when this procedure is used to create sampling schemes for different datasets. In the first row we show results where the latent frames are fixed to be those of the Autoreg sampling scheme, and in the second row the latent frames are fixed to match those of Hierarchy-2. On two of the three datasets the best results in Table 1 are improved upon, showing the utility of this optimization procedure. + +Table 2: FVD scores for our sampling schemes with observed indices optimized offline as described in Section 4. We mark with an asterisk $( ^ { * } )$ the eight numbers which improve on the corresponding non-optimized sampling schemes and highlight in bold those that are better than any in Table 1. + +
GQN-MazesMineRLCARLA Town01
Sampling schemeFVDAccuracyFVDFVDWDOP
Opt. autoreg53.6±1.2*80.2±1.2*257±6.8*146*0.452*0.65
Opt. hierarchy-251.1 ± 1.3*84.6±0.7*320 ± 7.01240.3494.11*
+ +Comparison with training on a single task Training a network with our distribution over training tasks could be expected to lead to worse performance on a single task than training specifically for that task. To test whether this is the case, we train an ablation of FDM with training tasks exclusively of the type used in our Autoreg sampling scheme, i.e. “predict ten consecutive frames given the previous ten.” Tested with the Autoreg sampling scheme, it obtained an FVD of 82.0 on GQN-Mazes and 234 on MineRL. As expected given the specialization to a single task, this is better than when FDM is run with the Autoreg sampling scheme (obtaining FVDs of 86.4 and 281 respectively). + +Ablation on training task distribution To test how important our proposed structured training distribution is to FDM’s performance, we perform an ablation with a different task distribution that samples $\mathcal { X }$ and $\mathcal { V }$ from uniform distributions instead of our proposed structured task distribution We provide full details in the appendix, but report here that switching away form our structured training distribution made the FVD scores worse on all five tested sampling schemes on both GQN-Mazes and MineRL. The reduction in the average FVD was $3 1 \%$ on GQN-Mazes and $5 2 \%$ on MineRL. This implies that our structured training distribution has a significant positive effect. + +# 7 Related work + +Some related work creates conditional models by adapting the sampling procedure of an unconditional DDPM [30, 18, 21, 16]. These approaches require approximations and the more direct approach that we use (explcitly training a conditional DDPM) was shown to have benefits by Tashiro et al. [32]. We consider further comparison of these competing approaches to be outside the focus of this work, which is on modeling a small portion of video frames at a time, essentially performing marginalization in addition to conditioning. + +There are a number of approaches in the literature which use VAEs rather than DDPMs for video modelling. Babaeizadeh et al. [2] use a VAE model which predicts frames autoregressively conditioned on a global time-invariant latent variable. A related approach by Denton and Fergus [7] also uses a VAE with convolutional LSTM architectures in both the encoder and decoder. Unlike Babaeizadeh et al. [2] the prior is learned and a different latent variable is sampled for each frame. Babaeizadeh et al. [3] use a VAE with one set of latent variables per frame and inter-frame dependencies tracked by a two-layer LSTM. Their architecture intentionally overfits to the training data, which when coupled with image augmentations techniques achieves SOTA on various video prediction tasks. Kim et al. [20] use a variational RNN [5] with a hierarchical latent space that includes binary indicator variables which specify how the video is divided into a series of subsequences. Both Villegas et al. [35] and Wichers et al. [37] target long-term video prediction using a hierarchical variational LSTM architecture, wherein high-level features such as landmarks are predicted first, then decoded into low-level pixel space. The two approaches differ in that Villegas et al. [35] requires ground truth landmark labels, while [37] removes this dependence using an unsupervised adversarial approach. Fully GAN-based video models have also been proposed [1, 6] but generally suffer from “low quality frames or low number of frames or both” [1]. + +# 8 Discussion + +We have defined and empirically explored a new method for generating photorealistic videos with long-range coherence that respects and efficiently uses fixed, finite computational resources. Our approach outperforms prior work on long-duration video modeling as measured by quantitative and semantically meaningful metrics and opens up several avenues for future research. For one, similar to using DDPMs for image generation, our method is slow to sample from (it takes approximately 16 minutes to generate a 300 frame video on a GPU). Ideas for making sampling faster by decreasing the number of integration steps [25, 29, 39] could be applied to our video model. + +On a different note, consider the datasets on which our artifact was trained. In each there was a policy for generating the sequences of actions that causally led to the frame-to-frame changes in camera pose. In MineRL the video was generated by agents that were trained to explore novel Minecraft worlds to find a goal block approximately 64 meters away [13]. The CARLA data was produced by a camera attached to an agent driven by a low level proportional–integral–derivative controller following waypoints laid down by a high level planner that was given new, random location goals to drive to intermittently. In both cases our video model had no access to either the policy or the specific actions taken by these agents and, so, in a formal sense, our models integrate or marginalize over actions drawn from the stochastic policy used to generate the videos in the first place. Near-term future work could involve adding other modalities (e.g. audio) to FDM as well as explicitly adding actions and rewards, transforming our video generative model into a vision-based world model in the reinforcement learning sense [17, 19]. Furthermore, we point out that FDM trained on CARLA Town01 is in theory capable of creating 100-second videos conditioned on both the first and final frame. Doing so can be interpreted as running a “visual” controller which proposes a path between a current state and a specified goal. Preliminary attempts to run in FDM in this way yielded inconsistent results but we believe that this could be a fruitful direction for further investigation. + +# Acknowledgments + +We would like to thank Inverted AI, and especially Alireza Morsali, for generating the CARLA Town01 dataset. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada CIFAR AI Chairs Program, and the Intel Parallel Computing Centers program. Additional support was provided by UBC’s Composites Research Network (CRN), and Data Science Institute (DSI). This research was enabled in part by technical support and computational resources provided by WestGrid (www.westgrid.ca), Compute Canada (www.computecanada.ca), and Advanced Research Computing at the University of British Columbia (arc.ubc.ca). WH acknowledges support by the University of British Columbia’s Four Year Doctoral Fellowship (4YF) program. + +# References + +[1] Nuha Aldausari, Arcot Sowmya, Nadine Marcus, and Gelareh Mohammadi. Video generative adversarial networks: a review. ACM Computing Surveys (CSUR), 55(2):1–25, 2022. +[2] Mohammad Babaeizadeh, Chelsea Finn, Dumitru Erhan, Roy H Campbell, and Sergey Levine. Stochastic variational video prediction. arXiv preprint arXiv:1710.11252, 2017. +[3] Mohammad Babaeizadeh, Mohammad Taghi Saffar, Suraj Nair, Sergey Levine, Chelsea Finn, and Dumitru Erhan. Fitvid: Overfitting in pixel-level video prediction. arXiv preprint arXiv:2106.13195, 2021. +[4] Rewon Child. Very deep vaes generalize autoregressive models and can outperform them on images. arXiv preprint arXiv:2011.10650, 2020. +[5] Junyoung Chung, Kyle Kastner, Laurent Dinh, Kratarth Goel, Aaron C Courville, and Yoshua Bengio. A recurrent latent variable model for sequential data. 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Hierarchical long-term video prediction without supervision. In International Conference on Machine Learning, pages 6038–6046. PMLR, 2018. + +[36] Dirk Weissenborn, Oscar Täckström, and Jakob Uszkoreit. Scaling autoregressive video models. arXiv preprint arXiv:1906.02634, 2019. + +[37] Nevan Wichers, Ruben Villegas, Dumitru Erhan, and Honglak Lee. Hierarchical Long-term Video Prediction without Supervision. June 2018. doi: 10.48550/arXiv.1806.04768. + +[38] Kan Wu, Houwen Peng, Minghao Chen, Jianlong Fu, and Hongyang Chao. Rethinking and improving relative position encoding for vision transformer. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 10033–10041, 2021. + +[39] Zhisheng Xiao, Karsten Kreis, and Arash Vahdat. Tackling the generative learning trilemma with denoising diffusion GANs. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id $\equiv$ JprM0p-q0Co. + +[40] Ruihan Yang, Prakhar Srivastava, and Stephan Mandt. Diffusion probabilistic modeling for video generation. arXiv preprint arXiv:2203.09481, 2022. + +[41] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 586–595, 2018. + +# Checklist + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] +(b) Did you describe the limitations of your work? [Yes] +(c) Did you discuss any potential negative societal impacts of your work? [Yes] +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] + +3. If you ran experiments... + +(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] +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] +(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] See appendix. + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] +(b) Did you mention the license of the assets? [Yes] +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/dev/0RTJcuvHtIu/0RTJcuvHtIu_content_list.json b/parse/dev/0RTJcuvHtIu/0RTJcuvHtIu_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..018917366e80bcb7fecd47293f2017fcba1499ef --- /dev/null +++ b/parse/dev/0RTJcuvHtIu/0RTJcuvHtIu_content_list.json @@ -0,0 +1,1137 @@ +[ + { + "type": "text", + "text": "Flexible Diffusion Modeling of Long Videos ", + "text_level": 1, + "bbox": [ + 235, + 122, + 764, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "William Harvey, Saeid Naderiparizi, Vaden Masrani, Christian Weilbach, Frank Wood∗ ", + "bbox": [ + 191, + 200, + 805, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Computer Science University of British Columbia Vancouver, Canada {wsgh,saeidnp,vadmas,weilbach,fwood}@cs.ubc.ca ", + "bbox": [ + 302, + 217, + 697, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 306, + 535, + 323 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present a framework for video modeling based on denoising diffusion probabilistic models that produces long-duration video completions in a variety of realistic environments. We introduce a generative model that can at test-time sample any arbitrary subset of video frames conditioned on any other subset and present an architecture adapted for this purpose. Doing so allows us to efficiently compare and optimize a variety of schedules for the order in which frames in a long video are sampled and use selective sparse and long-range conditioning on previously sampled frames. We demonstrate improved video modeling over prior work on a number of datasets and sample temporally coherent videos over 25 minutes in length. We additionally release a new video modeling dataset and semantically meaningful metrics based on videos generated in the CARLA autonomous driving simulator. ", + "bbox": [ + 233, + 338, + 766, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 529, + 310, + 545 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative modeling of photo-realistic videos is at the frontier of what is possible with deep learning on currently-available hardware. Although related work has demonstrated modeling of short photorealistic videos (e.g. 30 frames [36], 48 frames [6] or 64 frames [16]), generating longer videos that are both coherent and photo-realistic remains an open challenge. A major difficulty is scaling: photorealistic image generative models [4, 8] are already close to the memory and processing limits of modern hardware. A long video is at very least a concatenation of many photorealistic frames, implying resource requirements, long-range coherence notwithstanding, that scale with frame count. ", + "bbox": [ + 174, + 559, + 825, + 657 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Attempting to model such long-range coherence makes the problem harder still, especially because in general every frame can have statistical dependencies on other frames arbitrarily far back in the video. Unfortunately fixed-lag autoregressive models impose unrealistic conditional independence assumptions (the next frame being independent of frames further back in time than the autoregressive lag is problematic for generating videos with long-range coherence). And while deep generative models based on recurrent neural networks (RNN) theoretically impose no such conditional independence assumptions, in practice they must be trained over short sequences [12, 26] or with truncated gradients [31]. Despite this, some RNN-based video generative models have demonstrated longer-range coherence, albeit without yet achieving convincing photorealistic video generation [26, 3, 7, 20, 2]. ", + "bbox": [ + 174, + 664, + 825, + 787 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work we embrace the fact that finite architectures will always impose conditional independences. The question we ask is: given an explicit limit $K$ on the number of video frames we can jointly model, how can we best allocate these frames to generate a video of length $N > K ?$ One option is to use the previously-described autoregressive model but, if $K = N / 4$ , we could instead follow Ho et al. [16] by training two models: one which first samples every 4th frame in the video, and another which (in multiple stages) infills the remaining frames conditioned on those. To enable efficient exploration of the space of such sampling schemes, we propose a flexible architecture based on the denoising diffusion probabilistic model (DDPM) framework. This can sample any subset of video frames conditioned on observed values of any other subset of video frames. It therefore lets us explore a wide variety of previously untested sampling schemes while being easily repurposed for different generation tasks such as unconditional generation, video completion, and generation of videos of different lengths. Since our model can be flexibly applied to sample any frames given any others we call it a Flexible Diffusion Model, or FDM. ", + "bbox": [ + 174, + 795, + 825, + 877 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d3896c398df1e076a51acfb5819f046231ccb319e84ddbaf5049df13dc706653.jpg", + "image_caption": [ + "Figure 1: A long video (25 minutes, or approximately 15 000 frames) generated by FDM for each of CARLA Town01 and MineRL, conditioned on 500 and 250 prior frames respectively. We show blocks of frames from three points within each video, starting from the final observed frame on the left. Blocks are marked with the time elapsed since the last observation and frames within them are one second apart. We observe no degradation in sample quality even after $> 1 5 0 0 0$ frames. " + ], + "image_footnote": [], + "bbox": [ + 176, + 89, + 821, + 210 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 319, + 825, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions (1) At the highest level, we claim to have concurrently developed one of the first denoising diffusion probabilistic model (DDPM)-based video generative models [16, 40]. To do so we augment a previously-used DDPM image architecture [15, 22] with a temporal attention mechanism including a novel relative (frame) position encoding network. (2) The principal contribution of this paper, regardless, is a “meta-learning” training objective that encourages learning of a video generative model that can (a) be flexibly conditioned on any number of frames (up to computational resource constraints) at any time in the past and future and (b) be flexibly marginalized (to achieve this within computational resource constraints). (3) We demonstrate that our model can be used to efficiently explore the space of resource constrained video generation schemes, leading to improvements over prior work on several long-range video modeling tasks. (4) Finally, we release a new autonomous driving video dataset along with a new video generative model performance metric that captures semantics more directly than the visual quality and comparison metrics currently in widespread use. ", + "bbox": [ + 174, + 436, + 825, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Sampling long videos ", + "text_level": 1, + "bbox": [ + 174, + 627, + 382, + 645 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our goal in this paper is to sample coherent photo-realistic videos v with thousands of frames (see Fig. 1). To sample an arbitrarily long video with a generative model that can sample or condition on only a small number of frames at once, we must use a sequential procedure. The simplest example of this is an autoregressive scheme, an example of which is shown in Fig. 2a for a video completion task. In this example it takes seven stages to sample a complete video, in that we must run the generative model’s sampling procedure seven times. At each stage three frames are sampled conditioned on the immediately preceding four frames. This scheme is appealing for its simplicity but imposes a strong assumption that, given the set of four frames that are conditioned on at a particular stage, all frames that come afterwards are conditionally independent of all frames that came before. This restriction can be partially ameliorated with the sampling scheme shown in Fig. 2b where, in the first three stages, every second frame is sampled and then, in the remaining four stages, the remaining frames are infilled. One way to implement this would be to train two different models operating at the two different temporal resolutions. In the language of Ho et al. [16], who use a similar approach, sampling would be carried out in the first three stages by a “frameskip-2” model and, in the remaining stages, by a “frameskip-1” model. Both this approach and the autoregressive approach are examples of what we call sampling schemes. More generally, we characterize a sampling scheme as a sequence of tuples $[ ( \\mathcal { X } _ { s } , \\mathcal { Y } _ { s } ) ] _ { s = 1 } ^ { \\bar { S } }$ , each containing a vector $\\mathcal { X } _ { s }$ of indices of frames to sample and a vector $\\mathcal { { D } } _ { s }$ of indices of frames to condition on for stages $s = 1 , \\ldots , S$ . ", + "bbox": [ + 174, + 661, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithm 1 Sample a video v given a sampling scheme $[ ( \\mathcal { X } _ { s } , \\mathcal { Y } _ { s } ) ] _ { s = 1 } ^ { S }$ . For unconditional generation, the input v can be a tensor of zeros. For conditional generation, the observed input frames should contain their observed values. ", + "bbox": [ + 173, + 92, + 825, + 133 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/754ba6c3882c936adc15285b6eb3db96a76d55df22201eb208337df69c833391.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 176, + 137, + 825, + 218 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/600712d51e1e42029bc1b4c4cd2cbf9b83301ebb8fae3df91ba6c5a7c1851508.jpg", + "image_caption": [ + "Figure 2: Sampling schemes to complete a video of length $N = 3 0$ conditioned on the first 10 frames, with access to at most $K = 7$ frames at a time. Each stage $s$ of the sampling procedure is represented by one row in the figure, going from top to bottom. Within each subfigure, one column represents one frame of the video, from frame one on the left to frame 30 on the right. At each stage, the values of frames marked in blue are sampled conditioned on the (observed or previously sampled) values of frames marked in red; frames marked in gray are ignored; and frames marked in white are yet to be sampled. For every sampling scheme, all video frames have been sampled after the final row. " + ], + "image_footnote": [], + "bbox": [ + 181, + 236, + 818, + 286 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 1 lays out how such a sampling scheme is used to sample a video. If the underlying generative model is trained specifically to model sequences of consecutive frames, or sequences of regularly-spaced frames, then the design space for sampling schemes compatible with these models is severely constrained. In this paper we take a different approach. We design and train a generative model to sample any arbitrarily-chosen subset of video frames conditioned on any other subset and train it using an entirely novel distribution of such tasks. In short, our model is trained to generate frames for any choice of $\\mathcal { X }$ and $\\mathcal { V }$ . The only constraint we impose on our sampling schemes is therefore a computational consideration that $| \\mathcal { X } _ { s } | + | \\mathcal { V } _ { s } | \\le K$ for all $s$ but, to generate meaningful videos, any valid sampling scheme must also satisfy two more constraints: (1) all frames are sampled at at least one stage and (2) frames are never conditioned upon before they are sampled. ", + "bbox": [ + 173, + 421, + 825, + 560 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Such a flexible generative model allows us to explore and use sampling schemes like those in Fig. 2c and Fig. 2d. We find in our experiments that the best video sampling scheme is dataset dependent. Accordingly, we have developed methodology to optimize such sampling schemes in a dataset dependent way, leading to improved video quality as measured by the Fréchet Video Distance [33] among other metrics. We now review conditional DDPMs (Section 3), before discussing the FDM’s architecture, the specific task distribution used to train it, and the choice and optimization of sampling schemes in Section 4. ", + "bbox": [ + 174, + 566, + 825, + 664 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 A review of conditional denoising diffusion probabilistic models ", + "text_level": 1, + "bbox": [ + 174, + 686, + 733, + 705 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Denoising diffusion probabilistic models, or DDPMs [28, 15, 22, 30], are a class of generative model for data $\\mathbf { x }$ , which throughout this paper will take the form of a 4-dimensional tensor representing multiple video frames. We will describe the conditional extension [32], in which the modeled $\\mathbf { x }$ is conditioned on observations $\\mathbf { y }$ . DDPMs simulate a diffusion process which transforms $\\mathbf { x }$ to noise, and generate data by learning the probabilistic inverse of the diffusion process. The diffusion process happens over timesteps $0 , \\ldots , T$ such that $\\mathbf { x } _ { 0 } = \\mathbf { x }$ is data without noise, $\\mathbf { x } _ { 1 }$ has a very small amount of noise added, and so on until $\\mathbf { x } _ { T }$ is almost independent of $\\mathbf { x } _ { \\mathrm { 0 } }$ and approximates a random sample from a unit Gaussian. In the diffusion process we consider, the distribution over $\\mathbf { x } _ { t }$ depends only on $\\mathbf { x } _ { t - 1 }$ : ", + "bbox": [ + 173, + 720, + 825, + 847 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dce8a1e7e71cd9f590aa26dda40e122c7848f76674f30f8b852c3c2dc30ec66a.jpg", + "text": "$$\n\\begin{array} { r } { q ( \\mathbf x _ { t } | \\mathbf x _ { t - 1 } ) = \\mathcal N ( \\mathbf x _ { t } ; \\sqrt { \\alpha _ { t } } \\mathbf x _ { t - 1 } , ( 1 - \\alpha _ { t } ) \\mathbf I ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 849, + 643, + 867 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hyperparameters $\\alpha _ { 1 } , \\ldots , \\alpha _ { T }$ are chosen to all be close to but slightly less than 1 so that the amount of noise added at each step is small. The combination of this diffusion process and a data distribution ", + "bbox": [ + 173, + 882, + 821, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/3fc8bc3620910017f4fb64feeca7720f8dbd1c2dc54a5bb68290e6287af2087d.jpg", + "image_caption": [ + "Figure 3: Left: Our DDPM iteratively transforms Gaussian noise $\\mathbf { x } _ { T }$ to video frames $\\mathbf { x } _ { \\mathrm { 0 } }$ (shown with blue borders), conditioning on observed frames $\\mathbf { y }$ (red borders) at every step. Right: The U-net architecture used within each DDPM step. It computes $\\epsilon _ { \\theta } ( \\mathbf x _ { t } , \\mathbf y , t )$ , with which the Gaussian transition $p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } )$ is parameterized. " + ], + "image_footnote": [], + "bbox": [ + 210, + 88, + 785, + 279 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$q ( \\mathbf { x } _ { 0 } , \\mathbf { y } )$ (recalling that $\\mathbf { x } _ { 0 } = \\mathbf { x }$ ) defines the joint distribution ", + "bbox": [ + 173, + 364, + 573, + 380 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8995c3cea64e569618e46fcb6860c8c4f159411ad6a0ebb7097797ae5c3c9c15.jpg", + "text": "$$\nq ( \\mathbf x _ { 0 : T } , \\mathbf y ) = q ( \\mathbf x _ { 0 } , \\mathbf y ) \\prod _ { t = 1 } ^ { T } q ( \\mathbf x _ { t } | \\mathbf x _ { t - 1 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 381, + 625, + 424 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "DDPMs work by “inverting” the diffusion process: given values of $\\mathbf { x } _ { t }$ and $\\mathbf { y }$ a neural network is used to parameterize $p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )$ , an approximation of $q ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )$ . This neural network lets us draw samples of $\\mathbf { x } _ { \\mathrm { 0 } }$ by first sampling $\\mathbf { x } _ { T }$ from a unit Gaussian (recall that the diffusion process was chosen so that $q ( \\mathbf { x } _ { T } )$ is well approximated by a unit Gaussian), and then iteratively sampling $\\mathbf { x } _ { t - 1 } \\sim p _ { \\theta } ( \\cdot | \\mathbf { x } _ { t } , \\mathbf { y } )$ for $t = T , T - 1 , \\dots , 1$ . The joint distribution of sampled $\\mathbf { x } _ { \\mathrm { 0 : } T }$ given $\\mathbf { y }$ is ", + "bbox": [ + 173, + 425, + 825, + 496 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b454d0380db4bb7fd3af07540825b197bbc1e8138044742368a5bd96a5555403.jpg", + "text": "$$\np _ { \\theta } ( \\mathbf { x } _ { 0 : T } | \\mathbf { y } ) = p ( \\mathbf { x } _ { T } ) \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )\n$$", + "text_format": "latex", + "bbox": [ + 366, + 496, + 632, + 539 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $p ( \\mathbf { x } _ { T } )$ is a unit Gaussian that does not depend on $\\theta$ . Training the conditional DDPM therefore involves fitting $p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )$ to approximate $q ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )$ for all choices of $t$ , $\\mathbf { x } _ { t }$ , and $\\mathbf { y }$ . ", + "bbox": [ + 173, + 540, + 823, + 570 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Several observations have been made in recent years which simplify the learning of $p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )$ . Sohl-Dickstein et al. [28] showed that when $\\alpha _ { t }$ is close to 1, $p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } )$ is approximately Gaussian [28]. Furthermore, Ho et al. [15] showed that this Gaussian’s variance can be modeled well with a non-learned function of $t$ , and that a good estimate of the Gaussian’s mean can be obtained from a “denoising model” as follows. Given data $\\mathbf { x } _ { \\mathrm { 0 } }$ and unit Gaussian noise $\\epsilon$ , the denoising model (in the form of a neural network) is fed “noisy” data $\\mathbf { x } _ { t } : = \\sqrt { \\tilde { \\alpha } _ { t } } \\mathbf { x } _ { 0 } + \\sqrt { 1 - \\tilde { \\alpha } _ { t } } \\epsilon$ and trained to recover $\\epsilon$ via a mean squared error loss. The parameters $\\begin{array} { r } { \\tilde { \\alpha } _ { t } : = \\prod _ { i = 1 } ^ { t } \\alpha _ { i } } \\end{array}$ are chosen to ensure that the marginal distribution of $\\mathbf { x } _ { t }$ given $\\mathbf { x } _ { \\mathrm { 0 } }$ is $q ( \\mathbf { x } _ { t } | \\mathbf { x } _ { 0 } )$ as derived from Eq. (1). Given a weighting function $\\lambda ( t )$ , the denoising loss is ", + "bbox": [ + 173, + 574, + 826, + 702 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/44c3ffa1c847398e4b684bdbf6916d12017a75017767ddfdf7c1534d3b8ab6ea.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) = \\mathbb { E } _ { q ( \\mathbf { x } _ { 0 } , \\mathbf { y } , \\epsilon ) } \\left[ \\sum _ { t = 1 } ^ { T } \\lambda ( t ) | | \\epsilon - \\epsilon _ { \\theta } ( \\mathbf { x } _ { t } , \\mathbf { y } , t ) | | _ { 2 } ^ { 2 } \\right] \\quad \\mathrm { w i t h } \\quad \\mathbf { x } _ { t } = \\sqrt { \\tilde { \\alpha } _ { t } } \\mathbf { x } _ { 0 } + \\sqrt { 1 - \\tilde { \\alpha } _ { t } } \\epsilon .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 703, + 781, + 747 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The mean of $p _ { \\theta } ( \\mathbf { x } _ { t - 1 } | \\mathbf { x } _ { t } , \\mathbf { y } )$ is obtained from the denoising model’s output $\\epsilon _ { \\theta } ( \\mathbf x _ { t } , \\mathbf y , t )$ as $\\begin{array} { r } { \\frac { 1 } { \\alpha _ { t } } \\mathbf { x } _ { t } - \\frac { 1 - \\alpha _ { t } } { \\sqrt { 1 - \\tilde { \\alpha } _ { t } } } \\epsilon _ { \\theta } \\big ( \\mathbf { x } _ { t } , \\mathbf { y } , t \\big ) } \\end{array}$ . If the weighting function $\\lambda ( t )$ is chosen appropriately, optimising Eq. (4) is equivalent to optimising a lower-bound on the data likelihood under $p _ { \\theta }$ . In practice, simply setting $\\lambda ( t ) : = 1$ for all $t$ can produce more visually compelling results in the image domain [15]. ", + "bbox": [ + 173, + 747, + 826, + 809 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In our proposed method, as in Tashiro et al. [32], the shapes of $\\mathbf { x } _ { \\mathrm { 0 } }$ and $\\mathbf { y }$ sampled from $q ( \\cdot )$ vary. This is because we want to train a model which can flexibly adapt to e.g. varying numbers of observed frames. To map Eq. (4) to this scenario, note that both $\\mathbf { x } _ { \\mathrm { 0 } }$ and $\\mathbf { y }$ implicitly contain information about which frames in the video they represent (via the index vectors $\\mathcal { X }$ and $\\mathcal { V }$ introduced in the previous section). This information is used inside the neural network $\\epsilon _ { \\theta } ( \\mathbf x _ { t } , \\mathbf y , t )$ so that interactions between frames can be conditioned on the distance between them (as described in the following section) and also to ensure that the sampled noise vector $\\epsilon$ has the same shape as $\\mathbf { x } _ { \\mathrm { 0 } }$ . ", + "bbox": [ + 173, + 813, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$: \\mathcal { X } : = \\{ \\} ; \\mathcal { Y } : = \\{ \\}$ \n2: while True do \n3: $n _ { \\mathrm { g r o u p } } \\sim$ UniformDiscrete $( 1 , K )$ \n4: $s _ { \\mathrm { g r o u p } } \\sim \\mathrm { L o g U n i f o r m } ( 1 , ( N - 1 ) / n _ { \\mathrm { g r o u p } } )$ \n5: $x _ { \\mathrm { g r o u p } } \\sim \\mathrm { U n i f o r m } ( 0 , N - ( n _ { \\mathrm { g r o u p } } - 1 ) \\cdot s _ { \\mathrm { g r o u p } } )$ \n6: ogroup ∼ Bernoulli(0.5) \n7: $\\tilde { \\mathcal { G } } : = \\{ \\lfloor x _ { \\mathrm { g r o u p } } + s _ { \\mathrm { g r o u p } } \\cdot i \\rfloor | i \\in \\{ 0 , \\ldots , n _ { \\mathrm { g r o u p } } - 1 \\} \\} \\backslash \\mathcal { X } \\backslash \\mathcal { Y }$ \n8: if $| \\mathcal { X } | + | \\mathcal { Y } | + | \\mathcal { G } | > K$ then \n9: return set2vector $( \\mathcal { X } )$ , set2vector $( { \\mathcal { V } } )$ \n10: else if $| \\mathcal { X } | = 0$ or $o _ { \\mathrm { g r o u p } } = 0$ then \n11: $\\mathcal { X } : = \\mathcal { X } \\cup \\mathcal { G }$ \n12: else \n13: $y : = \\mathcal { y } \\cup \\mathcal { G }$ ", + "bbox": [ + 415, + 125, + 820, + 294 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/5026656df62c5d1435c78fbe9221db88072c7b53140f572331883e0731269e40.jpg", + "image_caption": [ + "Figure 4: Left: Samples from $u ( \\mathcal { X } , \\mathcal { Y } )$ with video length $N = 3 0$ and limit $K = 1 0$ on the number of sampled indices. Each row shows one sample and columns map to frames, with frame 1 on the left and frame $N$ on the right. Blue and red denote latent and observed frames respectively. All other frames are ignored and shown as white. Right: Pseudocode for drawing these samples. The while loop iterates over a series of regularly-spaced groups of latent variables. Each group is parameterized by: the number of indices in it, $n _ { \\mathrm { g r o u p } }$ ; the spacing between indices in it, $s _ { \\mathrm { g r o u p } }$ ; the position of the first frame in it, $x _ { \\mathrm { g r o u p } }$ , and an indicator variable for whether this group is observed, $O _ { \\mathrm { { g r o u p } } }$ (which is ignored on line 10 if $\\mathcal { X }$ is empty to ensure that the returned value of $\\mathcal { X }$ is never empty). These quantities are sampled in a continuous space and then discretized to make a set of integer coordinates on line 7. The process repeats until a group is sampled which, if added to $\\mathcal { X }$ or $\\mathcal { V }$ , will cause the number of frames to exceed $K$ . That group is then discarded and $\\mathcal { X }$ and $\\mathcal { V }$ are returned as vectors. The FDM’s training objective forces it to work well for any $( \\mathcal { X } , \\mathcal { Y } )$ pair from this broad distribution. " + ], + "image_footnote": [], + "bbox": [ + 174, + 103, + 403, + 292 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Training procedure and architecture ", + "text_level": 1, + "bbox": [ + 174, + 492, + 511, + 510 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training task distribution Different choices of latent and observed indices $\\mathcal { X }$ and $\\mathcal { V }$ can be regarded as defining different conditional generation tasks. In this sense, we aim to learn a model which can work well on any task (i.e. any choice of $\\mathcal { X }$ and $\\mathcal { V }$ ) and so we randomly sample these vectors of indices during training. We do so with the distribution $u ( \\mathcal { X } , \\mathcal { Y } )$ described in Fig. 4. This provides a broad distribution covering many plausible test-time use cases while still providing sufficient structure to improve learning (see ablation in Section 6 and more details in Appendix C). To cope with constrained computational resources, the distribution is designed such that $\\left| \\mathcal { X } \\right| + \\left| \\mathcal { V } \\right|$ is upper-bounded by some pre-specified $K$ . Sampling from $q ( \\mathbf { x } _ { 0 } , \\mathbf { y } )$ in Eq. (4) is then accomplished by randomly selecting both a full training video $\\mathbf { v }$ and indices $\\mathcal { X } , \\mathcal { Y } \\sim u ( \\cdot , \\cdot )$ . We then extract the specified frames $\\mathbf { x } = \\mathbf { v } [ \\mathcal { X } ]$ and $\\mathbf { y } = \\mathbf { v } [ \\mathcal { V } ]$ (where we use $\\mathbf { v } [ \\mathcal { X } ]$ to denote the concatenation of all frames in $\\mathbf { v }$ with indices in $\\mathcal { X }$ and and $\\mathbf { v } [ \\mathcal { V } ]$ similarly). ", + "bbox": [ + 173, + 523, + 826, + 676 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Architecture DDPM image models [15, 22] typically use a U-net architecture [24]. Its distinguishing feature is a series of spatial downsampling layers followed by a series of upsampling layers, and these are interspersed with convolutional res-net blocks [14] and spatial attention layers. Since we require an architecture which operates on 4-D video tensors rather than 3-D image tensors we add an extra frame dimension to its input, output and hidden state, resulting in the architecture shown on the right of Fig. 3. We create the input to this architecture as a concatenation $\\mathbf x _ { t } \\oplus \\mathbf y$ , adding an extra input channel which is all ones for observed frames and all zeros for latent frames. For RGB video, the input shape is therefore ( $K$ , image height, image width, 4). Since the output should have the same shape as $\\mathbf { x } _ { t }$ we only return outputs corresponding to the latent frames, giving output shape $\\vert \\mathcal { X } \\vert$ , image height, image width, 3). We run all layers from the original model (including convolution, resizing, group normalization, and spatial attention) independently for each of the $K$ frames. To allow communication between the frames, we add a temporal attention layer after each spatial attention layer, described in more detail in the appendix. The spatial attention layer allows each spatial location to attend to all other spatial locations within the same frame, while the temporal attention layer allows each spatial location to attend to the same spatial location across all other frames. This combination of a temporal attention layer with a spatial attention layer is sometimes referred to as factorized attention [32, 16]. We found that, when using this architecture in conjunction with our meta-learning approach, performance could be improved by using a novel form of relative position encoding [27, 38]. This is included in our released source code but we leave its exposition to the supplementary material. ", + "bbox": [ + 174, + 690, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training batch padding Although the size $| \\mathcal { X } \\oplus \\mathcal { Y } |$ of index vectors sampled from our training distribution is bounded above by $K$ , it can vary. To fit examples with various sizes of index vectors into the same batch, one option would be to pad them all to length $K$ with zeros and use masks so that the zeros cannot affect the loss. This, however, would waste computation on processing tensors of zeros. We instead use this computation to obtain a lower-variance loss estimate by processing additional data with “training batch padding”. This means that, for training examples where $| x \\oplus y | < K$ , we concatenate frames uniformly sampled from a second video to increase the length along the frame-dimension to $K$ . Masks are applied to the temporal attention mechanisms so that frames from different videos cannot attend to eachother and the output for each is the same as that achieved by processing the videos in different batches. ", + "bbox": [ + 174, + 162, + 825, + 301 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sampling schemes Before describing the sampling schemes we explore experimentally, we emphasize that the relative performance of each is dataset-dependent and there is no single best choice. A central benefit of FDM is that it can be used at test-time with different sampling schemes without retraining. Our simplest sampling scheme, Autoreg, samples ten consecutives frames at each stage conditioned on the previous ten frames. Long-range is similar to Autoreg but conditions on only the five most recent frames as well as five of the original 36 observed frames. Hierarchy-2 uses a multi-level sampling procedure. In the first level, ten evenly spaced frames spanning the non-observed portion of the video are sampled (conditioned on ten observed frames). In the second level, groups of consecutive frames are sampled conditioned on the closest past and future frames until all frames have been sampled. Hierarchy-3 adds an intermediate stage where several groups of variables with an intermediate spacing between them are sampled. We include adaptive hierarchy-2, abbreviated Ad. hierarchy-2, as a demonstration of a sampling scheme only possible with a model like FDM. It samples the same frames at each stage as Hierarchy-2 but selects which frames to condition on adaptively at test-time with a heuristic aimed at collecting the maximally diverse set of frames, as measured by the pairwise LPIPS distance [41] between them. ", + "bbox": [ + 173, + 318, + 825, + 523 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Optimizing sampling schemes An appealing alternative to the heuristic sampling schemes described in the previous paragraph would be to find a sampling scheme that is, in some sense, optimal for a given model and video generation/completion task. While it is unclear how to tractably choose which frames should be sampled at each stage, we suggest that the frames to condition on at each stage can be chosen by greedily optimizing the diffusion model loss which, as mentioned in Section 3, is closely related to the data log-likelihood. Given a fixed sequence of frames to sample at each stage $[ \\mathcal { X } _ { s } ] _ { s = 1 } ^ { S }$ we select $\\mathcal { \\partial } _ { s }$ for each $s$ to minimize Eq. (4). This is estimated using a set of 100 training videos and by iterating over 10 evenly-spaced values of $t$ (which reduced variance relative to random sampling of $t$ ). See the appendix for further details. We create two optimized sampling schemes: one with the same latent indices as Autoreg, and one with the same latent indices as Hierarchy-2. We call the corresponding optimized schemes Opt. autoreg and Opt. hierarchy-2. ", + "bbox": [ + 174, + 540, + 825, + 693 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 CARLA Town01 Dataset ", + "text_level": 1, + "bbox": [ + 176, + 713, + 413, + 731 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In addition to our methodological contributions, we propose a new video-modeling dataset and benchmark which provides an interpretable measure of video completion quality. The dataset consists of videos of a car driving with a first-person view, produced using the CARLA autonomous driving simulator [9]. All 408 training and 100 test videos (of length 1000 frames and resolution $1 2 8 \\times 1 2 8 )$ ) are produced within a single small town, CARLA’s Town01. As such, when a sufficiently expressive video model is trained on this dataset it memorizes the layout of the town and videos sampled from the model will be recognisable as corresponding to routes travelled within the town. We train a regression model in the form of a neural network which maps with high accuracy from any single rendered frame to $( x , y )$ coordinates representing the car’s position. Doing so allows us to plot the routes corresponding to sampled videos (see left of Fig. 5) and compute semantically-meaningful yet quantitative measures of the validity of these routes. Specifically, we compute histograms of speeds, where each speed is estimated by measuring the distance between the regressed locations for frames spaced ten apart (1 second at the dataset’s frame rate). Sampled videos occasionally “jump” between disparate locations in the town, resulting in unrealistically large estimated speeds. To measure the frequency of these events for each method, we compute the percentage of our point-speed estimates that exceed a threshold of $1 0 \\mathrm { m / s }$ (the dataset was generated with a maximum simulated speed of $\\mathrm { 3 m / s }$ ). We report this metric as the outlier percentage (OP). After filtering out these outliers, we compute the Wasserstein distance (WD) between the resulting empirical distribution and that of the original dataset, giving a measure of how well generated videos match the speed of videos in the dataset. We release the CARLA Town 01 dataset along with code and our trained regression model to allow future comparisons.2 ", + "bbox": [ + 173, + 746, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/91d03113bf431ff2ce8593f9dcf8c30de45d678fc436461bb3511e1f415c4d07.jpg", + "image_caption": [ + "Figure 5: Left: Map of the town featured in the CARLA Town01 dataset. We visualize two video completions by FDM by showing coordinates output by our regressor (discussed in Section 5) for each frame. Those corresponding to the initial 36 observed frames are shown in red and those for the 964 sampled frames are shown in blue. Right: For each completion, we show one of the initially observed frames followed by four of the sampled frames (at positions chosen to show the progression with respect to visible landmarks and marked by black dots on the map). The town’s landmarks are usually sampled with high-fidelity, which is key to allowing the regressor to produce a coherent trajectory on the left. However there are sometimes failures: a blue square near the top-right of the map shows where the video model “jumped” to a wrong location for a single frame. " + ], + "image_footnote": [], + "bbox": [ + 174, + 90, + 820, + 210 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 371, + 825, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 515, + 312, + 532 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We perform our main comparisons on the video completion task. In keeping with Saxena et al. [26], we condition on the first 36 frames of each video and sample the remainder. We present results on three datasets: GQN-Mazes [10], in which videos are 300 frames long; MineRL Navigate [13, 26] (which we will from now on refer to as simply MineRL), in which videos are 500 frames long; and the CARLA Town01 dataset we release, for which videos are 1000 frames long. We train FDM in all cases with the maximum number of represented frames $K = 2 0$ . We host non-cherry-picked video samples (both conditional and unconditional) from FDM and all baselines online3. ", + "bbox": [ + 173, + 546, + 825, + 643 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison of sampling schemes The relative performance of different sampling schemes varies significantly between datasets as shown in Table 1. We report Fréchet Video Distances (FVDs) [33], a measure of how similar sampled completions are to the test set, on all datasets. In addition on GQN-Mazes we we report the accuracy metric [26], which classifies videos based on which rooms are visited and measures how often a completion is given the same class as the corresponding test video. For CARLA Town01 we report the previously described percentage outliers (PO) and Wasserstein distance (WD) metrics. ", + "bbox": [ + 174, + 660, + 524, + 824 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/20181a2ed912845712556e621993f86259d7ed3f838de77c7ab7cd3878c59243.jpg", + "image_caption": [ + "Figure 6: Speed distributions measured from sampled and ground-truth dataset videos. " + ], + "image_footnote": [], + "bbox": [ + 545, + 676, + 808, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We can broadly consider the aforementioned sampling schemes as either being in the “autoregressive” family (Autoreg and Long-range) or in the “hierarchical” ", + "bbox": [ + 173, + 832, + 524, + 873 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f4fd07bb7c01804ebaf569cda8a5523542a1ee9451fda5a2a392cb477928e63f.jpg", + "table_caption": [ + "Table 1: Evaluation on video completion with various modes of our method along with several baselines from the literature. Error bars denote the standard error computed with 5 random seeds. Higher is better for the accuracy metric [26] and lower is better for all other metrics shown. " + ], + "table_footnote": [], + "table_body": "
ModelSampling schemeGQN-MazesMineRLCARLA Town01
FVD AccuracyFVDFVDWDOP
CWVAE [26]CWVAE837±882.6± 0.51573±511610.66644.4
TATS [11]TATS163 ± 2.677.0± 0.8807±143291.64842.4
VDM [16]VDM66.7 ± 1.577.8± 0.5271±8.81690.50116.9
FDM (ours)Autoreg86.4± 5.269.6 ± 1.3281±102220.5790.51
Long-range64.5 ± 1.977.0 ± 1.4267 ±4.02130.6530.47
Hierarchy-253.1 ± 1.182.8± 0.7275±7.71200.3183.28
Hierarchy-353.7 ± 1.983.8 ± 1.1311± 6.81490.3634.53
Ad. hierarchy-255.0 ± 1.483.2 ± 1.3316 ±8.91170.3113.44
", + "bbox": [ + 173, + 137, + 825, + 308 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "family (the remainder). Those in the hierarchical family achieve significantly better FVDs [33] on GQN-Mazes. Our samples in the appendix suggest that this is related to the autoregressive methods “forgetting” the colors of walls after looking away from them for a short time. In contrast, for MineRL the autoregressive methods tend to achieve the best FVDs. This may relate to the fact that trajectories in MineRL tend to travel in straight lines through procedurally-generated “worlds”[13, 26], limiting the number of long-range dependencies. Finally on CARLA Town01 we notice qualitatively different behaviours from our autoregressive and hierarchical sampling schemes. The hierarchical sampling schemes have a tendency to occasionally lose coherence and “jump” to different locations in the town. This is reflected by higher outlier percentages (OP) in Table 1. On the other hand the autoregressive schemes often stay stationary for unrealistically long times at traffic lights. This is reflected in the histogram of speeds in Fig. 6, which has a larger peak around zero than the ground truth. The high variance of the sampling scheme’s relative performance over different datasets points to a strength of our method, which need only be trained once and then used to explore a variety of sampling schemes. Furthermore, we point out that the best FVDs in Table 1 on all datasets were obtained using sampling schemes that could not be implemented using models trained in prior work, or over evenly spaced frames. ", + "bbox": [ + 173, + 334, + 825, + 554 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Comparison with baselines The related work most relevant to ours is the concurrent work of Ho et al. [16], who model 64-frame videos using two trained DDPMs. The first is a “frameskip-4” model trained to generate every fourth frame and the second is a “frameskip-1” model trained on sequences of nine consecutive frames and used to “fill in” the gaps between frames generated in the first stage. To compare against this approach, which we denote VDM, we train both a “frameskip-4” and a “frameskip-1” model with architectures identical to our own.4 Since VDM requires two trained DDPMs, we train it for more GPU-hours than FDM despite the fact that FDM is meta-learning over a far broader task distribution. We also compare against TATS [11], which embeds videos into a discrete latent space before modelling them with a transformers, and the clockwork VAE (CWVAE) [26], a VAE-based model specifically designed to maintain long-range dependencies within video. ", + "bbox": [ + 173, + 571, + 826, + 724 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Both the diffusion-based methods, FDM and VDM, achieve significantly higher FVD scores than TATS and CWVAE. This may point toward the utility of diffusion models in general for modeling images and video. Table 1 also makes clear the main benefit of FDM over VDM: although there is no sampling scheme for FDM which always outperforms VDM, there is at least one sampling scheme that outperforms it on each dataset. This speaks to the utility of learning a flexible model like FDM that allows different sampling schemes to be experimented with after training. ", + "bbox": [ + 174, + 731, + 825, + 814 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Optimized sampling schemes As mentioned in Section 4, another advantage of FDM is that it makes possible a model- and dataset-specific optimization procedure to determine on which frames to condition. Table 2 shows the results when this procedure is used to create sampling schemes for different datasets. In the first row we show results where the latent frames are fixed to be those of the Autoreg sampling scheme, and in the second row the latent frames are fixed to match those of Hierarchy-2. On two of the three datasets the best results in Table 1 are improved upon, showing the utility of this optimization procedure. ", + "bbox": [ + 176, + 830, + 823, + 859 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/abab72ee95103acbfb30e20e81038ca8ab2dcfdd4907e58e3942bc60dd057563.jpg", + "table_caption": [ + "Table 2: FVD scores for our sampling schemes with observed indices optimized offline as described in Section 4. We mark with an asterisk $( ^ { * } )$ the eight numbers which improve on the corresponding non-optimized sampling schemes and highlight in bold those that are better than any in Table 1. " + ], + "table_footnote": [], + "table_body": "
GQN-MazesMineRLCARLA Town01
Sampling schemeFVDAccuracyFVDFVDWDOP
Opt. autoreg53.6±1.2*80.2±1.2*257±6.8*146*0.452*0.65
Opt. hierarchy-251.1 ± 1.3*84.6±0.7*320 ± 7.01240.3494.11*
", + "bbox": [ + 209, + 145, + 787, + 219 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 247, + 825, + 316 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Comparison with training on a single task Training a network with our distribution over training tasks could be expected to lead to worse performance on a single task than training specifically for that task. To test whether this is the case, we train an ablation of FDM with training tasks exclusively of the type used in our Autoreg sampling scheme, i.e. “predict ten consecutive frames given the previous ten.” Tested with the Autoreg sampling scheme, it obtained an FVD of 82.0 on GQN-Mazes and 234 on MineRL. As expected given the specialization to a single task, this is better than when FDM is run with the Autoreg sampling scheme (obtaining FVDs of 86.4 and 281 respectively). ", + "bbox": [ + 174, + 334, + 825, + 431 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Ablation on training task distribution To test how important our proposed structured training distribution is to FDM’s performance, we perform an ablation with a different task distribution that samples $\\mathcal { X }$ and $\\mathcal { V }$ from uniform distributions instead of our proposed structured task distribution We provide full details in the appendix, but report here that switching away form our structured training distribution made the FVD scores worse on all five tested sampling schemes on both GQN-Mazes and MineRL. The reduction in the average FVD was $3 1 \\%$ on GQN-Mazes and $5 2 \\%$ on MineRL. This implies that our structured training distribution has a significant positive effect. ", + "bbox": [ + 174, + 449, + 825, + 546 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Related work ", + "text_level": 1, + "bbox": [ + 174, + 568, + 316, + 584 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Some related work creates conditional models by adapting the sampling procedure of an unconditional DDPM [30, 18, 21, 16]. These approaches require approximations and the more direct approach that we use (explcitly training a conditional DDPM) was shown to have benefits by Tashiro et al. [32]. We consider further comparison of these competing approaches to be outside the focus of this work, which is on modeling a small portion of video frames at a time, essentially performing marginalization in addition to conditioning. ", + "bbox": [ + 174, + 601, + 825, + 684 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "There are a number of approaches in the literature which use VAEs rather than DDPMs for video modelling. Babaeizadeh et al. [2] use a VAE model which predicts frames autoregressively conditioned on a global time-invariant latent variable. A related approach by Denton and Fergus [7] also uses a VAE with convolutional LSTM architectures in both the encoder and decoder. Unlike Babaeizadeh et al. [2] the prior is learned and a different latent variable is sampled for each frame. Babaeizadeh et al. [3] use a VAE with one set of latent variables per frame and inter-frame dependencies tracked by a two-layer LSTM. Their architecture intentionally overfits to the training data, which when coupled with image augmentations techniques achieves SOTA on various video prediction tasks. Kim et al. [20] use a variational RNN [5] with a hierarchical latent space that includes binary indicator variables which specify how the video is divided into a series of subsequences. Both Villegas et al. [35] and Wichers et al. [37] target long-term video prediction using a hierarchical variational LSTM architecture, wherein high-level features such as landmarks are predicted first, then decoded into low-level pixel space. The two approaches differ in that Villegas et al. [35] requires ground truth landmark labels, while [37] removes this dependence using an unsupervised adversarial approach. Fully GAN-based video models have also been proposed [1, 6] but generally suffer from “low quality frames or low number of frames or both” [1]. ", + "bbox": [ + 173, + 690, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 Discussion ", + "text_level": 1, + "bbox": [ + 173, + 89, + 292, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have defined and empirically explored a new method for generating photorealistic videos with long-range coherence that respects and efficiently uses fixed, finite computational resources. Our approach outperforms prior work on long-duration video modeling as measured by quantitative and semantically meaningful metrics and opens up several avenues for future research. For one, similar to using DDPMs for image generation, our method is slow to sample from (it takes approximately 16 minutes to generate a 300 frame video on a GPU). Ideas for making sampling faster by decreasing the number of integration steps [25, 29, 39] could be applied to our video model. ", + "bbox": [ + 174, + 121, + 825, + 218 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "On a different note, consider the datasets on which our artifact was trained. In each there was a policy for generating the sequences of actions that causally led to the frame-to-frame changes in camera pose. In MineRL the video was generated by agents that were trained to explore novel Minecraft worlds to find a goal block approximately 64 meters away [13]. The CARLA data was produced by a camera attached to an agent driven by a low level proportional–integral–derivative controller following waypoints laid down by a high level planner that was given new, random location goals to drive to intermittently. In both cases our video model had no access to either the policy or the specific actions taken by these agents and, so, in a formal sense, our models integrate or marginalize over actions drawn from the stochastic policy used to generate the videos in the first place. Near-term future work could involve adding other modalities (e.g. audio) to FDM as well as explicitly adding actions and rewards, transforming our video generative model into a vision-based world model in the reinforcement learning sense [17, 19]. Furthermore, we point out that FDM trained on CARLA Town01 is in theory capable of creating 100-second videos conditioned on both the first and final frame. Doing so can be interpreted as running a “visual” controller which proposes a path between a current state and a specified goal. Preliminary attempts to run in FDM in this way yielded inconsistent results but we believe that this could be a fruitful direction for further investigation. ", + "bbox": [ + 174, + 224, + 825, + 444 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments ", + "text_level": 1, + "bbox": [ + 176, + 462, + 303, + 477 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Inverted AI, and especially Alireza Morsali, for generating the CARLA Town01 dataset. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada CIFAR AI Chairs Program, and the Intel Parallel Computing Centers program. Additional support was provided by UBC’s Composites Research Network (CRN), and Data Science Institute (DSI). This research was enabled in part by technical support and computational resources provided by WestGrid (www.westgrid.ca), Compute Canada (www.computecanada.ca), and Advanced Research Computing at the University of British Columbia (arc.ubc.ca). WH acknowledges support by the University of British Columbia’s Four Year Doctoral Fellowship (4YF) program. ", + "bbox": [ + 174, + 486, + 825, + 611 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 632, + 266, + 647 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Nuha Aldausari, Arcot Sowmya, Nadine Marcus, and Gelareh Mohammadi. Video generative adversarial networks: a review. ACM Computing Surveys (CSUR), 55(2):1–25, 2022. \n[2] Mohammad Babaeizadeh, Chelsea Finn, Dumitru Erhan, Roy H Campbell, and Sergey Levine. 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", + "bbox": [ + 174, + 859, + 825, + 904 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 174, + 89, + 254, + 106 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 116, + 339, + 131 + ], + "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] \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 135, + 825, + 227 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 231, + 493, + 244 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ", + "bbox": [ + 238, + 248, + 738, + 280 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 212, + 285, + 393, + 299 + ], + "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] \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] \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] See appendix. ", + "bbox": [ + 238, + 303, + 825, + 421 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 214, + 426, + 823, + 440 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [Yes] \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 444, + 823, + 551 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 556, + 705, + 570 + ], + "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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Although related work has demonstrated modeling of short photo-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "realistic videos (e.g. 30 frames [36], 48 frames [6] or 64 frames [16]), generating longer videos", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 475, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 507, + 491 + ], + "score": 1.0, + "content": "that are both coherent and photo-realistic remains an open challenge. A major difficulty is scaling:", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "photorealistic image generative models [4, 8] are already close to the memory and processing limits", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 498, + 507, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 507, + 512 + ], + "score": 1.0, + "content": "of modern hardware. A long video is at very least a concatenation of many photorealistic frames,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 510, + 507, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 507, + 522 + ], + "score": 1.0, + "content": "implying resource requirements, long-range coherence notwithstanding, that scale with frame count.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 442, + 507, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "Attempting to model such long-range coherence makes the problem harder still, especially because in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 536, + 507, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 507, + 550 + ], + "score": 1.0, + "content": "general every frame can have statistical dependencies on other frames arbitrarily far back in the video.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 546, + 507, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 507, + 562 + ], + "score": 1.0, + "content": "Unfortunately fixed-lag autoregressive models impose unrealistic conditional independence assump-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "tions (the next frame being independent of frames further back in time than the autoregressive lag is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "score": 1.0, + "content": "problematic for generating videos with long-range coherence). And while deep generative models", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "based on recurrent neural networks (RNN) theoretically impose no such conditional independence", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "assumptions, in practice they must be trained over short sequences [12, 26] or with truncated gradi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 601, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 616 + ], + "score": 1.0, + "content": "ents [31]. Despite this, some RNN-based video generative models have demonstrated longer-range", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 613, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 504, + 626 + ], + "score": 1.0, + "content": "coherence, albeit without yet achieving convincing photorealistic video generation [26, 3, 7, 20, 2].", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 527, + 507, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "In this work we embrace the fact that finite architectures will always impose conditional indepen-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 333, + 653 + ], + "score": 1.0, + "content": "dences. The question we ask is: given an explicit limit", + "type": "text" + }, + { + "bbox": [ + 333, + 641, + 344, + 650 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "on the number of video frames we can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 444, + 664 + ], + "score": 1.0, + "content": "jointly model, how can we best allocate these frames to generate a video of length", + "type": "text" + }, + { + "bbox": [ + 445, + 651, + 483, + 662 + ], + "score": 0.89, + "content": "N > K ?", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "One", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 388, + 674 + ], + "score": 1.0, + "content": "option is to use the previously-described autoregressive model but, if", + "type": "text" + }, + { + "bbox": [ + 389, + 662, + 430, + 674 + ], + "score": 0.92, + "content": "K = N / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 663, + 505, + 674 + ], + "score": 1.0, + "content": ", we could instead", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "score": 1.0, + "content": "follow Ho et al. [16] by training two models: one which first samples every 4th frame in the video,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 696 + ], + "score": 1.0, + "content": "and another which (in multiple stages) infills the remaining frames conditioned on those. To enable", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "efficient exploration of the space of such sampling schemes, we propose a flexible architecture based", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "on the denoising diffusion probabilistic model (DDPM) framework. This can sample any subset of", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "video frames conditioned on observed values of any other subset of video frames. It therefore lets", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "us explore a wide variety of previously untested sampling schemes while being easily repurposed", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "for different generation tasks such as unconditional generation, video completion, and generation of", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "score": 1.0, + "content": "videos of different lengths. Since our model can be flexibly applied to sample any frames given any", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 318, + 322, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 322, + 330 + ], + "score": 1.0, + "content": "others we call it a Flexible Diffusion Model, or FDM.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 38.5, + "bbox_fs": [ + 104, + 629, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 71, + 503, + 167 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 71, + 503, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 71, + 503, + 167 + ], + "spans": [ + { + "bbox": [ + 108, + 71, + 503, + 167 + ], + "score": 0.968, + "type": "image", + "image_path": "d3896c398df1e076a51acfb5819f046231ccb319e84ddbaf5049df13dc706653.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 71, + 503, + 103.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 103.0, + 503, + 135.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 135.0, + 503, + 167.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 172, + 505, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "Figure 1: A long video (25 minutes, or approximately 15 000 frames) generated by FDM for each", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "of CARLA Town01 and MineRL, conditioned on 500 and 250 prior frames respectively. We show", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "blocks of frames from three points within each video, starting from the final observed frame on the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "left. Blocks are marked with the time elapsed since the last observation and frames within them are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 215, + 474, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 402, + 228 + ], + "score": 1.0, + "content": "one second apart. We observe no degradation in sample quality even after", + "type": "text" + }, + { + "bbox": [ + 402, + 216, + 441, + 226 + ], + "score": 0.75, + "content": "> 1 5 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 215, + 474, + 228 + ], + "score": 1.0, + "content": "frames.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "efficient exploration of the space of such sampling schemes, we propose a flexible architecture based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "on the denoising diffusion probabilistic model (DDPM) framework. This can sample any subset of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "video frames conditioned on observed values of any other subset of video frames. It therefore lets", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "us explore a wide variety of previously untested sampling schemes while being easily repurposed", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "for different generation tasks such as unconditional generation, video completion, and generation of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "score": 1.0, + "content": "videos of different lengths. Since our model can be flexibly applied to sample any frames given any", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 318, + 322, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 322, + 330 + ], + "score": 1.0, + "content": "others we call it a Flexible Diffusion Model, or FDM.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 107, + 346, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 107, + 346, + 506, + 358 + ], + "score": 1.0, + "content": "Contributions (1) At the highest level, we claim to have concurrently developed one of the first", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "denoising diffusion probabilistic model (DDPM)-based video generative models [16, 40]. To do so we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "augment a previously-used DDPM image architecture [15, 22] with a temporal attention mechanism", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "including a novel relative (frame) position encoding network. (2) The principal contribution of this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "paper, regardless, is a “meta-learning” training objective that encourages learning of a video generative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "model that can (a) be flexibly conditioned on any number of frames (up to computational resource", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "constraints) at any time in the past and future and (b) be flexibly marginalized (to achieve this within", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "computational resource constraints). (3) We demonstrate that our model can be used to efficiently", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "explore the space of resource constrained video generation schemes, leading to improvements over", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "prior work on several long-range video modeling tasks. (4) Finally, we release a new autonomous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "driving video dataset along with a new video generative model performance metric that captures", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "semantics more directly than the visual quality and comparison metrics currently in widespread use.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 497, + 234, + 511 + ], + "lines": [ + { + "bbox": [ + 104, + 495, + 236, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 236, + 515 + ], + "score": 1.0, + "content": "2 Sampling long videos", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "Our goal in this paper is to sample coherent photo-realistic videos v with thousands of frames (see", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "Fig. 1). To sample an arbitrarily long video with a generative model that can sample or condition on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "only a small number of frames at once, we must use a sequential procedure. The simplest example of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 557, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 507, + 572 + ], + "score": 1.0, + "content": "this is an autoregressive scheme, an example of which is shown in Fig. 2a for a video completion task.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "In this example it takes seven stages to sample a complete video, in that we must run the generative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "model’s sampling procedure seven times. At each stage three frames are sampled conditioned on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 507, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 507, + 605 + ], + "score": 1.0, + "content": "the immediately preceding four frames. This scheme is appealing for its simplicity but imposes a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 601, + 507, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 507, + 616 + ], + "score": 1.0, + "content": "strong assumption that, given the set of four frames that are conditioned on at a particular stage,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "all frames that come afterwards are conditionally independent of all frames that came before. This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "restriction can be partially ameliorated with the sampling scheme shown in Fig. 2b where, in the first", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 649 + ], + "score": 1.0, + "content": "three stages, every second frame is sampled and then, in the remaining four stages, the remaining", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "frames are infilled. One way to implement this would be to train two different models operating at", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "the two different temporal resolutions. In the language of Ho et al. [16], who use a similar approach,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "sampling would be carried out in the first three stages by a “frameskip-2” model and, in the remaining", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "stages, by a “frameskip-1” model. Both this approach and the autoregressive approach are examples", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "of what we call sampling schemes. More generally, we characterize a sampling scheme as a sequence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 504, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 144, + 714 + ], + "score": 1.0, + "content": "of tuples", + "type": "text" + }, + { + "bbox": [ + 145, + 699, + 199, + 712 + ], + "score": 0.93, + "content": "[ ( \\mathcal { X } _ { s } , \\mathcal { Y } _ { s } ) ] _ { s = 1 } ^ { \\bar { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 699, + 303, + 714 + ], + "score": 1.0, + "content": ", each containing a vector", + "type": "text" + }, + { + "bbox": [ + 304, + 702, + 315, + 711 + ], + "score": 0.9, + "content": "\\mathcal { X } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 699, + 492, + 714 + ], + "score": 1.0, + "content": "of indices of frames to sample and a vector", + "type": "text" + }, + { + "bbox": [ + 492, + 700, + 504, + 711 + ], + "score": 0.87, + "content": "\\mathcal { { D } } _ { s }", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 349, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 292, + 723 + ], + "score": 1.0, + "content": "of indices of frames to condition on for stages", + "type": "text" + }, + { + "bbox": [ + 293, + 713, + 345, + 722 + ], + "score": 0.93, + "content": "s = 1 , \\ldots , S", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 710, + 349, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 36.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 71, + 503, + 167 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 71, + 503, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 71, + 503, + 167 + ], + "spans": [ + { + "bbox": [ + 108, + 71, + 503, + 167 + ], + "score": 0.968, + "type": "image", + "image_path": "d3896c398df1e076a51acfb5819f046231ccb319e84ddbaf5049df13dc706653.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 71, + 503, + 103.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 103.0, + 503, + 135.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 135.0, + 503, + 167.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 172, + 505, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "Figure 1: A long video (25 minutes, or approximately 15 000 frames) generated by FDM for each", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "of CARLA Town01 and MineRL, conditioned on 500 and 250 prior frames respectively. We show", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "blocks of frames from three points within each video, starting from the final observed frame on the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "left. Blocks are marked with the time elapsed since the last observation and frames within them are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 215, + 474, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 402, + 228 + ], + "score": 1.0, + "content": "one second apart. We observe no degradation in sample quality even after", + "type": "text" + }, + { + "bbox": [ + 402, + 216, + 441, + 226 + ], + "score": 0.75, + "content": "> 1 5 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 215, + 474, + 228 + ], + "score": 1.0, + "content": "frames.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 330 + ], + "lines": [], + "index": 11, + "bbox_fs": [ + 105, + 252, + 506, + 330 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 107, + 346, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 107, + 346, + 506, + 358 + ], + "score": 1.0, + "content": "Contributions (1) At the highest level, we claim to have concurrently developed one of the first", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "denoising diffusion probabilistic model (DDPM)-based video generative models [16, 40]. To do so we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "augment a previously-used DDPM image architecture [15, 22] with a temporal attention mechanism", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "including a novel relative (frame) position encoding network. (2) The principal contribution of this", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "paper, regardless, is a “meta-learning” training objective that encourages learning of a video generative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 414 + ], + "score": 1.0, + "content": "model that can (a) be flexibly conditioned on any number of frames (up to computational resource", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "constraints) at any time in the past and future and (b) be flexibly marginalized (to achieve this within", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "computational resource constraints). (3) We demonstrate that our model can be used to efficiently", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "explore the space of resource constrained video generation schemes, leading to improvements over", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "prior work on several long-range video modeling tasks. (4) Finally, we release a new autonomous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "driving video dataset along with a new video generative model performance metric that captures", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "semantics more directly than the visual quality and comparison metrics currently in widespread use.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 346, + 506, + 479 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 497, + 234, + 511 + ], + "lines": [ + { + "bbox": [ + 104, + 495, + 236, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 236, + 515 + ], + "score": 1.0, + "content": "2 Sampling long videos", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "Our goal in this paper is to sample coherent photo-realistic videos v with thousands of frames (see", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "Fig. 1). To sample an arbitrarily long video with a generative model that can sample or condition on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "only a small number of frames at once, we must use a sequential procedure. The simplest example of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 557, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 507, + 572 + ], + "score": 1.0, + "content": "this is an autoregressive scheme, an example of which is shown in Fig. 2a for a video completion task.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "In this example it takes seven stages to sample a complete video, in that we must run the generative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "model’s sampling procedure seven times. At each stage three frames are sampled conditioned on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 507, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 507, + 605 + ], + "score": 1.0, + "content": "the immediately preceding four frames. This scheme is appealing for its simplicity but imposes a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 601, + 507, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 507, + 616 + ], + "score": 1.0, + "content": "strong assumption that, given the set of four frames that are conditioned on at a particular stage,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "all frames that come afterwards are conditionally independent of all frames that came before. This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "restriction can be partially ameliorated with the sampling scheme shown in Fig. 2b where, in the first", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 649 + ], + "score": 1.0, + "content": "three stages, every second frame is sampled and then, in the remaining four stages, the remaining", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "frames are infilled. One way to implement this would be to train two different models operating at", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "the two different temporal resolutions. In the language of Ho et al. [16], who use a similar approach,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "sampling would be carried out in the first three stages by a “frameskip-2” model and, in the remaining", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "stages, by a “frameskip-1” model. Both this approach and the autoregressive approach are examples", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "of what we call sampling schemes. 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For unconditional generation,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 85, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 85, + 505, + 97 + ], + "score": 1.0, + "content": "the input v can be a tensor of zeros. 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For every sampling scheme, all video frames have been sampled after the final row.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "Algorithm 1 lays out how such a sampling scheme is used to sample a video. 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The only constraint we impose on our sampling schemes is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 288, + 424 + ], + "score": 1.0, + "content": "therefore a computational consideration that", + "type": "text" + }, + { + "bbox": [ + 288, + 411, + 356, + 423 + ], + "score": 0.93, + "content": "| \\mathcal { X } _ { s } | + | \\mathcal { V } _ { s } | \\le K", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 411, + 384, + 424 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 385, + 412, + 390, + 421 + ], + "score": 0.69, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "but, to generate meaningful", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "videos, any valid sampling scheme must also satisfy two more constraints: (1) all frames are sampled", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 433, + 458, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 458, + 446 + ], + "score": 1.0, + "content": "at at least one stage and (2) frames are never conditioned upon before they are sampled.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "Such a flexible generative model allows us to explore and use sampling schemes like those in Fig. 2c", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "and Fig. 2d. We find in our experiments that the best video sampling scheme is dataset dependent.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Accordingly, we have developed methodology to optimize such sampling schemes in a dataset", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 481, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 495 + ], + "score": 1.0, + "content": "dependent way, leading to improved video quality as measured by the Fréchet Video Distance [33]", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "among other metrics. 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We find in our experiments that the best video sampling scheme is dataset dependent.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Accordingly, we have developed methodology to optimize such sampling schemes in a dataset", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 481, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 495 + ], + "score": 1.0, + "content": "dependent way, leading to improved video quality as measured by the Fréchet Video Distance [33]", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "among other metrics. 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That group is then discarded and", + "type": "text" + }, + { + "bbox": [ + 372, + 349, + 382, + 358 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 349, + 400, + 361 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 400, + 349, + 410, + 359 + ], + "score": 0.83, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 349, + 506, + 361 + ], + "score": 1.0, + "content": "are returned as vectors.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 343, + 372 + ], + "score": 1.0, + "content": "The FDM’s training objective forces it to work well for any", + "type": "text" + }, + { + "bbox": [ + 344, + 360, + 372, + 372 + ], + "score": 0.92, + "content": "( \\mathcal { X } , \\mathcal { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "pair from this broad distribution.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + } + ], + "index": 12.75 + }, + { + "type": "title", + "bbox": [ + 107, + 390, + 313, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 314, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 314, + 407 + ], + "score": 1.0, + "content": "4 Training procedure and architecture", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 506, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 434, + 428 + ], + "score": 1.0, + "content": "Training task distribution Different choices of latent and observed indices", + "type": "text" + }, + { + "bbox": [ + 434, + 416, + 444, + 425 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 415, + 464, + 428 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 464, + 415, + 473, + 426 + ], + "score": 0.83, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "regarded as defining different conditional generation tasks. In this sense, we aim to learn a model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 322, + 450 + ], + "score": 1.0, + "content": "which can work well on any task (i.e. any choice of", + "type": "text" + }, + { + "bbox": [ + 322, + 437, + 332, + 447 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 436, + 350, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 437, + 360, + 448 + ], + "score": 0.73, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 436, + 506, + 450 + ], + "score": 1.0, + "content": ") and so we randomly sample these", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 385, + 461 + ], + "score": 1.0, + "content": "vectors of indices during training. We do so with the distribution", + "type": "text" + }, + { + "bbox": [ + 385, + 448, + 420, + 460 + ], + "score": 0.93, + "content": "u ( \\mathcal { X } , \\mathcal { Y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "described in Fig. 4.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 457, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 506, + 472 + ], + "score": 1.0, + "content": "This provides a broad distribution covering many plausible test-time use cases while still providing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "sufficient structure to improve learning (see ablation in Section 6 and more details in Appendix C).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 454, + 493 + ], + "score": 1.0, + "content": "To cope with constrained computational resources, the distribution is designed such that", + "type": "text" + }, + { + "bbox": [ + 455, + 480, + 495, + 492 + ], + "score": 0.91, + "content": "\\left| \\mathcal { X } \\right| + \\left| \\mathcal { V } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 262, + 504 + ], + "score": 1.0, + "content": "upper-bounded by some pre-specified", + "type": "text" + }, + { + "bbox": [ + 262, + 492, + 272, + 502 + ], + "score": 0.7, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 491, + 340, + 504 + ], + "score": 1.0, + "content": ". 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Its distinguish-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "ing feature is a series of spatial downsampling layers followed by a series of upsampling layers, and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "these are interspersed with convolutional res-net blocks [14] and spatial attention layers. Since we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "require an architecture which operates on 4-D video tensors rather than 3-D image tensors we add", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "an extra frame dimension to its input, output and hidden state, resulting in the architecture shown", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 440, + 614 + ], + "score": 1.0, + "content": "on the right of Fig. 3. We create the input to this architecture as a concatenation", + "type": "text" + }, + { + "bbox": [ + 440, + 603, + 470, + 613 + ], + "score": 0.89, + "content": "\\mathbf x _ { t } \\oplus \\mathbf y", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 601, + 505, + 614 + ], + "score": 1.0, + "content": ", adding", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "an extra input channel which is all ones for observed frames and all zeros for latent frames. For", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 272, + 636 + ], + "score": 1.0, + "content": "RGB video, the input shape is therefore (", + "type": "text" + }, + { + "bbox": [ + 272, + 624, + 282, + 634 + ], + "score": 0.61, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 624, + 505, + 636 + ], + "score": 1.0, + "content": ", image height, image width, 4). Since the output should", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 201, + 647 + ], + "score": 1.0, + "content": "have the same shape as", + "type": "text" + }, + { + "bbox": [ + 201, + 636, + 213, + 645 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "we only return outputs corresponding to the latent frames, giving output", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 136, + 659 + ], + "score": 1.0, + "content": "shape", + "type": "text" + }, + { + "bbox": [ + 136, + 645, + 152, + 657 + ], + "score": 0.82, + "content": "\\vert \\mathcal { X } \\vert", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 645, + 506, + 659 + ], + "score": 1.0, + "content": ", image height, image width, 3). We run all layers from the original model (including", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 656, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 493, + 668 + ], + "score": 1.0, + "content": "convolution, resizing, group normalization, and spatial attention) independently for each of the", + "type": "text" + }, + { + "bbox": [ + 493, + 657, + 504, + 666 + ], + "score": 0.73, + "content": "K", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "frames. To allow communication between the frames, we add a temporal attention layer after each", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "spatial attention layer, described in more detail in the appendix. The spatial attention layer allows", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "each spatial location to attend to all other spatial locations within the same frame, while the temporal", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "attention layer allows each spatial location to attend to the same spatial location across all other", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "frames. This combination of a temporal attention layer with a spatial attention layer is sometimes", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "referred to as factorized attention [32, 16]. We found that, when using this architecture in conjunction", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "with our meta-learning approach, performance could be improved by using a novel form of relative", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 506, + 107 + ], + "score": 1.0, + "content": "position encoding [27, 38]. This is included in our released source code but we leave its exposition to", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 219, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 219, + 117 + ], + "score": 1.0, + "content": "the supplementary material.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 547, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "referred to as factorized attention [32, 16]. We found that, when using this architecture in conjunction", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "with our meta-learning approach, performance could be improved by using a novel form of relative", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 506, + 107 + ], + "score": 1.0, + "content": "position encoding [27, 38]. This is included in our released source code but we leave its exposition to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 219, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 219, + 117 + ], + "score": 1.0, + "content": "the supplementary material.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 128, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 293, + 143 + ], + "score": 1.0, + "content": "Training batch padding Although the size", + "type": "text" + }, + { + "bbox": [ + 293, + 130, + 328, + 141 + ], + "score": 0.92, + "content": "| \\mathcal { X } \\oplus \\mathcal { Y } |", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 128, + 505, + 143 + ], + "score": 1.0, + "content": "of index vectors sampled from our training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 239, + 153 + ], + "score": 1.0, + "content": "distribution is bounded above by", + "type": "text" + }, + { + "bbox": [ + 240, + 141, + 250, + 150 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 140, + 505, + 153 + ], + "score": 1.0, + "content": ", it can vary. To fit examples with various sizes of index vectors", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 152, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 385, + 163 + ], + "score": 1.0, + "content": "into the same batch, one option would be to pad them all to length", + "type": "text" + }, + { + "bbox": [ + 385, + 152, + 396, + 161 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 152, + 505, + 163 + ], + "score": 1.0, + "content": "with zeros and use masks", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "so that the zeros cannot affect the loss. This, however, would waste computation on processing", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "tensors of zeros. We instead use this computation to obtain a lower-variance loss estimate by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 184, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 197 + ], + "score": 1.0, + "content": "processing additional data with “training batch padding”. This means that, for training examples", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 133, + 207 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 195, + 190, + 207 + ], + "score": 0.91, + "content": "| x \\oplus y | < K", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 194, + 505, + 207 + ], + "score": 1.0, + "content": ", we concatenate frames uniformly sampled from a second video to increase the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 253, + 219 + ], + "score": 1.0, + "content": "length along the frame-dimension to", + "type": "text" + }, + { + "bbox": [ + 253, + 207, + 264, + 216 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 206, + 506, + 219 + ], + "score": 1.0, + "content": ". Masks are applied to the temporal attention mechanisms so", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "that frames from different videos cannot attend to eachother and the output for each is the same as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 228, + 343, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 343, + 240 + ], + "score": 1.0, + "content": "that achieved by processing the videos in different batches.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "Sampling schemes Before describing the sampling schemes we explore experimentally, we empha-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "size that the relative performance of each is dataset-dependent and there is no single best choice. A", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "central benefit of FDM is that it can be used at test-time with different sampling schemes without", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "retraining. Our simplest sampling scheme, Autoreg, samples ten consecutives frames at each stage", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "conditioned on the previous ten frames. Long-range is similar to Autoreg but conditions on only", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "score": 1.0, + "content": "the five most recent frames as well as five of the original 36 observed frames. Hierarchy-2 uses a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "multi-level sampling procedure. In the first level, ten evenly spaced frames spanning the non-observed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "portion of the video are sampled (conditioned on ten observed frames). In the second level, groups of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "consecutive frames are sampled conditioned on the closest past and future frames until all frames", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "have been sampled. Hierarchy-3 adds an intermediate stage where several groups of variables with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 362, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 373 + ], + "score": 1.0, + "content": "an intermediate spacing between them are sampled. We include adaptive hierarchy-2, abbreviated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "Ad. hierarchy-2, as a demonstration of a sampling scheme only possible with a model like FDM.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "It samples the same frames at each stage as Hierarchy-2 but selects which frames to condition on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "adaptively at test-time with a heuristic aimed at collecting the maximally diverse set of frames, as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 405, + 354, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 354, + 416 + ], + "score": 1.0, + "content": "measured by the pairwise LPIPS distance [41] between them.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "Optimizing sampling schemes An appealing alternative to the heuristic sampling schemes de-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "scribed in the previous paragraph would be to find a sampling scheme that is, in some sense, optimal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "for a given model and video generation/completion task. While it is unclear how to tractably choose", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "which frames should be sampled at each stage, we suggest that the frames to condition on at each", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "stage can be chosen by greedily optimizing the diffusion model loss which, as mentioned in Section 3,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "is closely related to the data log-likelihood. Given a fixed sequence of frames to sample at each stage", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 491, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 493, + 139, + 506 + ], + "score": 0.91, + "content": "[ \\mathcal { X } _ { s } ] _ { s = 1 } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 491, + 181, + 510 + ], + "score": 1.0, + "content": "we select", + "type": "text" + }, + { + "bbox": [ + 181, + 494, + 192, + 505 + ], + "score": 0.88, + "content": "\\mathcal { \\partial } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 491, + 230, + 510 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 230, + 496, + 236, + 504 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 491, + 505, + 510 + ], + "score": 1.0, + "content": "to minimize Eq. (4). This is estimated using a set of 100 training", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 326, + 517 + ], + "score": 1.0, + "content": "videos and by iterating over 10 evenly-spaced values of", + "type": "text" + }, + { + "bbox": [ + 326, + 506, + 331, + 515 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "(which reduced variance relative to random", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 155, + 529 + ], + "score": 1.0, + "content": "sampling of", + "type": "text" + }, + { + "bbox": [ + 155, + 517, + 161, + 526 + ], + "score": 0.48, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "). See the appendix for further details. We create two optimized sampling schemes: one", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "with the same latent indices as Autoreg, and one with the same latent indices as Hierarchy-2. We call", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 538, + 409, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 409, + 550 + ], + "score": 1.0, + "content": "the corresponding optimized schemes Opt. autoreg and Opt. hierarchy-2.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 108, + 565, + 253, + 579 + ], + "lines": [ + { + "bbox": [ + 104, + 564, + 254, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 254, + 581 + ], + "score": 1.0, + "content": "5 CARLA Town01 Dataset", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "In addition to our methodological contributions, we propose a new video-modeling dataset and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "benchmark which provides an interpretable measure of video completion quality. The dataset consists", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "of videos of a car driving with a first-person view, produced using the CARLA autonomous driving", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 457, + 636 + ], + "score": 1.0, + "content": "simulator [9]. All 408 training and 100 test videos (of length 1000 frames and resolution", + "type": "text" + }, + { + "bbox": [ + 458, + 624, + 503, + 635 + ], + "score": 0.86, + "content": "1 2 8 \\times 1 2 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 624, + 506, + 636 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "are produced within a single small town, CARLA’s Town01. As such, when a sufficiently expressive", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "video model is trained on this dataset it memorizes the layout of the town and videos sampled from", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "the model will be recognisable as corresponding to routes travelled within the town. We train a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "regression model in the form of a neural network which maps with high accuracy from any single", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 181, + 691 + ], + "score": 1.0, + "content": "rendered frame to", + "type": "text" + }, + { + "bbox": [ + 181, + 678, + 205, + 690 + ], + "score": 0.93, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "coordinates representing the car’s position. Doing so allows us to plot the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "routes corresponding to sampled videos (see left of Fig. 5) and compute semantically-meaningful yet", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "quantitative measures of the validity of these routes. Specifically, we compute histograms of speeds,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "where each speed is estimated by measuring the distance between the regressed locations for frames", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 46.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 506, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 128, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 293, + 143 + ], + "score": 1.0, + "content": "Training batch padding Although the size", + "type": "text" + }, + { + "bbox": [ + 293, + 130, + 328, + 141 + ], + "score": 0.92, + "content": "| \\mathcal { X } \\oplus \\mathcal { Y } |", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 128, + 505, + 143 + ], + "score": 1.0, + "content": "of index vectors sampled from our training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 239, + 153 + ], + "score": 1.0, + "content": "distribution is bounded above by", + "type": "text" + }, + { + "bbox": [ + 240, + 141, + 250, + 150 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 140, + 505, + 153 + ], + "score": 1.0, + "content": ", it can vary. To fit examples with various sizes of index vectors", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 152, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 385, + 163 + ], + "score": 1.0, + "content": "into the same batch, one option would be to pad them all to length", + "type": "text" + }, + { + "bbox": [ + 385, + 152, + 396, + 161 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 152, + 505, + 163 + ], + "score": 1.0, + "content": "with zeros and use masks", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "so that the zeros cannot affect the loss. This, however, would waste computation on processing", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "tensors of zeros. We instead use this computation to obtain a lower-variance loss estimate by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 184, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 197 + ], + "score": 1.0, + "content": "processing additional data with “training batch padding”. This means that, for training examples", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 133, + 207 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 195, + 190, + 207 + ], + "score": 0.91, + "content": "| x \\oplus y | < K", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 194, + 505, + 207 + ], + "score": 1.0, + "content": ", we concatenate frames uniformly sampled from a second video to increase the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 253, + 219 + ], + "score": 1.0, + "content": "length along the frame-dimension to", + "type": "text" + }, + { + "bbox": [ + 253, + 207, + 264, + 216 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 206, + 506, + 219 + ], + "score": 1.0, + "content": ". Masks are applied to the temporal attention mechanisms so", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "that frames from different videos cannot attend to eachother and the output for each is the same as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 228, + 343, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 343, + 240 + ], + "score": 1.0, + "content": "that achieved by processing the videos in different batches.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 128, + 506, + 240 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "Sampling schemes Before describing the sampling schemes we explore experimentally, we empha-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "size that the relative performance of each is dataset-dependent and there is no single best choice. A", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "central benefit of FDM is that it can be used at test-time with different sampling schemes without", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "retraining. Our simplest sampling scheme, Autoreg, samples ten consecutives frames at each stage", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "conditioned on the previous ten frames. Long-range is similar to Autoreg but conditions on only", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 319 + ], + "score": 1.0, + "content": "the five most recent frames as well as five of the original 36 observed frames. Hierarchy-2 uses a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "multi-level sampling procedure. In the first level, ten evenly spaced frames spanning the non-observed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "portion of the video are sampled (conditioned on ten observed frames). In the second level, groups of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "consecutive frames are sampled conditioned on the closest past and future frames until all frames", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "have been sampled. Hierarchy-3 adds an intermediate stage where several groups of variables with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 362, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 373 + ], + "score": 1.0, + "content": "an intermediate spacing between them are sampled. We include adaptive hierarchy-2, abbreviated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "Ad. hierarchy-2, as a demonstration of a sampling scheme only possible with a model like FDM.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "It samples the same frames at each stage as Hierarchy-2 but selects which frames to condition on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "adaptively at test-time with a heuristic aimed at collecting the maximally diverse set of frames, as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 405, + 354, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 354, + 416 + ], + "score": 1.0, + "content": "measured by the pairwise LPIPS distance [41] between them.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 252, + 506, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "Optimizing sampling schemes An appealing alternative to the heuristic sampling schemes de-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "scribed in the previous paragraph would be to find a sampling scheme that is, in some sense, optimal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "for a given model and video generation/completion task. While it is unclear how to tractably choose", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "which frames should be sampled at each stage, we suggest that the frames to condition on at each", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "stage can be chosen by greedily optimizing the diffusion model loss which, as mentioned in Section 3,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "is closely related to the data log-likelihood. Given a fixed sequence of frames to sample at each stage", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 491, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 107, + 493, + 139, + 506 + ], + "score": 0.91, + "content": "[ \\mathcal { X } _ { s } ] _ { s = 1 } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 491, + 181, + 510 + ], + "score": 1.0, + "content": "we select", + "type": "text" + }, + { + "bbox": [ + 181, + 494, + 192, + 505 + ], + "score": 0.88, + "content": "\\mathcal { \\partial } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 491, + 230, + 510 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 230, + 496, + 236, + 504 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 491, + 505, + 510 + ], + "score": 1.0, + "content": "to minimize Eq. (4). This is estimated using a set of 100 training", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 326, + 517 + ], + "score": 1.0, + "content": "videos and by iterating over 10 evenly-spaced values of", + "type": "text" + }, + { + "bbox": [ + 326, + 506, + 331, + 515 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "(which reduced variance relative to random", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 155, + 529 + ], + "score": 1.0, + "content": "sampling of", + "type": "text" + }, + { + "bbox": [ + 155, + 517, + 161, + 526 + ], + "score": 0.48, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "). See the appendix for further details. We create two optimized sampling schemes: one", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "with the same latent indices as Autoreg, and one with the same latent indices as Hierarchy-2. We call", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 538, + 409, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 409, + 550 + ], + "score": 1.0, + "content": "the corresponding optimized schemes Opt. autoreg and Opt. hierarchy-2.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 429, + 506, + 550 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 565, + 253, + 579 + ], + "lines": [ + { + "bbox": [ + 104, + 564, + 254, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 254, + 581 + ], + "score": 1.0, + "content": "5 CARLA Town01 Dataset", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "In addition to our methodological contributions, we propose a new video-modeling dataset and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "benchmark which provides an interpretable measure of video completion quality. The dataset consists", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "of videos of a car driving with a first-person view, produced using the CARLA autonomous driving", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 457, + 636 + ], + "score": 1.0, + "content": "simulator [9]. All 408 training and 100 test videos (of length 1000 frames and resolution", + "type": "text" + }, + { + "bbox": [ + 458, + 624, + 503, + 635 + ], + "score": 0.86, + "content": "1 2 8 \\times 1 2 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 624, + 506, + 636 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "are produced within a single small town, CARLA’s Town01. As such, when a sufficiently expressive", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "video model is trained on this dataset it memorizes the layout of the town and videos sampled from", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "the model will be recognisable as corresponding to routes travelled within the town. We train a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "regression model in the form of a neural network which maps with high accuracy from any single", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 181, + 691 + ], + "score": 1.0, + "content": "rendered frame to", + "type": "text" + }, + { + "bbox": [ + 181, + 678, + 205, + 690 + ], + "score": 0.93, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "coordinates representing the car’s position. Doing so allows us to plot the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "routes corresponding to sampled videos (see left of Fig. 5) and compute semantically-meaningful yet", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "quantitative measures of the validity of these routes. Specifically, we compute histograms of speeds,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 724 + ], + "score": 1.0, + "content": "where each speed is estimated by measuring the distance between the regressed locations for frames", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "spaced ten apart (1 second at the dataset’s frame rate). Sampled videos occasionally “jump” between", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "disparate locations in the town, resulting in unrealistically large estimated speeds. To measure the", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "frequency of these events for each method, we compute the percentage of our point-speed estimates", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 215, + 340 + ], + "score": 1.0, + "content": "that exceed a threshold of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 215, + 327, + 241, + 338 + ], + "score": 0.81, + "content": "1 0 \\mathrm { m / s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 242, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "(the dataset was generated with a maximum simulated speed of", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 128, + 349 + ], + "score": 0.54, + "content": "\\mathrm { 3 m / s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 128, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "). We report this metric as the outlier percentage (OP). After filtering out these outliers, we", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "compute the Wasserstein distance (WD) between the resulting empirical distribution and that of the", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "original dataset, giving a measure of how well generated videos match the speed of videos in the", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "dataset. We release the CARLA Town 01 dataset along with code and our trained regression model to", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 382, + 216, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 216, + 394 + ], + "score": 1.0, + "content": "allow future comparisons.2", + "type": "text", + "cross_page": true + } + ], + "index": 20 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 591, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 72, + 502, + 167 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 72, + 502, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 72, + 502, + 167 + ], + "spans": [ + { + "bbox": [ + 107, + 72, + 502, + 167 + ], + "score": 0.969, + "type": "image", + "image_path": "91d03113bf431ff2ce8593f9dcf8c30de45d678fc436461bb3511e1f415c4d07.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 72, + 502, + 103.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 103.66666666666667, + 502, + 135.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 135.33333333333334, + 502, + 167.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 173, + 505, + 273 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "Figure 5: Left: Map of the town featured in the CARLA Town01 dataset. We visualize two video", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "completions by FDM by showing coordinates output by our regressor (discussed in Section 5) for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "each frame. Those corresponding to the initial 36 observed frames are shown in red and those for the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "964 sampled frames are shown in blue. Right: For each completion, we show one of the initially", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "observed frames followed by four of the sampled frames (at positions chosen to show the progression", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "with respect to visible landmarks and marked by black dots on the map). The town’s landmarks", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "are usually sampled with high-fidelity, which is key to allowing the regressor to produce a coherent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "trajectory on the left. However there are sometimes failures: a blue square near the top-right of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 261, + 443, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 443, + 275 + ], + "score": 1.0, + "content": "map shows where the video model “jumped” to a wrong location for a single frame.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "spaced ten apart (1 second at the dataset’s frame rate). Sampled videos occasionally “jump” between", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "disparate locations in the town, resulting in unrealistically large estimated speeds. To measure the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "frequency of these events for each method, we compute the percentage of our point-speed estimates", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 215, + 340 + ], + "score": 1.0, + "content": "that exceed a threshold of", + "type": "text" + }, + { + "bbox": [ + 215, + 327, + 241, + 338 + ], + "score": 0.81, + "content": "1 0 \\mathrm { m / s }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "(the dataset was generated with a maximum simulated speed of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 128, + 349 + ], + "score": 0.54, + "content": "\\mathrm { 3 m / s }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "). We report this metric as the outlier percentage (OP). After filtering out these outliers, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "compute the Wasserstein distance (WD) between the resulting empirical distribution and that of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "original dataset, giving a measure of how well generated videos match the speed of videos in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "dataset. We release the CARLA Town 01 dataset along with code and our trained regression model to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 382, + 216, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 216, + 394 + ], + "score": 1.0, + "content": "allow future comparisons.2", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 408, + 191, + 422 + ], + "lines": [ + { + "bbox": [ + 104, + 406, + 193, + 425 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 193, + 425 + ], + "score": 1.0, + "content": "6 Experiments", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 447 + ], + "score": 1.0, + "content": "We perform our main comparisons on the video completion task. In keeping with Saxena et al. [26],", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "we condition on the first 36 frames of each video and sample the remainder. We present results on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "three datasets: GQN-Mazes [10], in which videos are 300 frames long; MineRL Navigate [13, 26]", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "(which we will from now on refer to as simply MineRL), in which videos are 500 frames long; and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "the CARLA Town01 dataset we release, for which videos are 1000 frames long. We train FDM in all", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 331, + 501 + ], + "score": 1.0, + "content": "cases with the maximum number of represented frames", + "type": "text" + }, + { + "bbox": [ + 331, + 488, + 365, + 498 + ], + "score": 0.9, + "content": "K = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 488, + 506, + 501 + ], + "score": 1.0, + "content": ". We host non-cherry-picked video", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 438, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 438, + 511 + ], + "score": 1.0, + "content": "samples (both conditional and unconditional) from FDM and all baselines online3.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 321, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 322, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 322, + 535 + ], + "score": 1.0, + "content": "Comparison of sampling schemes The relative per-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 322, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 322, + 546 + ], + "score": 1.0, + "content": "formance of different sampling schemes varies sig-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 544, + 321, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 321, + 556 + ], + "score": 1.0, + "content": "nificantly between datasets as shown in Table 1. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 322, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 322, + 568 + ], + "score": 1.0, + "content": "report Fréchet Video Distances (FVDs) [33], a mea-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 321, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 321, + 578 + ], + "score": 1.0, + "content": "sure of how similar sampled completions are to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 576, + 321, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 321, + 589 + ], + "score": 1.0, + "content": "test set, on all datasets. In addition on GQN-Mazes we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 321, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 321, + 600 + ], + "score": 1.0, + "content": "we report the accuracy metric [26], which classifies", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 598, + 321, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 321, + 611 + ], + "score": 1.0, + "content": "videos based on which rooms are visited and measures", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 610, + 321, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 321, + 622 + ], + "score": 1.0, + "content": "how often a completion is given the same class as the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 322, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 322, + 633 + ], + "score": 1.0, + "content": "corresponding test video. 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ModelSampling schemeGQN-MazesMineRLCARLA Town01
FVD AccuracyFVDFVDWDOP
CWVAE [26]CWVAE837±882.6± 0.51573±511610.66644.4
TATS [11]TATS163 ± 2.677.0± 0.8807±143291.64842.4
VDM [16]VDM66.7 ± 1.577.8± 0.5271±8.81690.50116.9
FDM (ours)Autoreg86.4± 5.269.6 ± 1.3281±102220.5790.51
Long-range64.5 ± 1.977.0 ± 1.4267 ±4.02130.6530.47
Hierarchy-253.1 ± 1.182.8± 0.7275±7.71200.3183.28
Hierarchy-353.7 ± 1.983.8 ± 1.1311± 6.81490.3634.53
Ad. hierarchy-255.0 ± 1.483.2 ± 1.3316 ±8.91170.3113.44
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ModelSampling schemeGQN-MazesMineRLCARLA Town01
FVD AccuracyFVDFVDWDOP
CWVAE [26]CWVAE837±882.6± 0.51573±511610.66644.4
TATS [11]TATS163 ± 2.677.0± 0.8807±143291.64842.4
VDM [16]VDM66.7 ± 1.577.8± 0.5271±8.81690.50116.9
FDM (ours)Autoreg86.4± 5.269.6 ± 1.3281±102220.5790.51
Long-range64.5 ± 1.977.0 ± 1.4267 ±4.02130.6530.47
Hierarchy-253.1 ± 1.182.8± 0.7275±7.71200.3183.28
Hierarchy-353.7 ± 1.983.8 ± 1.1311± 6.81490.3634.53
Ad. hierarchy-255.0 ± 1.483.2 ± 1.3316 ±8.91170.3113.44
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To compare against this approach, which we denote VDM, we train both a “frameskip-4”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "and a “frameskip-1” model with architectures identical to our own.4 Since VDM requires two trained", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "DDPMs, we train it for more GPU-hours than FDM despite the fact that FDM is meta-learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "over a far broader task distribution. We also compare against TATS [11], which embeds videos", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "into a discrete latent space before modelling them with a transformers, and the clockwork VAE", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "(CWVAE) [26], a VAE-based model specifically designed to maintain long-range dependencies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 564, + 162, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 162, + 574 + ], + "score": 1.0, + "content": "within video.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 453, + 507, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "Both the diffusion-based methods, FDM and VDM, achieve significantly higher FVD scores than", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "TATS and CWVAE. This may point toward the utility of diffusion models in general for modeling", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "images and video. Table 1 also makes clear the main benefit of FDM over VDM: although there is no", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "sampling scheme for FDM which always outperforms VDM, there is at least one sampling scheme", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 636 + ], + "score": 1.0, + "content": "that outperforms it on each dataset. This speaks to the utility of learning a flexible model like FDM", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 419, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 419, + 647 + ], + "score": 1.0, + "content": "that allows different sampling schemes to be experimented with after training.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 578, + 506, + 647 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 658, + 504, + 681 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "Optimized sampling schemes As mentioned in Section 4, another advantage of FDM is that it", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 670, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 681 + ], + "score": 1.0, + "content": "makes possible a model- and dataset-specific optimization procedure to determine on which frames", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "to condition. Table 2 shows the results when this procedure is used to create sampling schemes for", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 506, + 219 + ], + "score": 1.0, + "content": "different datasets. In the first row we show results where the latent frames are fixed to be those of", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "the Autoreg sampling scheme, and in the second row the latent frames are fixed to match those of", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "Hierarchy-2. On two of the three datasets the best results in Table 1 are improved upon, showing the", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 240, + 258, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 258, + 253 + ], + "score": 1.0, + "content": "utility of this optimization procedure.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 39.5, + "bbox_fs": [ + 106, + 657, + 505, + 681 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 115, + 482, + 174 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 77, + 505, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "score": 1.0, + "content": "Table 2: FVD scores for our sampling schemes with observed indices optimized offline as described", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 88, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 265, + 101 + ], + "score": 1.0, + "content": "in Section 4. We mark with an asterisk", + "type": "text" + }, + { + "bbox": [ + 266, + 89, + 278, + 100 + ], + "score": 0.58, + "content": "( ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 88, + 505, + 101 + ], + "score": 1.0, + "content": "the eight numbers which improve on the corresponding", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 99, + 491, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 491, + 111 + ], + "score": 1.0, + "content": "non-optimized sampling schemes and highlight in bold those that are better than any in Table 1.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 128, + 115, + 482, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 115, + 482, + 174 + ], + "spans": [ + { + "bbox": [ + 128, + 115, + 482, + 174 + ], + "score": 0.973, + "html": "
GQN-MazesMineRLCARLA Town01
Sampling schemeFVDAccuracyFVDFVDWDOP
Opt. autoreg53.6±1.2*80.2±1.2*257±6.8*146*0.452*0.65
Opt. hierarchy-251.1 ± 1.3*84.6±0.7*320 ± 7.01240.3494.11*
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On two of the three datasets the best results in Table 1 are improved upon, showing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 240, + 258, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 258, + 253 + ], + "score": 1.0, + "content": "utility of this optimization procedure.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 279 + ], + "score": 1.0, + "content": "Comparison with training on a single task Training a network with our distribution over training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "tasks could be expected to lead to worse performance on a single task than training specifically for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "that task. To test whether this is the case, we train an ablation of FDM with training tasks exclusively", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "of the type used in our Autoreg sampling scheme, i.e. “predict ten consecutive frames given the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "previous ten.” Tested with the Autoreg sampling scheme, it obtained an FVD of 82.0 on GQN-Mazes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "and 234 on MineRL. As expected given the specialization to a single task, this is better than when", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 330, + 488, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 488, + 344 + ], + "score": 1.0, + "content": "FDM is run with the Autoreg sampling scheme (obtaining FVDs of 86.4 and 281 respectively).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "score": 1.0, + "content": "Ablation on training task distribution To test how important our proposed structured training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "distribution is to FDM’s performance, we perform an ablation with a different task distribution that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 141, + 390 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 141, + 378, + 151, + 388 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 377, + 168, + 390 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 378, + 178, + 388 + ], + "score": 0.8, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "from uniform distributions instead of our proposed structured task distribution We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "provide full details in the appendix, but report here that switching away form our structured training", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "distribution made the FVD scores worse on all five tested sampling schemes on both GQN-Mazes", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 313, + 423 + ], + "score": 1.0, + "content": "and MineRL. The reduction in the average FVD was", + "type": "text" + }, + { + "bbox": [ + 313, + 410, + 333, + 421 + ], + "score": 0.88, + "content": "3 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 410, + 414, + 423 + ], + "score": 1.0, + "content": "on GQN-Mazes and", + "type": "text" + }, + { + "bbox": [ + 415, + 410, + 434, + 421 + ], + "score": 0.88, + "content": "5 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "on MineRL. This", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 421, + 424, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 424, + 434 + ], + "score": 1.0, + "content": "implies that our structured training distribution has a significant positive effect.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 194, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 196, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 196, + 465 + ], + "score": 1.0, + "content": "7 Related work", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "Some related work creates conditional models by adapting the sampling procedure of an unconditional", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "DDPM [30, 18, 21, 16]. These approaches require approximations and the more direct approach", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "that we use (explcitly training a conditional DDPM) was shown to have benefits by Tashiro et al.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "[32]. We consider further comparison of these competing approaches to be outside the focus of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "this work, which is on modeling a small portion of video frames at a time, essentially performing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 282, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 282, + 543 + ], + "score": 1.0, + "content": "marginalization in addition to conditioning.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "There are a number of approaches in the literature which use VAEs rather than DDPMs for video mod-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "elling. Babaeizadeh et al. [2] use a VAE model which predicts frames autoregressively conditioned", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "on a global time-invariant latent variable. A related approach by Denton and Fergus [7] also uses a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "VAE with convolutional LSTM architectures in both the encoder and decoder. Unlike Babaeizadeh", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "et al. [2] the prior is learned and a different latent variable is sampled for each frame. Babaeizadeh", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "et al. [3] use a VAE with one set of latent variables per frame and inter-frame dependencies tracked", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "by a two-layer LSTM. Their architecture intentionally overfits to the training data, which when", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "coupled with image augmentations techniques achieves SOTA on various video prediction tasks. Kim", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "score": 1.0, + "content": "et al. [20] use a variational RNN [5] with a hierarchical latent space that includes binary indicator", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "variables which specify how the video is divided into a series of subsequences. Both Villegas et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "[35] and Wichers et al. [37] target long-term video prediction using a hierarchical variational LSTM", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "architecture, wherein high-level features such as landmarks are predicted first, then decoded into", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "low-level pixel space. The two approaches differ in that Villegas et al. [35] requires ground truth", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "landmark labels, while [37] removes this dependence using an unsupervised adversarial approach.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Fully GAN-based video models have also been proposed [1, 6] but generally suffer from “low quality", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 289, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 289, + 723 + ], + "score": 1.0, + "content": "frames or low number of frames or both” [1].", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 39.5 + } + ], + "page_idx": 8, + "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": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 115, + 482, + 174 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 77, + 505, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "score": 1.0, + "content": "Table 2: FVD scores for our sampling schemes with observed indices optimized offline as described", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 88, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 265, + 101 + ], + "score": 1.0, + "content": "in Section 4. We mark with an asterisk", + "type": "text" + }, + { + "bbox": [ + 266, + 89, + 278, + 100 + ], + "score": 0.58, + "content": "( ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 88, + 505, + 101 + ], + "score": 1.0, + "content": "the eight numbers which improve on the corresponding", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 99, + 491, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 491, + 111 + ], + "score": 1.0, + "content": "non-optimized sampling schemes and highlight in bold those that are better than any in Table 1.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 128, + 115, + 482, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 115, + 482, + 174 + ], + "spans": [ + { + "bbox": [ + 128, + 115, + 482, + 174 + ], + "score": 0.973, + "html": "
GQN-MazesMineRLCARLA Town01
Sampling schemeFVDAccuracyFVDFVDWDOP
Opt. autoreg53.6±1.2*80.2±1.2*257±6.8*146*0.452*0.65
Opt. hierarchy-251.1 ± 1.3*84.6±0.7*320 ± 7.01240.3494.11*
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To test whether this is the case, we train an ablation of FDM with training tasks exclusively", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "of the type used in our Autoreg sampling scheme, i.e. “predict ten consecutive frames given the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "previous ten.” Tested with the Autoreg sampling scheme, it obtained an FVD of 82.0 on GQN-Mazes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "and 234 on MineRL. As expected given the specialization to a single task, this is better than when", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 330, + 488, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 488, + 344 + ], + "score": 1.0, + "content": "FDM is run with the Autoreg sampling scheme (obtaining FVDs of 86.4 and 281 respectively).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 263, + 506, + 344 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "score": 1.0, + "content": "Ablation on training task distribution To test how important our proposed structured training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "distribution is to FDM’s performance, we perform an ablation with a different task distribution that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 141, + 390 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 141, + 378, + 151, + 388 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 377, + 168, + 390 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 378, + 178, + 388 + ], + "score": 0.8, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "from uniform distributions instead of our proposed structured task distribution We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "provide full details in the appendix, but report here that switching away form our structured training", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "distribution made the FVD scores worse on all five tested sampling schemes on both GQN-Mazes", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 313, + 423 + ], + "score": 1.0, + "content": "and MineRL. The reduction in the average FVD was", + "type": "text" + }, + { + "bbox": [ + 313, + 410, + 333, + 421 + ], + "score": 0.88, + "content": "3 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 410, + 414, + 423 + ], + "score": 1.0, + "content": "on GQN-Mazes and", + "type": "text" + }, + { + "bbox": [ + 415, + 410, + 434, + 421 + ], + "score": 0.88, + "content": "5 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "on MineRL. This", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 421, + 424, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 424, + 434 + ], + "score": 1.0, + "content": "implies that our structured training distribution has a significant positive effect.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 354, + 506, + 434 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 194, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 196, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 196, + 465 + ], + "score": 1.0, + "content": "7 Related work", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 476, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "Some related work creates conditional models by adapting the sampling procedure of an unconditional", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "DDPM [30, 18, 21, 16]. These approaches require approximations and the more direct approach", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "that we use (explcitly training a conditional DDPM) was shown to have benefits by Tashiro et al.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "[32]. We consider further comparison of these competing approaches to be outside the focus of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "this work, which is on modeling a small portion of video frames at a time, essentially performing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 282, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 282, + 543 + ], + "score": 1.0, + "content": "marginalization in addition to conditioning.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 475, + 506, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "There are a number of approaches in the literature which use VAEs rather than DDPMs for video mod-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "elling. Babaeizadeh et al. [2] use a VAE model which predicts frames autoregressively conditioned", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "on a global time-invariant latent variable. A related approach by Denton and Fergus [7] also uses a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "VAE with convolutional LSTM architectures in both the encoder and decoder. Unlike Babaeizadeh", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "et al. [2] the prior is learned and a different latent variable is sampled for each frame. Babaeizadeh", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "et al. [3] use a VAE with one set of latent variables per frame and inter-frame dependencies tracked", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "by a two-layer LSTM. Their architecture intentionally overfits to the training data, which when", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "coupled with image augmentations techniques achieves SOTA on various video prediction tasks. Kim", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "score": 1.0, + "content": "et al. [20] use a variational RNN [5] with a hierarchical latent space that includes binary indicator", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "variables which specify how the video is divided into a series of subsequences. Both Villegas et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "[35] and Wichers et al. [37] target long-term video prediction using a hierarchical variational LSTM", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "architecture, wherein high-level features such as landmarks are predicted first, then decoded into", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "low-level pixel space. The two approaches differ in that Villegas et al. [35] requires ground truth", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "landmark labels, while [37] removes this dependence using an unsupervised adversarial approach.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Fully GAN-based video models have also been proposed [1, 6] but generally suffer from “low quality", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 289, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 289, + 723 + ], + "score": 1.0, + "content": "frames or low number of frames or both” [1].", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 547, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 71, + 179, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 181, + 87 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 181, + 87 + ], + "score": 1.0, + "content": "8 Discussion", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 505, + 110 + ], + "score": 1.0, + "content": "We have defined and empirically explored a new method for generating photorealistic videos with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 108, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 120 + ], + "score": 1.0, + "content": "long-range coherence that respects and efficiently uses fixed, finite computational resources. Our", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 131 + ], + "score": 1.0, + "content": "approach outperforms prior work on long-duration video modeling as measured by quantitative and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "semantically meaningful metrics and opens up several avenues for future research. For one, similar", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "to using DDPMs for image generation, our method is slow to sample from (it takes approximately 16", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 165 + ], + "score": 1.0, + "content": "minutes to generate a 300 frame video on a GPU). Ideas for making sampling faster by decreasing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 430, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 430, + 174 + ], + "score": 1.0, + "content": "the number of integration steps [25, 29, 39] could be applied to our video model.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "On a different note, consider the datasets on which our artifact was trained. In each there was a policy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "for generating the sequences of actions that causally led to the frame-to-frame changes in camera", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "pose. In MineRL the video was generated by agents that were trained to explore novel Minecraft", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "worlds to find a goal block approximately 64 meters away [13]. The CARLA data was produced", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "by a camera attached to an agent driven by a low level proportional–integral–derivative controller", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "following waypoints laid down by a high level planner that was given new, random location goals to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "drive to intermittently. 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Near-term", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "future work could involve adding other modalities (e.g. audio) to FDM as well as explicitly adding", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "actions and rewards, transforming our video generative model into a vision-based world model in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "the reinforcement learning sense [17, 19]. Furthermore, we point out that FDM trained on CARLA", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "Town01 is in theory capable of creating 100-second videos conditioned on both the first and final", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "frame. Doing so can be interpreted as running a “visual” controller which proposes a path between a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "current state and a specified goal. 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We acknowledge the support of the Natural Sciences and Engineering Research", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "Council of Canada (NSERC), the Canada CIFAR AI Chairs Program, and the Intel Parallel Com-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "score": 1.0, + "content": "puting Centers program. Additional support was provided by UBC’s Composites Research Net-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "work (CRN), and Data Science Institute (DSI). This research was enabled in part by technical", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "support and computational resources provided by WestGrid (www.westgrid.ca), Compute Canada", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "(www.computecanada.ca), and Advanced Research Computing at the University of British Columbia", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "(arc.ubc.ca). WH acknowledges support by the University of British Columbia’s Four Year Doctoral", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 471, + 218, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 218, + 486 + ], + "score": 1.0, + "content": "Fellowship (4YF) program.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 501, + 163, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 499, + 165, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 165, + 515 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 111, + 520, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 111, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 111, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "[1] Nuha Aldausari, Arcot Sowmya, Nadine Marcus, and Gelareh Mohammadi. 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Ideas for making sampling faster by decreasing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 430, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 430, + 174 + ], + "score": 1.0, + "content": "the number of integration steps [25, 29, 39] could be applied to our video model.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 96, + 506, + 174 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "On a different note, consider the datasets on which our artifact was trained. In each there was a policy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "for generating the sequences of actions that causally led to the frame-to-frame changes in camera", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "pose. In MineRL the video was generated by agents that were trained to explore novel Minecraft", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "worlds to find a goal block approximately 64 meters away [13]. 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In both cases our video model had no access to either the policy or the specific", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "actions taken by these agents and, so, in a formal sense, our models integrate or marginalize over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "actions drawn from the stochastic policy used to generate the videos in the first place. Near-term", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "future work could involve adding other modalities (e.g. audio) to FDM as well as explicitly adding", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "actions and rewards, transforming our video generative model into a vision-based world model in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "the reinforcement learning sense [17, 19]. Furthermore, we point out that FDM trained on CARLA", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "Town01 is in theory capable of creating 100-second videos conditioned on both the first and final", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "frame. Doing so can be interpreted as running a “visual” controller which proposes a path between a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "current state and a specified goal. Preliminary attempts to run in FDM in this way yielded inconsistent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 341, + 441, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 441, + 355 + ], + "score": 1.0, + "content": "results but we believe that this could be a fruitful direction for further investigation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 177, + 506, + 355 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 366, + 186, + 378 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 187, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 187, + 379 + ], + "score": 1.0, + "content": "Acknowledgments", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "We would like to thank Inverted AI, and especially Alireza Morsali, for generating the CARLA", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "Town01 dataset. We acknowledge the support of the Natural Sciences and Engineering Research", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "Council of Canada (NSERC), the Canada CIFAR AI Chairs Program, and the Intel Parallel Com-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "score": 1.0, + "content": "puting Centers program. Additional support was provided by UBC’s Composites Research Net-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "work (CRN), and Data Science Institute (DSI). This research was enabled in part by technical", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "support and computational resources provided by WestGrid (www.westgrid.ca), Compute Canada", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "(www.computecanada.ca), and Advanced Research Computing at the University of British Columbia", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "(arc.ubc.ca). 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ModelSampling schemeGQN-MazesMineRLCARLA Town01
FVD AccuracyFVDFVDWDOP
CWVAE [26]CWVAE837±882.6± 0.51573±511610.66644.4
TATS [11]TATS163 ± 2.677.0± 0.8807±143291.64842.4
VDM [16]VDM66.7 ± 1.577.8± 0.5271±8.81690.50116.9
FDM (ours)Autoreg86.4± 5.269.6 ± 1.3281±102220.5790.51
Long-range64.5 ± 1.977.0 ± 1.4267 ±4.02130.6530.47
Hierarchy-253.1 ± 1.182.8± 0.7275±7.71200.3183.28
Hierarchy-353.7 ± 1.983.8 ± 1.1311± 6.81490.3634.53
Ad. hierarchy-255.0 ± 1.483.2 ± 1.3316 ±8.91170.3113.44
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GQN-MazesMineRLCARLA Town01
Sampling schemeFVDAccuracyFVDFVDWDOP
Opt. autoreg53.6±1.2*80.2±1.2*257±6.8*146*0.452*0.65
Opt. hierarchy-251.1 ± 1.3*84.6±0.7*320 ± 7.01240.3494.11*
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sha256:48d73b4ba97d996be70dd6b49effd4010f53a077b49a913292312297a79280b9 +size 10618 diff --git a/parse/dev/K48UYo0glaJ/K48UYo0glaJ.md b/parse/dev/K48UYo0glaJ/K48UYo0glaJ.md new file mode 100644 index 0000000000000000000000000000000000000000..a7b95f4251a57acfdc1b5d192a0b1c27aac6552b --- /dev/null +++ b/parse/dev/K48UYo0glaJ/K48UYo0glaJ.md @@ -0,0 +1,406 @@ +# Theseus: A Library for Differentiable Nonlinear Optimization + +Luis Pineda1, Taosha Fan1, Maurizio Monge2, Shobha Venkataraman1, Paloma Sodhi1, Ricky T. Q. Chen1, Joseph Ortiz1, Daniel DeTone2, Austin Wang1, Stuart Anderson1, Jing Dong2, Brandon Amos1, Mustafa Mukadam1 + +1Meta AI, 2Reality Labs Research + +# Abstract + +We present Theseus, an efficient application-agnostic open source library for differentiable nonlinear least squares (DNLS) optimization built on PyTorch, providing a common framework for end-to-end structured learning in robotics and vision. Existing DNLS implementations are application specific and do not always incorporate many ingredients important for efficiency. Theseus is application-agnostic, as we illustrate with several example applications that are built using the same underlying differentiable components, such as second-order optimizers, standard costs functions, and Lie groups. For efficiency, Theseus incorporates support for sparse solvers, automatic vectorization, batching, GPU acceleration, and gradient computation with implicit differentiation and direct loss minimization. We do extensive performance evaluation in a set of applications, demonstrating significant efficiency gains and better scalability when these features are incorporated. Project page: https://sites.google.com/view/theseus-ai/ + +# 1 Introduction + +Reconciling traditional approaches with deep learning to leverage their complementary strengths is a common thread in a large body of recent work in robotics. In particular, an emerging trend is to differentiate through nonlinear least squares (NLS) [1] which is a second-order optimization formulation at the heart of many problems in robotics [2–7] and vision [8–13]. Optimization layers as inductive priors in neural models have been explored in machine learning with convex optimization [14, 15] and in meta learning with gradient descent [16, 17] based first-order optimization. + +Differentiable nonlinear least squares (DNLS) provides a general scheme to encode inductive priors, as the objective function can be partly parameterized by neural models and partly with engineered domain-specific differentiable models. Here, as illustrated in Fig. 1, input tensors define a sum of weighted squares objective function and output tensors are minima of that objective. Such implicit layers [18] are in contrast to typical (explicit) layers that take input tensors through a linear transformation and some element-wise nonlinear activation function. + +![](images/f32571828e106eb15cfc6d5807fd15ee3907f8a37214f72613464968ff0d3eb6.jpg) +Figure 1: Theseus enables building custom, efficient DNLS layers that support end-to-end structured learning. + +36th Conference on Neural Information Processing Systems (NeurIPS 2022). + +The ability to compute gradients end-to-end is retained by differentiating through the optimizer which allows neural models to train on the final task loss, while also taking advantage of priors captured by the optimizer. The flexibility of such a scheme has led to promising state-of-the-art results in a wide range of applications such as structure from motion [19], motion planning [20], SLAM [21, 22], bundle adjustment [23], state estimation [24, 25], image alignment [26] with other applications like manipulation and tactile sensing [27, 28], control [29], human pose tracking [30, 31] to be explored. However, existing implementations from above are application specific, common underlying tools like optimizers get reimplemented, and features like sparse solvers, batching, and GPU support that impact efficiency are not always included. This has led to a fragmented literature where it is difficult to start work on new ideas or to build on the progress of prior work. + +To address this gap, we present Theseus, an open source library for differentiable nonlinear least squares optimization built on PyTorch. Theseus provides an efficient application-agnostic interface that consolidates recent efforts and catalyzes future progress in the domain of structured end-to-end learning for robotics and vision. Our contributions are summarized below. + +Application agnostic interface. Our implementation provides an easy to use interface to build custom optimization layers and plug them into any neural architecture. (i) The layer can be constructed from a set of available second-order optimizers like Gauss-Newton, Levenberg–Marquardt (with adaptive damping) and Dogleg, and a nonlinear least squares objective. (ii) The objective can be constructed with learnable or hand-specified cost functions, either by applying one of many common costs already provided in the library, or by building custom costs in-place with support for automatic differentiation through PyTorch [32]. (iii) We also provide differentiable Lie groups for representing 2D/3D positions and rotations [33], and differentiable kinematics wrapping over an existing library [34] for representing robot models. More details are described in Sec. 3. + +Efficiency based design. Efficiency is a central design consideration and we make several advancements in improving computation times and memory consumption. (i) As common in prior work, an optimizer implementation using PyTorch’s native linear solver would use a dense representation for solving the linear system within the nonlinear optimization. In practice, these optimization problems often have a considerable amount of sparsity that can be exploited [35–38]. In Theseus, we implement sparse linear solvers that are differentiable end-to-end and make them efficient with custom CPU and CUDA backends to support batching and GPU acceleration. (ii) Beyond sparse solvers, we extend batching and GPU support to all features in the library and add automatic vectorization of cost functions and other operations to significantly boost efficiency. (iii) Finally, we introduce implicit differentiation [39] and direct loss minimization [40, 41], which have been previously applied to only first order optimizers like gradient descent and convex optimization, to a new class of second-order optimizers. This goes beyond prior work with nonlinear least squares that currently only support differentiation with standard unrolling, which is known to have challenges with compute, memory, and vanishing gradients. More details are described in Sec. 4. + +Highlights of results. Together, the application-agnostic features let users easily set up a variety of problems like pose graph optimization, tactile state estimation, bundle adjustment, motion planning, and homography estimation, all of which are included as examples in the open source code and described in Sec. 3.1. In evaluations, we find that on a standard GPU, Theseus with a sparse solver is much faster and requires significantly less memory than a dense solver, and when solving a batch of large problems the forward pass of Theseus is up to $2 0 \mathrm { x }$ faster than state-of-the-art $\mathrm { C } { + } { + }$ based solver Ceres that has limited GPU support and does not support batching and end-to-end learning. We also compare all backward modes to find that with increasing number of optimization iterations, compute and memory increases linearly for unrolling and stays constant for implicit differentiation, while the latter also provides better gradients. More details are described in Sec. 5. + +# 2 Background and related work + +Nonlinear least squares (NLS) is an optimization problem [1] that finds optimization variables $\theta$ + +$$ +\theta ^ { \star } = \operatorname * { a r g m i n } _ { \theta } S ( \theta ) , \qquad S ( \theta ) = \frac { 1 } { 2 } \sum _ { i } | | r _ { i } ( \theta ^ { i } ) | | ^ { 2 } = \frac { 1 } { 2 } \sum _ { i } | | w _ { i } c _ { i } ( \theta ^ { i } ) | | ^ { 2 } +$$ + +where the objective $S ( \theta )$ is a sum of squared vector-valued residual terms $r _ { i }$ , each a function of $\theta ^ { i } \subset \theta$ that are (non-disjoint) subsets of the optimization variables $\theta = \{ \theta _ { j } \}$ . Any variable $\theta _ { j }$ is a manifold object; for example, a Euclidean vector or a matrix Lie group. For flexibility, we represent a residual $\dot { r _ { i } } ( \theta ^ { i } ) = w _ { i } c _ { i } ( \bar { \theta } ^ { i } )$ as a product of a matrix weight $w _ { i }$ and vector cost $c _ { i }$ . Robotics and vision have used this general optimization formulation to tackle many applications [3, 4]. For example, costs capture sensor measurement errors and physical constraints to optimize camera, robot, object, or human poses in estimation and tracking problems like simultaneous localization and mapping (SLAM) [42], structure from motion [13], bundle adjustment [8], visual inertial odometry [2], articulated tracking [12], contact odometry in legged locomotion [27], 3D pose and shape reconstruction of humans [30, 31] or objects [10]. Similarly, costs can also capture constraints and desired future goals to find robot states or actions in motion planning [5], dynamics [6], and control [29] problems. + +Solving NLS. Problems represented by Eq. (1) are solved by iteratively linearizing the nonlinear objective around the current variables to get the linear system $\begin{array} { r } { ( \sum _ { i } J _ { i } ^ { \top } J _ { i } ) \delta \theta = ( \bar { \sum _ { i } } J _ { i } ^ { \top } r _ { i } ) } \end{array}$ , then solving the linear system to find the update $\delta \theta$ , and finally updating the variables $\theta \theta - \delta \theta$ , until convergence. Note that in the update the minus operation is more generally a retraction mapping for non-Euclidean variables. In the linear system, $\bar { J } _ { i } = [ \partial r _ { i } / \partial \theta ^ { i } ]$ are the Jacobians of residuals with respect to the variables and the iterative method above, called Gauss-Newton (GN), is a nonlinear optimizer that is (approximately) second-order, since $\begin{array} { r } { \dot { H } = ( \sum _ { i } J _ { i } ^ { \top } J _ { i } ) } \end{array}$ represents the approximate Hessian. To improve robustness and convergence, variations like Levenberg–Marquardt (LM) damp the linear system, while others use a trust region and adjust step size for the update with line search (e.g., Dogleg). Please refer to [1, 43] for an in-depth exploration. In most applications discussed above the objective structure gives rise to a sparse Hessian, since not all costs depend on all variables. Several general purpose frameworks [35–38] have been built that leverage this sparsity property to efficiently solve the sparse linear system in every iteration of the nonlinear optimization. While these frameworks were not built for deep learning, they are highly efficient and performant on CPU. + +NLS with learning. Data driven learning has been explored to address challenges in hand crafting costs or features for costs, finding weights to balance different costs, or to find initializations that lead to better convergence. Some examples include, learning object shape code [44] or environment depth code [45] for SLAM [46], learning motion priors for planning to manipulate articulated objects [47], learning relative pose from tactile images to estimate object state during pushing [28], and semantic 2D segmentation fused in 3D mesh for semantic SLAM [7]. These approaches only train features on a surrogate or intermediate loss and then apply optimization at inference where the true downstream task loss is available but not utilized. To take full advantage of end-to-end learning, latest approaches thus are redesigning the optimization to be differentiable. + +Differentiable NLS (DNLS) solves the optimization in Eq. (1) and also provides gradients of the solution $\theta ^ { \star }$ with respect to any upstream neural model parameters $\phi$ that parameterize the objective $S ( \theta ; \phi )$ and in turn any costs $\boxed { c _ { i } ( \theta ^ { i } ; \phi ) }$ , weights $w _ { i } ( \phi )$ , or initialization for variables $\theta _ { i n i t } ( \phi )$ . The goal is to learn these parameters $\phi$ end-to-end with a downstream learning objective $L$ defined as a function of $\theta ^ { \star }$ . This results in a bilevel optimization setup as shown in Fig. 1 + +$$ +\theta ^ { \star } ( \phi ) = \operatorname * { a r g m i n } _ { \theta } S ( \theta ; \phi ) , \qquad \mathrm { o u t e r ~ l o o p : ~ } \phi ^ { \star } = \operatorname * { a r g m i n } _ { \phi } L ( \theta ^ { \star } ( \phi ) ) +$$ + +where the inner loop is DNLS and the outer loop is gradient descent class of optimization that is standard in deep learning. The outer loop performs update $\phi \phi + \delta \phi$ by computing $\delta \phi$ using gradients ${ \partial \theta ^ { \star } } / { \bar { \partial } \phi }$ through inner loop DNLS. Note that more generally the learning objective i.e. outer loss $L$ can also depend on other quantities like neural model parameters downstream of $\theta ^ { * }$ , but we omit them here for clarity. + +Recent works with DNLS have outperformed optimization only or learning only methods by combining the strengths of classical methods with deep learning. For example, learning features for costs to represent depth in bundle adjustment [23] and monocular stereo [48] where an initialization network also learns to predict depth and pose, learning cost weights like motion model weights in video to depth estimation [19], obstacle avoidance weights in 2D motion planning from occupancy images [20], learning robust loss weights in image alignment [26] and state-of-the-art dense SLAM [22], and confidence weights for feature matching to optimize camera pose [49]. Other works, backpropagate reconstruction error to sensor model in a SLAM system [21], solve large scale bundle adjustment on a GPU [50], and learn sensor and dynamics models for 2D visual object tracking and visual odometry [24]. These implementations however, are application specific which has led to repeated work in building DNLS where features like learnable costs and weights, Lie groups, and kinematics are not always present. Additionally, features that have a significant impact on performance, like sparsity and vectorization of costs are only considered by some [24, 50, 51] or in the case of implicit differentiation for NLS optimization, have not yet been explored. + +# 3 Application agnostic interface + +Given the lack of a common and efficient framework for DNLS an important goal of Theseus is to provide an application-agnostic interface. In this section, we describe how we enable this with an easy-to-use core API, standard cost functions, and features like Lie groups and kinematics, and illustrate several examples using this interface. We discuss design for efficiency in the next section. + +![](images/4adcd5b9bfba8d359b492ca749686c0e143e3f7523ae53c2ee192ad37fecb287.jpg) + +The core API lets users focus on describing the DNLS problem and their interaction with the outer loss $L$ and parameters $\phi$ within any broader PyTorch model, while the solution and differentiation are seamlessly taken care of under-the-hood. The basic components of the core API are described below with the help of a simple example in Listing 1 (see App. B for more details on the example): + +• Variable: refers to either optimization variables, $\theta$ , or auxiliary variables (those constant with respect to $S$ , e.g., parameters $\phi$ or data tensors), which are named wrappers of torch batched tensors stored in Variable.tensor (lines 3-5). +• CostFunction: defines costs $c _ { i }$ (lines 12-14) and are also responsible for declaring which of its variables are optimization and which are auxiliary (lines 8-9), +• CostWeight: defines weights $w _ { i }$ associated with cost $c _ { i }$ (line 14). +• Objective: defines $S ( \theta ; \phi )$ , and thus the structure of an optimization problem (lines 11, 15) by holding all cost functions and weights, and their associated variables. These are implicitly obtained when a CostFunction is added to the Objective, and Variable names are used to infer which are shared by one or more CostFunction. +• Optimizer: is the inner loop optimization algorithm (e.g. Gauss-Newton) that finds the solution $\theta ^ { \star }$ given objective $S$ (line 16). +• TheseusLayer: encapsulates Optimizer and Objective, and serves as the interface between the DNLS block and other torch modules upstream or downstream (line 16). + +The interface between the inner loop optimization and the outer loop’s parameters and loss occurs via TheseusLayer.forward (lines 21-23). This receives as input a dictionary mapping variable names to torch tensors, which Theseus then uses to populate the corresponding Variable with the tensor mapped to its name. With the input dictionary users can provide initial values for the optimization variables, data tensors, or current values for parameters $\phi$ before running the inner loop optimization. The output of forward is another dictionary that maps variable names to tensors with their optimal values found in the inner loop (lines 21, 24); auxiliary variables are not modified during the forward pass. The output tensors can then be combined with other torch modules downstream to compute $L$ while maintaining the full differentiable computation graph (lines 24-26). + +We currently provide Gauss-Newton, Levenberg–Marquardt (with adaptive damping), and Dogleg as nonlinear Optimizer for the inner loop, with the ability to easily add support for more optimizers in the future. Listing 1 uses AutoDiffCostFunction to construct an in-place CostFunction (line 12) which allows automatically calculating Jacobians $J _ { i }$ with PyTorch (see App. C). Beyond this, in the library we include standard cost functions with analytical Jacobians broadly used in many applications, like Gaussian measurements, reprojection error, relative pose measurement, motion models, and collision costs. We also include a variety of robust loss functions, useful for example in handling outliers [52], which can be easily integrated with CostFunction. Next we describe support for Lie groups and kinematics. + +Differentiable Lie groups. Lie groups are widely used in robotics and vision to represent 2D/3D positions and rotations [33]. Due to their non-Euclidean geometry, it is difficult to apply them to deep learning, which primarily operates with Euclidean tensors, but recently there is growing interest in making them compatible [24, 53–57]. LieTorch [54] generalizes automatic differentiation on the Lie group tangent space through local parameterization around the identity, but the implementation is complex since every operation requires a custom kernel. In contrast, Theseus computes common Lie group operators, e.g., the exponential and logarithm map, inverse, composition, etc., in closed form, and provides their corresponding analytical derivatives on the tangent space. Following [58], we also implement a projection operator that allows us to project gradients computed by PyTorch’s autodiff to the tangent space and use them to easily compute Jacobians and update Lie group variables correctly; a similar strategy has also been implemented in [59]. Additionally, our Lie group implementation includes a heuristic extension that allows using any of PyTorch’s first-order optimizers on nonEuclidean manifolds with minimal code changes. All of these make it easy and straightforward to run optimization and train neural networks with Lie groups variables. More details in App. D. + +Differentiable kinematics. Many problems such as motion planning or state estimation on high degree of freedom robots like arms or mobile manipulators, involve computation of robot kinematics for collision avoidance or computing distance of end effector to goal. Theseus provides a differentiable implementation of forward kinematics by wrapping over Differentiable Robot Model [34], which builds a differentiable kinematics function from a standard robot model file. Gradients are computed through autodiff, while we also provide a more efficient, analytical manipulator Jacobian. This module can be used within any CostFunction in Theseus. + +# 3.1 Example applications + +To illustrate the versatility of Theseus, we include a number of example DNLS applications below with more details in App. E. Crucially, to implement these with Theseus, most of the effort is only in defining application-specific components such as data management, neural models, or custom CostFunction. With these defined, putting the full DNLS block together is a few lines of code to setup a TheseusLayer and an outer loop, similar to the simple example in Listing 1. + +Pose graph optimization (PGO) estimates poses from their noisy relative measurements [60]. With DNLS we learn the radius of a Welsh robust cost function for outlier rejection, using the difference between estimated and ground truth poses as the outer loss on a synthetic dataset. + +Tactile state estimation follows [28], which estimates 2D poses of an object pushed by a robot hand with an image-based tactile sensor [61]. A neural network that predicts relative pose between hand and object from tactile images is learned end-to-end through the TheseusLayer. + +Bundle adjustment is the problem of optimizing a 3D reconstruction formed by a set of camera images and a set of landmarks observed and matched across the images [62]. We learn the radius of a soft-kernel that penalizes outlier observations, using the average frame pose error as outer loss. + +Motion planning considers a differentiable version of the GPMP2 planning algorithm, inspired by [20], where the outer loss tries to match expert demonstrations. Here we learn a model for initializing optimization variables, and we include the inner loop objective as a term in the outer loss. + +Homography estimation. Homography is a linear transformation between corresponding points in two images and can be solved by minimising a dense photometric loss. Robustness to lighting and viewpoint change can be improved with a feature-metric loss based on CNN features [63–68]. In our outer loop, we train a CNN to produce robust features for image alignment. + +# 4 Efficiency based design + +Theseus enables several different applications with a general interface. Compute and memory efficiency are central to making its usage practical. Next, we explain design considerations to support batching and vectorization, sparsity, and backward modes for differentiation, which we demonstrate boost performance in the evaluations section. + +# 4.1 Batching and vectorization + +Parallel processing is important to improve computational efficiency in machine learning and optimization. In Theseus, we enable two levels of parallelization. First, Theseus natively supports solving a batch of DNLS in parallel, thus fitting seamlessly in the PyTorch framework, where training and inferences on batches is the standard. Second, inspired by DeepLM [50], and noting that lots of the operations such as costs, gradients/Jacobian computation, and variable + +![](images/15fbb0913b91156eb3a69e3af4fdd5d459dc9750a0150f6fea435a8603a80e2f.jpg) +Figure 2: Speedup with automatic vectorization on PGO. Black dotted line is without vectorization. + +updates only differ from each other in terms of the input data, we make use of the single-instructionmultiple-data (SIMD) protocol by automatically detecting and vectorizing operations of the same type, significantly reducing computation overhead. Using the PGO example, Fig. 2 shows that Theseus achieves significant speedup with automatic vectorization both for forward and backward pass. Note that there is an application-dependent trade-off between memory and speed; here the memory use increases by up to $\sim 8 2 \%$ for forward and $\sim 5 5 \%$ for backward. + +# 4.2 Handling sparsity with linear solvers beyond PyTorch + +Solving NLS requires solving a sequence of linear systems to obtain descent directions. As discussed in Sec. 2, these systems are generally sparse and can be solved much more efficiently if not treated as dense. Theseus includes differentiable sparse solvers that take advantage of the sparsity, complementing PyTorch’s native dense solvers. Importantly, Theseus seamlessly takes care of assembling the cost functions and variables in the objective into sparse data structures that our linear solvers can consume, without any extra burden on the user. Currently, we provide three sparse solvers: (i) a CPU-based solver that relies on CHOLMOD [69], (ii) cudaLU, which is based on the cuSolverRF package that is part of Nvidia’s cuSolver library provided with CUDA, and (iii) BaSpaCho, our novel batched sparse Cholesky solver with GPU support. As a bonus feature, we provide access to these solvers as standalone PyTorch functions, so they can be used to solve sparse matrices arising outside of NLS or DNLS optimization. + +CHOLMOD-based solver. CHOLMOD [69] achieves state-of-the-art performance on computation of the Cholesky decomposition of sparse matrices. It exploits parallelism by grouping sparse entries to take advantage of high-performance multi-threaded dense matrix operations in BLAS/LAPACK libraries. CHOLMOD has some limited support for GPU for some of its operations, but the algorithm is strongly CPU-based, and the user is expected to provide matrix data on the CPU. One convenient feature is computing the symbolic analysis of a sparse matrix pattern as a separate step and creating a symbolic decomposition object that can be used for all subsequent factorizations. We also take advantage of builtin functionality for sparse multiplication and only provide the Jacobian matrix $J$ to solve for the Hessian $H = j ^ { \top } J$ . Two limitations of the library with respect to Theseus are, first, the lack of proper GPU support, which forces us to provide matrix data on the CPU, and, second, the lack of batching, which requires us to loop to solve every problem in the batch independently. On the other hand, since it runs on CPU, it has less memory restrictions than GPU-based solvers (see Sec. 5.1). + +cudaLU solver. cuSolverRF is designed to accelerate the solution of sets of linear systems by fast LU refactorization when given new coefficients for the same sparsity pattern. To take advantage of this, we implemented custom CUDA kernels for batched sparse matrix-matrix and matrix-vector products, and for solving a batch of sparse linear systems using LU factorization from cuSolverRF. Although this solver leads to a substantial performance boost over PyTorch’s dense solver (see Sec. 5.1), the closed-source nature of cuSOLVER results in some challenges and limitations: (i) cuSolverRF does not support separate symbolic decomposition and numeric contexts, so it’s not possible to use the same symbolic decomposition to hold in memory separate factors. Since this is necessary in Theseus for unrolling of the inner loop, we work around this limitation by creating a pool of contexts, and we use the least recently used context for factorization. As a consequence, the number of contexts must be set according to the number of iterations that need to be unrolled; (ii) The batch size is fixed once a context is created. Since recreating the contexts is an expensive operation, it means that the batch size has to be constant over the course of outer loop optimization; (iii) It relies on LU factorization, which for symmetric matrices (the case of Theseus) is less efficient than using Cholesky decomposition. + +BaSpaCho solver. Batched Sparse Cholesky (BaSpaCho) is a novel open-source sparse Cholesky solver designed for Theseus with support for batching (https://github.com/facebookresearch/ baspacho). BaSpaCho implements the supernodal Cholesky algorithm [70] to achieve state-of-the art performance by exploiting dense operations via BLAS/cuBLAS. This is achieved by building an elimination tree and then clustering column blocks with similar sparsity patterns. These blocks form nodes of the elimination tree and allow dense operations. In BaSpaCho, the dense operations are dispatched to BLAS (on CPU) or cuBLAS (on GPU), with additional support added on top for batching matrix operations with the same sparsity patterns. In problems with very sparse matrices, like bundle adjustment [8], the supernodal algorithm employed in state-of-the-art solvers [37] is unable to eliminate columns of parameter blocks simultaneously. Thus, past work has resorted to the Schur complement trick [71] to send a reduced problem to the sparse solver. However, this logic adds extra complexity to the nonlinear optimization, while essentially duplicating the work of the (mathematically equivalent) Cholesky decomposition. In BaSpaCho, we instead complement the supernodal algorithm with sparse elimination that removes the need to externally handle Schur complement as a workaround to the limitation of the supernodal algorithm. More details are described in App. F. + +Backward for custom linear solvers. Obtaining derivatives of the linear system solve with respect to the parameters is a crucial operation for DNLS. In particular, we consider optimizing the parameters $A$ and $b$ of a linear system $y = A ^ { - 1 } b$ to minimize a downstream function $f ( y )$ . The derivatives of the loss with respect to the parameters of the linear system can be obtained with implicit differentiation, $\begin{array} { r } { \frac { \partial f } { \partial b } = A ^ { - 1 } \frac { \partial \mathbf { \bar { f } } } { \partial y } } \end{array}$ and $\begin{array} { r } { \frac { \partial f } { \partial A } = - A ^ { - 1 } \frac { \partial f } { \partial y } y ^ { \top } } \end{array}$ 1 @f@y y>, as done in Barron and Poole [72]. In Theseus, we implement this by connecting the Python interface of our sparse solvers with PyTorch’s autograd.Function classes that implement the gradients above in their backward methods. This connects the computation graph between the downstream function and any upstream parameters that modify the system via auxiliary variables or values for optimization variables. Furthermore, since the gradients require solving linear systems that use the same matrix as the forward pass, our backward pass can cache factorizations, resulting in it being significantly faster than the forward pass (see Fig. 3). + +# 4.3 Backward modes for DNLS + +The parameters $\phi$ upstream of DNLS can be learned end-to-end through the solution $\theta ^ { \star } ( \phi )$ by using the adjoint derivatives ${ { \partial \theta ^ { \star } ( \phi ) } / { \partial \phi } }$ . We include four methods for computing them in Theseus. + +Unrolling is the standard way in which past work in DNLS has computed the adjoint derivatives. This is often referred to as backpropagation through time or unrolled optimization and is explored in [16, 20, 73–82]. In practice, often only a few steps of unrolling are performed due to challenges with compute, memory, and vanishing gradients. + +Truncated differentiation. Aside from unrolling a few steps, another way of approximating the derivatives is to use truncated backpropagation through time (TBPTT) [83, 84]. Truncation unfortunately results in biased derivatives and many works [85–89] seek to further theoretically understand the properties of TBPTT, including the bias of the estimator and how to unbias it. + +Implicit differentiation. If $\theta ^ { \star }$ can be computed exactly, then the implicit function theorem provides a way of computing the adjoint derivatives, as done in related work in convex optimization and first-order gradient descent methods [14, 15, 90–95]. We apply the implicit function theorem from Dontchev and Rockafellar [39, Theorem 1B.1] (see App. H) to Eq. (2) to perform implicit differentiation on a new class of second-order NLS optimization. This first requires that we transform Eq. (2) into an implicit function that finds the roots. We do this via the first-order optimality condition, resulting in $\bar { g } ( \theta ; \phi ) : = \nabla _ { \theta } S ( \theta ; \phi )$ . Finding $\Theta ^ { \star } ( \phi ) : = \{ \theta \mid g ( \theta ; \phi ) = 0 \}$ corresponds to solving Eq. (2). Under mild assumptions, the theorem above gives the adjoint derivative at $\bar { \phi }$ + +$$ +\mathrm { D } _ { \phi } \theta ^ { \star } ( \bar { \phi } ) = - \mathrm { D } _ { \theta } ^ { - 1 } g ( \theta ^ { \star } ( \bar { \phi } ) ; \bar { \phi } ) \mathrm { D } _ { \phi } g ( \theta ^ { \star } ( \bar { \phi } ) ; \bar { \phi } ) . +$$ + +As Theseus internally uses a (Gauss-)Newton solver, the following proposition provided in App. H shows how we can compute Eq. (3) by differentiating a single Newton step at an optimal solution. + +Proposition 1. The implicit derivative (Eq. (3)) can be computed by differentiating a Newton step $\bar { h ( \theta ; \phi ) } : = \theta - [ \nabla _ { \theta } ^ { 2 } S ( \hat { \theta } ; \phi ) ] _ { \mathrm { { s t o p } } } ^ { - 1 } \nabla _ { \theta } S ( \theta ; \phi )$ at an optimal $\theta ^ { \star }$ , where $[ \cdot ] _ { \mathrm { s t o p } }$ zeros the derivative. + +![](images/34be76bb347a8d76fcfa01017b987d15fb2306443580a73d8087400d22648ccd.jpg) +Figure 3: Forward/backward times of Theseus with sparse and dense solvers on different PGO problem scales. + +Direct loss minimization. Suppose we have an outer loss as in Eq. (2). The direct loss minimization (DLM) approach uses this loss to augment the inner-loop optimization problem in order to define a finite difference scheme that approaches the true gradient $\begin{array} { r } { \nabla _ { \phi } L = \operatorname* { l i m } _ { \varepsilon \to 0 } g _ { \mathrm { D L M } } ^ { \varepsilon } } \end{array}$ , where $g _ { \mathrm { D L M } } ^ { \varepsilon } \triangleq$ $\begin{array} { r } { \frac { 1 } { \varepsilon } \{ \frac { \partial } { \partial \phi } S ( \theta ^ { * } ; \phi ) - \frac { \partial } { \partial \phi } S ( \theta _ { \mathrm { d i r e c t } } ; \phi ) \} } \end{array}$ ! . This was used in prior works that solve optimization problems on structured discrete domains [40, 41, 96, 97], but has so far not seen much use in structured continuous settings. We modify the original DLM formulation to better suit its implementation within Theseus + +$$ +\begin{array} { r l } { \theta ^ { \star } = \underset { \hat { \theta } } { \arg \operatorname* { m i n } } S ( \hat { \theta } ; \phi ) , \quad } & { \theta _ { \mathrm { d i r e c t } } = \underset { \hat { \theta } } { \arg \operatorname* { m i n } } S ( \hat { \theta } ; \phi ) + \left\| \varepsilon \hat { \theta } - \frac { 1 } { 2 } \nabla _ { \theta } L ( \theta ^ { * } ) \right\| ^ { 2 } . } \end{array} +$$ + +This is different from the original formulation in two ways: (i) we only assume access to the gradient vector $\nabla _ { { \boldsymbol { \theta } } } L ( { \boldsymbol { \theta } } ^ { * } )$ , which helps formulate DLM as an algorithm for computing vector-Jacobian products, and (ii) we add a small regularization term to ensure the modified objective for $\theta _ { \mathrm { d i r e c t } }$ is a sum-of-squares without affecting the limit as $\varepsilon \to 0$ . See App. H for more details. + +# 5 Evaluation + +We evaluate the performance of Theseus under different settings with PGO and tactile state estimation applications from Sec. 3.1. PGO allows us to easily control the problem scales for performance evaluation; in Sec. 5.1 we profile time and memory consumption of Theseus in an end-to-end setup and in Sec. 5.2 we evaluate timings of Theseus as a stand-alone NLS optimizer and compare with state-of-the-art Ceres [37]. The tactile state estimation application involves a more complex outer loop model that is useful for comparing all different backward modes, which we present in Sec. 5.3. + +# 5.1 Profiling forward and backward pass of Theseus for DNLS + +We study the performance of Theseus for DNLS on the PGO problem [60] with the synthetic Cube dataset, as described in App. E. We run 10 inner loop iterations and 20 outer loop epochs, and use implicit differentiation to compute gradients of the inner NLS optimization. For these experiments we used an Nvidia V100 GPU with 32GBs of memory for all Python computation, and Intel Xeon 2.2GHz CPU with 20 threads for the CPU-based CHOLMOD linear solver. We evaluate performance using our sparse solvers in Theseus and using PyTorch’s Cholesky dense solver. + +Fig. 3 shows the average time of a full forward and backward pass for a given batch size, taken by Theseus with different solvers (cudaLU, CHOLMOD, BaSpaCho and dense), for different problem scales (number of poses and batch size). The two left plots show time as a function of number of poses for a batch size of 128, while the two right plots show time as a function of batch size for 2048 poses. We find that dense does not scale well with poses or batch size. For a batch size of 128, the largest problem that it can solve before running out of GPU memory has 256 poses (left two plots). With 2048 poses, dense is unable to solve the problem regardless of batch size (right two plots). On the other hand with a batch size of 128, our solvers BaSpaCho scale to 2048 poses and cudaLU scale to 4096 poses. CHOLMOD can solve problems even larger, since the linear system is solved on CPU and we have successfully tested up to 8192 poses and batch size 256 (see App. G), for a total of 22GBs of GPU usage for residuals and Jacobian blocks computation. + +In addition to being more memory efficient, running times of our sparse solvers are also smaller for large enough number of poses/batch size, especially for the backward pass. Even though dense’s total time for forward+backward is comparable to cudaLU and faster than CHOLMOD for smaller problems: e.g., 1.47s (dense) vs. 1.32s (cudaLU) and 2.82s (CHOLMOD) for batch size 128 and 64 poses, dense is significantly slower or out of memory for larger problems. For the largest problem that dense can solve (batch size 128 and 256 poses) its total time is already much slower than all others methods: + +20.81s (dense) vs. 10.96s (CHOLMOD), 2.86s (cudaLU), and 2.25s (BaSpaCho). Furthermore, BaSpaCho outperforms dense for any problem scale and is up to one order of magnitude faster, including for smaller batch sizes and number of poses (see App. G for more results and details). For the largest problem that we consider (batch size 256 and 2048 poses), the total times for our sparse solvers are 170.28s for cudaLU, 239.07s for CHOLMOD, and 57.67s for BaSpaCho. + +# 5.2 Profiling Theseus as stand-alone NLS optimizer + +DNLS typically involves solving numerous optimization problems each epoch where a fast NLS optimizer is essential. We compare Theseus as a stand-alone NLS optimizer with the state-of-the-art Ceres [37] library for solving a batch of PGO problems without any learning involved. We compare all solvers in terms of the total time required to perform 10 iterations on a set of 256 PGO problems. CPU/GPU configurations are same as before. For CHOLMOD, we also include a configuration that runs everything on CPU, including Jacobians and residual computation (labelled CHOLMOD-allcpu). + +Fig. 4 shows speedup obtained by Theseus with batching, vectorization and sparse solvers, over Ceres as a function of increasing number of poses or batch size. We vary the number of poses for a fixed batch size of 256, and vary the batch size for a fixed number of poses of 2048. Although Ceres is faster than all of our solvers when the number of poses and batch size are small (for instance, Ceres is $2 5 \mathrm { x }$ faster with 256 poses and 16 batch size, see App. G), as these increase Theseus shows signif + +![](images/ea54f83cdfabe19990c310cda65475bf4791b5a3adff46b06d1784c65e347701.jpg) +Figure 4: Speedup of Theseus (forward pass) over Ceres (black dashed) on different PGO problem scales. + +icant speedup by being able to solve larger batches of problems in parallel. For the largest setup that all our solvers can scale to (2048 poses, 256 batch size), BaSpaCho is ${ \sim } 2 3 \mathbf { x }$ faster than Ceres, and our other solvers are ${ \sim } 4 \mathbf { x }$ faster. CHOLMOD has a 6x speedup for its largest setting (4096 poses, 256 batch size). + +Since typical use case of Theseus involves large batches and number of variables during end-to-end learning with DNLS, the speedups in this setting against a performant NLS solver highlights the significance of our efficiency-based design choices. See App. G for additional results of smaller fixed batch size and number of poses. + +# 5.3 Backward modes analysis + +We explore the trade-offs between our different backward modes using the tactile state estimation application in Sec. 3.1. The learnable components here include a neural network, and thus closely follow the type of applications that motivate Theseus. We compare the following backward modes: derivative unrolling (Unroll), implicit differentiation (Implicit), truncated differentiation (Trunc), and direct loss minimization (DLM); for Trunc we include results when truncating 5 and 10 steps. We compare all modes along 3 axis of performance: validation loss after 100 epochs (outer loop), run time during training, and peak GPU memory consumption of TheseusLayer. For these experiments we used Quadro GP100 GPUs with 16GB of memory. For time and memory we present separate results for forward and backward pass, and all numbers are averaged over 700 (7 batches for 100 epochs). Below we discuss our main findings from this analysis, and more results and details can be found in App. H. + +Fig. 5 shows average run times for all backward modes as a function of the maximum number of iterations in the inner loop optimization. We observe that the time used in the forward pass (Fig. 5, far left) increases roughly linearly for all modes, all having similar times except for Unroll, which is slower than other modes. On the other hand, we observe stark differences in the backward pass time (Fig. 5, center left), where Unroll is the only method that has a linear dependence on the number of inner loop iterations. All other methods have a constant footprint for computing derivatives, independent of the number of inner loop iterations. As expected, increasing the number of iterations through which we backprop (5 or 10 for Trunc, all iterations for Unroll) increases the time necessary for a backward pass $( \mathtt { I m p l i c i t } = \mathtt { D L M } < \mathtt { T r u n c - 5 } < \mathtt { T r u n c - 1 } \theta < < \mathtt { U n r o l 1 } )$ . + +Figure 5 (center right) shows the average peak memory consumption of the backward modes. In this case, the trends observed for the backward pass memory consumption is similar to the trends in time. In particular, Unroll’s memory footprint increases linearly with the number of inner loop iterations, from $\mathrm { \sim 3 4 M B s }$ to $\sim \mathrm { 2 6 2 M B s }$ ; for all other methods the memory consumption remains constant. The best memory profiles in this example is obtained with Implicit and DLM backward modes, with $\sim \mathrm { 2 8 M B s }$ and $\sim \mathrm { 2 9 M B s }$ , respectively. These trends also hold for the forward pass memory consumption. + +![](images/76832a8958ee7b5e173871e737d1c489533f2a9e26449e6214b37904f103dae4.jpg) +Figure 5: Time and memory consumption of different backward modes in tactile state estimation. + +Figure 5 also shows the validation losses obtained with all backward modes (far right). The best validation loss, after 100 epochs of training, is obtained using Implicit, followed by Trunc variants. We notice that both variants of Trunc keep improving with increasing number of inner loop iterations, and that Unroll and Implicit achieve the best results with 20 iterations. One exception is DLM, which doesn’t improve much with the number of iterations, but is also the best method when only 2 inner loop iterations are performed. As a point of caution, we stress that, unlike the timing and memory results, the relative training performance between different backward modes is likely to be application dependent, and is affected by hyperparameters such as the step size used for the inner loop optimizer (0.05 in this example), and the outer optimizer’s learning rate. Our experiments suggest that implicit differentiation is a good default to use for differentiable optimization, considering its low time/memory footprint, and potential for better end-to-end performance with proper hyperparameter tuning. + +# 6 Discussion + +Summary. Theseus provides nonlinear least squares as a differentiable layer and enables easily building and training end-to-end architectures for robotics and vision applications. We illustrate several example applications using the same application-agnostic interface and demonstrate significant improvements in performance with our efficiency-based design. Following how autodiff and GPU acceleration (among others) have led to the evolution of PyTorch in contrast to NumPy [98], we can similarly view sparsity and implicit differentiation on top of autodiff and GPU acceleration as the key ingredients that power Theseus, in contrast to solvers like Ceres that typically only support sparsity. When solving a batch of large problems the forward pass of Theseus is up to $2 0 \mathrm { x }$ faster than Ceres. + +Limitations. Theseus currently has a few limitations. The nonlinear solvers we currently support apply constraints in a soft manner (i.e., using weighted costs). Hard constraints can be handled with methods like augmented Lagrangian or sequential quadratic programs [99, 100], and differentiating through them are active research topics. The current implementation of LM does not support damping to be learnable. Some limitations and trade-offs with the sparse linear solvers are discussed in Sec. 4.2, and with backward modes are discussed in App. H. Online learning applications may require frequently editing the objective and depending on the problem size there may be a nontrivial overhead that is not currently optimized as we explored only non-incremental settings in this work. Additional performance gains can be extracted by moving some of our Python implementation to $\mathrm { C } { + } { + }$ but we prioritized flexibility in evolving the API in the short-term. We do not yet support distributed training beyond what PyTorch natively supports. We will explore these features and optimizations in the future as the library continues to evolve. + +# Acknowledgments and Disclosure of Funding + +The authors would like to thank Dhruv Batra, Olivier Delalleau, Jessica Hodgins, and Mary Williamson for guidance and support on the project, Dhruv Batra and Sal Candido for feedback on early drafts of the paper, Franziska Meier for help with the differentiable robot model library, Paul-Edouard Sarlin for helpful discussion on the homography example, Horace He, Richard Zou and Samantha Andow for help with functorch [101] library, Terran Washington, Gopika Jhala and Chantal Mora for designing the Theseus logo, and Oliver Libaw, Christine Gibson, Orialis Valentin, Alyssa Newcomb and Eric Kaplan for help with the blog post. The authors also thank community members for contributions to the open source code. Work by PS and JO was done while at Meta AI. + +References [1] Jorge Nocedal and Stephen Wright. Numerical optimization. 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[No] We do not foresee any beyond what the field of differentiable optimization already have. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] + +3. If you ran experiments... + +(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] +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] +(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] + +4. 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[N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/dev/K48UYo0glaJ/K48UYo0glaJ_content_list.json b/parse/dev/K48UYo0glaJ/K48UYo0glaJ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cc15bd30c0d0b0c4dc12f039f505578b33c006b0 --- /dev/null +++ b/parse/dev/K48UYo0glaJ/K48UYo0glaJ_content_list.json @@ -0,0 +1,2084 @@ +[ + { + "type": "text", + "text": "Theseus: A Library for Differentiable Nonlinear Optimization ", + "text_level": 1, + "bbox": [ + 181, + 122, + 818, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Luis Pineda1, Taosha Fan1, Maurizio Monge2, Shobha Venkataraman1, Paloma Sodhi1, Ricky T. Q. Chen1, Joseph Ortiz1, Daniel DeTone2, Austin Wang1, Stuart Anderson1, Jing Dong2, Brandon Amos1, Mustafa Mukadam1 ", + "bbox": [ + 214, + 224, + 785, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Meta AI, 2Reality Labs Research ", + "bbox": [ + 387, + 275, + 611, + 290 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 308, + 535, + 324 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present Theseus, an efficient application-agnostic open source library for differentiable nonlinear least squares (DNLS) optimization built on PyTorch, providing a common framework for end-to-end structured learning in robotics and vision. Existing DNLS implementations are application specific and do not always incorporate many ingredients important for efficiency. Theseus is application-agnostic, as we illustrate with several example applications that are built using the same underlying differentiable components, such as second-order optimizers, standard costs functions, and Lie groups. For efficiency, Theseus incorporates support for sparse solvers, automatic vectorization, batching, GPU acceleration, and gradient computation with implicit differentiation and direct loss minimization. We do extensive performance evaluation in a set of applications, demonstrating significant efficiency gains and better scalability when these features are incorporated. Project page: https://sites.google.com/view/theseus-ai/ ", + "bbox": [ + 232, + 332, + 767, + 510 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 530, + 312, + 547 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Reconciling traditional approaches with deep learning to leverage their complementary strengths is a common thread in a large body of recent work in robotics. In particular, an emerging trend is to differentiate through nonlinear least squares (NLS) [1] which is a second-order optimization formulation at the heart of many problems in robotics [2–7] and vision [8–13]. Optimization layers as inductive priors in neural models have been explored in machine learning with convex optimization [14, 15] and in meta learning with gradient descent [16, 17] based first-order optimization. ", + "bbox": [ + 174, + 553, + 825, + 637 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Differentiable nonlinear least squares (DNLS) provides a general scheme to encode inductive priors, as the objective function can be partly parameterized by neural models and partly with engineered domain-specific differentiable models. Here, as illustrated in Fig. 1, input tensors define a sum of weighted squares objective function and output tensors are minima of that objective. Such implicit layers [18] are in contrast to typical (explicit) layers that take input tensors through a linear transformation and some element-wise nonlinear activation function. ", + "bbox": [ + 173, + 642, + 825, + 726 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/f32571828e106eb15cfc6d5807fd15ee3907f8a37214f72613464968ff0d3eb6.jpg", + "image_caption": [ + "Figure 1: Theseus enables building custom, efficient DNLS layers that support end-to-end structured learning. " + ], + "image_footnote": [], + "bbox": [ + 184, + 731, + 816, + 883 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "36th Conference on Neural Information Processing Systems (NeurIPS 2022). ", + "bbox": [ + 173, + 922, + 629, + 936 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The ability to compute gradients end-to-end is retained by differentiating through the optimizer which allows neural models to train on the final task loss, while also taking advantage of priors captured by the optimizer. The flexibility of such a scheme has led to promising state-of-the-art results in a wide range of applications such as structure from motion [19], motion planning [20], SLAM [21, 22], bundle adjustment [23], state estimation [24, 25], image alignment [26] with other applications like manipulation and tactile sensing [27, 28], control [29], human pose tracking [30, 31] to be explored. However, existing implementations from above are application specific, common underlying tools like optimizers get reimplemented, and features like sparse solvers, batching, and GPU support that impact efficiency are not always included. This has led to a fragmented literature where it is difficult to start work on new ideas or to build on the progress of prior work. ", + "bbox": [ + 174, + 90, + 825, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To address this gap, we present Theseus, an open source library for differentiable nonlinear least squares optimization built on PyTorch. Theseus provides an efficient application-agnostic interface that consolidates recent efforts and catalyzes future progress in the domain of structured end-to-end learning for robotics and vision. Our contributions are summarized below. ", + "bbox": [ + 174, + 236, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Application agnostic interface. Our implementation provides an easy to use interface to build custom optimization layers and plug them into any neural architecture. (i) The layer can be constructed from a set of available second-order optimizers like Gauss-Newton, Levenberg–Marquardt (with adaptive damping) and Dogleg, and a nonlinear least squares objective. (ii) The objective can be constructed with learnable or hand-specified cost functions, either by applying one of many common costs already provided in the library, or by building custom costs in-place with support for automatic differentiation through PyTorch [32]. (iii) We also provide differentiable Lie groups for representing 2D/3D positions and rotations [33], and differentiable kinematics wrapping over an existing library [34] for representing robot models. More details are described in Sec. 3. ", + "bbox": [ + 174, + 297, + 825, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Efficiency based design. Efficiency is a central design consideration and we make several advancements in improving computation times and memory consumption. (i) As common in prior work, an optimizer implementation using PyTorch’s native linear solver would use a dense representation for solving the linear system within the nonlinear optimization. In practice, these optimization problems often have a considerable amount of sparsity that can be exploited [35–38]. In Theseus, we implement sparse linear solvers that are differentiable end-to-end and make them efficient with custom CPU and CUDA backends to support batching and GPU acceleration. (ii) Beyond sparse solvers, we extend batching and GPU support to all features in the library and add automatic vectorization of cost functions and other operations to significantly boost efficiency. (iii) Finally, we introduce implicit differentiation [39] and direct loss minimization [40, 41], which have been previously applied to only first order optimizers like gradient descent and convex optimization, to a new class of second-order optimizers. This goes beyond prior work with nonlinear least squares that currently only support differentiation with standard unrolling, which is known to have challenges with compute, memory, and vanishing gradients. More details are described in Sec. 4. ", + "bbox": [ + 174, + 429, + 825, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Highlights of results. Together, the application-agnostic features let users easily set up a variety of problems like pose graph optimization, tactile state estimation, bundle adjustment, motion planning, and homography estimation, all of which are included as examples in the open source code and described in Sec. 3.1. In evaluations, we find that on a standard GPU, Theseus with a sparse solver is much faster and requires significantly less memory than a dense solver, and when solving a batch of large problems the forward pass of Theseus is up to $2 0 \\mathrm { x }$ faster than state-of-the-art $\\mathrm { C } { + } { + }$ based solver Ceres that has limited GPU support and does not support batching and end-to-end learning. We also compare all backward modes to find that with increasing number of optimization iterations, compute and memory increases linearly for unrolling and stays constant for implicit differentiation, while the latter also provides better gradients. More details are described in Sec. 5. ", + "bbox": [ + 173, + 628, + 825, + 767 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Background and related work ", + "text_level": 1, + "bbox": [ + 174, + 775, + 455, + 792 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Nonlinear least squares (NLS) is an optimization problem [1] that finds optimization variables $\\theta$ ", + "bbox": [ + 173, + 796, + 812, + 811 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4621c4ec2789c19bcb3d12812a3b12a64e8684a0a4ed1ed4a58a5921898caae9.jpg", + "text": "$$\n\\theta ^ { \\star } = \\operatorname * { a r g m i n } _ { \\theta } S ( \\theta ) , \\qquad S ( \\theta ) = \\frac { 1 } { 2 } \\sum _ { i } | | r _ { i } ( \\theta ^ { i } ) | | ^ { 2 } = \\frac { 1 } { 2 } \\sum _ { i } | | w _ { i } c _ { i } ( \\theta ^ { i } ) | | ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 261, + 814, + 735, + 852 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where the objective $S ( \\theta )$ is a sum of squared vector-valued residual terms $r _ { i }$ , each a function of $\\theta ^ { i } \\subset \\theta$ that are (non-disjoint) subsets of the optimization variables $\\theta = \\{ \\theta _ { j } \\}$ . Any variable $\\theta _ { j }$ is a manifold object; for example, a Euclidean vector or a matrix Lie group. For flexibility, we represent a residual $\\dot { r _ { i } } ( \\theta ^ { i } ) = w _ { i } c _ { i } ( \\bar { \\theta } ^ { i } )$ as a product of a matrix weight $w _ { i }$ and vector cost $c _ { i }$ . Robotics and vision have used this general optimization formulation to tackle many applications [3, 4]. For example, costs capture sensor measurement errors and physical constraints to optimize camera, robot, object, or human poses in estimation and tracking problems like simultaneous localization and mapping (SLAM) [42], structure from motion [13], bundle adjustment [8], visual inertial odometry [2], articulated tracking [12], contact odometry in legged locomotion [27], 3D pose and shape reconstruction of humans [30, 31] or objects [10]. Similarly, costs can also capture constraints and desired future goals to find robot states or actions in motion planning [5], dynamics [6], and control [29] problems. ", + "bbox": [ + 174, + 854, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Solving NLS. Problems represented by Eq. (1) are solved by iteratively linearizing the nonlinear objective around the current variables to get the linear system $\\begin{array} { r } { ( \\sum _ { i } J _ { i } ^ { \\top } J _ { i } ) \\delta \\theta = ( \\bar { \\sum _ { i } } J _ { i } ^ { \\top } r _ { i } ) } \\end{array}$ , then solving the linear system to find the update $\\delta \\theta$ , and finally updating the variables $\\theta \\theta - \\delta \\theta$ , until convergence. Note that in the update the minus operation is more generally a retraction mapping for non-Euclidean variables. In the linear system, $\\bar { J } _ { i } = [ \\partial r _ { i } / \\partial \\theta ^ { i } ]$ are the Jacobians of residuals with respect to the variables and the iterative method above, called Gauss-Newton (GN), is a nonlinear optimizer that is (approximately) second-order, since $\\begin{array} { r } { \\dot { H } = ( \\sum _ { i } J _ { i } ^ { \\top } J _ { i } ) } \\end{array}$ represents the approximate Hessian. To improve robustness and convergence, variations like Levenberg–Marquardt (LM) damp the linear system, while others use a trust region and adjust step size for the update with line search (e.g., Dogleg). Please refer to [1, 43] for an in-depth exploration. In most applications discussed above the objective structure gives rise to a sparse Hessian, since not all costs depend on all variables. Several general purpose frameworks [35–38] have been built that leverage this sparsity property to efficiently solve the sparse linear system in every iteration of the nonlinear optimization. While these frameworks were not built for deep learning, they are highly efficient and performant on CPU. ", + "bbox": [ + 174, + 207, + 825, + 400 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "NLS with learning. Data driven learning has been explored to address challenges in hand crafting costs or features for costs, finding weights to balance different costs, or to find initializations that lead to better convergence. Some examples include, learning object shape code [44] or environment depth code [45] for SLAM [46], learning motion priors for planning to manipulate articulated objects [47], learning relative pose from tactile images to estimate object state during pushing [28], and semantic 2D segmentation fused in 3D mesh for semantic SLAM [7]. These approaches only train features on a surrogate or intermediate loss and then apply optimization at inference where the true downstream task loss is available but not utilized. To take full advantage of end-to-end learning, latest approaches thus are redesigning the optimization to be differentiable. ", + "bbox": [ + 173, + 405, + 825, + 530 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Differentiable NLS (DNLS) solves the optimization in Eq. (1) and also provides gradients of the solution $\\theta ^ { \\star }$ with respect to any upstream neural model parameters $\\phi$ that parameterize the objective $S ( \\theta ; \\phi )$ and in turn any costs $\\boxed { c _ { i } ( \\theta ^ { i } ; \\phi ) }$ , weights $w _ { i } ( \\phi )$ , or initialization for variables $\\theta _ { i n i t } ( \\phi )$ . The goal is to learn these parameters $\\phi$ end-to-end with a downstream learning objective $L$ defined as a function of $\\theta ^ { \\star }$ . This results in a bilevel optimization setup as shown in Fig. 1 ", + "bbox": [ + 174, + 534, + 825, + 603 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b847b2ba2c4eaf7a7abdd247eaf40b630f08a5f580d591eeaddbca4ec19b42b2.jpg", + "text": "$$\n\\theta ^ { \\star } ( \\phi ) = \\operatorname * { a r g m i n } _ { \\theta } S ( \\theta ; \\phi ) , \\qquad \\mathrm { o u t e r ~ l o o p : ~ } \\phi ^ { \\star } = \\operatorname * { a r g m i n } _ { \\phi } L ( \\theta ^ { \\star } ( \\phi ) )\n$$", + "text_format": "latex", + "bbox": [ + 313, + 609, + 756, + 637 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the inner loop is DNLS and the outer loop is gradient descent class of optimization that is standard in deep learning. The outer loop performs update $\\phi \\phi + \\delta \\phi$ by computing $\\delta \\phi$ using gradients ${ \\partial \\theta ^ { \\star } } / { \\bar { \\partial } \\phi }$ through inner loop DNLS. Note that more generally the learning objective i.e. outer loss $L$ can also depend on other quantities like neural model parameters downstream of $\\theta ^ { * }$ , but we omit them here for clarity. ", + "bbox": [ + 174, + 643, + 825, + 713 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recent works with DNLS have outperformed optimization only or learning only methods by combining the strengths of classical methods with deep learning. For example, learning features for costs to represent depth in bundle adjustment [23] and monocular stereo [48] where an initialization network also learns to predict depth and pose, learning cost weights like motion model weights in video to depth estimation [19], obstacle avoidance weights in 2D motion planning from occupancy images [20], learning robust loss weights in image alignment [26] and state-of-the-art dense SLAM [22], and confidence weights for feature matching to optimize camera pose [49]. Other works, backpropagate reconstruction error to sensor model in a SLAM system [21], solve large scale bundle adjustment on a GPU [50], and learn sensor and dynamics models for 2D visual object tracking and visual odometry [24]. These implementations however, are application specific which has led to repeated work in building DNLS where features like learnable costs and weights, Lie groups, and kinematics are not always present. Additionally, features that have a significant impact on performance, like sparsity and vectorization of costs are only considered by some [24, 50, 51] or in the case of implicit differentiation for NLS optimization, have not yet been explored. ", + "bbox": [ + 173, + 717, + 826, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Application agnostic interface ", + "text_level": 1, + "bbox": [ + 176, + 89, + 454, + 107 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given the lack of a common and efficient framework for DNLS an important goal of Theseus is to provide an application-agnostic interface. In this section, we describe how we enable this with an easy-to-use core API, standard cost functions, and features like Lie groups and kinematics, and illustrate several examples using this interface. We discuss design for efficiency in the next section. ", + "bbox": [ + 174, + 111, + 825, + 166 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4adcd5b9bfba8d359b492ca749686c0e143e3f7523ae53c2ee192ad37fecb287.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 176, + 178, + 790, + 512 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The core API lets users focus on describing the DNLS problem and their interaction with the outer loss $L$ and parameters $\\phi$ within any broader PyTorch model, while the solution and differentiation are seamlessly taken care of under-the-hood. The basic components of the core API are described below with the help of a simple example in Listing 1 (see App. B for more details on the example): ", + "bbox": [ + 174, + 537, + 825, + 593 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Variable: refers to either optimization variables, $\\theta$ , or auxiliary variables (those constant with respect to $S$ , e.g., parameters $\\phi$ or data tensors), which are named wrappers of torch batched tensors stored in Variable.tensor (lines 3-5). \n• CostFunction: defines costs $c _ { i }$ (lines 12-14) and are also responsible for declaring which of its variables are optimization and which are auxiliary (lines 8-9), \n• CostWeight: defines weights $w _ { i }$ associated with cost $c _ { i }$ (line 14). \n• Objective: defines $S ( \\theta ; \\phi )$ , and thus the structure of an optimization problem (lines 11, 15) by holding all cost functions and weights, and their associated variables. These are implicitly obtained when a CostFunction is added to the Objective, and Variable names are used to infer which are shared by one or more CostFunction. \n• Optimizer: is the inner loop optimization algorithm (e.g. Gauss-Newton) that finds the solution $\\theta ^ { \\star }$ given objective $S$ (line 16). \n• TheseusLayer: encapsulates Optimizer and Objective, and serves as the interface between the DNLS block and other torch modules upstream or downstream (line 16). ", + "bbox": [ + 187, + 597, + 826, + 809 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The interface between the inner loop optimization and the outer loop’s parameters and loss occurs via TheseusLayer.forward (lines 21-23). This receives as input a dictionary mapping variable names to torch tensors, which Theseus then uses to populate the corresponding Variable with the tensor mapped to its name. With the input dictionary users can provide initial values for the optimization variables, data tensors, or current values for parameters $\\phi$ before running the inner loop optimization. The output of forward is another dictionary that maps variable names to tensors with their optimal values found in the inner loop (lines 21, 24); auxiliary variables are not modified during the forward pass. The output tensors can then be combined with other torch modules downstream to compute $L$ while maintaining the full differentiable computation graph (lines 24-26). ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 825, + 119 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We currently provide Gauss-Newton, Levenberg–Marquardt (with adaptive damping), and Dogleg as nonlinear Optimizer for the inner loop, with the ability to easily add support for more optimizers in the future. Listing 1 uses AutoDiffCostFunction to construct an in-place CostFunction (line 12) which allows automatically calculating Jacobians $J _ { i }$ with PyTorch (see App. C). Beyond this, in the library we include standard cost functions with analytical Jacobians broadly used in many applications, like Gaussian measurements, reprojection error, relative pose measurement, motion models, and collision costs. We also include a variety of robust loss functions, useful for example in handling outliers [52], which can be easily integrated with CostFunction. Next we describe support for Lie groups and kinematics. ", + "bbox": [ + 174, + 126, + 825, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Differentiable Lie groups. Lie groups are widely used in robotics and vision to represent 2D/3D positions and rotations [33]. Due to their non-Euclidean geometry, it is difficult to apply them to deep learning, which primarily operates with Euclidean tensors, but recently there is growing interest in making them compatible [24, 53–57]. LieTorch [54] generalizes automatic differentiation on the Lie group tangent space through local parameterization around the identity, but the implementation is complex since every operation requires a custom kernel. In contrast, Theseus computes common Lie group operators, e.g., the exponential and logarithm map, inverse, composition, etc., in closed form, and provides their corresponding analytical derivatives on the tangent space. Following [58], we also implement a projection operator that allows us to project gradients computed by PyTorch’s autodiff to the tangent space and use them to easily compute Jacobians and update Lie group variables correctly; a similar strategy has also been implemented in [59]. Additionally, our Lie group implementation includes a heuristic extension that allows using any of PyTorch’s first-order optimizers on nonEuclidean manifolds with minimal code changes. All of these make it easy and straightforward to run optimization and train neural networks with Lie groups variables. More details in App. D. ", + "bbox": [ + 174, + 257, + 825, + 449 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Differentiable kinematics. Many problems such as motion planning or state estimation on high degree of freedom robots like arms or mobile manipulators, involve computation of robot kinematics for collision avoidance or computing distance of end effector to goal. Theseus provides a differentiable implementation of forward kinematics by wrapping over Differentiable Robot Model [34], which builds a differentiable kinematics function from a standard robot model file. Gradients are computed through autodiff, while we also provide a more efficient, analytical manipulator Jacobian. This module can be used within any CostFunction in Theseus. ", + "bbox": [ + 174, + 457, + 825, + 553 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 Example applications ", + "text_level": 1, + "bbox": [ + 176, + 565, + 362, + 580 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To illustrate the versatility of Theseus, we include a number of example DNLS applications below with more details in App. E. Crucially, to implement these with Theseus, most of the effort is only in defining application-specific components such as data management, neural models, or custom CostFunction. With these defined, putting the full DNLS block together is a few lines of code to setup a TheseusLayer and an outer loop, similar to the simple example in Listing 1. ", + "bbox": [ + 174, + 587, + 825, + 656 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Pose graph optimization (PGO) estimates poses from their noisy relative measurements [60]. With DNLS we learn the radius of a Welsh robust cost function for outlier rejection, using the difference between estimated and ground truth poses as the outer loss on a synthetic dataset. ", + "bbox": [ + 174, + 662, + 825, + 704 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Tactile state estimation follows [28], which estimates 2D poses of an object pushed by a robot hand with an image-based tactile sensor [61]. A neural network that predicts relative pose between hand and object from tactile images is learned end-to-end through the TheseusLayer. ", + "bbox": [ + 174, + 710, + 825, + 752 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Bundle adjustment is the problem of optimizing a 3D reconstruction formed by a set of camera images and a set of landmarks observed and matched across the images [62]. We learn the radius of a soft-kernel that penalizes outlier observations, using the average frame pose error as outer loss. ", + "bbox": [ + 174, + 758, + 825, + 801 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Motion planning considers a differentiable version of the GPMP2 planning algorithm, inspired by [20], where the outer loss tries to match expert demonstrations. Here we learn a model for initializing optimization variables, and we include the inner loop objective as a term in the outer loss. ", + "bbox": [ + 174, + 808, + 825, + 849 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Homography estimation. Homography is a linear transformation between corresponding points in two images and can be solved by minimising a dense photometric loss. Robustness to lighting and viewpoint change can be improved with a feature-metric loss based on CNN features [63–68]. In our outer loop, we train a CNN to produce robust features for image alignment. ", + "bbox": [ + 176, + 856, + 823, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Efficiency based design ", + "text_level": 1, + "bbox": [ + 174, + 89, + 398, + 106 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theseus enables several different applications with a general interface. Compute and memory efficiency are central to making its usage practical. Next, we explain design considerations to support batching and vectorization, sparsity, and backward modes for differentiation, which we demonstrate boost performance in the evaluations section. ", + "bbox": [ + 174, + 111, + 825, + 165 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Batching and vectorization ", + "text_level": 1, + "bbox": [ + 174, + 175, + 400, + 189 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Parallel processing is important to improve computational efficiency in machine learning and optimization. In Theseus, we enable two levels of parallelization. First, Theseus natively supports solving a batch of DNLS in parallel, thus fitting seamlessly in the PyTorch framework, where training and inferences on batches is the standard. Second, inspired by DeepLM [50], and noting that lots of the operations such as costs, gradients/Jacobian computation, and variable ", + "bbox": [ + 174, + 194, + 485, + 332 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/15fbb0913b91156eb3a69e3af4fdd5d459dc9750a0150f6fea435a8603a80e2f.jpg", + "image_caption": [ + "Figure 2: Speedup with automatic vectorization on PGO. Black dotted line is without vectorization. " + ], + "image_footnote": [], + "bbox": [ + 504, + 166, + 808, + 291 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "updates only differ from each other in terms of the input data, we make use of the single-instructionmultiple-data (SIMD) protocol by automatically detecting and vectorizing operations of the same type, significantly reducing computation overhead. Using the PGO example, Fig. 2 shows that Theseus achieves significant speedup with automatic vectorization both for forward and backward pass. Note that there is an application-dependent trade-off between memory and speed; here the memory use increases by up to $\\sim 8 2 \\%$ for forward and $\\sim 5 5 \\%$ for backward. ", + "bbox": [ + 174, + 333, + 825, + 415 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 Handling sparsity with linear solvers beyond PyTorch ", + "text_level": 1, + "bbox": [ + 173, + 422, + 584, + 438 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Solving NLS requires solving a sequence of linear systems to obtain descent directions. As discussed in Sec. 2, these systems are generally sparse and can be solved much more efficiently if not treated as dense. Theseus includes differentiable sparse solvers that take advantage of the sparsity, complementing PyTorch’s native dense solvers. Importantly, Theseus seamlessly takes care of assembling the cost functions and variables in the objective into sparse data structures that our linear solvers can consume, without any extra burden on the user. Currently, we provide three sparse solvers: (i) a CPU-based solver that relies on CHOLMOD [69], (ii) cudaLU, which is based on the cuSolverRF package that is part of Nvidia’s cuSolver library provided with CUDA, and (iii) BaSpaCho, our novel batched sparse Cholesky solver with GPU support. As a bonus feature, we provide access to these solvers as standalone PyTorch functions, so they can be used to solve sparse matrices arising outside of NLS or DNLS optimization. ", + "bbox": [ + 174, + 443, + 825, + 594 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CHOLMOD-based solver. CHOLMOD [69] achieves state-of-the-art performance on computation of the Cholesky decomposition of sparse matrices. It exploits parallelism by grouping sparse entries to take advantage of high-performance multi-threaded dense matrix operations in BLAS/LAPACK libraries. CHOLMOD has some limited support for GPU for some of its operations, but the algorithm is strongly CPU-based, and the user is expected to provide matrix data on the CPU. One convenient feature is computing the symbolic analysis of a sparse matrix pattern as a separate step and creating a symbolic decomposition object that can be used for all subsequent factorizations. We also take advantage of builtin functionality for sparse multiplication and only provide the Jacobian matrix $J$ to solve for the Hessian $H = j ^ { \\top } J$ . Two limitations of the library with respect to Theseus are, first, the lack of proper GPU support, which forces us to provide matrix data on the CPU, and, second, the lack of batching, which requires us to loop to solve every problem in the batch independently. On the other hand, since it runs on CPU, it has less memory restrictions than GPU-based solvers (see Sec. 5.1). ", + "bbox": [ + 173, + 601, + 825, + 766 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "cudaLU solver. cuSolverRF is designed to accelerate the solution of sets of linear systems by fast LU refactorization when given new coefficients for the same sparsity pattern. To take advantage of this, we implemented custom CUDA kernels for batched sparse matrix-matrix and matrix-vector products, and for solving a batch of sparse linear systems using LU factorization from cuSolverRF. Although this solver leads to a substantial performance boost over PyTorch’s dense solver (see Sec. 5.1), the closed-source nature of cuSOLVER results in some challenges and limitations: (i) cuSolverRF does not support separate symbolic decomposition and numeric contexts, so it’s not possible to use the same symbolic decomposition to hold in memory separate factors. Since this is necessary in Theseus for unrolling of the inner loop, we work around this limitation by creating a pool of contexts, and we use the least recently used context for factorization. As a consequence, the number of contexts must be set according to the number of iterations that need to be unrolled; (ii) The batch size is fixed once a context is created. Since recreating the contexts is an expensive operation, it means that the batch size has to be constant over the course of outer loop optimization; (iii) It relies on LU factorization, which for symmetric matrices (the case of Theseus) is less efficient than using Cholesky decomposition. ", + "bbox": [ + 174, + 772, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "BaSpaCho solver. Batched Sparse Cholesky (BaSpaCho) is a novel open-source sparse Cholesky solver designed for Theseus with support for batching (https://github.com/facebookresearch/ baspacho). BaSpaCho implements the supernodal Cholesky algorithm [70] to achieve state-of-the art performance by exploiting dense operations via BLAS/cuBLAS. This is achieved by building an elimination tree and then clustering column blocks with similar sparsity patterns. These blocks form nodes of the elimination tree and allow dense operations. In BaSpaCho, the dense operations are dispatched to BLAS (on CPU) or cuBLAS (on GPU), with additional support added on top for batching matrix operations with the same sparsity patterns. In problems with very sparse matrices, like bundle adjustment [8], the supernodal algorithm employed in state-of-the-art solvers [37] is unable to eliminate columns of parameter blocks simultaneously. Thus, past work has resorted to the Schur complement trick [71] to send a reduced problem to the sparse solver. However, this logic adds extra complexity to the nonlinear optimization, while essentially duplicating the work of the (mathematically equivalent) Cholesky decomposition. In BaSpaCho, we instead complement the supernodal algorithm with sparse elimination that removes the need to externally handle Schur complement as a workaround to the limitation of the supernodal algorithm. More details are described in App. F. ", + "bbox": [ + 173, + 154, + 825, + 375 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Backward for custom linear solvers. Obtaining derivatives of the linear system solve with respect to the parameters is a crucial operation for DNLS. In particular, we consider optimizing the parameters $A$ and $b$ of a linear system $y = A ^ { - 1 } b$ to minimize a downstream function $f ( y )$ . The derivatives of the loss with respect to the parameters of the linear system can be obtained with implicit differentiation, $\\begin{array} { r } { \\frac { \\partial f } { \\partial b } = A ^ { - 1 } \\frac { \\partial \\mathbf { \\bar { f } } } { \\partial y } } \\end{array}$ and $\\begin{array} { r } { \\frac { \\partial f } { \\partial A } = - A ^ { - 1 } \\frac { \\partial f } { \\partial y } y ^ { \\top } } \\end{array}$ 1 @f@y y>, as done in Barron and Poole [72]. In Theseus, we implement this by connecting the Python interface of our sparse solvers with PyTorch’s autograd.Function classes that implement the gradients above in their backward methods. This connects the computation graph between the downstream function and any upstream parameters that modify the system via auxiliary variables or values for optimization variables. Furthermore, since the gradients require solving linear systems that use the same matrix as the forward pass, our backward pass can cache factorizations, resulting in it being significantly faster than the forward pass (see Fig. 3). ", + "bbox": [ + 173, + 381, + 825, + 536 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Backward modes for DNLS ", + "text_level": 1, + "bbox": [ + 174, + 541, + 405, + 556 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The parameters $\\phi$ upstream of DNLS can be learned end-to-end through the solution $\\theta ^ { \\star } ( \\phi )$ by using the adjoint derivatives ${ { \\partial \\theta ^ { \\star } ( \\phi ) } / { \\partial \\phi } }$ . We include four methods for computing them in Theseus. ", + "bbox": [ + 174, + 560, + 821, + 588 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Unrolling is the standard way in which past work in DNLS has computed the adjoint derivatives. This is often referred to as backpropagation through time or unrolled optimization and is explored in [16, 20, 73–82]. In practice, often only a few steps of unrolling are performed due to challenges with compute, memory, and vanishing gradients. ", + "bbox": [ + 174, + 594, + 825, + 650 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Truncated differentiation. Aside from unrolling a few steps, another way of approximating the derivatives is to use truncated backpropagation through time (TBPTT) [83, 84]. Truncation unfortunately results in biased derivatives and many works [85–89] seek to further theoretically understand the properties of TBPTT, including the bias of the estimator and how to unbias it. ", + "bbox": [ + 174, + 656, + 825, + 712 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implicit differentiation. If $\\theta ^ { \\star }$ can be computed exactly, then the implicit function theorem provides a way of computing the adjoint derivatives, as done in related work in convex optimization and first-order gradient descent methods [14, 15, 90–95]. We apply the implicit function theorem from Dontchev and Rockafellar [39, Theorem 1B.1] (see App. H) to Eq. (2) to perform implicit differentiation on a new class of second-order NLS optimization. This first requires that we transform Eq. (2) into an implicit function that finds the roots. We do this via the first-order optimality condition, resulting in $\\bar { g } ( \\theta ; \\phi ) : = \\nabla _ { \\theta } S ( \\theta ; \\phi )$ . Finding $\\Theta ^ { \\star } ( \\phi ) : = \\{ \\theta \\mid g ( \\theta ; \\phi ) = 0 \\}$ corresponds to solving Eq. (2). Under mild assumptions, the theorem above gives the adjoint derivative at $\\bar { \\phi }$ ", + "bbox": [ + 173, + 718, + 825, + 830 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/ecd675f1a807fecc35ad872135a26962235f0807291bc492f97a4892825501d9.jpg", + "text": "$$\n\\mathrm { D } _ { \\phi } \\theta ^ { \\star } ( \\bar { \\phi } ) = - \\mathrm { D } _ { \\theta } ^ { - 1 } g ( \\theta ^ { \\star } ( \\bar { \\phi } ) ; \\bar { \\phi } ) \\mathrm { D } _ { \\phi } g ( \\theta ^ { \\star } ( \\bar { \\phi } ) ; \\bar { \\phi } ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 833, + 650, + 852 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As Theseus internally uses a (Gauss-)Newton solver, the following proposition provided in App. H shows how we can compute Eq. (3) by differentiating a single Newton step at an optimal solution. ", + "bbox": [ + 174, + 856, + 821, + 883 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proposition 1. The implicit derivative (Eq. (3)) can be computed by differentiating a Newton step $\\bar { h ( \\theta ; \\phi ) } : = \\theta - [ \\nabla _ { \\theta } ^ { 2 } S ( \\hat { \\theta } ; \\phi ) ] _ { \\mathrm { { s t o p } } } ^ { - 1 } \\nabla _ { \\theta } S ( \\theta ; \\phi )$ at an optimal $\\theta ^ { \\star }$ , where $[ \\cdot ] _ { \\mathrm { s t o p } }$ zeros the derivative. ", + "bbox": [ + 173, + 886, + 825, + 915 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/34be76bb347a8d76fcfa01017b987d15fb2306443580a73d8087400d22648ccd.jpg", + "image_caption": [ + "Figure 3: Forward/backward times of Theseus with sparse and dense solvers on different PGO problem scales. " + ], + "image_footnote": [], + "bbox": [ + 174, + 90, + 820, + 222 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Direct loss minimization. Suppose we have an outer loss as in Eq. (2). The direct loss minimization (DLM) approach uses this loss to augment the inner-loop optimization problem in order to define a finite difference scheme that approaches the true gradient $\\begin{array} { r } { \\nabla _ { \\phi } L = \\operatorname* { l i m } _ { \\varepsilon \\to 0 } g _ { \\mathrm { D L M } } ^ { \\varepsilon } } \\end{array}$ , where $g _ { \\mathrm { D L M } } ^ { \\varepsilon } \\triangleq$ $\\begin{array} { r } { \\frac { 1 } { \\varepsilon } \\{ \\frac { \\partial } { \\partial \\phi } S ( \\theta ^ { * } ; \\phi ) - \\frac { \\partial } { \\partial \\phi } S ( \\theta _ { \\mathrm { d i r e c t } } ; \\phi ) \\} } \\end{array}$ ! . This was used in prior works that solve optimization problems on structured discrete domains [40, 41, 96, 97], but has so far not seen much use in structured continuous settings. We modify the original DLM formulation to better suit its implementation within Theseus ", + "bbox": [ + 173, + 244, + 825, + 334 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/0971aa339764aa817296b759aa89c7611a50966052b52d71947a99e416f87b69.jpg", + "text": "$$\n\\begin{array} { r l } { \\theta ^ { \\star } = \\underset { \\hat { \\theta } } { \\arg \\operatorname* { m i n } } S ( \\hat { \\theta } ; \\phi ) , \\quad } & { \\theta _ { \\mathrm { d i r e c t } } = \\underset { \\hat { \\theta } } { \\arg \\operatorname* { m i n } } S ( \\hat { \\theta } ; \\phi ) + \\left\\| \\varepsilon \\hat { \\theta } - \\frac { 1 } { 2 } \\nabla _ { \\theta } L ( \\theta ^ { * } ) \\right\\| ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 340, + 750, + 372 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This is different from the original formulation in two ways: (i) we only assume access to the gradient vector $\\nabla _ { { \\boldsymbol { \\theta } } } L ( { \\boldsymbol { \\theta } } ^ { * } )$ , which helps formulate DLM as an algorithm for computing vector-Jacobian products, and (ii) we add a small regularization term to ensure the modified objective for $\\theta _ { \\mathrm { d i r e c t } }$ is a sum-of-squares without affecting the limit as $\\varepsilon \\to 0$ . See App. H for more details. ", + "bbox": [ + 173, + 378, + 825, + 435 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 Evaluation ", + "text_level": 1, + "bbox": [ + 173, + 444, + 297, + 460 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We evaluate the performance of Theseus under different settings with PGO and tactile state estimation applications from Sec. 3.1. PGO allows us to easily control the problem scales for performance evaluation; in Sec. 5.1 we profile time and memory consumption of Theseus in an end-to-end setup and in Sec. 5.2 we evaluate timings of Theseus as a stand-alone NLS optimizer and compare with state-of-the-art Ceres [37]. The tactile state estimation application involves a more complex outer loop model that is useful for comparing all different backward modes, which we present in Sec. 5.3. ", + "bbox": [ + 173, + 468, + 825, + 553 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.1 Profiling forward and backward pass of Theseus for DNLS ", + "text_level": 1, + "bbox": [ + 174, + 561, + 624, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We study the performance of Theseus for DNLS on the PGO problem [60] with the synthetic Cube dataset, as described in App. E. We run 10 inner loop iterations and 20 outer loop epochs, and use implicit differentiation to compute gradients of the inner NLS optimization. For these experiments we used an Nvidia V100 GPU with 32GBs of memory for all Python computation, and Intel Xeon 2.2GHz CPU with 20 threads for the CPU-based CHOLMOD linear solver. We evaluate performance using our sparse solvers in Theseus and using PyTorch’s Cholesky dense solver. ", + "bbox": [ + 174, + 579, + 825, + 664 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Fig. 3 shows the average time of a full forward and backward pass for a given batch size, taken by Theseus with different solvers (cudaLU, CHOLMOD, BaSpaCho and dense), for different problem scales (number of poses and batch size). The two left plots show time as a function of number of poses for a batch size of 128, while the two right plots show time as a function of batch size for 2048 poses. We find that dense does not scale well with poses or batch size. For a batch size of 128, the largest problem that it can solve before running out of GPU memory has 256 poses (left two plots). With 2048 poses, dense is unable to solve the problem regardless of batch size (right two plots). On the other hand with a batch size of 128, our solvers BaSpaCho scale to 2048 poses and cudaLU scale to 4096 poses. CHOLMOD can solve problems even larger, since the linear system is solved on CPU and we have successfully tested up to 8192 poses and batch size 256 (see App. G), for a total of 22GBs of GPU usage for residuals and Jacobian blocks computation. ", + "bbox": [ + 173, + 669, + 826, + 821 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In addition to being more memory efficient, running times of our sparse solvers are also smaller for large enough number of poses/batch size, especially for the backward pass. Even though dense’s total time for forward+backward is comparable to cudaLU and faster than CHOLMOD for smaller problems: e.g., 1.47s (dense) vs. 1.32s (cudaLU) and 2.82s (CHOLMOD) for batch size 128 and 64 poses, dense is significantly slower or out of memory for larger problems. For the largest problem that dense can solve (batch size 128 and 256 poses) its total time is already much slower than all others methods: ", + "bbox": [ + 174, + 828, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "20.81s (dense) vs. 10.96s (CHOLMOD), 2.86s (cudaLU), and 2.25s (BaSpaCho). Furthermore, BaSpaCho outperforms dense for any problem scale and is up to one order of magnitude faster, including for smaller batch sizes and number of poses (see App. G for more results and details). For the largest problem that we consider (batch size 256 and 2048 poses), the total times for our sparse solvers are 170.28s for cudaLU, 239.07s for CHOLMOD, and 57.67s for BaSpaCho. ", + "bbox": [ + 174, + 92, + 825, + 160 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.2 Profiling Theseus as stand-alone NLS optimizer ", + "text_level": 1, + "bbox": [ + 176, + 166, + 542, + 180 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "DNLS typically involves solving numerous optimization problems each epoch where a fast NLS optimizer is essential. We compare Theseus as a stand-alone NLS optimizer with the state-of-the-art Ceres [37] library for solving a batch of PGO problems without any learning involved. We compare all solvers in terms of the total time required to perform 10 iterations on a set of 256 PGO problems. CPU/GPU configurations are same as before. For CHOLMOD, we also include a configuration that runs everything on CPU, including Jacobians and residual computation (labelled CHOLMOD-allcpu). ", + "bbox": [ + 173, + 184, + 825, + 267 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Fig. 4 shows speedup obtained by Theseus with batching, vectorization and sparse solvers, over Ceres as a function of increasing number of poses or batch size. We vary the number of poses for a fixed batch size of 256, and vary the batch size for a fixed number of poses of 2048. Although Ceres is faster than all of our solvers when the number of poses and batch size are small (for instance, Ceres is $2 5 \\mathrm { x }$ faster with 256 poses and 16 batch size, see App. G), as these increase Theseus shows signif", + "bbox": [ + 174, + 273, + 433, + 452 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/ea54f83cdfabe19990c310cda65475bf4791b5a3adff46b06d1784c65e347701.jpg", + "image_caption": [ + "Figure 4: Speedup of Theseus (forward pass) over Ceres (black dashed) on different PGO problem scales. " + ], + "image_footnote": [], + "bbox": [ + 450, + 279, + 818, + 411 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "icant speedup by being able to solve larger batches of problems in parallel. For the largest setup that all our solvers can scale to (2048 poses, 256 batch size), BaSpaCho is ${ \\sim } 2 3 \\mathbf { x }$ faster than Ceres, and our other solvers are ${ \\sim } 4 \\mathbf { x }$ faster. CHOLMOD has a 6x speedup for its largest setting (4096 poses, 256 batch size). ", + "bbox": [ + 174, + 453, + 825, + 508 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Since typical use case of Theseus involves large batches and number of variables during end-to-end learning with DNLS, the speedups in this setting against a performant NLS solver highlights the significance of our efficiency-based design choices. See App. G for additional results of smaller fixed batch size and number of poses. ", + "bbox": [ + 174, + 515, + 825, + 570 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 Backward modes analysis ", + "text_level": 1, + "bbox": [ + 174, + 575, + 393, + 590 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We explore the trade-offs between our different backward modes using the tactile state estimation application in Sec. 3.1. The learnable components here include a neural network, and thus closely follow the type of applications that motivate Theseus. We compare the following backward modes: derivative unrolling (Unroll), implicit differentiation (Implicit), truncated differentiation (Trunc), and direct loss minimization (DLM); for Trunc we include results when truncating 5 and 10 steps. We compare all modes along 3 axis of performance: validation loss after 100 epochs (outer loop), run time during training, and peak GPU memory consumption of TheseusLayer. For these experiments we used Quadro GP100 GPUs with 16GB of memory. For time and memory we present separate results for forward and backward pass, and all numbers are averaged over 700 (7 batches for 100 epochs). Below we discuss our main findings from this analysis, and more results and details can be found in App. H. ", + "bbox": [ + 173, + 593, + 825, + 746 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Fig. 5 shows average run times for all backward modes as a function of the maximum number of iterations in the inner loop optimization. We observe that the time used in the forward pass (Fig. 5, far left) increases roughly linearly for all modes, all having similar times except for Unroll, which is slower than other modes. On the other hand, we observe stark differences in the backward pass time (Fig. 5, center left), where Unroll is the only method that has a linear dependence on the number of inner loop iterations. All other methods have a constant footprint for computing derivatives, independent of the number of inner loop iterations. As expected, increasing the number of iterations through which we backprop (5 or 10 for Trunc, all iterations for Unroll) increases the time necessary for a backward pass $( \\mathtt { I m p l i c i t } = \\mathtt { D L M } < \\mathtt { T r u n c - 5 } < \\mathtt { T r u n c - 1 } \\theta < < \\mathtt { U n r o l 1 } )$ . ", + "bbox": [ + 173, + 752, + 825, + 877 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Figure 5 (center right) shows the average peak memory consumption of the backward modes. In this case, the trends observed for the backward pass memory consumption is similar to the trends in time. In particular, Unroll’s memory footprint increases linearly with the number of inner loop iterations, from $\\mathrm { \\sim 3 4 M B s }$ to $\\sim \\mathrm { 2 6 2 M B s }$ ; for all other methods the memory consumption remains constant. The best memory profiles in this example is obtained with Implicit and DLM backward modes, with $\\sim \\mathrm { 2 8 M B s }$ and $\\sim \\mathrm { 2 9 M B s }$ , respectively. These trends also hold for the forward pass memory consumption. ", + "bbox": [ + 173, + 883, + 821, + 911 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/76832a8958ee7b5e173871e737d1c489533f2a9e26449e6214b37904f103dae4.jpg", + "image_caption": [ + "Figure 5: Time and memory consumption of different backward modes in tactile state estimation. " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 820, + 208 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 233, + 825, + 303 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Figure 5 also shows the validation losses obtained with all backward modes (far right). The best validation loss, after 100 epochs of training, is obtained using Implicit, followed by Trunc variants. We notice that both variants of Trunc keep improving with increasing number of inner loop iterations, and that Unroll and Implicit achieve the best results with 20 iterations. One exception is DLM, which doesn’t improve much with the number of iterations, but is also the best method when only 2 inner loop iterations are performed. As a point of caution, we stress that, unlike the timing and memory results, the relative training performance between different backward modes is likely to be application dependent, and is affected by hyperparameters such as the step size used for the inner loop optimizer (0.05 in this example), and the outer optimizer’s learning rate. Our experiments suggest that implicit differentiation is a good default to use for differentiable optimization, considering its low time/memory footprint, and potential for better end-to-end performance with proper hyperparameter tuning. ", + "bbox": [ + 173, + 309, + 825, + 460 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Discussion ", + "text_level": 1, + "bbox": [ + 174, + 468, + 294, + 484 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Summary. Theseus provides nonlinear least squares as a differentiable layer and enables easily building and training end-to-end architectures for robotics and vision applications. We illustrate several example applications using the same application-agnostic interface and demonstrate significant improvements in performance with our efficiency-based design. Following how autodiff and GPU acceleration (among others) have led to the evolution of PyTorch in contrast to NumPy [98], we can similarly view sparsity and implicit differentiation on top of autodiff and GPU acceleration as the key ingredients that power Theseus, in contrast to solvers like Ceres that typically only support sparsity. When solving a batch of large problems the forward pass of Theseus is up to $2 0 \\mathrm { x }$ faster than Ceres. ", + "bbox": [ + 174, + 489, + 825, + 601 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations. Theseus currently has a few limitations. The nonlinear solvers we currently support apply constraints in a soft manner (i.e., using weighted costs). Hard constraints can be handled with methods like augmented Lagrangian or sequential quadratic programs [99, 100], and differentiating through them are active research topics. The current implementation of LM does not support damping to be learnable. Some limitations and trade-offs with the sparse linear solvers are discussed in Sec. 4.2, and with backward modes are discussed in App. H. Online learning applications may require frequently editing the objective and depending on the problem size there may be a nontrivial overhead that is not currently optimized as we explored only non-incremental settings in this work. Additional performance gains can be extracted by moving some of our Python implementation to $\\mathrm { C } { + } { + }$ but we prioritized flexibility in evolving the API in the short-term. We do not yet support distributed training beyond what PyTorch natively supports. We will explore these features and optimizations in the future as the library continues to evolve. ", + "bbox": [ + 173, + 606, + 825, + 771 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 779, + 553, + 795 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The authors would like to thank Dhruv Batra, Olivier Delalleau, Jessica Hodgins, and Mary Williamson for guidance and support on the project, Dhruv Batra and Sal Candido for feedback on early drafts of the paper, Franziska Meier for help with the differentiable robot model library, Paul-Edouard Sarlin for helpful discussion on the homography example, Horace He, Richard Zou and Samantha Andow for help with functorch [101] library, Terran Washington, Gopika Jhala and Chantal Mora for designing the Theseus logo, and Oliver Libaw, Christine Gibson, Orialis Valentin, Alyssa Newcomb and Eric Kaplan for help with the blog post. The authors also thank community members for contributions to the open source code. 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[No] We do not foresee any beyond what the field of differentiable optimization already have. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 136, + 825, + 239 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 243, + 493, + 258 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ", + "bbox": [ + 238, + 262, + 738, + 294 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 212, + 299, + 393, + 313 + ], + "page_idx": 17 + }, + { + "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] \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] \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": [ + 238, + 316, + 825, + 435 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 220, + 439, + 823, + 454 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [Yes] \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 458, + 823, + 565 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 569, + 705, + 584 + ], + "page_idx": 17 + }, + { + "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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This has led to a fragmented literature where it is difficult", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 170, + 378, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 378, + 183 + ], + "score": 1.0, + "content": "to start work on new ideas or to build on the progress of prior work.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "To address this gap, we present Theseus, an open source library for differentiable nonlinear least", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "squares optimization built on PyTorch. 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Our implementation provides an easy to use interface to build", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "custom optimization layers and plug them into any neural architecture. (i) The layer can be constructed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 504, + 270 + ], + "score": 1.0, + "content": "from a set of available second-order optimizers like Gauss-Newton, Levenberg–Marquardt (with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "adaptive damping) and Dogleg, and a nonlinear least squares objective. (ii) The objective can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "be constructed with learnable or hand-specified cost functions, either by applying one of many", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "common costs already provided in the library, or by building custom costs in-place with support", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "for automatic differentiation through PyTorch [32]. (iii) We also provide differentiable Lie groups", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "score": 1.0, + "content": "for representing 2D/3D positions and rotations [33], and differentiable kinematics wrapping over an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 460, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 460, + 336 + ], + "score": 1.0, + "content": "existing library [34] for representing robot models. More details are described in Sec. 3.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "Efficiency based design. 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In practice, these optimization problems", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "often have a considerable amount of sparsity that can be exploited [35–38]. In Theseus, we imple-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "ment sparse linear solvers that are differentiable end-to-end and make them efficient with custom", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "CPU and CUDA backends to support batching and GPU acceleration. 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This goes beyond prior work with nonlinear least squares that currently only support", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 469, + 507, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 507, + 485 + ], + "score": 1.0, + "content": "differentiation with standard unrolling, which is known to have challenges with compute, memory,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 482, + 354, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 354, + 493 + ], + "score": 1.0, + "content": "and vanishing gradients. More details are described in Sec. 4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "Highlights of results. 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The flexibility of such a scheme has led to promising state-of-the-art results in a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "wide range of applications such as structure from motion [19], motion planning [20], SLAM [21, 22],", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "bundle adjustment [23], state estimation [24, 25], image alignment [26] with other applications like", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "manipulation and tactile sensing [27, 28], control [29], human pose tracking [30, 31] to be explored.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "However, existing implementations from above are application specific, common underlying tools", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "like optimizers get reimplemented, and features like sparse solvers, batching, and GPU support that", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "impact efficiency are not always included. This has led to a fragmented literature where it is difficult", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 170, + 378, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 378, + 183 + ], + "score": 1.0, + "content": "to start work on new ideas or to build on the progress of prior work.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 73, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "To address this gap, we present Theseus, an open source library for differentiable nonlinear least", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "squares optimization built on PyTorch. 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Our implementation provides an easy to use interface to build", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "custom optimization layers and plug them into any neural architecture. (i) The layer can be constructed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 504, + 270 + ], + "score": 1.0, + "content": "from a set of available second-order optimizers like Gauss-Newton, Levenberg–Marquardt (with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "adaptive damping) and Dogleg, and a nonlinear least squares objective. (ii) The objective can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "be constructed with learnable or hand-specified cost functions, either by applying one of many", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "common costs already provided in the library, or by building custom costs in-place with support", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "for automatic differentiation through PyTorch [32]. (iii) We also provide differentiable Lie groups", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "score": 1.0, + "content": "for representing 2D/3D positions and rotations [33], and differentiable kinematics wrapping over an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 460, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 460, + 336 + ], + "score": 1.0, + "content": "existing library [34] for representing robot models. More details are described in Sec. 3.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 236, + 506, + 336 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "Efficiency based design. Efficiency is a central design consideration and we make several advance-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "ments in improving computation times and memory consumption. (i) As common in prior work, an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "optimizer implementation using PyTorch’s native linear solver would use a dense representation for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 373, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 384 + ], + "score": 1.0, + "content": "solving the linear system within the nonlinear optimization. In practice, these optimization problems", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "often have a considerable amount of sparsity that can be exploited [35–38]. In Theseus, we imple-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "ment sparse linear solvers that are differentiable end-to-end and make them efficient with custom", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "CPU and CUDA backends to support batching and GPU acceleration. (ii) Beyond sparse solvers, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 428 + ], + "score": 1.0, + "content": "extend batching and GPU support to all features in the library and add automatic vectorization of cost", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "functions and other operations to significantly boost efficiency. 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Together, the application-agnostic features let users easily set up a variety of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 508, + 507, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 507, + 523 + ], + "score": 1.0, + "content": "problems like pose graph optimization, tactile state estimation, bundle adjustment, motion planning,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "and homography estimation, all of which are included as examples in the open source code and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "described in Sec. 3.1. 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For", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "example, costs capture sensor measurement errors and physical constraints to optimize camera,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "robot, object, or human poses in estimation and tracking problems like simultaneous localization", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "and mapping (SLAM) [42], structure from motion [13], bundle adjustment [8], visual inertial", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "odometry [2], articulated tracking [12], contact odometry in legged locomotion [27], 3D pose and", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "score": 1.0, + "content": "shape reconstruction of humans [30, 31] or objects [10]. Similarly, costs can also capture constraints", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "and desired future goals to find robot states or actions in motion planning [5], dynamics [6], and", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 199, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 199, + 162 + ], + "score": 1.0, + "content": "control [29] problems.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 677, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "vision have used this general optimization formulation to tackle many applications [3, 4]. For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "example, costs capture sensor measurement errors and physical constraints to optimize camera,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "robot, object, or human poses in estimation and tracking problems like simultaneous localization", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "and mapping (SLAM) [42], structure from motion [13], bundle adjustment [8], visual inertial", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "odometry [2], articulated tracking [12], contact odometry in legged locomotion [27], 3D pose and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "score": 1.0, + "content": "shape reconstruction of humans [30, 31] or objects [10]. Similarly, costs can also capture constraints", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "and desired future goals to find robot states or actions in motion planning [5], dynamics [6], and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 199, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 199, + 162 + ], + "score": 1.0, + "content": "control [29] problems.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "score": 1.0, + "content": "Solving NLS. Problems represented by Eq. (1) are solved by iteratively linearizing the nonlinear", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 174, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 365, + 189 + ], + "score": 1.0, + "content": "objective around the current variables to get the linear system", + "type": "text" + }, + { + "bbox": [ + 365, + 174, + 480, + 187 + ], + "score": 0.9, + "content": "\\begin{array} { r } { ( \\sum _ { i } J _ { i } ^ { \\top } J _ { i } ) \\delta \\theta = ( \\bar { \\sum _ { i } } J _ { i } ^ { \\top } r _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 174, + 507, + 189 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 281, + 199 + ], + "score": 1.0, + "content": "solving the linear system to find the update", + "type": "text" + }, + { + "bbox": [ + 281, + 186, + 292, + 196 + ], + "score": 0.81, + "content": "\\delta \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 185, + 432, + 199 + ], + "score": 1.0, + "content": ", and finally updating the variables", + "type": "text" + }, + { + "bbox": [ + 432, + 187, + 480, + 196 + ], + "score": 0.87, + "content": "\\theta \\theta - \\delta \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 185, + 506, + 199 + ], + "score": 1.0, + "content": ", until", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "score": 1.0, + "content": "convergence. Note that in the update the minus operation is more generally a retraction mapping for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 298, + 220 + ], + "score": 1.0, + "content": "non-Euclidean variables. In the linear system,", + "type": "text" + }, + { + "bbox": [ + 298, + 207, + 360, + 220 + ], + "score": 0.94, + "content": "\\bar { J } _ { i } = [ \\partial r _ { i } / \\partial \\theta ^ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "are the Jacobians of residuals with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "respect to the variables and the iterative method above, called Gauss-Newton (GN), is a nonlinear", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 323, + 243 + ], + "score": 1.0, + "content": "optimizer that is (approximately) second-order, since", + "type": "text" + }, + { + "bbox": [ + 324, + 228, + 393, + 242 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\dot { H } = ( \\sum _ { i } J _ { i } ^ { \\top } J _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 228, + 506, + 243 + ], + "score": 1.0, + "content": "represents the approximate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "Hessian. To improve robustness and convergence, variations like Levenberg–Marquardt (LM) damp", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "the linear system, while others use a trust region and adjust step size for the update with line search", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "(e.g., Dogleg). Please refer to [1, 43] for an in-depth exploration. In most applications discussed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 285 + ], + "score": 1.0, + "content": "above the objective structure gives rise to a sparse Hessian, since not all costs depend on all variables.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "Several general purpose frameworks [35–38] have been built that leverage this sparsity property to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "score": 1.0, + "content": "efficiently solve the sparse linear system in every iteration of the nonlinear optimization. While these", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 305, + 484, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 484, + 318 + ], + "score": 1.0, + "content": "frameworks were not built for deep learning, they are highly efficient and performant on CPU.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 321, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "NLS with learning. Data driven learning has been explored to address challenges in hand crafting", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 332, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 343 + ], + "score": 1.0, + "content": "costs or features for costs, finding weights to balance different costs, or to find initializations that lead", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "to better convergence. Some examples include, learning object shape code [44] or environment depth", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "code [45] for SLAM [46], learning motion priors for planning to manipulate articulated objects [47],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "learning relative pose from tactile images to estimate object state during pushing [28], and semantic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "2D segmentation fused in 3D mesh for semantic SLAM [7]. These approaches only train features on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "a surrogate or intermediate loss and then apply optimization at inference where the true downstream", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "task loss is available but not utilized. To take full advantage of end-to-end learning, latest approaches", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 408, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 338, + 420 + ], + "score": 1.0, + "content": "thus are redesigning the optimization to be differentiable.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "Differentiable NLS (DNLS) solves the optimization in Eq. 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The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 238, + 469 + ], + "score": 1.0, + "content": "goal is to learn these parameters", + "type": "text" + }, + { + "bbox": [ + 238, + 457, + 246, + 467 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 455, + 446, + 469 + ], + "score": 1.0, + "content": "end-to-end with a downstream learning objective", + "type": "text" + }, + { + "bbox": [ + 446, + 457, + 454, + 466 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "defined as a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 465, + 414, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 153, + 480 + ], + "score": 1.0, + "content": "function of", + "type": "text" + }, + { + "bbox": [ + 153, + 467, + 163, + 477 + ], + "score": 0.86, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 465, + 414, + 480 + ], + "score": 1.0, + "content": ". This results in a bilevel optimization setup as shown in Fig. 1", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 483, + 463, + 505 + ], + "lines": [ + { + "bbox": [ + 192, + 483, + 463, + 505 + ], + "spans": [ + { + "bbox": [ + 192, + 483, + 463, + 505 + ], + "score": 0.84, + "content": "\\theta ^ { \\star } ( \\phi ) = \\operatorname * { a r g m i n } _ { \\theta } S ( \\theta ; \\phi ) , \\qquad \\mathrm { o u t e r ~ l o o p : ~ } \\phi ^ { \\star } = \\operatorname * { a r g m i n } _ { \\phi } L ( \\theta ^ { \\star } ( \\phi ) )", + "type": "interline_equation", + "image_path": "b847b2ba2c4eaf7a7abdd247eaf40b630f08a5f580d591eeaddbca4ec19b42b2.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 192, + 483, + 463, + 505 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "where the inner loop is DNLS and the outer loop is gradient descent class of optimization that is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 351, + 535 + ], + "score": 1.0, + "content": "standard in deep learning. The outer loop performs update", + "type": "text" + }, + { + "bbox": [ + 352, + 522, + 406, + 533 + ], + "score": 0.92, + "content": "\\phi \\phi + \\delta \\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 520, + 466, + 535 + ], + "score": 1.0, + "content": "by computing", + "type": "text" + }, + { + "bbox": [ + 467, + 522, + 479, + 533 + ], + "score": 0.89, + "content": "\\delta \\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 520, + 506, + 535 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 531, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 146, + 546 + ], + "score": 1.0, + "content": "gradients", + "type": "text" + }, + { + "bbox": [ + 147, + 532, + 180, + 544 + ], + "score": 0.92, + "content": "{ \\partial \\theta ^ { \\star } } / { \\bar { \\partial } \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 531, + 507, + 546 + ], + "score": 1.0, + "content": "through inner loop DNLS. Note that more generally the learning objective i.e.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 147, + 556 + ], + "score": 1.0, + "content": "outer loss", + "type": "text" + }, + { + "bbox": [ + 147, + 544, + 155, + 553 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 543, + 476, + 556 + ], + "score": 1.0, + "content": "can also depend on other quantities like neural model parameters downstream of", + "type": "text" + }, + { + "bbox": [ + 476, + 543, + 486, + 553 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 543, + 506, + 556 + ], + "score": 1.0, + "content": ", but", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 553, + 226, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 226, + 567 + ], + "score": 1.0, + "content": "we omit them here for clarity.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "Recent works with DNLS have outperformed optimization only or learning only methods by com-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "bining the strengths of classical methods with deep learning. For example, learning features for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "costs to represent depth in bundle adjustment [23] and monocular stereo [48] where an initialization", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "network also learns to predict depth and pose, learning cost weights like motion model weights", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "score": 1.0, + "content": "in video to depth estimation [19], obstacle avoidance weights in 2D motion planning from occu-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "pancy images [20], learning robust loss weights in image alignment [26] and state-of-the-art dense", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "SLAM [22], and confidence weights for feature matching to optimize camera pose [49]. Other works,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "backpropagate reconstruction error to sensor model in a SLAM system [21], solve large scale bundle", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "adjustment on a GPU [50], and learn sensor and dynamics models for 2D visual object tracking", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "and visual odometry [24]. These implementations however, are application specific which has led", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 677, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 507, + 692 + ], + "score": 1.0, + "content": "to repeated work in building DNLS where features like learnable costs and weights, Lie groups,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "and kinematics are not always present. Additionally, features that have a significant impact on", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "performance, like sparsity and vectorization of costs are only considered by some [24, 50, 51] or in", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 447, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 447, + 723 + ], + "score": 1.0, + "content": "the case of implicit differentiation for NLS optimization, have not yet been explored.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 48.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 160 + ], + "lines": [], + "index": 3.5, + "bbox_fs": [ + 105, + 72, + 506, + 162 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "score": 1.0, + "content": "Solving NLS. Problems represented by Eq. (1) are solved by iteratively linearizing the nonlinear", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 174, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 365, + 189 + ], + "score": 1.0, + "content": "objective around the current variables to get the linear system", + "type": "text" + }, + { + "bbox": [ + 365, + 174, + 480, + 187 + ], + "score": 0.9, + "content": "\\begin{array} { r } { ( \\sum _ { i } J _ { i } ^ { \\top } J _ { i } ) \\delta \\theta = ( \\bar { \\sum _ { i } } J _ { i } ^ { \\top } r _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 174, + 507, + 189 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 281, + 199 + ], + "score": 1.0, + "content": "solving the linear system to find the update", + "type": "text" + }, + { + "bbox": [ + 281, + 186, + 292, + 196 + ], + "score": 0.81, + "content": "\\delta \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 185, + 432, + 199 + ], + "score": 1.0, + "content": ", and finally updating the variables", + "type": "text" + }, + { + "bbox": [ + 432, + 187, + 480, + 196 + ], + "score": 0.87, + "content": "\\theta \\theta - \\delta \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 185, + 506, + 199 + ], + "score": 1.0, + "content": ", until", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "score": 1.0, + "content": "convergence. Note that in the update the minus operation is more generally a retraction mapping for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 298, + 220 + ], + "score": 1.0, + "content": "non-Euclidean variables. In the linear system,", + "type": "text" + }, + { + "bbox": [ + 298, + 207, + 360, + 220 + ], + "score": 0.94, + "content": "\\bar { J } _ { i } = [ \\partial r _ { i } / \\partial \\theta ^ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "are the Jacobians of residuals with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "respect to the variables and the iterative method above, called Gauss-Newton (GN), is a nonlinear", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 323, + 243 + ], + "score": 1.0, + "content": "optimizer that is (approximately) second-order, since", + "type": "text" + }, + { + "bbox": [ + 324, + 228, + 393, + 242 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\dot { H } = ( \\sum _ { i } J _ { i } ^ { \\top } J _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 228, + 506, + 243 + ], + "score": 1.0, + "content": "represents the approximate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "Hessian. To improve robustness and convergence, variations like Levenberg–Marquardt (LM) damp", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "the linear system, while others use a trust region and adjust step size for the update with line search", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "(e.g., Dogleg). Please refer to [1, 43] for an in-depth exploration. In most applications discussed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 285 + ], + "score": 1.0, + "content": "above the objective structure gives rise to a sparse Hessian, since not all costs depend on all variables.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "Several general purpose frameworks [35–38] have been built that leverage this sparsity property to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "score": 1.0, + "content": "efficiently solve the sparse linear system in every iteration of the nonlinear optimization. While these", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 305, + 484, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 484, + 318 + ], + "score": 1.0, + "content": "frameworks were not built for deep learning, they are highly efficient and performant on CPU.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 163, + 507, + 318 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 321, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "NLS with learning. Data driven learning has been explored to address challenges in hand crafting", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 332, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 343 + ], + "score": 1.0, + "content": "costs or features for costs, finding weights to balance different costs, or to find initializations that lead", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "to better convergence. Some examples include, learning object shape code [44] or environment depth", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "code [45] for SLAM [46], learning motion priors for planning to manipulate articulated objects [47],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "learning relative pose from tactile images to estimate object state during pushing [28], and semantic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "2D segmentation fused in 3D mesh for semantic SLAM [7]. These approaches only train features on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "a surrogate or intermediate loss and then apply optimization at inference where the true downstream", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "task loss is available but not utilized. To take full advantage of end-to-end learning, latest approaches", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 408, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 338, + 420 + ], + "score": 1.0, + "content": "thus are redesigning the optimization to be differentiable.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 320, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "Differentiable NLS (DNLS) solves the optimization in Eq. (1) and also provides gradients of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 141, + 447 + ], + "score": 1.0, + "content": "solution", + "type": "text" + }, + { + "bbox": [ + 141, + 434, + 152, + 444 + ], + "score": 0.87, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 433, + 371, + 447 + ], + "score": 1.0, + "content": "with respect to any upstream neural model parameters", + "type": "text" + }, + { + "bbox": [ + 372, + 434, + 379, + 445 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "that parameterize the objective", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 107, + 445, + 137, + 457 + ], + "score": 0.92, + "content": "S ( \\theta ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 444, + 226, + 458 + ], + "score": 1.0, + "content": "and in turn any costs", + "type": "text" + }, + { + "bbox": [ + 226, + 444, + 261, + 457 + ], + "score": 0.93, + "content": "\\boxed { c _ { i } ( \\theta ^ { i } ; \\phi ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 444, + 299, + 458 + ], + "score": 1.0, + "content": ", weights", + "type": "text" + }, + { + "bbox": [ + 300, + 445, + 325, + 457 + ], + "score": 0.93, + "content": "w _ { i } ( \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 444, + 448, + 458 + ], + "score": 1.0, + "content": ", or initialization for variables", + "type": "text" + }, + { + "bbox": [ + 448, + 445, + 482, + 457 + ], + "score": 0.91, + "content": "\\theta _ { i n i t } ( \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 444, + 505, + 458 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 238, + 469 + ], + "score": 1.0, + "content": "goal is to learn these parameters", + "type": "text" + }, + { + "bbox": [ + 238, + 457, + 246, + 467 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 455, + 446, + 469 + ], + "score": 1.0, + "content": "end-to-end with a downstream learning objective", + "type": "text" + }, + { + "bbox": [ + 446, + 457, + 454, + 466 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "defined as a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 465, + 414, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 153, + 480 + ], + "score": 1.0, + "content": "function of", + "type": "text" + }, + { + "bbox": [ + 153, + 467, + 163, + 477 + ], + "score": 0.86, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 465, + 414, + 480 + ], + "score": 1.0, + "content": ". This results in a bilevel optimization setup as shown in Fig. 1", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 423, + 507, + 480 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 483, + 463, + 505 + ], + "lines": [ + { + "bbox": [ + 192, + 483, + 463, + 505 + ], + "spans": [ + { + "bbox": [ + 192, + 483, + 463, + 505 + ], + "score": 0.84, + "content": "\\theta ^ { \\star } ( \\phi ) = \\operatorname * { a r g m i n } _ { \\theta } S ( \\theta ; \\phi ) , \\qquad \\mathrm { o u t e r ~ l o o p : ~ } \\phi ^ { \\star } = \\operatorname * { a r g m i n } _ { \\phi } L ( \\theta ^ { \\star } ( \\phi ) )", + "type": "interline_equation", + "image_path": "b847b2ba2c4eaf7a7abdd247eaf40b630f08a5f580d591eeaddbca4ec19b42b2.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 192, + 483, + 463, + 505 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "where the inner loop is DNLS and the outer loop is gradient descent class of optimization that is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 351, + 535 + ], + "score": 1.0, + "content": "standard in deep learning. The outer loop performs update", + "type": "text" + }, + { + "bbox": [ + 352, + 522, + 406, + 533 + ], + "score": 0.92, + "content": "\\phi \\phi + \\delta \\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 520, + 466, + 535 + ], + "score": 1.0, + "content": "by computing", + "type": "text" + }, + { + "bbox": [ + 467, + 522, + 479, + 533 + ], + "score": 0.89, + "content": "\\delta \\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 520, + 506, + 535 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 531, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 146, + 546 + ], + "score": 1.0, + "content": "gradients", + "type": "text" + }, + { + "bbox": [ + 147, + 532, + 180, + 544 + ], + "score": 0.92, + "content": "{ \\partial \\theta ^ { \\star } } / { \\bar { \\partial } \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 531, + 507, + 546 + ], + "score": 1.0, + "content": "through inner loop DNLS. Note that more generally the learning objective i.e.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 147, + 556 + ], + "score": 1.0, + "content": "outer loss", + "type": "text" + }, + { + "bbox": [ + 147, + 544, + 155, + 553 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 543, + 476, + 556 + ], + "score": 1.0, + "content": "can also depend on other quantities like neural model parameters downstream of", + "type": "text" + }, + { + "bbox": [ + 476, + 543, + 486, + 553 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 543, + 506, + 556 + ], + "score": 1.0, + "content": ", but", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 553, + 226, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 226, + 567 + ], + "score": 1.0, + "content": "we omit them here for clarity.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 510, + 507, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "Recent works with DNLS have outperformed optimization only or learning only methods by com-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "bining the strengths of classical methods with deep learning. For example, learning features for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "costs to represent depth in bundle adjustment [23] and monocular stereo [48] where an initialization", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "network also learns to predict depth and pose, learning cost weights like motion model weights", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 507, + 625 + ], + "score": 1.0, + "content": "in video to depth estimation [19], obstacle avoidance weights in 2D motion planning from occu-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "pancy images [20], learning robust loss weights in image alignment [26] and state-of-the-art dense", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "SLAM [22], and confidence weights for feature matching to optimize camera pose [49]. Other works,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "backpropagate reconstruction error to sensor model in a SLAM system [21], solve large scale bundle", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "adjustment on a GPU [50], and learn sensor and dynamics models for 2D visual object tracking", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "and visual odometry [24]. These implementations however, are application specific which has led", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 677, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 507, + 692 + ], + "score": 1.0, + "content": "to repeated work in building DNLS where features like learnable costs and weights, Lie groups,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "and kinematics are not always present. Additionally, features that have a significant impact on", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "performance, like sparsity and vectorization of costs are only considered by some [24, 50, 51] or in", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 711, + 447, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 447, + 723 + ], + "score": 1.0, + "content": "the case of implicit differentiation for NLS optimization, have not yet been explored.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 48.5, + "bbox_fs": [ + 104, + 568, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 278, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 280, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 280, + 88 + ], + "score": 1.0, + "content": "3 Application agnostic interface", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 88, + 505, + 132 + ], + "lines": [ + { + "bbox": [ + 106, + 88, + 506, + 100 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 506, + 100 + ], + "score": 1.0, + "content": "Given the lack of a common and efficient framework for DNLS an important goal of Theseus is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 99, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 505, + 111 + ], + "score": 1.0, + "content": "to provide an application-agnostic interface. In this section, we describe how we enable this with", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "score": 1.0, + "content": "an easy-to-use core API, standard cost functions, and features like Lie groups and kinematics, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 504, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 504, + 134 + ], + "score": 1.0, + "content": "illustrate several examples using this interface. 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The basic components of the core API are described", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 458, + 502, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 502, + 471 + ], + "score": 1.0, + "content": "below with the help of a simple example in Listing 1 (see App. B for more details on the example):", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 115, + 473, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 115, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 115, + 473, + 324, + 486 + ], + "score": 1.0, + "content": "• Variable: refers to either optimization variables,", + "type": "text" + }, + { + "bbox": [ + 324, + 473, + 330, + 483 + ], + "score": 0.64, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 473, + 505, + 486 + ], + "score": 1.0, + "content": ", or auxiliary variables (those constant with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 122, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 122, + 484, + 165, + 496 + ], + "score": 1.0, + "content": "respect to", + "type": "text" + }, + { + "bbox": [ + 165, + 485, + 173, + 494 + ], + "score": 0.78, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 484, + 244, + 496 + ], + "score": 1.0, + "content": ", e.g., parameters", + "type": "text" + }, + { + "bbox": [ + 244, + 485, + 252, + 496 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "or data tensors), which are named wrappers of torch batched", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 123, + 495, + 315, + 507 + ], + "spans": [ + { + "bbox": [ + 123, + 495, + 315, + 507 + ], + "score": 1.0, + "content": "tensors stored in Variable.tensor (lines 3-5).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 115, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 115, + 509, + 241, + 521 + ], + "score": 1.0, + "content": "• CostFunction: defines costs", + "type": "text" + }, + { + "bbox": [ + 242, + 510, + 251, + 520 + ], + "score": 0.84, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "(lines 12-14) and are also responsible for declaring which of its", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 122, + 519, + 372, + 532 + ], + "spans": [ + { + "bbox": [ + 122, + 519, + 372, + 532 + ], + "score": 1.0, + "content": "variables are optimization and which are auxiliary (lines 8-9),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 114, + 533, + 387, + 546 + ], + "spans": [ + { + "bbox": [ + 114, + 533, + 243, + 546 + ], + "score": 1.0, + "content": "• CostWeight: defines weights", + "type": "text" + }, + { + "bbox": [ + 243, + 535, + 255, + 544 + ], + "score": 0.85, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 533, + 338, + 546 + ], + "score": 1.0, + "content": "associated with cost", + "type": "text" + }, + { + "bbox": [ + 338, + 534, + 348, + 545 + ], + "score": 0.84, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 533, + 387, + 546 + ], + "score": 1.0, + "content": "(line 14).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 113, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 113, + 545, + 207, + 560 + ], + "score": 1.0, + "content": "• Objective: defines", + "type": "text" + }, + { + "bbox": [ + 207, + 546, + 237, + 559 + ], + "score": 0.93, + "content": "S ( \\theta ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 545, + 506, + 560 + ], + "score": 1.0, + "content": ", and thus the structure of an optimization problem (lines 11, 15)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 123, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 123, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "by holding all cost functions and weights, and their associated variables. These are implicitly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 123, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 123, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "obtained when a CostFunction is added to the Objective, and Variable names are used to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 122, + 579, + 342, + 592 + ], + "spans": [ + { + "bbox": [ + 122, + 579, + 342, + 592 + ], + "score": 1.0, + "content": "infer which are shared by one or more CostFunction.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 115, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "• Optimizer: is the inner loop optimization algorithm (e.g. Gauss-Newton) that finds the solution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 123, + 603, + 247, + 617 + ], + "spans": [ + { + "bbox": [ + 123, + 604, + 134, + 614 + ], + "score": 0.81, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 603, + 198, + 617 + ], + "score": 1.0, + "content": "given objective", + "type": "text" + }, + { + "bbox": [ + 199, + 604, + 207, + 614 + ], + "score": 0.71, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 603, + 247, + 617 + ], + "score": 1.0, + "content": "(line 16).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 114, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "• TheseusLayer: encapsulates Optimizer and Objective, and serves as the interface between", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 123, + 629, + 434, + 641 + ], + "spans": [ + { + "bbox": [ + 123, + 629, + 434, + 641 + ], + "score": 1.0, + "content": "the DNLS block and other torch modules upstream or downstream (line 16).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "The interface between the inner loop optimization and the outer loop’s parameters and loss occurs via", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "TheseusLayer.forward (lines 21-23). This receives as input a dictionary mapping variable names", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "to torch tensors, which Theseus then uses to populate the corresponding Variable with the tensor", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "mapped to its name. 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In this section, we describe how we enable this with", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 122 + ], + "score": 1.0, + "content": "an easy-to-use core API, standard cost functions, and features like Lie groups and kinematics, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 504, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 504, + 134 + ], + "score": 1.0, + "content": "illustrate several examples using this interface. 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Listing 1 uses AutoDiffCostFunction to construct an in-place CostFunction (line", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 325, + 145 + ], + "score": 1.0, + "content": "12) which allows automatically calculating Jacobians", + "type": "text" + }, + { + "bbox": [ + 326, + 133, + 336, + 144 + ], + "score": 0.87, + "content": "J _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "with PyTorch (see App. C). Beyond this,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "in the library we include standard cost functions with analytical Jacobians broadly used in many", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "applications, like Gaussian measurements, reprojection error, relative pose measurement, motion", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "models, and collision costs. We also include a variety of robust loss functions, useful for example in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "score": 1.0, + "content": "handling outliers [52], which can be easily integrated with CostFunction. Next we describe support", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 231, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 231, + 200 + ], + "score": 1.0, + "content": "for Lie groups and kinematics.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "score": 1.0, + "content": "Differentiable Lie groups. 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LieTorch [54] generalizes automatic differentiation on the Lie", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "group tangent space through local parameterization around the identity, but the implementation is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "complex since every operation requires a custom kernel. In contrast, Theseus computes common Lie", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 507, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 507, + 282 + ], + "score": 1.0, + "content": "group operators, e.g., the exponential and logarithm map, inverse, composition, etc., in closed form,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 281, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 292 + ], + "score": 1.0, + "content": "and provides their corresponding analytical derivatives on the tangent space. Following [58], we also", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "implement a projection operator that allows us to project gradients computed by PyTorch’s autodiff to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "the tangent space and use them to easily compute Jacobians and update Lie group variables correctly;", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "a similar strategy has also been implemented in [59]. Additionally, our Lie group implementation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "includes a heuristic extension that allows using any of PyTorch’s first-order optimizers on non-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "Euclidean manifolds with minimal code changes. All of these make it easy and straightforward to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 345, + 483, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 483, + 358 + ], + "score": 1.0, + "content": "run optimization and train neural networks with Lie groups variables. More details in App. D.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 375 + ], + "score": 1.0, + "content": "Differentiable kinematics. Many problems such as motion planning or state estimation on high", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "degree of freedom robots like arms or mobile manipulators, involve computation of robot kinematics", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "for collision avoidance or computing distance of end effector to goal. Theseus provides a differen-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "tiable implementation of forward kinematics by wrapping over Differentiable Robot Model [34],", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "which builds a differentiable kinematics function from a standard robot model file. Gradients are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "computed through autodiff, while we also provide a more efficient, analytical manipulator Jacobian.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 427, + 365, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 365, + 440 + ], + "score": 1.0, + "content": "This module can be used within any CostFunction in Theseus.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 222, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 223, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 223, + 462 + ], + "score": 1.0, + "content": "3.1 Example applications", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "To illustrate the versatility of Theseus, we include a number of example DNLS applications below", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "with more details in App. E. Crucially, to implement these with Theseus, most of the effort is only", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "in defining application-specific components such as data management, neural models, or custom", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "CostFunction. With these defined, putting the full DNLS block together is a few lines of code to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 509, + 444, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 444, + 522 + ], + "score": 1.0, + "content": "setup a TheseusLayer and an outer loop, similar to the simple example in Listing 1.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "Pose graph optimization (PGO) estimates poses from their noisy relative measurements [60]. With", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "DNLS we learn the radius of a Welsh robust cost function for outlier rejection, using the difference", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 547, + 434, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 434, + 559 + ], + "score": 1.0, + "content": "between estimated and ground truth poses as the outer loss on a synthetic dataset.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "Tactile state estimation follows [28], which estimates 2D poses of an object pushed by a robot hand", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "with an image-based tactile sensor [61]. A neural network that predicts relative pose between hand", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 585, + 430, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 430, + 597 + ], + "score": 1.0, + "content": "and object from tactile images is learned end-to-end through the TheseusLayer.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "Bundle adjustment is the problem of optimizing a 3D reconstruction formed by a set of camera", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "images and a set of landmarks observed and matched across the images [62]. We learn the radius of a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 623, + 487, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 487, + 636 + ], + "score": 1.0, + "content": "soft-kernel that penalizes outlier observations, using the average frame pose error as outer loss.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Motion planning considers a differentiable version of the GPMP2 planning algorithm, inspired", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "by [20], where the outer loss tries to match expert demonstrations. Here we learn a model for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 662, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 507, + 674 + ], + "score": 1.0, + "content": "initializing optimization variables, and we include the inner loop objective as a term in the outer loss.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "Homography estimation. Homography is a linear transformation between corresponding points in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "two images and can be solved by minimising a dense photometric loss. Robustness to lighting and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "viewpoint change can be improved with a feature-metric loss based on CNN features [63–68]. In our", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 711, + 410, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 410, + 723 + ], + "score": 1.0, + "content": "outer loop, we train a CNN to produce robust features for image alignment.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 502, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "We currently provide Gauss-Newton, Levenberg–Marquardt (with adaptive damping), and Dogleg as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "nonlinear Optimizer for the inner loop, with the ability to easily add support for more optimizers", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "in the future. Listing 1 uses AutoDiffCostFunction to construct an in-place CostFunction (line", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 325, + 145 + ], + "score": 1.0, + "content": "12) which allows automatically calculating Jacobians", + "type": "text" + }, + { + "bbox": [ + 326, + 133, + 336, + 144 + ], + "score": 0.87, + "content": "J _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "with PyTorch (see App. C). Beyond this,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "in the library we include standard cost functions with analytical Jacobians broadly used in many", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "applications, like Gaussian measurements, reprojection error, relative pose measurement, motion", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "models, and collision costs. We also include a variety of robust loss functions, useful for example in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "score": 1.0, + "content": "handling outliers [52], which can be easily integrated with CostFunction. Next we describe support", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 231, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 231, + 200 + ], + "score": 1.0, + "content": "for Lie groups and kinematics.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 100, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "score": 1.0, + "content": "Differentiable Lie groups. Lie groups are widely used in robotics and vision to represent 2D/3D", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "positions and rotations [33]. Due to their non-Euclidean geometry, it is difficult to apply them to deep", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "learning, which primarily operates with Euclidean tensors, but recently there is growing interest in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "making them compatible [24, 53–57]. LieTorch [54] generalizes automatic differentiation on the Lie", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "group tangent space through local parameterization around the identity, but the implementation is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "complex since every operation requires a custom kernel. In contrast, Theseus computes common Lie", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 507, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 507, + 282 + ], + "score": 1.0, + "content": "group operators, e.g., the exponential and logarithm map, inverse, composition, etc., in closed form,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 281, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 292 + ], + "score": 1.0, + "content": "and provides their corresponding analytical derivatives on the tangent space. Following [58], we also", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "implement a projection operator that allows us to project gradients computed by PyTorch’s autodiff to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "the tangent space and use them to easily compute Jacobians and update Lie group variables correctly;", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "a similar strategy has also been implemented in [59]. Additionally, our Lie group implementation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "includes a heuristic extension that allows using any of PyTorch’s first-order optimizers on non-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "Euclidean manifolds with minimal code changes. All of these make it easy and straightforward to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 345, + 483, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 483, + 358 + ], + "score": 1.0, + "content": "run optimization and train neural networks with Lie groups variables. More details in App. D.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 202, + 507, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 375 + ], + "score": 1.0, + "content": "Differentiable kinematics. Many problems such as motion planning or state estimation on high", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "degree of freedom robots like arms or mobile manipulators, involve computation of robot kinematics", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "for collision avoidance or computing distance of end effector to goal. Theseus provides a differen-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "tiable implementation of forward kinematics by wrapping over Differentiable Robot Model [34],", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "which builds a differentiable kinematics function from a standard robot model file. Gradients are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "computed through autodiff, while we also provide a more efficient, analytical manipulator Jacobian.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 427, + 365, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 365, + 440 + ], + "score": 1.0, + "content": "This module can be used within any CostFunction in Theseus.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 360, + 506, + 440 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 222, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 223, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 223, + 462 + ], + "score": 1.0, + "content": "3.1 Example applications", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "To illustrate the versatility of Theseus, we include a number of example DNLS applications below", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "with more details in App. E. Crucially, to implement these with Theseus, most of the effort is only", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "in defining application-specific components such as data management, neural models, or custom", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "CostFunction. With these defined, putting the full DNLS block together is a few lines of code to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 509, + 444, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 444, + 522 + ], + "score": 1.0, + "content": "setup a TheseusLayer and an outer loop, similar to the simple example in Listing 1.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 465, + 506, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "Pose graph optimization (PGO) estimates poses from their noisy relative measurements [60]. With", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "DNLS we learn the radius of a Welsh robust cost function for outlier rejection, using the difference", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 547, + 434, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 434, + 559 + ], + "score": 1.0, + "content": "between estimated and ground truth poses as the outer loss on a synthetic dataset.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 525, + 505, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "Tactile state estimation follows [28], which estimates 2D poses of an object pushed by a robot hand", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "with an image-based tactile sensor [61]. A neural network that predicts relative pose between hand", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 585, + 430, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 430, + 597 + ], + "score": 1.0, + "content": "and object from tactile images is learned end-to-end through the TheseusLayer.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 563, + 505, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "Bundle adjustment is the problem of optimizing a 3D reconstruction formed by a set of camera", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "images and a set of landmarks observed and matched across the images [62]. We learn the radius of a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 623, + 487, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 487, + 636 + ], + "score": 1.0, + "content": "soft-kernel that penalizes outlier observations, using the average frame pose error as outer loss.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 601, + 505, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Motion planning considers a differentiable version of the GPMP2 planning algorithm, inspired", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "by [20], where the outer loss tries to match expert demonstrations. Here we learn a model for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 662, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 507, + 674 + ], + "score": 1.0, + "content": "initializing optimization variables, and we include the inner loop objective as a term in the outer loss.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 639, + 507, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "Homography estimation. Homography is a linear transformation between corresponding points in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "two images and can be solved by minimising a dense photometric loss. Robustness to lighting and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "viewpoint change can be improved with a feature-metric loss based on CNN features [63–68]. In our", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 711, + 410, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 410, + 723 + ], + "score": 1.0, + "content": "outer loop, we train a CNN to produce robust features for image alignment.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 244, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 245, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 245, + 88 + ], + "score": 1.0, + "content": "4 Efficiency based design", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 88, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 87, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 505, + 102 + ], + "score": 1.0, + "content": "Theseus enables several different applications with a general interface. Compute and memory", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 98, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 506, + 113 + ], + "score": 1.0, + "content": "efficiency are central to making its usage practical. Next, we explain design considerations to support", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "batching and vectorization, sparsity, and backward modes for differentiation, which we demonstrate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 289, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 289, + 133 + ], + "score": 1.0, + "content": "boost performance in the evaluations section.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 139, + 245, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 245, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 245, + 152 + ], + "score": 1.0, + "content": "4.1 Batching and vectorization", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 297, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 298, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 298, + 166 + ], + "score": 1.0, + "content": "Parallel processing is important to improve com-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 298, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 298, + 178 + ], + "score": 1.0, + "content": "putational efficiency in machine learning and op-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 298, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 298, + 187 + ], + "score": 1.0, + "content": "timization. In Theseus, we enable two levels of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 298, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 298, + 199 + ], + "score": 1.0, + "content": "parallelization. First, Theseus natively supports", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 298, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 298, + 210 + ], + "score": 1.0, + "content": "solving a batch of DNLS in parallel, thus fitting", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 297, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 297, + 220 + ], + "score": 1.0, + "content": "seamlessly in the PyTorch framework, where", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 298, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 298, + 231 + ], + "score": 1.0, + "content": "training and inferences on batches is the stan-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 298, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 298, + 242 + ], + "score": 1.0, + "content": "dard. Second, inspired by DeepLM [50], and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 298, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 298, + 253 + ], + "score": 1.0, + "content": "noting that lots of the operations such as costs,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 297, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 297, + 264 + ], + "score": 1.0, + "content": "gradients/Jacobian computation, and variable", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "image", + "bbox": [ + 309, + 132, + 495, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 309, + 132, + 495, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 309, + 132, + 495, + 231 + ], + "spans": [ + { + "bbox": [ + 309, + 132, + 495, + 231 + ], + "score": 0.97, + "type": "image", + "image_path": "15fbb0913b91156eb3a69e3af4fdd5d459dc9750a0150f6fea435a8603a80e2f.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 309, + 132, + 495, + 146.14285714285714 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 309, + 146.14285714285714, + 495, + 160.28571428571428 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 309, + 160.28571428571428, + 495, + 174.42857142857142 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 309, + 174.42857142857142, + 495, + 188.57142857142856 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 309, + 188.57142857142856, + 495, + 202.7142857142857 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 309, + 202.7142857142857, + 495, + 216.85714285714283 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 309, + 216.85714285714283, + 495, + 230.99999999999997 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 237, + 505, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 237, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 304, + 237, + 506, + 248 + ], + "score": 1.0, + "content": "Figure 2: Speedup with automatic vectorization on PGO.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 303, + 246, + 457, + 258 + ], + "spans": [ + { + "bbox": [ + 303, + 246, + 457, + 258 + ], + "score": 1.0, + "content": "Black dotted line is without vectorization.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + } + ], + "index": 21.25 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "updates only differ from each other in terms of the input data, we make use of the single-instruction-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "multiple-data (SIMD) protocol by automatically detecting and vectorizing operations of the same", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "type, significantly reducing computation overhead. Using the PGO example, Fig. 2 shows that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "Theseus achieves significant speedup with automatic vectorization both for forward and backward", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "pass. Note that there is an application-dependent trade-off between memory and speed; here the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 317, + 417, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 232, + 329 + ], + "score": 1.0, + "content": "memory use increases by up to", + "type": "text" + }, + { + "bbox": [ + 232, + 317, + 262, + 328 + ], + "score": 0.89, + "content": "\\sim 8 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 317, + 328, + 329 + ], + "score": 1.0, + "content": "for forward and", + "type": "text" + }, + { + "bbox": [ + 328, + 317, + 358, + 328 + ], + "score": 0.9, + "content": "\\sim 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 317, + 417, + 329 + ], + "score": 1.0, + "content": "for backward.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 106, + 335, + 358, + 347 + ], + "lines": [ + { + "bbox": [ + 104, + 333, + 359, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 359, + 351 + ], + "score": 1.0, + "content": "4.2 Handling sparsity with linear solvers beyond PyTorch", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "Solving NLS requires solving a sequence of linear systems to obtain descent directions. As discussed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "in Sec. 2, these systems are generally sparse and can be solved much more efficiently if not treated as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "dense. Theseus includes differentiable sparse solvers that take advantage of the sparsity, comple-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "menting PyTorch’s native dense solvers. Importantly, Theseus seamlessly takes care of assembling", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 395, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 406 + ], + "score": 1.0, + "content": "the cost functions and variables in the objective into sparse data structures that our linear solvers", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "can consume, without any extra burden on the user. Currently, we provide three sparse solvers: (i)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "a CPU-based solver that relies on CHOLMOD [69], (ii) cudaLU, which is based on the cuSolverRF", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "package that is part of Nvidia’s cuSolver library provided with CUDA, and (iii) BaSpaCho, our novel", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "batched sparse Cholesky solver with GPU support. As a bonus feature, we provide access to these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "solvers as standalone PyTorch functions, so they can be used to solve sparse matrices arising outside", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 460, + 233, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 233, + 472 + ], + "score": 1.0, + "content": "of NLS or DNLS optimization.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "CHOLMOD-based solver. CHOLMOD [69] achieves state-of-the-art performance on computation of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "Cholesky decomposition of sparse matrices. It exploits parallelism by grouping sparse entries to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "take advantage of high-performance multi-threaded dense matrix operations in BLAS/LAPACK libraries.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "CHOLMOD has some limited support for GPU for some of its operations, but the algorithm is strongly", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "CPU-based, and the user is expected to provide matrix data on the CPU. One convenient feature is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "computing the symbolic analysis of a sparse matrix pattern as a separate step and creating a symbolic", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "decomposition object that can be used for all subsequent factorizations. We also take advantage of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 447, + 565 + ], + "score": 1.0, + "content": "builtin functionality for sparse multiplication and only provide the Jacobian matrix", + "type": "text" + }, + { + "bbox": [ + 447, + 553, + 455, + 563 + ], + "score": 0.76, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "to solve for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 155, + 576 + ], + "score": 1.0, + "content": "the Hessian", + "type": "text" + }, + { + "bbox": [ + 156, + 563, + 199, + 574 + ], + "score": 0.91, + "content": "H = j ^ { \\top } J", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 562, + 506, + 576 + ], + "score": 1.0, + "content": ". Two limitations of the library with respect to Theseus are, first, the lack of", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "proper GPU support, which forces us to provide matrix data on the CPU, and, second, the lack of", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "batching, which requires us to loop to solve every problem in the batch independently. On the other", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 596, + 499, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 499, + 609 + ], + "score": 1.0, + "content": "hand, since it runs on CPU, it has less memory restrictions than GPU-based solvers (see Sec. 5.1).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "cudaLU solver. cuSolverRF is designed to accelerate the solution of sets of linear systems by fast LU", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "score": 1.0, + "content": "refactorization when given new coefficients for the same sparsity pattern. To take advantage of this,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "we implemented custom CUDA kernels for batched sparse matrix-matrix and matrix-vector products,", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "and for solving a batch of sparse linear systems using LU factorization from cuSolverRF. Although", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "this solver leads to a substantial performance boost over PyTorch’s dense solver (see Sec. 5.1), the", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "closed-source nature of cuSOLVER results in some challenges and limitations: (i) cuSolverRF does", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "not support separate symbolic decomposition and numeric contexts, so it’s not possible to use the", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 690, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 701 + ], + "score": 1.0, + "content": "same symbolic decomposition to hold in memory separate factors. Since this is necessary in Theseus", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "for unrolling of the inner loop, we work around this limitation by creating a pool of contexts, and we", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "use the least recently used context for factorization. As a consequence, the number of contexts must", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 59.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 244, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 245, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 245, + 88 + ], + "score": 1.0, + "content": "4 Efficiency based design", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 88, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 87, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 505, + 102 + ], + "score": 1.0, + "content": "Theseus enables several different applications with a general interface. 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Next, we explain design considerations to support", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "batching and vectorization, sparsity, and backward modes for differentiation, which we demonstrate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 120, + 289, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 289, + 133 + ], + "score": 1.0, + "content": "boost performance in the evaluations section.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 87, + 506, + 133 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 139, + 245, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 245, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 245, + 152 + ], + "score": 1.0, + "content": "4.1 Batching and vectorization", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 297, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 298, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 298, + 166 + ], + "score": 1.0, + "content": "Parallel processing is important to improve com-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 298, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 298, + 178 + ], + "score": 1.0, + "content": "putational efficiency in machine learning and op-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 298, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 298, + 187 + ], + "score": 1.0, + "content": "timization. In Theseus, we enable two levels of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 298, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 298, + 199 + ], + "score": 1.0, + "content": "parallelization. First, Theseus natively supports", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 298, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 298, + 210 + ], + "score": 1.0, + "content": "solving a batch of DNLS in parallel, thus fitting", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 297, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 297, + 220 + ], + "score": 1.0, + "content": "seamlessly in the PyTorch framework, where", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 298, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 298, + 231 + ], + "score": 1.0, + "content": "training and inferences on batches is the stan-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 298, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 298, + 242 + ], + "score": 1.0, + "content": "dard. Second, inspired by DeepLM [50], and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 298, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 298, + 253 + ], + "score": 1.0, + "content": "noting that lots of the operations such as costs,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 297, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 297, + 264 + ], + "score": 1.0, + "content": "gradients/Jacobian computation, and variable", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 154, + 298, + 264 + ] + }, + { + "type": "image", + "bbox": [ + 309, + 132, + 495, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 309, + 132, + 495, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 309, + 132, + 495, + 231 + ], + "spans": [ + { + "bbox": [ + 309, + 132, + 495, + 231 + ], + "score": 0.97, + "type": "image", + "image_path": "15fbb0913b91156eb3a69e3af4fdd5d459dc9750a0150f6fea435a8603a80e2f.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 309, + 132, + 495, + 146.14285714285714 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 309, + 146.14285714285714, + 495, + 160.28571428571428 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 309, + 160.28571428571428, + 495, + 174.42857142857142 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 309, + 174.42857142857142, + 495, + 188.57142857142856 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 309, + 188.57142857142856, + 495, + 202.7142857142857 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 309, + 202.7142857142857, + 495, + 216.85714285714283 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 309, + 216.85714285714283, + 495, + 230.99999999999997 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 237, + 505, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 237, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 304, + 237, + 506, + 248 + ], + "score": 1.0, + "content": "Figure 2: Speedup with automatic vectorization on PGO.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 303, + 246, + 457, + 258 + ], + "spans": [ + { + "bbox": [ + 303, + 246, + 457, + 258 + ], + "score": 1.0, + "content": "Black dotted line is without vectorization.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + } + ], + "index": 21.25 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "updates only differ from each other in terms of the input data, we make use of the single-instruction-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "multiple-data (SIMD) protocol by automatically detecting and vectorizing operations of the same", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "type, significantly reducing computation overhead. Using the PGO example, Fig. 2 shows that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "Theseus achieves significant speedup with automatic vectorization both for forward and backward", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "pass. Note that there is an application-dependent trade-off between memory and speed; here the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 317, + 417, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 232, + 329 + ], + "score": 1.0, + "content": "memory use increases by up to", + "type": "text" + }, + { + "bbox": [ + 232, + 317, + 262, + 328 + ], + "score": 0.89, + "content": "\\sim 8 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 317, + 328, + 329 + ], + "score": 1.0, + "content": "for forward and", + "type": "text" + }, + { + "bbox": [ + 328, + 317, + 358, + 328 + ], + "score": 0.9, + "content": "\\sim 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 317, + 417, + 329 + ], + "score": 1.0, + "content": "for backward.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 263, + 506, + 329 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 335, + 358, + 347 + ], + "lines": [ + { + "bbox": [ + 104, + 333, + 359, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 359, + 351 + ], + "score": 1.0, + "content": "4.2 Handling sparsity with linear solvers beyond PyTorch", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "Solving NLS requires solving a sequence of linear systems to obtain descent directions. As discussed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "in Sec. 2, these systems are generally sparse and can be solved much more efficiently if not treated as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "dense. Theseus includes differentiable sparse solvers that take advantage of the sparsity, comple-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "menting PyTorch’s native dense solvers. Importantly, Theseus seamlessly takes care of assembling", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 395, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 406 + ], + "score": 1.0, + "content": "the cost functions and variables in the objective into sparse data structures that our linear solvers", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "can consume, without any extra burden on the user. Currently, we provide three sparse solvers: (i)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "a CPU-based solver that relies on CHOLMOD [69], (ii) cudaLU, which is based on the cuSolverRF", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "package that is part of Nvidia’s cuSolver library provided with CUDA, and (iii) BaSpaCho, our novel", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "batched sparse Cholesky solver with GPU support. As a bonus feature, we provide access to these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "solvers as standalone PyTorch functions, so they can be used to solve sparse matrices arising outside", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 460, + 233, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 233, + 472 + ], + "score": 1.0, + "content": "of NLS or DNLS optimization.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 351, + 506, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "CHOLMOD-based solver. CHOLMOD [69] achieves state-of-the-art performance on computation of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "Cholesky decomposition of sparse matrices. It exploits parallelism by grouping sparse entries to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "take advantage of high-performance multi-threaded dense matrix operations in BLAS/LAPACK libraries.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "CHOLMOD has some limited support for GPU for some of its operations, but the algorithm is strongly", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "CPU-based, and the user is expected to provide matrix data on the CPU. One convenient feature is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "computing the symbolic analysis of a sparse matrix pattern as a separate step and creating a symbolic", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "decomposition object that can be used for all subsequent factorizations. We also take advantage of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 447, + 565 + ], + "score": 1.0, + "content": "builtin functionality for sparse multiplication and only provide the Jacobian matrix", + "type": "text" + }, + { + "bbox": [ + 447, + 553, + 455, + 563 + ], + "score": 0.76, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "to solve for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 155, + 576 + ], + "score": 1.0, + "content": "the Hessian", + "type": "text" + }, + { + "bbox": [ + 156, + 563, + 199, + 574 + ], + "score": 0.91, + "content": "H = j ^ { \\top } J", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 562, + 506, + 576 + ], + "score": 1.0, + "content": ". Two limitations of the library with respect to Theseus are, first, the lack of", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "proper GPU support, which forces us to provide matrix data on the CPU, and, second, the lack of", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "batching, which requires us to loop to solve every problem in the batch independently. On the other", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 596, + 499, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 499, + 609 + ], + "score": 1.0, + "content": "hand, since it runs on CPU, it has less memory restrictions than GPU-based solvers (see Sec. 5.1).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 475, + 506, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "cudaLU solver. cuSolverRF is designed to accelerate the solution of sets of linear systems by fast LU", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 636 + ], + "score": 1.0, + "content": "refactorization when given new coefficients for the same sparsity pattern. To take advantage of this,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "we implemented custom CUDA kernels for batched sparse matrix-matrix and matrix-vector products,", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "and for solving a batch of sparse linear systems using LU factorization from cuSolverRF. Although", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "this solver leads to a substantial performance boost over PyTorch’s dense solver (see Sec. 5.1), the", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "closed-source nature of cuSOLVER results in some challenges and limitations: (i) cuSolverRF does", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "not support separate symbolic decomposition and numeric contexts, so it’s not possible to use the", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 690, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 701 + ], + "score": 1.0, + "content": "same symbolic decomposition to hold in memory separate factors. Since this is necessary in Theseus", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "for unrolling of the inner loop, we work around this limitation by creating a pool of contexts, and we", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "use the least recently used context for factorization. As a consequence, the number of contexts must", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "be set according to the number of iterations that need to be unrolled; (ii) The batch size is fixed once a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "context is created. Since recreating the contexts is an expensive operation, it means that the batch size", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "has to be constant over the course of outer loop optimization; (iii) It relies on LU factorization, which", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 499, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 499, + 118 + ], + "score": 1.0, + "content": "for symmetric matrices (the case of Theseus) is less efficient than using Cholesky decomposition.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 59.5, + "bbox_fs": [ + 105, + 612, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "be set according to the number of iterations that need to be unrolled; (ii) The batch size is fixed once a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "context is created. Since recreating the contexts is an expensive operation, it means that the batch size", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "has to be constant over the course of outer loop optimization; (iii) It relies on LU factorization, which", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 499, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 499, + 118 + ], + "score": 1.0, + "content": "for symmetric matrices (the case of Theseus) is less efficient than using Cholesky decomposition.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "score": 1.0, + "content": "BaSpaCho solver. Batched Sparse Cholesky (BaSpaCho) is a novel open-source sparse Cholesky", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "solver designed for Theseus with support for batching (https://github.com/facebookresearch/", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "baspacho). BaSpaCho implements the supernodal Cholesky algorithm [70] to achieve state-of-the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "art performance by exploiting dense operations via BLAS/cuBLAS. This is achieved by building an", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "elimination tree and then clustering column blocks with similar sparsity patterns. These blocks", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "form nodes of the elimination tree and allow dense operations. In BaSpaCho, the dense operations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "are dispatched to BLAS (on CPU) or cuBLAS (on GPU), with additional support added on top for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "batching matrix operations with the same sparsity patterns. In problems with very sparse matrices,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "like bundle adjustment [8], the supernodal algorithm employed in state-of-the-art solvers [37] is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "unable to eliminate columns of parameter blocks simultaneously. Thus, past work has resorted to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "the Schur complement trick [71] to send a reduced problem to the sparse solver. However, this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "logic adds extra complexity to the nonlinear optimization, while essentially duplicating the work", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "of the (mathematically equivalent) Cholesky decomposition. In BaSpaCho, we instead complement", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the supernodal algorithm with sparse elimination that removes the need to externally handle Schur", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "complement as a workaround to the limitation of the supernodal algorithm. More details are described", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 150, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 150, + 299 + ], + "score": 1.0, + "content": "in App. F.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "Backward for custom linear solvers. Obtaining derivatives of the linear system solve with respect", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "to the parameters is a crucial operation for DNLS. In particular, we consider optimizing the parameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 322, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 107, + 324, + 115, + 334 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 322, + 132, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 133, + 324, + 138, + 334 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 322, + 210, + 337 + ], + "score": 1.0, + "content": "of a linear system", + "type": "text" + }, + { + "bbox": [ + 211, + 323, + 253, + 335 + ], + "score": 0.92, + "content": "y = A ^ { - 1 } b", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 322, + 394, + 337 + ], + "score": 1.0, + "content": "to minimize a downstream function", + "type": "text" + }, + { + "bbox": [ + 394, + 324, + 414, + 336 + ], + "score": 0.93, + "content": "f ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 322, + 505, + 337 + ], + "score": 1.0, + "content": ". The derivatives of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "loss with respect to the parameters of the linear system can be obtained with implicit differentiation,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 108, + 343, + 508, + 363 + ], + "spans": [ + { + "bbox": [ + 108, + 345, + 162, + 361 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { \\partial f } { \\partial b } = A ^ { - 1 } \\frac { \\partial \\mathbf { \\bar { f } } } { \\partial y } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 345, + 181, + 362 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 345, + 257, + 361 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial f } { \\partial A } = - A ^ { - 1 } \\frac { \\partial f } { \\partial y } y ^ { \\top } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 343, + 508, + 363 + ], + "score": 1.0, + "content": "1 @f@y y>, as done in Barron and Poole [72]. In Theseus, we implement", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "this by connecting the Python interface of our sparse solvers with PyTorch’s autograd.Function", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "classes that implement the gradients above in their backward methods. This connects the computation", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "graph between the downstream function and any upstream parameters that modify the system via", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "auxiliary variables or values for optimization variables. Furthermore, since the gradients require", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 404, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 415 + ], + "score": 1.0, + "content": "solving linear systems that use the same matrix as the forward pass, our backward pass can cache", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 414, + 460, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 460, + 426 + ], + "score": 1.0, + "content": "factorizations, resulting in it being significantly faster than the forward pass (see Fig. 3).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 248, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 249, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 249, + 442 + ], + "score": 1.0, + "content": "4.3 Backward modes for DNLS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 503, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 169, + 458 + ], + "score": 1.0, + "content": "The parameters", + "type": "text" + }, + { + "bbox": [ + 170, + 444, + 177, + 456 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 442, + 443, + 458 + ], + "score": 1.0, + "content": "upstream of DNLS can be learned end-to-end through the solution", + "type": "text" + }, + { + "bbox": [ + 443, + 444, + 467, + 456 + ], + "score": 0.92, + "content": "\\theta ^ { \\star } ( \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 442, + 505, + 458 + ], + "score": 1.0, + "content": "by using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 453, + 484, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 198, + 469 + ], + "score": 1.0, + "content": "the adjoint derivatives", + "type": "text" + }, + { + "bbox": [ + 198, + 455, + 245, + 467 + ], + "score": 0.93, + "content": "{ { \\partial \\theta ^ { \\star } ( \\phi ) } / { \\partial \\phi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 453, + 484, + 469 + ], + "score": 1.0, + "content": ". We include four methods for computing them in Theseus.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Unrolling is the standard way in which past work in DNLS has computed the adjoint derivatives.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "This is often referred to as backpropagation through time or unrolled optimization and is explored in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "[16, 20, 73–82]. In practice, often only a few steps of unrolling are performed due to challenges with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 505, + 283, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 283, + 516 + ], + "score": 1.0, + "content": "compute, memory, and vanishing gradients.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "Truncated differentiation. Aside from unrolling a few steps, another way of approximating the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 531, + 507, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 544 + ], + "score": 1.0, + "content": "derivatives is to use truncated backpropagation through time (TBPTT) [83, 84]. Truncation unfortu-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 504, + 554 + ], + "score": 1.0, + "content": "nately results in biased derivatives and many works [85–89] seek to further theoretically understand", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 554, + 433, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 433, + 564 + ], + "score": 1.0, + "content": "the properties of TBPTT, including the bias of the estimator and how to unbias it.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 222, + 582 + ], + "score": 1.0, + "content": "Implicit differentiation. If", + "type": "text" + }, + { + "bbox": [ + 222, + 570, + 233, + 580 + ], + "score": 0.86, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "can be computed exactly, then the implicit function theorem pro-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "vides a way of computing the adjoint derivatives, as done in related work in convex optimization", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "and first-order gradient descent methods [14, 15, 90–95]. We apply the implicit function theorem", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "from Dontchev and Rockafellar [39, Theorem 1B.1] (see App. H) to Eq. (2) to perform implicit", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "differentiation on a new class of second-order NLS optimization. This first requires that we trans-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "form Eq. (2) into an implicit function that finds the roots. We do this via the first-order optimality", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 199, + 648 + ], + "score": 1.0, + "content": "condition, resulting in", + "type": "text" + }, + { + "bbox": [ + 199, + 635, + 288, + 647 + ], + "score": 0.93, + "content": "\\bar { g } ( \\theta ; \\phi ) : = \\nabla _ { \\theta } S ( \\theta ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 635, + 327, + 648 + ], + "score": 1.0, + "content": ". Finding", + "type": "text" + }, + { + "bbox": [ + 327, + 635, + 442, + 647 + ], + "score": 0.93, + "content": "\\Theta ^ { \\star } ( \\phi ) : = \\{ \\theta \\mid g ( \\theta ; \\phi ) = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 645, + 476, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 469, + 658 + ], + "score": 1.0, + "content": "solving Eq. (2). Under mild assumptions, the theorem above gives the adjoint derivative at", + "type": "text" + }, + { + "bbox": [ + 469, + 645, + 476, + 658 + ], + "score": 0.84, + "content": "\\bar { \\phi }", + "type": "inline_equation" + } + ], + "index": 49 + } + ], + "index": 45.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 660, + 398, + 675 + ], + "lines": [ + { + "bbox": [ + 212, + 660, + 398, + 675 + ], + "spans": [ + { + "bbox": [ + 212, + 660, + 398, + 675 + ], + "score": 0.93, + "content": "\\mathrm { D } _ { \\phi } \\theta ^ { \\star } ( \\bar { \\phi } ) = - \\mathrm { D } _ { \\theta } ^ { - 1 } g ( \\theta ^ { \\star } ( \\bar { \\phi } ) ; \\bar { \\phi } ) \\mathrm { D } _ { \\phi } g ( \\theta ^ { \\star } ( \\bar { \\phi } ) ; \\bar { \\phi } ) .", + "type": "interline_equation", + "image_path": "ecd675f1a807fecc35ad872135a26962235f0807291bc492f97a4892825501d9.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 212, + 660, + 398, + 675 + ], + "spans": [], + "index": 50 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 503, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "As Theseus internally uses a (Gauss-)Newton solver, the following proposition provided in App. H", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 690, + 499, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 499, + 701 + ], + "score": 1.0, + "content": "shows how we can compute Eq. (3) by differentiating a single Newton step at an optimal solution.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + }, + { + "type": "text", + "bbox": [ + 106, + 702, + 505, + 725 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 716 + ], + "score": 1.0, + "content": "Proposition 1. The implicit derivative (Eq. (3)) can be computed by differentiating a Newton step", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 710, + 483, + 727 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 275, + 725 + ], + "score": 0.9, + "content": "\\bar { h ( \\theta ; \\phi ) } : = \\theta - [ \\nabla _ { \\theta } ^ { 2 } S ( \\hat { \\theta } ; \\phi ) ] _ { \\mathrm { { s t o p } } } ^ { - 1 } \\nabla _ { \\theta } S ( \\theta ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 710, + 333, + 727 + ], + "score": 1.0, + "content": "at an optimal", + "type": "text" + }, + { + "bbox": [ + 333, + 714, + 342, + 722 + ], + "score": 0.87, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 710, + 374, + 727 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 374, + 713, + 398, + 725 + ], + "score": 0.89, + "content": "[ \\cdot ] _ { \\mathrm { s t o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 710, + 483, + 727 + ], + "score": 1.0, + "content": "zeros the derivative.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53.5 + } + ], + "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": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 506, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 504, + 133 + ], + "score": 1.0, + "content": "BaSpaCho solver. Batched Sparse Cholesky (BaSpaCho) is a novel open-source sparse Cholesky", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "solver designed for Theseus with support for batching (https://github.com/facebookresearch/", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "baspacho). BaSpaCho implements the supernodal Cholesky algorithm [70] to achieve state-of-the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 506, + 168 + ], + "score": 1.0, + "content": "art performance by exploiting dense operations via BLAS/cuBLAS. This is achieved by building an", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "elimination tree and then clustering column blocks with similar sparsity patterns. These blocks", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "form nodes of the elimination tree and allow dense operations. In BaSpaCho, the dense operations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "are dispatched to BLAS (on CPU) or cuBLAS (on GPU), with additional support added on top for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "batching matrix operations with the same sparsity patterns. In problems with very sparse matrices,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "like bundle adjustment [8], the supernodal algorithm employed in state-of-the-art solvers [37] is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "unable to eliminate columns of parameter blocks simultaneously. Thus, past work has resorted to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "the Schur complement trick [71] to send a reduced problem to the sparse solver. However, this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "score": 1.0, + "content": "logic adds extra complexity to the nonlinear optimization, while essentially duplicating the work", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "of the (mathematically equivalent) Cholesky decomposition. In BaSpaCho, we instead complement", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the supernodal algorithm with sparse elimination that removes the need to externally handle Schur", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "complement as a workaround to the limitation of the supernodal algorithm. More details are described", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 150, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 150, + 299 + ], + "score": 1.0, + "content": "in App. F.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 122, + 506, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "Backward for custom linear solvers. Obtaining derivatives of the linear system solve with respect", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "to the parameters is a crucial operation for DNLS. In particular, we consider optimizing the parameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 322, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 107, + 324, + 115, + 334 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 322, + 132, + 337 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 133, + 324, + 138, + 334 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 322, + 210, + 337 + ], + "score": 1.0, + "content": "of a linear system", + "type": "text" + }, + { + "bbox": [ + 211, + 323, + 253, + 335 + ], + "score": 0.92, + "content": "y = A ^ { - 1 } b", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 322, + 394, + 337 + ], + "score": 1.0, + "content": "to minimize a downstream function", + "type": "text" + }, + { + "bbox": [ + 394, + 324, + 414, + 336 + ], + "score": 0.93, + "content": "f ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 322, + 505, + 337 + ], + "score": 1.0, + "content": ". The derivatives of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "loss with respect to the parameters of the linear system can be obtained with implicit differentiation,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 108, + 343, + 508, + 363 + ], + "spans": [ + { + "bbox": [ + 108, + 345, + 162, + 361 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { \\partial f } { \\partial b } = A ^ { - 1 } \\frac { \\partial \\mathbf { \\bar { f } } } { \\partial y } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 345, + 181, + 362 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 345, + 257, + 361 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { \\partial f } { \\partial A } = - A ^ { - 1 } \\frac { \\partial f } { \\partial y } y ^ { \\top } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 343, + 508, + 363 + ], + "score": 1.0, + "content": "1 @f@y y>, as done in Barron and Poole [72]. In Theseus, we implement", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "this by connecting the Python interface of our sparse solvers with PyTorch’s autograd.Function", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "classes that implement the gradients above in their backward methods. This connects the computation", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "graph between the downstream function and any upstream parameters that modify the system via", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "auxiliary variables or values for optimization variables. Furthermore, since the gradients require", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 404, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 415 + ], + "score": 1.0, + "content": "solving linear systems that use the same matrix as the forward pass, our backward pass can cache", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 414, + 460, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 460, + 426 + ], + "score": 1.0, + "content": "factorizations, resulting in it being significantly faster than the forward pass (see Fig. 3).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 300, + 508, + 426 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 429, + 248, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 249, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 249, + 442 + ], + "score": 1.0, + "content": "4.3 Backward modes for DNLS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 503, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 169, + 458 + ], + "score": 1.0, + "content": "The parameters", + "type": "text" + }, + { + "bbox": [ + 170, + 444, + 177, + 456 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 442, + 443, + 458 + ], + "score": 1.0, + "content": "upstream of DNLS can be learned end-to-end through the solution", + "type": "text" + }, + { + "bbox": [ + 443, + 444, + 467, + 456 + ], + "score": 0.92, + "content": "\\theta ^ { \\star } ( \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 442, + 505, + 458 + ], + "score": 1.0, + "content": "by using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 453, + 484, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 198, + 469 + ], + "score": 1.0, + "content": "the adjoint derivatives", + "type": "text" + }, + { + "bbox": [ + 198, + 455, + 245, + 467 + ], + "score": 0.93, + "content": "{ { \\partial \\theta ^ { \\star } ( \\phi ) } / { \\partial \\phi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 453, + 484, + 469 + ], + "score": 1.0, + "content": ". We include four methods for computing them in Theseus.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 442, + 505, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Unrolling is the standard way in which past work in DNLS has computed the adjoint derivatives.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "This is often referred to as backpropagation through time or unrolled optimization and is explored in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "[16, 20, 73–82]. In practice, often only a few steps of unrolling are performed due to challenges with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 505, + 283, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 283, + 516 + ], + "score": 1.0, + "content": "compute, memory, and vanishing gradients.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 471, + 506, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 533 + ], + "score": 1.0, + "content": "Truncated differentiation. Aside from unrolling a few steps, another way of approximating the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 531, + 507, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 544 + ], + "score": 1.0, + "content": "derivatives is to use truncated backpropagation through time (TBPTT) [83, 84]. Truncation unfortu-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 504, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 504, + 554 + ], + "score": 1.0, + "content": "nately results in biased derivatives and many works [85–89] seek to further theoretically understand", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 554, + 433, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 433, + 564 + ], + "score": 1.0, + "content": "the properties of TBPTT, including the bias of the estimator and how to unbias it.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 519, + 507, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 222, + 582 + ], + "score": 1.0, + "content": "Implicit differentiation. If", + "type": "text" + }, + { + "bbox": [ + 222, + 570, + 233, + 580 + ], + "score": 0.86, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "can be computed exactly, then the implicit function theorem pro-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "vides a way of computing the adjoint derivatives, as done in related work in convex optimization", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "and first-order gradient descent methods [14, 15, 90–95]. We apply the implicit function theorem", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "from Dontchev and Rockafellar [39, Theorem 1B.1] (see App. H) to Eq. (2) to perform implicit", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "differentiation on a new class of second-order NLS optimization. This first requires that we trans-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "form Eq. (2) into an implicit function that finds the roots. We do this via the first-order optimality", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 199, + 648 + ], + "score": 1.0, + "content": "condition, resulting in", + "type": "text" + }, + { + "bbox": [ + 199, + 635, + 288, + 647 + ], + "score": 0.93, + "content": "\\bar { g } ( \\theta ; \\phi ) : = \\nabla _ { \\theta } S ( \\theta ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 635, + 327, + 648 + ], + "score": 1.0, + "content": ". Finding", + "type": "text" + }, + { + "bbox": [ + 327, + 635, + 442, + 647 + ], + "score": 0.93, + "content": "\\Theta ^ { \\star } ( \\phi ) : = \\{ \\theta \\mid g ( \\theta ; \\phi ) = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 645, + 476, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 469, + 658 + ], + "score": 1.0, + "content": "solving Eq. (2). Under mild assumptions, the theorem above gives the adjoint derivative at", + "type": "text" + }, + { + "bbox": [ + 469, + 645, + 476, + 658 + ], + "score": 0.84, + "content": "\\bar { \\phi }", + "type": "inline_equation" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 569, + 506, + 658 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 660, + 398, + 675 + ], + "lines": [ + { + "bbox": [ + 212, + 660, + 398, + 675 + ], + "spans": [ + { + "bbox": [ + 212, + 660, + 398, + 675 + ], + "score": 0.93, + "content": "\\mathrm { D } _ { \\phi } \\theta ^ { \\star } ( \\bar { \\phi } ) = - \\mathrm { D } _ { \\theta } ^ { - 1 } g ( \\theta ^ { \\star } ( \\bar { \\phi } ) ; \\bar { \\phi } ) \\mathrm { D } _ { \\phi } g ( \\theta ^ { \\star } ( \\bar { \\phi } ) ; \\bar { \\phi } ) .", + "type": "interline_equation", + "image_path": "ecd675f1a807fecc35ad872135a26962235f0807291bc492f97a4892825501d9.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 212, + 660, + 398, + 675 + ], + "spans": [], + "index": 50 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 503, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "As Theseus internally uses a (Gauss-)Newton solver, the following proposition provided in App. H", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 690, + 499, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 499, + 701 + ], + "score": 1.0, + "content": "shows how we can compute Eq. (3) by differentiating a single Newton step at an optimal solution.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5, + "bbox_fs": [ + 106, + 678, + 506, + 701 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 702, + 505, + 725 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 716 + ], + "score": 1.0, + "content": "Proposition 1. The implicit derivative (Eq. (3)) can be computed by differentiating a Newton step", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 710, + 483, + 727 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 275, + 725 + ], + "score": 0.9, + "content": "\\bar { h ( \\theta ; \\phi ) } : = \\theta - [ \\nabla _ { \\theta } ^ { 2 } S ( \\hat { \\theta } ; \\phi ) ] _ { \\mathrm { { s t o p } } } ^ { - 1 } \\nabla _ { \\theta } S ( \\theta ; \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 710, + 333, + 727 + ], + "score": 1.0, + "content": "at an optimal", + "type": "text" + }, + { + "bbox": [ + 333, + 714, + 342, + 722 + ], + "score": 0.87, + "content": "\\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 710, + 374, + 727 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 374, + 713, + 398, + 725 + ], + "score": 0.89, + "content": "[ \\cdot ] _ { \\mathrm { s t o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 710, + 483, + 727 + ], + "score": 1.0, + "content": "zeros the derivative.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 700, + 505, + 727 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 72, + 502, + 176 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 72, + 502, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 72, + 502, + 176 + ], + "spans": [ + { + "bbox": [ + 107, + 72, + 502, + 176 + ], + "score": 0.97, + "type": "image", + "image_path": "34be76bb347a8d76fcfa01017b987d15fb2306443580a73d8087400d22648ccd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 72, + 502, + 106.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 106.66666666666666, + 502, + 141.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 141.33333333333331, + 502, + 175.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 178, + 503, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "Figure 3: Forward/backward times of Theseus with sparse and dense solvers on different PGO problem scales.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 194, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "Direct loss minimization. Suppose we have an outer loss as in Eq. (2). The direct loss minimization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 204, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 218 + ], + "score": 1.0, + "content": "(DLM) approach uses this loss to augment the inner-loop optimization problem in order to define", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 102, + 214, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 102, + 214, + 351, + 233 + ], + "score": 1.0, + "content": "a finite difference scheme that approaches the true gradient", + "type": "text" + }, + { + "bbox": [ + 351, + 217, + 439, + 230 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nabla _ { \\phi } L = \\operatorname* { l i m } _ { \\varepsilon \\to 0 } g _ { \\mathrm { D L M } } ^ { \\varepsilon } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 214, + 470, + 233 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 471, + 216, + 505, + 230 + ], + "score": 0.9, + "content": "g _ { \\mathrm { D L M } } ^ { \\varepsilon } \\triangleq", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 224, + 508, + 246 + ], + "spans": [ + { + "bbox": [ + 107, + 228, + 240, + 244 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { \\varepsilon } \\{ \\frac { \\partial } { \\partial \\phi } S ( \\theta ^ { * } ; \\phi ) - \\frac { \\partial } { \\partial \\phi } S ( \\theta _ { \\mathrm { d i r e c t } } ; \\phi ) \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 224, + 508, + 246 + ], + "score": 1.0, + "content": "! . This was used in prior works that solve optimization problems on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 243, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 504, + 254 + ], + "score": 1.0, + "content": "structured discrete domains [40, 41, 96, 97], but has so far not seen much use in structured continuous", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 253, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 266 + ], + "score": 1.0, + "content": "settings. We modify the original DLM formulation to better suit its implementation within Theseus", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 270, + 459, + 295 + ], + "lines": [ + { + "bbox": [ + 151, + 270, + 459, + 295 + ], + "spans": [ + { + "bbox": [ + 151, + 270, + 459, + 295 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\theta ^ { \\star } = \\underset { \\hat { \\theta } } { \\arg \\operatorname* { m i n } } S ( \\hat { \\theta } ; \\phi ) , \\quad } & { \\theta _ { \\mathrm { d i r e c t } } = \\underset { \\hat { \\theta } } { \\arg \\operatorname* { m i n } } S ( \\hat { \\theta } ; \\phi ) + \\left\\| \\varepsilon \\hat { \\theta } - \\frac { 1 } { 2 } \\nabla _ { \\theta } L ( \\theta ^ { * } ) \\right\\| ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "0971aa339764aa817296b759aa89c7611a50966052b52d71947a99e416f87b69.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 151, + 270, + 459, + 295 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "This is different from the original formulation in two ways: (i) we only assume access to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 167, + 324 + ], + "score": 1.0, + "content": "gradient vector", + "type": "text" + }, + { + "bbox": [ + 168, + 311, + 205, + 324 + ], + "score": 0.92, + "content": "\\nabla _ { { \\boldsymbol { \\theta } } } L ( { \\boldsymbol { \\theta } } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 311, + 506, + 324 + ], + "score": 1.0, + "content": ", which helps formulate DLM as an algorithm for computing vector-Jacobian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 465, + 335 + ], + "score": 1.0, + "content": "products, and (ii) we add a small regularization term to ensure the modified objective for", + "type": "text" + }, + { + "bbox": [ + 466, + 322, + 488, + 334 + ], + "score": 0.9, + "content": "\\theta _ { \\mathrm { d i r e c t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 435, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 288, + 345 + ], + "score": 1.0, + "content": "sum-of-squares without affecting the limit as", + "type": "text" + }, + { + "bbox": [ + 288, + 334, + 314, + 344 + ], + "score": 0.89, + "content": "\\varepsilon \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 334, + 435, + 345 + ], + "score": 1.0, + "content": ". See App. H for more details.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 106, + 352, + 182, + 365 + ], + "lines": [ + { + "bbox": [ + 104, + 350, + 183, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 183, + 368 + ], + "score": 1.0, + "content": "5 Evaluation", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "We evaluate the performance of Theseus under different settings with PGO and tactile state estimation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "applications from Sec. 3.1. PGO allows us to easily control the problem scales for performance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "evaluation; in Sec. 5.1 we profile time and memory consumption of Theseus in an end-to-end setup", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "and in Sec. 5.2 we evaluate timings of Theseus as a stand-alone NLS optimizer and compare with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "state-of-the-art Ceres [37]. The tactile state estimation application involves a more complex outer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "loop model that is useful for comparing all different backward modes, which we present in Sec. 5.3.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 445, + 382, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 383, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 383, + 459 + ], + "score": 1.0, + "content": "5.1 Profiling forward and backward pass of Theseus for DNLS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 504, + 471 + ], + "score": 1.0, + "content": "We study the performance of Theseus for DNLS on the PGO problem [60] with the synthetic Cube", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "dataset, as described in App. E. We run 10 inner loop iterations and 20 outer loop epochs, and use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "implicit differentiation to compute gradients of the inner NLS optimization. For these experiments", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "we used an Nvidia V100 GPU with 32GBs of memory for all Python computation, and Intel Xeon", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "2.2GHz CPU with 20 threads for the CPU-based CHOLMOD linear solver. 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For a batch size of 128, the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "largest problem that it can solve before running out of GPU memory has 256 poses (left two plots).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "With 2048 poses, dense is unable to solve the problem regardless of batch size (right two plots). On", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "the other hand with a batch size of 128, our solvers BaSpaCho scale to 2048 poses and cudaLU scale", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "to 4096 poses. 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This was used in prior works that solve optimization problems on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 243, + 504, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 504, + 254 + ], + "score": 1.0, + "content": "structured discrete domains [40, 41, 96, 97], but has so far not seen much use in structured continuous", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 253, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 266 + ], + "score": 1.0, + "content": "settings. 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See App. H for more details.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 299, + 506, + 345 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 352, + 182, + 365 + ], + "lines": [ + { + "bbox": [ + 104, + 350, + 183, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 183, + 368 + ], + "score": 1.0, + "content": "5 Evaluation", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "We evaluate the performance of Theseus under different settings with PGO and tactile state estimation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "applications from Sec. 3.1. 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We evaluate performance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 514, + 433, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 433, + 527 + ], + "score": 1.0, + "content": "using our sparse solvers in Theseus and using PyTorch’s Cholesky dense solver.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 460, + 506, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 506, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "Fig. 3 shows the average time of a full forward and backward pass for a given batch size, taken", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "by Theseus with different solvers (cudaLU, CHOLMOD, BaSpaCho and dense), for different problem", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "scales (number of poses and batch size). The two left plots show time as a function of number of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "poses for a batch size of 128, while the two right plots show time as a function of batch size for 2048", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "poses. We find that dense does not scale well with poses or batch size. For a batch size of 128, the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "largest problem that it can solve before running out of GPU memory has 256 poses (left two plots).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "With 2048 poses, dense is unable to solve the problem regardless of batch size (right two plots). On", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "the other hand with a batch size of 128, our solvers BaSpaCho scale to 2048 poses and cudaLU scale", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "to 4096 poses. CHOLMOD can solve problems even larger, since the linear system is solved on CPU and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "we have successfully tested up to 8192 poses and batch size 256 (see App. G), for a total of 22GBs of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 639, + 343, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 343, + 652 + ], + "score": 1.0, + "content": "GPU usage for residuals and Jacobian blocks computation.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 531, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "In addition to being more memory efficient, running times of our sparse solvers are also smaller for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "large enough number of poses/batch size, especially for the backward pass. Even though dense’s total", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "time for forward+backward is comparable to cudaLU and faster than CHOLMOD for smaller problems:", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "e.g., 1.47s (dense) vs. 1.32s (cudaLU) and 2.82s (CHOLMOD) for batch size 128 and 64 poses, dense", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "is significantly slower or out of memory for larger problems. For the largest problem that dense can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "solve (batch size 128 and 256 poses) its total time is already much slower than all others methods:", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 656, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "20.81s (dense) vs. 10.96s (CHOLMOD), 2.86s (cudaLU), and 2.25s (BaSpaCho). Furthermore, BaSpaCho", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "outperforms dense for any problem scale and is up to one order of magnitude faster, including for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "smaller batch sizes and number of poses (see App. G for more results and details). For the largest", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "problem that we consider (batch size 256 and 2048 poses), the total times for our sparse solvers are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 380, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 380, + 128 + ], + "score": 1.0, + "content": "170.28s for cudaLU, 239.07s for CHOLMOD, and 57.67s for BaSpaCho.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 132, + 332, + 143 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 334, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 334, + 146 + ], + "score": 1.0, + "content": "5.2 Profiling Theseus as stand-alone NLS optimizer", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 159 + ], + "score": 1.0, + "content": "DNLS typically involves solving numerous optimization problems each epoch where a fast NLS", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "optimizer is essential. We compare Theseus as a stand-alone NLS optimizer with the state-of-the-art", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "Ceres [37] library for solving a batch of PGO problems without any learning involved. We compare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "all solvers in terms of the total time required to perform 10 iterations on a set of 256 PGO problems.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "CPU/GPU configurations are same as before. For CHOLMOD, we also include a configuration that runs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 488, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 488, + 213 + ], + "score": 1.0, + "content": "everything on CPU, including Jacobians and residual computation (labelled CHOLMOD-allcpu).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 265, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 265, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 265, + 230 + ], + "score": 1.0, + "content": "Fig. 4 shows speedup obtained by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 227, + 266, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 266, + 241 + ], + "score": 1.0, + "content": "Theseus with batching, vectorization", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 239, + 266, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 266, + 250 + ], + "score": 1.0, + "content": "and sparse solvers, over Ceres as a func-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 249, + 266, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 266, + 262 + ], + "score": 1.0, + "content": "tion of increasing number of poses or", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 260, + 266, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 266, + 272 + ], + "score": 1.0, + "content": "batch size. We vary the number of poses", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 266, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 266, + 284 + ], + "score": 1.0, + "content": "for a fixed batch size of 256, and vary", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 282, + 266, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 266, + 294 + ], + "score": 1.0, + "content": "the batch size for a fixed number of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 293, + 266, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 266, + 305 + ], + "score": 1.0, + "content": "poses of 2048. Although Ceres is faster", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 304, + 266, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 266, + 316 + ], + "score": 1.0, + "content": "than all of our solvers when the number", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 316, + 266, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 266, + 327 + ], + "score": 1.0, + "content": "of poses and batch size are small (for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 326, + 265, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 182, + 337 + ], + "score": 1.0, + "content": "instance, Ceres is", + "type": "text" + }, + { + "bbox": [ + 183, + 326, + 200, + 337 + ], + "score": 0.65, + "content": "2 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 326, + 265, + 337 + ], + "score": 1.0, + "content": "faster with 256", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 337, + 266, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 266, + 349 + ], + "score": 1.0, + "content": "poses and 16 batch size, see App. G),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 348, + 267, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 267, + 359 + ], + "score": 1.0, + "content": "as these increase Theseus shows signif-", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18 + }, + { + "type": "image", + "bbox": [ + 276, + 221, + 501, + 326 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 276, + 221, + 501, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 276, + 221, + 501, + 326 + ], + "spans": [ + { + "bbox": [ + 276, + 221, + 501, + 326 + ], + "score": 0.969, + "type": "image", + "image_path": "ea54f83cdfabe19990c310cda65475bf4791b5a3adff46b06d1784c65e347701.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 276, + 221, + 501, + 234.125 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 276, + 234.125, + 501, + 247.25 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 276, + 247.25, + 501, + 260.375 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 276, + 260.375, + 501, + 273.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 276, + 273.5, + 501, + 286.625 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 276, + 286.625, + 501, + 299.75 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 276, + 299.75, + 501, + 312.875 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 276, + 312.875, + 501, + 326.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 272, + 334, + 504, + 355 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 272, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 272, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "Figure 4: Speedup of Theseus (forward pass) over Ceres (black", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 272, + 343, + 425, + 356 + ], + "spans": [ + { + "bbox": [ + 272, + 343, + 425, + 356 + ], + "score": 1.0, + "content": "dashed) on different PGO problem scales.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "index": 31.0 + }, + { + "type": "text", + "bbox": [ + 107, + 359, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "icant speedup by being able to solve larger batches of problems in parallel. For the largest setup that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 388, + 382 + ], + "score": 1.0, + "content": "all our solvers can scale to (2048 poses, 256 batch size), BaSpaCho is", + "type": "text" + }, + { + "bbox": [ + 389, + 370, + 414, + 380 + ], + "score": 0.88, + "content": "{ \\sim } 2 3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "faster than Ceres, and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 191, + 393 + ], + "score": 1.0, + "content": "our other solvers are", + "type": "text" + }, + { + "bbox": [ + 192, + 381, + 211, + 391 + ], + "score": 0.78, + "content": "{ \\sim } 4 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "faster. CHOLMOD has a 6x speedup for its largest setting (4096 poses, 256", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 391, + 156, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 156, + 405 + ], + "score": 1.0, + "content": "batch size).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "Since typical use case of Theseus involves large batches and number of variables during end-to-end", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "learning with DNLS, the speedups in this setting against a performant NLS solver highlights the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "significance of our efficiency-based design choices. See App. G for additional results of smaller fixed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 440, + 236, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 236, + 453 + ], + "score": 1.0, + "content": "batch size and number of poses.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 107, + 456, + 241, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 241, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 241, + 469 + ], + "score": 1.0, + "content": "5.3 Backward modes analysis", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "We explore the trade-offs between our different backward modes using the tactile state estimation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "application in Sec. 3.1. The learnable components here include a neural network, and thus closely", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "follow the type of applications that motivate Theseus. We compare the following backward modes:", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "derivative unrolling (Unroll), implicit differentiation (Implicit), truncated differentiation (Trunc),", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "and direct loss minimization (DLM); for Trunc we include results when truncating 5 and 10 steps. We", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 526, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 504, + 537 + ], + "score": 1.0, + "content": "compare all modes along 3 axis of performance: validation loss after 100 epochs (outer loop), run", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "time during training, and peak GPU memory consumption of TheseusLayer. For these experiments", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "we used Quadro GP100 GPUs with 16GB of memory. For time and memory we present separate", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "results for forward and backward pass, and all numbers are averaged over 700 (7 batches for 100", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "epochs). Below we discuss our main findings from this analysis, and more results and details can be", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 580, + 176, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 176, + 592 + ], + "score": 1.0, + "content": "found in App. H.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "Fig. 5 shows average run times for all backward modes as a function of the maximum number of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "iterations in the inner loop optimization. We observe that the time used in the forward pass (Fig. 5,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "far left) increases roughly linearly for all modes, all having similar times except for Unroll, which is", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "slower than other modes. On the other hand, we observe stark differences in the backward pass time", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "score": 1.0, + "content": "(Fig. 5, center left), where Unroll is the only method that has a linear dependence on the number", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "of inner loop iterations. All other methods have a constant footprint for computing derivatives,", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "independent of the number of inner loop iterations. As expected, increasing the number of iterations", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "score": 1.0, + "content": "through which we backprop (5 or 10 for Trunc, all iterations for Unroll) increases the time necessary", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 684, + 416, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 190, + 695 + ], + "score": 1.0, + "content": "for a backward pass", + "type": "text" + }, + { + "bbox": [ + 190, + 684, + 411, + 694 + ], + "score": 0.5, + "content": "( \\mathtt { I m p l i c i t } = \\mathtt { D L M } < \\mathtt { T r u n c - 5 } < \\mathtt { T r u n c - 1 } \\theta < < \\mathtt { U n r o l 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 684, + 416, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 59 + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "Figure 5 (center right) shows the average peak memory consumption of the backward modes. In", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 711, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 504, + 722 + ], + "score": 1.0, + "content": "this case, the trends observed for the backward pass memory consumption is similar to the trends", + "type": "text" + } + ], + "index": 65 + } + ], + "index": 64.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "20.81s (dense) vs. 10.96s (CHOLMOD), 2.86s (cudaLU), and 2.25s (BaSpaCho). Furthermore, BaSpaCho", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "outperforms dense for any problem scale and is up to one order of magnitude faster, including for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "smaller batch sizes and number of poses (see App. G for more results and details). For the largest", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "problem that we consider (batch size 256 and 2048 poses), the total times for our sparse solvers are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 380, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 380, + 128 + ], + "score": 1.0, + "content": "170.28s for cudaLU, 239.07s for CHOLMOD, and 57.67s for BaSpaCho.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 73, + 506, + 128 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 132, + 332, + 143 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 334, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 334, + 146 + ], + "score": 1.0, + "content": "5.2 Profiling Theseus as stand-alone NLS optimizer", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 505, + 159 + ], + "score": 1.0, + "content": "DNLS typically involves solving numerous optimization problems each epoch where a fast NLS", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "optimizer is essential. We compare Theseus as a stand-alone NLS optimizer with the state-of-the-art", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "Ceres [37] library for solving a batch of PGO problems without any learning involved. We compare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "all solvers in terms of the total time required to perform 10 iterations on a set of 256 PGO problems.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "CPU/GPU configurations are same as before. For CHOLMOD, we also include a configuration that runs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 488, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 488, + 213 + ], + "score": 1.0, + "content": "everything on CPU, including Jacobians and residual computation (labelled CHOLMOD-allcpu).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 146, + 506, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 265, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 265, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 265, + 230 + ], + "score": 1.0, + "content": "Fig. 4 shows speedup obtained by", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 227, + 266, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 266, + 241 + ], + "score": 1.0, + "content": "Theseus with batching, vectorization", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 239, + 266, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 266, + 250 + ], + "score": 1.0, + "content": "and sparse solvers, over Ceres as a func-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 249, + 266, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 266, + 262 + ], + "score": 1.0, + "content": "tion of increasing number of poses or", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 260, + 266, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 266, + 272 + ], + "score": 1.0, + "content": "batch size. We vary the number of poses", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 271, + 266, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 266, + 284 + ], + "score": 1.0, + "content": "for a fixed batch size of 256, and vary", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 282, + 266, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 266, + 294 + ], + "score": 1.0, + "content": "the batch size for a fixed number of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 293, + 266, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 266, + 305 + ], + "score": 1.0, + "content": "poses of 2048. Although Ceres is faster", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 304, + 266, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 266, + 316 + ], + "score": 1.0, + "content": "than all of our solvers when the number", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 316, + 266, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 266, + 327 + ], + "score": 1.0, + "content": "of poses and batch size are small (for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 326, + 265, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 182, + 337 + ], + "score": 1.0, + "content": "instance, Ceres is", + "type": "text" + }, + { + "bbox": [ + 183, + 326, + 200, + 337 + ], + "score": 0.65, + "content": "2 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 326, + 265, + 337 + ], + "score": 1.0, + "content": "faster with 256", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 337, + 266, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 266, + 349 + ], + "score": 1.0, + "content": "poses and 16 batch size, see App. G),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 348, + 267, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 267, + 359 + ], + "score": 1.0, + "content": "as these increase Theseus shows signif-", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 216, + 267, + 359 + ] + }, + { + "type": "image", + "bbox": [ + 276, + 221, + 501, + 326 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 276, + 221, + 501, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 276, + 221, + 501, + 326 + ], + "spans": [ + { + "bbox": [ + 276, + 221, + 501, + 326 + ], + "score": 0.969, + "type": "image", + "image_path": "ea54f83cdfabe19990c310cda65475bf4791b5a3adff46b06d1784c65e347701.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 276, + 221, + 501, + 234.125 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 276, + 234.125, + 501, + 247.25 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 276, + 247.25, + 501, + 260.375 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 276, + 260.375, + 501, + 273.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 276, + 273.5, + 501, + 286.625 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 276, + 286.625, + 501, + 299.75 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 276, + 299.75, + 501, + 312.875 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 276, + 312.875, + 501, + 326.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 272, + 334, + 504, + 355 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 272, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 272, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "Figure 4: Speedup of Theseus (forward pass) over Ceres (black", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 272, + 343, + 425, + 356 + ], + "spans": [ + { + "bbox": [ + 272, + 343, + 425, + 356 + ], + "score": 1.0, + "content": "dashed) on different PGO problem scales.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "index": 31.0 + }, + { + "type": "text", + "bbox": [ + 107, + 359, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "icant speedup by being able to solve larger batches of problems in parallel. For the largest setup that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 388, + 382 + ], + "score": 1.0, + "content": "all our solvers can scale to (2048 poses, 256 batch size), BaSpaCho is", + "type": "text" + }, + { + "bbox": [ + 389, + 370, + 414, + 380 + ], + "score": 0.88, + "content": "{ \\sim } 2 3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "faster than Ceres, and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 191, + 393 + ], + "score": 1.0, + "content": "our other solvers are", + "type": "text" + }, + { + "bbox": [ + 192, + 381, + 211, + 391 + ], + "score": 0.78, + "content": "{ \\sim } 4 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "faster. CHOLMOD has a 6x speedup for its largest setting (4096 poses, 256", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 391, + 156, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 156, + 405 + ], + "score": 1.0, + "content": "batch size).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 358, + 506, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "Since typical use case of Theseus involves large batches and number of variables during end-to-end", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "learning with DNLS, the speedups in this setting against a performant NLS solver highlights the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "significance of our efficiency-based design choices. See App. G for additional results of smaller fixed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 440, + 236, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 236, + 453 + ], + "score": 1.0, + "content": "batch size and number of poses.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 408, + 506, + 453 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 456, + 241, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 241, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 241, + 469 + ], + "score": 1.0, + "content": "5.3 Backward modes analysis", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "We explore the trade-offs between our different backward modes using the tactile state estimation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "application in Sec. 3.1. The learnable components here include a neural network, and thus closely", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "follow the type of applications that motivate Theseus. We compare the following backward modes:", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "derivative unrolling (Unroll), implicit differentiation (Implicit), truncated differentiation (Trunc),", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "and direct loss minimization (DLM); for Trunc we include results when truncating 5 and 10 steps. We", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 526, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 504, + 537 + ], + "score": 1.0, + "content": "compare all modes along 3 axis of performance: validation loss after 100 epochs (outer loop), run", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "time during training, and peak GPU memory consumption of TheseusLayer. For these experiments", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "we used Quadro GP100 GPUs with 16GB of memory. For time and memory we present separate", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "results for forward and backward pass, and all numbers are averaged over 700 (7 batches for 100", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "epochs). Below we discuss our main findings from this analysis, and more results and details can be", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 580, + 176, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 176, + 592 + ], + "score": 1.0, + "content": "found in App. H.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 470, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "Fig. 5 shows average run times for all backward modes as a function of the maximum number of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "iterations in the inner loop optimization. We observe that the time used in the forward pass (Fig. 5,", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "far left) increases roughly linearly for all modes, all having similar times except for Unroll, which is", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "slower than other modes. On the other hand, we observe stark differences in the backward pass time", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "score": 1.0, + "content": "(Fig. 5, center left), where Unroll is the only method that has a linear dependence on the number", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "of inner loop iterations. All other methods have a constant footprint for computing derivatives,", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "independent of the number of inner loop iterations. As expected, increasing the number of iterations", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "score": 1.0, + "content": "through which we backprop (5 or 10 for Trunc, all iterations for Unroll) increases the time necessary", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 684, + 416, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 190, + 695 + ], + "score": 1.0, + "content": "for a backward pass", + "type": "text" + }, + { + "bbox": [ + 190, + 684, + 411, + 694 + ], + "score": 0.5, + "content": "( \\mathtt { I m p l i c i t } = \\mathtt { D L M } < \\mathtt { T r u n c - 5 } < \\mathtt { T r u n c - 1 } \\theta < < \\mathtt { U n r o l 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 684, + 416, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 59, + "bbox_fs": [ + 105, + 596, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "Figure 5 (center right) shows the average peak memory consumption of the backward modes. In", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 711, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 504, + 722 + ], + "score": 1.0, + "content": "this case, the trends observed for the backward pass memory consumption is similar to the trends", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "in time. In particular, Unroll’s memory footprint increases linearly with the number of inner loop", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 171, + 208 + ], + "score": 1.0, + "content": "iterations, from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 172, + 196, + 213, + 207 + ], + "score": 0.69, + "content": "\\mathrm { \\sim 3 4 M B s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 214, + 195, + 225, + 208 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 225, + 196, + 272, + 207 + ], + "score": 0.65, + "content": "\\sim \\mathrm { 2 6 2 M B s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 272, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "; for all other methods the memory consumption remains", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "constant. The best memory profiles in this example is obtained with Implicit and DLM backward", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 159, + 230 + ], + "score": 1.0, + "content": "modes, with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 159, + 218, + 203, + 228 + ], + "score": 0.6, + "content": "\\sim \\mathrm { 2 8 M B s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 203, + 217, + 221, + 230 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 222, + 218, + 264, + 228 + ], + "score": 0.59, + "content": "\\sim \\mathrm { 2 9 M B s }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 264, + 217, + 506, + 230 + ], + "score": 1.0, + "content": ", respectively. These trends also hold for the forward pass", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 229, + 198, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 198, + 242 + ], + "score": 1.0, + "content": "memory consumption.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 64.5, + "bbox_fs": [ + 106, + 700, + 505, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 72, + 502, + 165 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 72, + 502, + 165 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 72, + 502, + 165 + ], + "spans": [ + { + "bbox": [ + 108, + 72, + 502, + 165 + ], + "score": 0.966, + "type": "image", + "image_path": "76832a8958ee7b5e173871e737d1c489533f2a9e26449e6214b37904f103dae4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 72, + 502, + 103.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 103.0, + 502, + 134.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 134.0, + 502, + 165.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 125, + 169, + 480, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 168, + 482, + 182 + ], + "spans": [ + { + "bbox": [ + 128, + 168, + 482, + 182 + ], + "score": 1.0, + "content": "Figure 5: Time and memory consumption of different backward modes in tactile state estimation.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 185, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "in time. In particular, Unroll’s memory footprint increases linearly with the number of inner loop", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 171, + 208 + ], + "score": 1.0, + "content": "iterations, from", + "type": "text" + }, + { + "bbox": [ + 172, + 196, + 213, + 207 + ], + "score": 0.69, + "content": "\\mathrm { \\sim 3 4 M B s }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 195, + 225, + 208 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 225, + 196, + 272, + 207 + ], + "score": 0.65, + "content": "\\sim \\mathrm { 2 6 2 M B s }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "; for all other methods the memory consumption remains", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "constant. The best memory profiles in this example is obtained with Implicit and DLM backward", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 159, + 230 + ], + "score": 1.0, + "content": "modes, with", + "type": "text" + }, + { + "bbox": [ + 159, + 218, + 203, + 228 + ], + "score": 0.6, + "content": "\\sim \\mathrm { 2 8 M B s }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 217, + 221, + 230 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 222, + 218, + 264, + 228 + ], + "score": 0.59, + "content": "\\sim \\mathrm { 2 9 M B s }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 217, + 506, + 230 + ], + "score": 1.0, + "content": ", respectively. These trends also hold for the forward pass", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 229, + 198, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 198, + 242 + ], + "score": 1.0, + "content": "memory consumption.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "Figure 5 also shows the validation losses obtained with all backward modes (far right). The best", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 255, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 268 + ], + "score": 1.0, + "content": "validation loss, after 100 epochs of training, is obtained using Implicit, followed by Trunc variants.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "score": 1.0, + "content": "We notice that both variants of Trunc keep improving with increasing number of inner loop iterations,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "and that Unroll and Implicit achieve the best results with 20 iterations. One exception is DLM, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "doesn’t improve much with the number of iterations, but is also the best method when only 2 inner", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "loop iterations are performed. As a point of caution, we stress that, unlike the timing and memory", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "results, the relative training performance between different backward modes is likely to be application", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "dependent, and is affected by hyperparameters such as the step size used for the inner loop optimizer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "(0.05 in this example), and the outer optimizer’s learning rate. Our experiments suggest that implicit", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "score": 1.0, + "content": "differentiation is a good default to use for differentiable optimization, considering its low time/memory", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 482, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 482, + 368 + ], + "score": 1.0, + "content": "footprint, and potential for better end-to-end performance with proper hyperparameter tuning.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 180, + 384 + ], + "lines": [ + { + "bbox": [ + 104, + 369, + 181, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 181, + 386 + ], + "score": 1.0, + "content": "6 Discussion", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "Summary. Theseus provides nonlinear least squares as a differentiable layer and enables easily", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "building and training end-to-end architectures for robotics and vision applications. We illustrate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "several example applications using the same application-agnostic interface and demonstrate significant", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "improvements in performance with our efficiency-based design. Following how autodiff and GPU", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "acceleration (among others) have led to the evolution of PyTorch in contrast to NumPy [98], we can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "similarly view sparsity and implicit differentiation on top of autodiff and GPU acceleration as the key", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "ingredients that power Theseus, in contrast to solvers like Ceres that typically only support sparsity.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 413, + 477 + ], + "score": 1.0, + "content": "When solving a batch of large problems the forward pass of Theseus is up to", + "type": "text" + }, + { + "bbox": [ + 414, + 465, + 430, + 475 + ], + "score": 0.52, + "content": "2 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "faster than Ceres.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "Limitations. Theseus currently has a few limitations. The nonlinear solvers we currently support", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "apply constraints in a soft manner (i.e., using weighted costs). Hard constraints can be handled with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "methods like augmented Lagrangian or sequential quadratic programs [99, 100], and differentiating", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "through them are active research topics. The current implementation of LM does not support", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "damping to be learnable. Some limitations and trade-offs with the sparse linear solvers are discussed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "in Sec. 4.2, and with backward modes are discussed in App. H. Online learning applications may", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 546, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 558 + ], + "score": 1.0, + "content": "require frequently editing the objective and depending on the problem size there may be a nontrivial", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "overhead that is not currently optimized as we explored only non-incremental settings in this work.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 486, + 580 + ], + "score": 1.0, + "content": "Additional performance gains can be extracted by moving some of our Python implementation to", + "type": "text" + }, + { + "bbox": [ + 486, + 567, + 506, + 578 + ], + "score": 0.86, + "content": "\\mathrm { C } { + } { + }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "but we prioritized flexibility in evolving the API in the short-term. We do not yet support distributed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "training beyond what PyTorch natively supports. 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The best", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 255, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 268 + ], + "score": 1.0, + "content": "validation loss, after 100 epochs of training, is obtained using Implicit, followed by Trunc variants.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "score": 1.0, + "content": "We notice that both variants of Trunc keep improving with increasing number of inner loop iterations,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "and that Unroll and Implicit achieve the best results with 20 iterations. One exception is DLM, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "doesn’t improve much with the number of iterations, but is also the best method when only 2 inner", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "loop iterations are performed. 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Our experiments suggest that implicit", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "score": 1.0, + "content": "differentiation is a good default to use for differentiable optimization, considering its low time/memory", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 353, + 482, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 482, + 368 + ], + "score": 1.0, + "content": "footprint, and potential for better end-to-end performance with proper hyperparameter tuning.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 244, + 506, + 368 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 180, + 384 + ], + "lines": [ + { + "bbox": [ + 104, + 369, + 181, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 181, + 386 + ], + "score": 1.0, + "content": "6 Discussion", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "Summary. Theseus provides nonlinear least squares as a differentiable layer and enables easily", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 412 + ], + "score": 1.0, + "content": "building and training end-to-end architectures for robotics and vision applications. We illustrate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "several example applications using the same application-agnostic interface and demonstrate significant", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "improvements in performance with our efficiency-based design. Following how autodiff and GPU", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "acceleration (among others) have led to the evolution of PyTorch in contrast to NumPy [98], we can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "similarly view sparsity and implicit differentiation on top of autodiff and GPU acceleration as the key", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "ingredients that power Theseus, in contrast to solvers like Ceres that typically only support sparsity.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 413, + 477 + ], + "score": 1.0, + "content": "When solving a batch of large problems the forward pass of Theseus is up to", + "type": "text" + }, + { + "bbox": [ + 414, + 465, + 430, + 475 + ], + "score": 0.52, + "content": "2 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "faster than Ceres.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 388, + 506, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "Limitations. Theseus currently has a few limitations. The nonlinear solvers we currently support", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "apply constraints in a soft manner (i.e., using weighted costs). Hard constraints can be handled with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "methods like augmented Lagrangian or sequential quadratic programs [99, 100], and differentiating", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "through them are active research topics. The current implementation of LM does not support", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "damping to be learnable. Some limitations and trade-offs with the sparse linear solvers are discussed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "in Sec. 4.2, and with backward modes are discussed in App. H. 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The authors also thank community", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "members for contributions to the open source code. Work by PS and JO was done while at Meta AI.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 633, + 507, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 63, + 507, + 725 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 115, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "[1] Jorge Nocedal and Stephen Wright. Numerical optimization. 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0000000000000000000000000000000000000000..5655b40f81fc1944baceaab393c3544d4837b4b3 --- /dev/null +++ b/parse/dev/PlKWVd2yBkY/PlKWVd2yBkY_content_list.json @@ -0,0 +1,2718 @@ +[ + { + "type": "text", + "text": "PSEUDO NUMERICAL METHODS FOR DIFFUSION MODELS ON MANIFOLDS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Luping Liu, Yi Ren, Zhijie Lin & Zhou Zhao∗ Zhejiang University {luping.liu,rayeren,linzhijie,zhaozhou}@zju.edu.cn ", + "bbox": [ + 183, + 169, + 671, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 248, + 544, + 263 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Denoising Diffusion Probabilistic Models (DDPMs) can generate high-quality samples such as image and audio samples. However, DDPMs require hundreds to thousands of iterations to produce final samples. Several prior works have successfully accelerated DDPMs through adjusting the variance schedule (e.g., Improved Denoising Diffusion Probabilistic Models) or the denoising equation (e.g., Denoising Diffusion Implicit Models (DDIMs)). However, these acceleration methods cannot maintain the quality of samples and even introduce new noise at a high speedup rate, which limit their practicability. To accelerate the inference process while keeping the sample quality, we provide a fresh perspective that DDPMs should be treated as solving differential equations on manifolds. Under such a perspective, we propose pseudo numerical methods for diffusion models (PNDMs). Specifically, we figure out how to solve differential equations on manifolds and show that DDIMs are simple cases of pseudo numerical methods. We change several classical numerical methods to corresponding pseudo numerical methods and find that the pseudo linear multi-step method is the best in most situations. According to our experiments, by directly using pre-trained models on Cifar10, CelebA and LSUN, PNDMs can generate higher quality synthetic images with only 50 steps compared with 1000-step DDIMs (20x speedup), significantly outperform DDIMs with 250 steps (by around 0.4 in FID) and have good generalization on different variance schedules.1 ", + "bbox": [ + 233, + 281, + 764, + 558 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 588, + 336, + 603 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Denoising Diffusion Probabilistic Models (DDPMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020) is a class of generative models which model the data distribution through an iterative denoising process reversing a multi-step noising process. DDPMs have been applied successfully to a variety of applications, including image generation (Ho et al., 2020; Song et al., 2020b), text generation (Hoogeboom et al., 2021; Austin et al., 2021), 3D point cloud generation (Luo & Hu, 2021), textto-speech (Kong et al., 2021; Chen et al., 2020) and image super-resolution (Saharia et al., 2021). ", + "bbox": [ + 174, + 619, + 825, + 704 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Unlike Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which require careful hyperparameter tuning according to different model structures and datasets, DDPMs can use similar model structures and be trained by a simple denoising objective which makes the models fit the noise in the data. To generate samples, the iterative denoising process starts from white noise and progressively denoises it into the target domain according to the noise predicted by the model at every step. However, a critical drawback of DDPMs is that DDPMs require hundreds to thousands of iterations to produce high-quality samples and need to pass through a network at least once at every step, which makes the generation of a large number of samples extremely slow and infeasible. In contrast, GANs only need one pass through a network. ", + "bbox": [ + 174, + 710, + 825, + 835 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There have been many recent works focusing on improving the speed of the denoising process. Some works search for better variance schedules, including Nichol & Dhariwal (2021) and Watson et al. (2021). Some works focus on changing the inference equation, including Song et al. (2020a) ", + "bbox": [ + 176, + 843, + 823, + 885 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "and Song et al. (2020b). Denoising Diffusion Implicit Models (DDIMs) (Song et al., 2020a) relying on a non-Markovian process accelerate the denoising process by taking multiple steps every iteration. Probability Flows (PFs) (Song et al., 2020b) build a connection between the denoising process and solving ordinary differential equations and use numerical methods of differential equations to accelerate the denoising process. Additionally, we introduce more related works in Appendix A.1. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, this direct connection between DDPMs and numerical methods (e.g., forward Euler method, linear multi-step method and RungeKutta method (Timothy, 2017)) has weaknesses in both speed and effect (see Section 3.1). Some numerical methods are straightforward, like the forward Euler method, but they can only trade quality for speed. Some numerical methods can accelerate the reverse process without loss of quality, like the Runge-Kutta method, but ", + "bbox": [ + 174, + 181, + 418, + 359 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/50dbf5ca5b09ab08a54e777730ea0a2143b4755a78b4ba24ccff58bfbeb6ff30.jpg", + "image_caption": [ + "Figure 1: 5, 10, 20, 50 and 100-steps generated results using DDIMs, classical numerical methods and PNDMs. " + ], + "image_footnote": [], + "bbox": [ + 434, + 181, + 821, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "they need to propagate forward more times along a neural network at every step. Furthermore, we also notice that numerical methods can introduce noticeable noise at a high speedup rate, which makes high-order numerical methods (e.g., Runge-Kutta method) even less effective than DDIMs. This phenomenon is also mentioned in Salimans & Ho (2022). ", + "bbox": [ + 174, + 361, + 825, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To figure out the reason for the performance degradation in classical numerical methods, we conduct some analyses and find that classical numerical methods may sample data far away from the main distribution area of the data, and the inference equations of DDPMs do not satisfy a necessary condition of numerical methods at the last several steps (see Section 3.2). ", + "bbox": [ + 174, + 422, + 825, + 478 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To tackle these problems, we design new numerical methods called pseudo numerical methods for diffusion models (PNDMs) to generate samples along a specific manifold in $\\mathbb { R } ^ { n }$ , which is the highdensity region of the data. We first compute the corresponding differential equations of diffusion models directly and self-consistently, which builds a theoretical connection between DDPMs and numerical methods. Considering that classical numerical methods cannot guarantee to generate samples on certain manifolds, we provide brand-new numerical methods called pseudo numerical methods based on our theoretical analyses. We also find that DDIMs are simple cases of pseudo numerical methods, which means that we also provide a new way to understand DDIMs better. Furthermore, we find that the pseudo linear multi-step method is the fastest method for diffusion models under similar generated quality. ", + "bbox": [ + 174, + 486, + 825, + 626 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Besides, we provide a detailed theoretical analysis of our new theory and give visualization results to support our theory intuitively. According to our experiments, our methods have several advantages: ", + "bbox": [ + 174, + 632, + 820, + 660 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Our methods combine the benefits of DDIMs and high-order numerical methods successfully. We theoretically prove that our new methods PNDMs are second-order convergent while DDIMs are first-order convergent, which makes PNDMs $2 0 \\mathrm { x }$ faster without loss of quality on Cifar10 and CelebA. \n• Our methods can reduce the best FID of pre-trained models with even shorter sampling time. With only 250 steps, our new denoising process can reduce the best FID by around 0.4 points Cifar10 and CelebA. We achieve a new SOTA FID score of 2.71 on CelebA. \n• Our methods work well with different variance schedules, which means that our methods have a good generalization and can be used together with those works introducing better variance schedules to accelerate the denoising process further. ", + "bbox": [ + 173, + 670, + 826, + 818 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 174, + 837, + 326, + 853 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we introduce some backgrounds. Firstly, we present the classical understanding of DDPMs. Then we provide another understanding based on Song et al. (2020b), which inspires us to use numerical methods to accelerate the denoising process of diffusion models. After that, we introduce some background on numerical methods used later in this paper. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 DENOISING DIFFUSION PROBABILISTIC MODELS ", + "text_level": 1, + "bbox": [ + 173, + 103, + 555, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "DDPMs model the data distribution from Gaussian distribution to image distribution through an iterative denoising process. Let $x _ { 0 }$ be an image, then the diffusion process is a Markov process and the reverse process has a similar form to the diffusion process, which satisfies: ", + "bbox": [ + 174, + 128, + 826, + 172 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/105742b9adc7361780afd973d8febdee597bffe2e8382ab67a184440b7746032.jpg", + "text": "$$\n\\begin{array} { r l } & { x _ { t + 1 } \\sim \\mathcal { N } ( \\sqrt { 1 - \\beta _ { t } } x _ { t } , \\beta _ { t } \\mathrm { I } ) , t = 0 , 1 , \\cdots , N - 1 . } \\\\ & { x _ { t - 1 } \\sim \\mathcal { N } ( \\mu _ { \\theta } ( x _ { t } , t ) , \\beta _ { \\theta } ( x _ { t } , t ) \\mathrm { I } ) , t = N , N - 1 , \\cdots , 1 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 316, + 175, + 681, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, $\\beta _ { t }$ controls the speed of adding noise to the data, calling them the variance schedule. $N$ is the total number of steps of the denoising process. $\\mu _ { \\theta }$ and $\\beta _ { \\theta }$ are two neural networks, and $\\theta$ are their parameters. ", + "bbox": [ + 173, + 218, + 825, + 260 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Ho et al. (2020) get some statistics estimations of $\\mu _ { \\theta }$ and $\\beta _ { \\theta }$ . According to the properties of the conditional Gaussian distribution, we have: ", + "bbox": [ + 176, + 266, + 823, + 295 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8e68e9baf8ab9a55c18ad8eaed6e2bcfc9f7ed07c17ddc0f31724ae945cff06c.jpg", + "text": "$$\n\\begin{array} { r l } & { q ( x _ { t } | x _ { 0 } ) = \\mathcal { N } ( \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } , ( 1 - \\bar { \\alpha } _ { t } ) \\mathrm { I } ) , } \\\\ & { q ( x _ { t - 1 } | x _ { t } , x _ { 0 } ) = \\mathcal { N } ( \\bar { \\mu } _ { t } ( x _ { t } , x _ { 0 } ) , \\bar { \\beta } _ { t } \\mathrm { I } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 297, + 624, + 338 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, $\\alpha _ { t } = 1 - \\beta _ { t }$ , $\\textstyle { \\bar { \\alpha } } _ { t } = \\prod _ { i = 1 } ^ { t } \\alpha _ { i }$ , $\\begin{array} { r } { \\bar { \\mu } _ { t } = \\frac { \\sqrt { \\bar { \\alpha } _ { t - 1 } } \\beta _ { t } } { 1 - \\bar { \\alpha } _ { t } } x _ { 0 } + \\frac { \\sqrt { \\alpha _ { t } } \\left( 1 - \\bar { \\alpha } _ { t - 1 } \\right) } { 1 - \\bar { \\alpha } _ { t } } x _ { t } } \\end{array}$ and $\\begin{array} { r } { \\bar { \\beta } _ { t } = \\frac { 1 - \\bar { \\alpha } _ { t - 1 } } { 1 - \\bar { \\alpha } _ { t } } \\beta _ { t } } \\end{array}$ 1−α¯t−1 βt. Then this paper sets $\\beta _ { \\theta } = \\bar { \\beta } _ { t }$ and designs a objective function to help neural networks to represent $\\mu _ { \\theta }$ . ", + "bbox": [ + 173, + 342, + 828, + 378 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Objective Function The objective function is defined by: ", + "bbox": [ + 174, + 383, + 553, + 398 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9492f3195a3db54a9c5d531b9776280d1ef443dfad9ded5975b4e88ac6d886ec.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\cal L } _ { t - 1 } = { \\mathbb { E } } _ { q } \\left[ | | \\bar { \\mu } _ { t } ( x _ { t } , x _ { 0 } ) - \\mu _ { \\theta } ( x _ { t } , t ) | | ^ { 2 } \\right] } } \\\\ { { \\displaystyle ~ = { \\mathbb { E } } _ { x _ { 0 } , \\epsilon } \\left[ | | \\frac { 1 } { \\sqrt { \\alpha _ { t } } } \\left( x _ { t } ( x _ { 0 } , \\epsilon ) - \\frac { \\beta _ { t } } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } \\epsilon \\right) - \\mu _ { \\theta } ( x _ { t } ( x _ { 0 } , \\epsilon ) , t ) | | ^ { 2 } \\right] } } \\\\ { { \\displaystyle ~ = { \\mathbb { E } } _ { x _ { 0 } , \\epsilon } \\left[ \\frac { \\beta _ { t } ^ { 2 } } { \\alpha _ { t } ( 1 - \\bar { \\alpha } _ { t } ) } | | \\epsilon - \\epsilon _ { \\theta } ( \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon , t ) | | ^ { 2 } \\right] . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 402, + 732, + 494 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, $x _ { t } ( x _ { 0 } , \\epsilon ) = \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon$ , $\\epsilon \\sim \\mathcal { N } ( 0 , 1 )$ , $\\epsilon _ { \\theta }$ is an estimate of the noise $\\epsilon$ . The relationship between µθ and ϵθ is µθ = √1α ( $\\begin{array} { r } { \\mu _ { \\theta } = \\frac { 1 } { \\sqrt { \\alpha _ { t } } } ( x _ { t } - \\frac { \\beta _ { t } } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } \\epsilon _ { \\theta } ) } \\end{array}$ . Because $\\epsilon \\sim \\mathcal { N } ( 0 , 1 )$ , we assume that the mean and variance of $\\epsilon _ { \\theta }$ are 0 and 1. ", + "bbox": [ + 173, + 497, + 826, + 547 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 STOCHASTIC DIFFERENTIAL EQUATION ", + "text_level": 1, + "bbox": [ + 176, + 563, + 490, + 578 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "According to Song et al. (2020b), there is another understanding of DDPMs. The diffusion process can be treated as solving a certain stochastic differential equation $d x = ( { \\sqrt { 1 - \\beta ( t ) } } - 1 ) x ( t ) d t +$ $\\sqrt { \\beta ( t ) } d w$ . According to Anderson (1982), the denoising process also satisfies a similar stochastic differential equation: ", + "bbox": [ + 173, + 588, + 825, + 651 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/4bbbbcaf46d19233959ed4eb69fad78e7e66600321a06fe3a40cdf4c4d4b0118.jpg", + "text": "$$\nd x = \\Big ( ( \\sqrt { 1 - \\beta ( t ) } - 1 ) x ( t ) - \\beta ( t ) \\epsilon _ { \\theta } ( x ( t ) , t ) \\Big ) d t + \\sqrt { \\beta ( t ) } d \\bar { w } .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 654, + 714, + 681 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This is Variance Preserving stochastic differential equations (VP-SDEs). Here, we change the domain of $t$ from $[ 1 , N ]$ to $[ 0 , 1 ]$ . When $N$ tends to infinity, $\\{ \\beta _ { i } \\} _ { i = 1 } ^ { N }$ , $\\{ x _ { i } \\} _ { i = 1 } ^ { N }$ become continuous functions $\\beta ( t )$ and $x ( t )$ on $[ 0 , 1 ]$ . Song et al. (2020b) also show that this equation has an ordinary differential equation (ODE) version with the same marginal probability density as Equation (4): ", + "bbox": [ + 174, + 684, + 825, + 741 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e18be762eb3829bcb789a5f399193aece8d023108e2be6c440b8eb8e2eaed2cb.jpg", + "text": "$$\nd x = \\left( ( { \\sqrt { 1 - \\beta ( t ) } } - 1 ) x ( t ) - { \\frac { 1 } { 2 } } \\beta ( t ) \\epsilon _ { \\theta } ( x ( t ) , t ) \\right) d t .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 744, + 678, + 780 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This different denoising equation with no random item and the same diffusion equation together is Probability Flows (PFs). These two denoising equations show us a new possibility that we can use numerical methods to accelerate the reverse process. As far as we know, DDIMs first try to remove this random item, so PFs can also be treated as an acceleration of DDIMs, while VP-SDEs are an acceleration of DDPMs. ", + "bbox": [ + 173, + 782, + 825, + 853 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 NUMERICAL METHOD ", + "text_level": 1, + "bbox": [ + 174, + 869, + 370, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Many classical numerical methods can be used to solve ODEs, including the forward Euler method, Runge-Kutta method and linear multi-step method (Timothy, 2017). ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Forward Euler Method For a certain differential equation satisfying $\\begin{array} { r } { \\frac { d x } { d t } = f ( x , t ) } \\end{array}$ . The trivial numerical method is forward Euler method satisfying $x _ { t + \\delta } = x _ { t } + \\delta f ( x _ { t } ^ { ^ { \\mathrm { a } \\delta } } t )$ . ", + "bbox": [ + 171, + 102, + 823, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Runge-Kutta Method Runge-Kutta method uses more information at every step, so it can achieve higher accuracy 2. Runge-Kutta method satisfies: ", + "bbox": [ + 169, + 138, + 823, + 167 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/10be13b41eedb3c348a243bd90aecd0c8d00cb3f3f5da9b53c8d26fb786c835d.jpg", + "text": "$$\n\\left\\{ \\begin{array} { r l } & { k _ { 1 } = f ( x _ { t } , t ) , \\quad \\quad k _ { 2 } = f ( x _ { t } + \\frac { \\delta } { 2 } k _ { 1 } , t + \\frac { \\delta } { 2 } ) } \\\\ & { k _ { 3 } = f ( x _ { t } + \\frac { \\delta } { 2 } k _ { 2 } , t + \\frac { \\delta } { 2 } ) , \\quad k _ { 4 } = f ( x _ { t } + \\delta k _ { 3 } , t + \\delta ) } \\\\ & { x _ { t + \\delta } = x _ { t } + \\frac { \\delta } { 6 } ( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } ) . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 303, + 174, + 694, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Linear Multi-Step Method Linear multi-step method is another numerical method and satisfies: ", + "bbox": [ + 169, + 242, + 808, + 257 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3f0434bbb9374a4e7e10c11e531cc8211374412b7f94e0c45a66e44af6b5c6de.jpg", + "text": "$$\nx _ { t + \\delta } = x _ { t } + \\frac { \\delta } { 2 4 } \\big ( 5 5 f _ { t } - 5 9 f _ { t - \\delta } + 3 7 f _ { t - 2 \\delta } - 9 f _ { t - 3 \\delta } \\big ) , f _ { t } = f ( x _ { t } , t ) .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 263, + 730, + 296 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PSEUDO NUMERICAL METHOD FOR DDPM ", + "text_level": 1, + "bbox": [ + 174, + 310, + 568, + 328 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we first compute the corresponding differential equations of diffusion models to build a direct connection between DDPMs and numerical methods. As a byproduct, we can directly use pre-trained models from DDPMs. After establishing this connection, we provide detailed analyses on the weakness of classical numerical methods. To solve the problems in classical numerical methods, we dive into the structure of numerical methods by dividing their equations into a gradient part and a transfer part and define pseudo numerical methods by introducing nonlinear transfer parts. We find that DDIMs can be regarded as simple pseudo numerical methods. Then, We explore the pros and cons of different numerical methods and choose the linear multi-step method to make numerical methods faster. Finally, we summarize our findings and analyses and safely propose our novel pseudo numerical methods for diffusion models (PNDMs), which combine our proposed transfer part and the gradient part of the linear multi-step method. Furthermore, we analyze the convergence order of pseudo numerical methods to demonstrate the effectiveness of our methods theoretically. ", + "bbox": [ + 173, + 343, + 826, + 511 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 FORMULA TRANSFORMATION ", + "text_level": 1, + "bbox": [ + 176, + 529, + 421, + 542 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "According to Song et al. (2020a), the reverse process of DDPMs and DDIMs satisfies: ", + "bbox": [ + 174, + 553, + 738, + 569 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f02079c70488e8c098c1e787f0af456fd5d016e5a3de26400fdf3aba98171010.jpg", + "text": "$$\nx _ { t - 1 } = \\sqrt { \\bar { \\alpha } _ { t - 1 } } \\left( \\frac { x _ { t } - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon _ { \\theta } ( x _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t - 1 } - \\sigma _ { t } ^ { 2 } } \\epsilon _ { \\theta } ( x _ { t } , t ) + \\sigma _ { t } \\epsilon _ { t } .\n$$", + "text_format": "latex", + "bbox": [ + 236, + 575, + 761, + 612 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, $\\sigma _ { t }$ controls the ratio of random noise. If $\\sigma _ { t }$ equals one, Equation (8) represents the reverse process of DDPMs; if $\\sigma _ { t }$ equals zero, this equation represents the reverse process of DDIMs. And only when $\\sigma _ { t }$ equals zero, this equation removes the random item and becomes a discrete form of a certain ODE. Theoretically, the numerical methods that can be used on differential equations with random items are limited. And Song et al. (2020b) have done enough research in this case. Empirically, Song et al. (2020a) have shown that DDIMs have a better acceleration effect when the number of total steps is relatively small. Therefore, our work concentrate on the case $\\sigma _ { t }$ equals zero. ", + "bbox": [ + 173, + 617, + 825, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To find the corresponding ODE of Equation (8), we replace discrete $t - 1$ with a continuous version $t - \\delta$ according to (Song et al., 2020a) and change this equation into a differential form, namely, subtract $x _ { t }$ from both sides of this equation: ", + "bbox": [ + 174, + 722, + 825, + 765 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/577c3ec5876a9604c90387e3d534299c1ae1f71d2d18a024141de10a63208af1.jpg", + "text": "$$\n\\mathfrak { z } _ { t - \\delta } - \\mathfrak { x } _ { t } = \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) \\left( \\frac { \\mathfrak { x } _ { t } } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { \\bar { \\alpha } _ { t - \\delta } } + \\sqrt { \\bar { \\alpha } _ { t } } \\right) } - \\frac { \\epsilon _ { \\theta } ( \\mathfrak { x } _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } \\right) } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 771, + 825, + 814 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Because $\\delta$ is a continuous variable from 0 to $t$ , we can now compute the derivative of the generation data $x _ { t }$ and get that $\\begin{array} { r } { \\underset { \\delta 0 } { \\operatorname* { l i m } } \\frac { x _ { t } - x _ { t - \\delta } } { \\delta } = - \\bar { \\alpha } ^ { \\prime } ( t ) ( \\frac { x ( t ) } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x ( t ) , t ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } ) } \\end{array}$ Here, $\\bar { \\alpha } ( t )$ is the continuous version of $\\{ \\bar { \\alpha } _ { i } \\} _ { i = 1 } ^ { N }$ like the definition of $x ( t )$ . Therefore, the corresponding ODE when $\\delta$ tends to ", + "bbox": [ + 173, + 824, + 825, + 887 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "zero of Equation (9) is: ", + "bbox": [ + 174, + 103, + 326, + 118 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b8e28e0672da3f0640c702d9463c85d624cbd79ee3f60d28c45f64be27d3b94d.jpg", + "text": "$$\n\\frac { d x } { d t } = - \\bar { \\alpha } ^ { \\prime } ( t ) \\left( \\frac { x ( t ) } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x ( t ) , t ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 126, + 651, + 167 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 CLASSICAL NUMERICAL METHOD ", + "text_level": 1, + "bbox": [ + 176, + 184, + 454, + 199 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "After getting the target ODE, the easiest way to solve it is through classical numerical methods. However, We notice that classical numerical methods can introduce noticeable noise at a high speedup rate, making high-order numerical methods (e.g., Runge-Kutta method) even less effective than DDIMs. This phenomenon is also mentioned in Salimans & Ho (2022). To make better use of numerical methods, we analyze the differences between Equation (10) and usual differential equations and find two main problems when we directly use numerical methods with diffusion models. ", + "bbox": [ + 173, + 210, + 825, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first problem is that the neural network $\\epsilon _ { \\theta }$ and Equation (10) are well-defined only in a limited area. Equation (2) shows that the data $x _ { t }$ is generated along a curve close to an arc. According to Figure 2, most of $x _ { t }$ is concentrated in a band with a width of around 0.1, namely the red area in Figure 2. This means that the neural network $\\epsilon _ { \\theta }$ cannot get enough examples to fit the noise successfully away from this area. Therefore, $\\epsilon _ { \\theta }$ and Equation (10), which contains $\\epsilon _ { \\theta }$ , are only well-defined in this limited area. However, all classical numerical methods generate results along a straight line instead of an arc. The generation process may generate samples away from the well-defined area and then introduce new errors. In Section 4.3 we will give more visualization results to support this. ", + "bbox": [ + 174, + 315, + 562, + 508 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/5d0a4adc964cc9dab17aeddee06c037ae60160c3540a2b0026d4c04a47a7047d.jpg", + "image_caption": [ + "Figure 2: the density distribution of the norm of the data. " + ], + "image_footnote": [], + "bbox": [ + 576, + 318, + 821, + 460 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The second problem is that Equation (10) is unbounded at most cases. We find that for most linear variance schedules $\\beta _ { t }$ , Equation (10) tends to infinity when $t$ tends to zero (see Appendix A.4), which does not satisfy the condition of numerical methods mentioned in Section 2.3. This is an apparent theoretical weakness that previous works have not explored. On the contrary, in the original DDPMs and DDIMs, the prediction of the sample $x _ { t }$ and the noise $\\epsilon _ { \\theta }$ in the data are more and more precise as the index $t$ tends to zero (see Appendix A.5). This means original diffusion models do not make a significant error in the last several steps, whereas using numerical methods on Equation (10) does. This explains why DDIMs are better than higher-order numerical methods. ", + "bbox": [ + 174, + 516, + 825, + 627 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 PSEUDO NUMERICAL METHOD ON MANIFOLD ", + "text_level": 1, + "bbox": [ + 174, + 646, + 537, + 660 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first problem above shows that we should try to solve our problems on certain manifolds. Here, target manifolds are the high-density region of the data √ $x _ { t }$ of DDPMs, which is defined by $x _ { t } ( x _ { 0 } , \\epsilon ) \\overset { } { = } \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon , \\epsilon \\sim \\bar { \\mathcal { N } } ( 0 , \\bar { 1 } )$ . Ernst & Gerhard (1996) show several numerical methods to solve differential equations on manifolds that have analytic expressions. Unfortunately, it’s challenging to use the above expression of manifolds. Because we do not know the target $x _ { 0 }$ in the reverse process and random items $\\epsilon$ are hard to handle, too. ", + "bbox": [ + 174, + 672, + 825, + 756 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this paper, we design a different way that we make our new equation of denoising process more fits with the equation of original DDIMs to make their results share similar data distribution. Firstly, we divide the classical numerical methods into two parts: gradient and transfer parts. The gradient part determines the gradient at each step, while the transfer part generates the result at the next step. For example, linear multi-step method can be divided into the gradient part $\\begin{array} { r } { f ^ { \\prime } = \\frac { \\delta } { 2 4 } ( 5 5 f _ { t } - 5 9 f _ { t - \\delta } ) ^ { 2 } } \\end{array}$ $3 7 f _ { t - 2 \\delta } - 9 f _ { t - 3 \\delta } )$ and the transfer part $x _ { t + \\delta } = x _ { t } + \\delta f ^ { \\prime }$ . All classical numerical methods have the same linear transfer part, while gradient parts are different. ", + "bbox": [ + 174, + 763, + 825, + 861 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We define those numerical methods which use a nonlinear transfer part as pseudo numerical methods. And an expected transfer part should have the property that when the result from the gradient part is precise, then the result of the transfer part is as close to the manifold as possible and the error of this result is as small as possible. We find that Equation (9) satisfies such property. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Property 3.1 If ϵ is the precise noise in $x _ { t }$ , then the result of $x _ { t - \\delta }$ from Equation (9) is also precise. ", + "bbox": [ + 174, + 103, + 823, + 119 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "And we put the proof of this property in Appendix A.5. Therefore, we use: ", + "bbox": [ + 174, + 127, + 668, + 143 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0505272c5972520578859285aff50d4ab192f035bd95522592e4f5b6ce879226.jpg", + "text": "$$\n\\phi ( x _ { t } , \\epsilon _ { t } , t , t - \\delta ) = \\frac { \\sqrt { \\bar { \\alpha } _ { t - \\delta } } } { \\sqrt { \\bar { \\alpha } _ { t } } } x _ { t } - \\frac { \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) } { \\sqrt { \\bar { \\alpha } _ { t } } ( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } ) } \\epsilon _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 243, + 159, + 756, + 195 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "as the transfer part and $\\epsilon _ { \\theta }$ as the gradient part. That if $\\epsilon _ { \\theta }$ is precise, the result of $x _ { t - \\delta }$ is also precise, which means that $\\epsilon _ { \\theta }$ can determine the direction of the denoising process to generate the final results. Therefore, such a choice also satisfies the definition of a gradient part. Now, we have our gradient part $\\epsilon _ { \\theta }$ and transfer part $\\phi$ . ", + "bbox": [ + 174, + 202, + 823, + 258 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This combination solves the two problems mentioned above successfully. Firstly, our new transfer parts do not introduce new errors. This property also means that it keeps the results at the next step on the target manifold because generating samples away is a kind of error. This shows that we solve the first problem. Secondly, we know that the prediction of $\\epsilon _ { \\theta }$ is more and more precise in the reverse process in the above subsection. And our new transfer part can generate precise results according to the precise prediction of $\\epsilon _ { \\theta }$ . Therefore, our generation results are more and more precise using pseudo numerical methods, while classical numerical methods can introduce obvious error at the last several steps. This shows that we solve the second problem, too. We also find that their combination $\\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } , t ) , t , t - 1 )$ is just the inference equation used by DDIMs, so DDIMs is a simple case of pseudo numerical methods. Here, we define DDIMs as DDIMs\\*, emphasizing that it is a pseudo numerical method. ", + "bbox": [ + 173, + 265, + 825, + 417 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 GRADIENT PART ", + "text_level": 1, + "bbox": [ + 176, + 434, + 331, + 449 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Because we split numerical methods into two parts, we can use the same gradient part from different classical numerical methods freely (e.g., linear multi-step method), although we change the transfer part of our inference equation. Our theoretical analyses and experiments show that the gradient part from different classical methods can work well with our new transfer part (see Section 3.6, 4.2). By using the same gradient part of the linear multi-step method, we have: ", + "bbox": [ + 173, + 460, + 825, + 531 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1588a7d1247e616e51fcf13ac73292868a0dd72cfb3203e0977ccf70dbeacb20.jpg", + "text": "$$\n\\left\\{ \\begin{array} { l l } & { e _ { t } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { e _ { t } ^ { \\prime } = \\frac { 1 } { 2 4 } ( 5 5 e _ { t } - 5 9 e _ { t - \\delta } + 3 7 e _ { t - 2 \\delta } - 9 e _ { t - 3 \\delta } ) } \\\\ & { x _ { t + \\delta } = \\phi ( x _ { t } , e _ { t } ^ { \\prime } , t , t + \\delta ) . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 183, + 540, + 517, + 594 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "By using the same gradient part of Runge-Kutta method, we have: ", + "bbox": [ + 173, + 606, + 526, + 632 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 DDIMs ", + "text_level": 1, + "bbox": [ + 535, + 545, + 676, + 559 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: $x _ { T } \\sim \\mathcal { N } ( 0 , I )$ \n2: for $t = T - 1 , \\cdots , 1 , 0$ do \n3: $x _ { t } = \\phi ( x _ { t + 1 } , \\epsilon _ { \\theta } ( x _ { t + 1 } , t + 1 ) , t + 1 , t )$ \n4: end for \n5: return $x _ { 0 }$ ", + "bbox": [ + 539, + 563, + 813, + 630 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 2 PNDMs ", + "text_level": 1, + "bbox": [ + 535, + 640, + 679, + 654 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1c6989a2d2d747c8bef31a38648ada89764f4d6bca4f2192e585be120a5c5f06.jpg", + "text": "$$\n\\left\\{ \\begin{array} { l l } { \\begin{array} { l l } { e _ { t } ^ { 1 } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ { x _ { t } ^ { 1 } = \\phi ( x _ { t } , e _ { t } ^ { 1 } , t , t + \\frac { \\delta } { 2 } ) } \\\\ { e _ { t } ^ { 2 } = \\epsilon _ { \\theta } ( x _ { t } ^ { 1 } , t + \\frac { \\delta } { 2 } ) } \\\\ { x _ { t } ^ { 2 } = \\phi ( x _ { t } , e _ { t } ^ { 2 } , t , t + \\frac { \\delta } { 2 } ) } \\\\ { e _ { t } ^ { 3 } = \\epsilon _ { \\theta } ( x _ { t } ^ { 2 } , t + \\frac { \\delta } { 2 } ) } \\\\ { x _ { t } ^ { 3 } = \\phi ( x _ { t } , e _ { t } ^ { 3 } , t , t + \\delta ) } \\\\ { e _ { t } ^ { 4 } = \\epsilon _ { \\theta } ( x _ { t } ^ { 4 } , t + \\delta ) } \\\\ { e _ { t } ^ { 2 } = \\frac { 1 } { 6 } ( e _ { t } ^ { 1 } + 2 e _ { t } ^ { 2 } + 2 e _ { t } ^ { 3 } + e _ { t } ^ { 4 } ) } \\\\ { x _ { t - \\delta } = \\phi ( x _ { t } , e _ { t } ^ { \\prime } , t , t + \\delta ) . } \\end{array} } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 233, + 640, + 467, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: $x _ { T } \\sim \\mathcal { N } ( 0 , I )$ \n2: for $t = T - 1 , T - 2 , T - 3 \\mathbf { d } \\mathbf { 0 }$ \n3: $x _ { t } , e _ { t } = \\mathrm { P R K } ( x _ { t + 1 } , t + 1 , t )$ \n4: end for \n5: for $t = T - 4 , \\cdots , 1 , 0$ do \n6: $x _ { t } , e _ { t } = \\operatorname { P L M S } ( x _ { t + 1 } , \\{ e _ { p } \\} _ { p > t } , t + 1 , t )$ \n7: end for \n8: return $x _ { 0 }$ ", + "bbox": [ + 539, + 659, + 816, + 762 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Abbreviate Equation (12) and (13) as $x _ { t + \\delta } , e _ { t } ~ = ~ \\mathrm { P L M S } ( x _ { t } , \\{ e _ { p } \\} _ { p < t } , t , t + \\delta ) , ~ x _ { t + \\delta } , e _ { t } ^ { 1 } ~ =$ $\\mathrm { P R K } ( x _ { t } , t , t + \\delta )$ . ", + "bbox": [ + 171, + 789, + 825, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here, we have provided three kinds of pseudo numerical methods. Although advanced numerical methods can accelerate the denoising process, some may have to compute the gradient part $\\epsilon _ { \\theta }$ more times at every step, like the Runge-Kutta method. Propagating forward four times along a neural network makes the denoising process slower. However, we find that the linear multi-step method can reuse the result of $\\epsilon _ { \\theta }$ four times and only compute $\\epsilon _ { \\theta }$ once at every step. And theoretical analyses tell us that the Runge-Kutta and linear multi-step method have the same convergence order and similar results. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Therefore, we use the gradient part of the linear multi-step method and our new transfer part as our main pseudo numerical methods for diffusion models (PNDMs). In Table 1, we show the relationship between different numerical methods. Here, we can see PNDMs combine the benefits of higher-order classical numerical methods (in the gradient part) and DDIMs (in the transfer part). ", + "bbox": [ + 174, + 103, + 473, + 229 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e2e05a3016a399de590fee2dda6265c4358cabe0f92622bba3cbeb6b75e1bca5.jpg", + "table_caption": [ + "Table 1: The relationship between different numerical methods. " + ], + "table_footnote": [], + "table_body": "
order+firstnon-first
linearforwardEulerlinearmulti-step,Runge-Kutta...
nonlinearDDIMPNDM
", + "bbox": [ + 486, + 106, + 818, + 172 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.5 ALGORITHM ", + "text_level": 1, + "bbox": [ + 176, + 246, + 302, + 261 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We can provide our whole algorithm of the denoising process of DDIMs now. According to Song et al. (2020a), the algorithm of the original method satisfies Algorithm 1. And our new algorithm of DNPMs uses the pseudo linear multi-step and pseudo Runge-Kutta method, which satisfies Algorithm 2. Here, we cannot use linear multi-step initially because the linear multi-step method cannot start automatically, which needs at least three previous steps’ information to generate results. So we use the Runge-Kutta method to compute the first three steps’ results and then use the linear multi-step method to calculate the remaining. ", + "bbox": [ + 174, + 272, + 825, + 369 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We also use the gradient parts of two second-order numerical methods to get another pseudo numerical method. We introduce the details of this method in Appendix A.3. We call it S-PNDMs, because its gradient part uses information from two steps at every step. Similarly, we also call our first PNDMs F-PNDMs, which use data from four steps, when we need to distinguish them. ", + "bbox": [ + 174, + 377, + 825, + 433 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.6 CONVERGENCE ORDER ", + "text_level": 1, + "bbox": [ + 176, + 449, + 377, + 464 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Change the transfer part of numerical methods may introduce unknown error. To determine the influence of our new transfer part theoretically, we compute the local and global error between the theoretical result of Equation (10) $x ( t + \\delta )$ and our new methods, we find that $x ( t + \\delta ) - x _ { \\mathrm { D D I M } } ( x +$ $\\delta ) = O ( \\delta ^ { 2 } )$ and ", + "bbox": [ + 173, + 476, + 823, + 531 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e92c1130506a0fe0d87ac0b98b06802b20ec41a0334893283ac0931d2035665c.jpg", + "text": "$$\nx ( t + \\delta ) - x _ { \\mathrm { S / F - P N D M } } ( x + \\delta ) = O ( \\delta ^ { 3 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 530, + 627, + 549 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "If the target ODE satisfies Lipschitz condition and local error $e _ { \\mathrm { l o c a l } } = O ( \\delta ^ { k } )$ , then there are $C$ and $h$ such that the global error $e _ { \\mathrm { g l o b a l } }$ satisfies $e _ { \\mathrm { g l o b a l } } \\leq C \\delta ^ { k } ( 1 + e ^ { h } + e ^ { 2 h } + \\dot { \\cdot } \\dot { \\cdot } \\dot { \\cdot } ) \\leq C ^ { \\prime } \\delta ^ { k - 1 }$ . And we have that the convergence order is equal to the order of the global error. The detailed proof can be found in Appendix A.6. Therefore, we get the following property: ", + "bbox": [ + 174, + 553, + 823, + 608 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Property 3.2 S/F-PNDMs have third-order local error and are second-order convergent. ", + "bbox": [ + 171, + 621, + 758, + 635 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENT ", + "text_level": 1, + "bbox": [ + 176, + 655, + 316, + 671 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 SETUP ", + "text_level": 1, + "bbox": [ + 174, + 686, + 261, + 700 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We conduct unconditional image generation experiments on four datasets: Cifar10 $( 3 2 \\times 3 2 )$ ) (Krizhevsky et al., 2009), CelebA $( 6 4 \\times 6 4 )$ (Liu et al., 2015), LSUN-church $( 2 5 6 \\times 2 5 6 )$ and LSUN-bedroom $( 2 5 6 \\times 2 5 6 )$ ) (Yu et al., 2016). According to the analysis in Section 3.1, we can use pre-trained models from prior works in our experiments. The pre-trained models for Cifar10, LSUNchurch and LSUN-bedroom are taken from Ho et al. (2020) and the pre-trained model for CelebA is taken from Song et al. (2020a). In these models, the number of total steps N is 1000 and the variance schedule is linear variance schedule. And we also use a pre-trained model for Cifar10, which uses a cosine variance schedule from improved denoising diffusion probabilistic models (iDDPMs (Nichol & Dhariwal, 2021)). ", + "bbox": [ + 174, + 713, + 825, + 838 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 SAMPLE EFFICIENCY AND QUALITY", + "text_level": 1, + "bbox": [ + 176, + 854, + 462, + 869 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To analyze the acceleration effect, we test Fenchel Inception Distance (FID (Heusel et al., 2018)) on different datasets under different steps and different numerical methods, including DDIMs, S-PNDMs, F-PNDMs and classical fourth-order numerical methods (FONs) (e.g., Runge-Kutta ", + "bbox": [ + 176, + 882, + 825, + 922 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/58a22667444088a7a7ff645d89f3778e4d527c0461b40ed261184e41dbb7de13.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
datasetFID step model1020501002501000time
Cifar10DDIM PF13.46.84 13.84.67 3.894.16 3.693.714.04 3.72
Cifar10 (linear)DDIM*18.510.96.995.524.524.000.337
FON13.17.415.264.654.123.710.390
S-PNDM11.67.565.184.343.913.800.344
Cifar10 (cosine)F-PNDM DDIM7.03 14.55.00 8.793.95 5.863.72 4.923.60 4.303.70 3.690.391 0.505
S-PNDM8.645.774.463.943.713.380.517
F-PNDM7.054.613.683.533.493.260.595
CelebADDIM17.313.79.176.533.51
CelebA (linear)DDIM*16.913.48.956.364.443.411.237
FON16.011.68.136.705.144.171.431
S-PNDM12.29.455.694.033.192.991.258
F-PNDM7.715.513.342.812.712.861.433
", + "bbox": [ + 173, + 103, + 647, + 333 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2: Image generation measured in FID on Cifar10 and CelebA. PFs use black box ODE solvers and we use the number of score function evaluations as the step of PFs. DDIM\\* is a retest of DDIM. The bold results mean the best ones using the same pretrained model. We use the 50-step, 512 batch size experiment on an RTX3090 to test the computational cost and the column time is the average computational cost per step in seconds. And we put the results of standard deviation in Appendix A.12 ", + "bbox": [ + 650, + 98, + 820, + 338 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "method and linear multi-step method). On Cifar10 and CelebA, we first provide the results of previous works DDIMs. Then, we use the same pre-trained models to test numerical methods mentioned in this paper and put the results in Cifar10 / CelebA (linear). We also use models from iDDPMs to test nonlinear variance schedules and put the results in Cifar10 (cosine). Song et al. (2020b) do not provide detailed FID results of probability flows (PFs) under different steps, so we retest the results using its pretrained models by ourselves. ", + "bbox": [ + 174, + 353, + 825, + 436 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Efficiency Our two baselines are DDIM and PF. DDIM is a simple case of pseudo numerical methods, and PF is a case of classical numerical methods. However, PF uses a much bigger model than DDIM and uses some tricks to improve the sample quality. To ensure the experiment’s fairness, we use fourth-order numerical methods on Equation (10) and the model from DDIM. In Table 2, we find that the performance of FON is limited when the number of steps is small. By contrast, our new methods, including S-PNDM and F-PNDM, can improve the generated results regardless of whether the number of steps used is large or small. According to Cifar10 / CelebA (linear), F-PNDM can achieve lower FID than 1000 steps DDIM using only 50 steps, making diffusion models $2 0 \\mathrm { x }$ faster without losing quality. ", + "bbox": [ + 173, + 443, + 825, + 569 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We draw a line chart of computation cost with FID according to the results of Cifar10 (linear) above in Figure 3. Because F-PNDM uses the pseudo Runge-Kutta method to generate the first three steps, it is slower than other methods at the first several steps. Therefore, S-PNDM can achieve the best FID initially, then F-PNDM becomes the best and the acceleration is significant. ", + "bbox": [ + 174, + 575, + 562, + 674 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Quality When the number of steps is relatively big, the results of FON become more and more similar to that of pseudo numerical methods. This is because all the methods are solving Equation (10), and their convergent results should be the same. However, pseudo numerical methods still work better using a large number of steps empirically. F-PNDM can improve the best FID around 0.4 using pretrained models and achieves a new SOTA FID score of 2.71 on CelebA, which shows that our work can not only accelerate diffusion models but also improve the sample quality topline. We also notice that the FID results of F-PNDM converge after more than 250 steps. The FID results will fluctuate around a value then. This phenomenon is more pronounced when we test our methods on LSUN (see Table 5, 6). ", + "bbox": [ + 174, + 680, + 560, + 791 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/988a0e4aa3b27ecdc0d0e15c4f94b23487b35295b934829b8f68160586957c92.jpg", + "image_caption": [ + "Figure 3: The FID results under different computation costs and different numerical methods on Cifar10. The unit of time is the computational cost of 1-step DDIM, which is 0.337s. " + ], + "image_footnote": [], + "bbox": [ + 575, + 563, + 820, + 708 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 791, + 825, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "According to Cifar10 (cosine), the cosine variance schedule can lower FID using a relatively large number of steps. More analyses about variance schedule can be found in Appendix A.7. What’s more, we test our methods on other datasets and provide our FID results in Appendix A.9 and image results in Appendix A.10. We can draw similar conclusions on our methods’ acceleration and sampling quality, regardless of the datasets and the size of the images. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/65efd211782730face9f7f6f648ca37c2a522e6bc77cdc1991a885b3333e5f3f.jpg", + "image_caption": [ + "Figure 4: The upper part shows the change of norm with the number of steps using different methods and different steps. The lower part shows the generation curves of two points using different methods and different steps. DDIM-n means $\\mathbf { n }$ -step DDIM method. Experiments in this subsection all use the Cifar10 dataset and we use the 1000-step DDIM’s result as our target result. " + ], + "image_footnote": [], + "bbox": [ + 173, + 92, + 825, + 338 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 SAMPLE ON MANIFOLDS ", + "text_level": 1, + "bbox": [ + 176, + 416, + 387, + 430 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Here, we design visualization experiments to show the effort of our new methods and support our analyses. Because it is hard to visualize high-dimensional data, we use the change of a global characteristic norm and a local characteristic pixel to show the change of the data under different steps. For pixel, we randomly choose two positions $p ^ { 1 } , p ^ { 2 }$ . Then for a series of images $x _ { T } , x _ { T - k } , \\cdot \\cdot \\cdot , x _ { 0 }$ derived from the reverse process, we denote $y _ { t } ^ { k }$ as the value of $x _ { t }$ at position $p ^ { k }$ . Then we draw a polyline $( y _ { t } ^ { 1 } , y _ { t } ^ { 2 } ) _ { t = T , \\cdots }$ in $\\mathbb { R } ^ { 2 }$ . For norm, we first count the distribution of the norm of the training datasets under different steps and use this to make a heat map as the background. After that, we draw the norm of our generated results using different methods and steps above this heat map. ", + "bbox": [ + 174, + 443, + 825, + 555 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Figure 4, we can see that the FON may run far away from the high-density area of the data, which explains why FON may introduce noticeable noise. However, PNDM can avoid this problem and appropriately fit the target result. More visualization results supporting our analysis can be found in Appendix A.11. What’s more, we design a toy example to test our new methods without the influence of neural networks and get similar conclusions as to the real cases above. We put the detailed results in Appendix A.8. ", + "bbox": [ + 174, + 561, + 825, + 645 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 667, + 310, + 684 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we provided DNPMs, a new numerical method suitable for solving the corresponding ODEs of DDPMs. DNPMs can generate high-quality images using fewer steps without loss of quality successfully. Based on the idea of this work, further improvement can be explored in our future works: 1) find a better variance schedule for DNPMs: although we tested DNPMs on linear variance schedule and cosine variance schedule in this work, there might be another variance schedule more suitable for our proposed numerical methods. 2) Find higher-order convergent pseudo numerical methods: we analyzed the convergence order of S/F-DNPMs, which are both second-order convergent. However, F-DNPMs achieve better FID than S-DNPMs in most cases. We think this is because the result between our transfer part and target ODE has a higher-order error, which limits the convergence order of F-DNPMs. This error from the change of the transfer part is theoretical but does not influence the quality of images according to the property of Equation (11). However, making the transfer part higher-order convergent and finding the influence of such change is still exciting and needs more research. 3) Extend PNDMs to more general applications: when we proved the convergence order of DNPMs, we found that other kinds of transfer parts can keep the convergence order unchanged too, which means that pseudo numerical methods can be used on more general applications, like certain neural ODEs (Chen et al., 2019; Dupont et al., 2019). 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Song et al. (2020a) introduce new inference equations to accelerate DDPMs. Nichol & Dhariwal (2021), Watson et al. (2021) and Kong & Ping (2021) choose to find better variance schedules to improve the images quality. Vahdat et al. (2021) combine the advantages of DDPMs and Variational Autoencoders and get better results. Furthermore, Kim et al. (2021) try to solve an existing bottleneck that the inference equations of DDPMs are unbounded in some situations. ", + "bbox": [ + 174, + 694, + 825, + 791 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Song et al. (2020b) find the similarity between DDPMs and noise conditional score networks (NCSNs (Song & Ermon, 2020b)), which is that they both use a process similar to Langevin dynamics to produce samples. Therefore, some works (Song & Ermon, 2020a; Kim et al., 2021) that can improve the results of NCSNs also can be used in DDPMs. Additional, Song et al. (2020b) combine DDPMs and NCSNs under the framework of neural differential equations (Chen et al., 2019; Dupont et al., 2019). Therefore, numerical methods widely used in neural differential equations can also be applied to accelerate DDPMs. Our work successfully combines the advantages of Song et al. (2020a) and Song et al. (2020b). We use a transfer part from DDIMs and use different gradient parts from different numerical methods. Although we and Song et al. (2020b) solve certain differential equations derived from DDPMs, we use different target different equations and different numerical methods, which get better results. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The application of DDPMs is not limited to unconditional image generation. Some works apply DDPMs to various types of data successfully, including text-to-speech (Chen et al., 2020; Lam et al., 2021), singing voice (Liu et al., 2021), 3D Point Cloud (Luo & Hu, 2021), text generation (Austin et al., 2021). Additionally, DDPMs can also be used to generate conditional samples, too (Jeong et al., 2021; Choi et al., 2021). ", + "bbox": [ + 173, + 138, + 825, + 209 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 CONVERGENT ORDER OF METHOD ", + "text_level": 1, + "bbox": [ + 176, + 224, + 462, + 239 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We use the forward Euler method and linear multi-step method to show what is the order of a method. Assume that $x _ { t }$ is precise and compute the error at $x _ { t + \\delta }$ . For forward Euler method, we have: ", + "bbox": [ + 176, + 251, + 823, + 291 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/83ad2575f9858c2190d2151954f85be69eb29b6b8608e859266b38b3deaf8035.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle e _ { t , \\delta } = \\boldsymbol { x } ( t + \\delta ) - \\boldsymbol { x } _ { t , \\delta } } \\\\ { \\displaystyle = \\bigg ( \\boldsymbol { x } ( t ) + \\delta f ( \\boldsymbol { x } ( t ) , t ) + \\frac { \\delta ^ { 2 } } { 2 } f ^ { \\prime \\prime } ( c ) \\bigg ) - ( \\boldsymbol { x } ( t ) + \\delta f ( \\boldsymbol { x } _ { t } , t ) ) } \\\\ { \\displaystyle = \\frac { \\delta ^ { 2 } } { 2 } f ^ { \\prime \\prime } ( c ) \\leq \\frac { \\delta ^ { 2 } } { 2 } M } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 290, + 702, + 378 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here, we assume that $f ^ { \\prime \\prime }$ is continuous, so $f ^ { \\prime \\prime }$ is bounded in a close area. $x ( t + \\delta )$ is the precise result and $x _ { t , \\delta }$ is the numerical result from $t$ to $t + \\delta$ . ", + "bbox": [ + 171, + 378, + 823, + 407 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For linear multi-step method, we have: ", + "bbox": [ + 173, + 414, + 429, + 429 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/a1924594e76b54f7037b6af8d94f6021242cd86fd9603ce9fbcc17d7a2ab0a8e.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle e _ { t , \\delta } = x ( t + \\delta ) - x _ { t , \\delta } } \\\\ { \\displaystyle = \\Bigg ( x ( t ) + \\frac { \\delta } { 1 ! } f ( x ( t ) , t ) + \\frac { \\delta ^ { 2 } } { 2 ! } f ^ { \\prime } ( x ( t ) , t ) + \\cdots + \\frac { \\delta ^ { 4 } } { 4 ! } f ^ { ( 3 ) } ( x ( t ) , t ) + O \\left( \\delta ^ { 5 } \\right) \\Bigg ) } \\\\ { \\displaystyle \\quad - \\left( x ( t ) + b _ { 1 } \\delta f ( x ( t ) , t ) + \\sum _ { s = 2 } ^ { 4 } b _ { s } \\delta \\left( \\sum _ { k = 0 } ^ { 4 } \\frac { ( - ( s - 1 ) \\delta ) ^ { k } } { ( k ) ! } f ^ { ( k - 1 ) } ( x ( t ) , t ) + O \\left( \\delta ^ { 5 } \\right) \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 429, + 789, + 531 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here, $\\{ b _ { s } \\}$ satisfy $\\textstyle \\sum _ { s = 1 } ^ { 4 } b _ { s } = 1$ and the following equations for $j \\in \\{ 1 , \\cdots , 3 \\}$ ", + "bbox": [ + 173, + 549, + 700, + 566 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/55885cd55f64a66acd21027b81db647a019bf5552d6c51d5d2788c357bfa80c2.jpg", + "text": "$$\n( - 1 ) ^ { j } b _ { 2 } + ( - 2 ) ^ { j } b _ { 3 } + ( - 3 ) ^ { j } b _ { 4 } = \\frac { 1 } { j + 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 568, + 635, + 601 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We call the error at $x _ { t , \\delta }$ local error, and the error at $x _ { t + M \\delta }$ $\\mathbf { M }$ is big enough but finite) global error. Assume the local error of our method has order $_ { \\mathrm { k + 1 } }$ and the target ODE satisfies Lipschitz condition, then: ", + "bbox": [ + 174, + 607, + 823, + 648 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/bbdf21d2145f10416c60b2785734376cc8056e32d32ef105bcfbf840e614051d.jpg", + "text": "$$\n\\begin{array} { r l } & { x ( t + M \\delta ) - x _ { t , M \\delta } \\leq \\vert x ( t + M \\delta ) - x _ { t + ( M - 1 ) \\delta , \\delta } \\vert + \\vert x _ { t + ( M - 1 ) \\delta , \\delta } - x _ { t , M \\delta } \\vert } \\\\ & { \\qquad \\leq e _ { t + ( M - 1 ) \\delta , \\delta } + e ^ { L \\delta } \\vert x ( t + ( M - 1 ) \\delta ) - x _ { t , ( M - 1 ) \\delta } \\vert } \\\\ & { \\qquad \\leq e _ { t + ( M - 1 ) \\delta , \\delta } + e ^ { L \\delta } e _ { t + ( M - 2 ) \\delta , \\delta } + \\cdots } \\\\ & { \\qquad \\leq C \\delta ^ { k + 1 } \\left( 1 + e ^ { L h } + \\cdots + e ^ { ( i - 1 ) L h } \\right) } \\\\ & { \\qquad = C h ^ { k + 1 } \\frac { e ^ { i L \\delta } - 1 } { e ^ { L \\delta } - 1 } \\leq C \\delta ^ { k + 1 } \\frac { e ^ { i L \\delta } - 1 } { L \\delta } } \\\\ & { \\qquad = O ( \\delta ^ { k } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 646, + 753, + 791 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Therefore, the global error will be one order lower than the local error. From Equation (15), we can see the forward Euler method has local error $O ( \\delta ^ { 2 } )$ and global error $O ( \\delta )$ , so we call it the first-order numerical method. And the linear multi-step method has local error ${ \\dot { O } } ( \\delta ^ { 5 } )$ and global error ${ \\cal O } ( \\delta ^ { 4 } )$ , and we call it the fourth-order numerical method. ", + "bbox": [ + 173, + 797, + 825, + 854 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In addition, assuming that a numerical method has kth-order global error, we can compute the convergence speed of this numerical method: ", + "bbox": [ + 174, + 861, + 821, + 888 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/685bb3df44c9b75e9f21adc907ec8f60ecdc68db6a506c09864fa43b7327c4b7.jpg", + "text": "$$\n\\operatorname* { l i m } _ { \\delta 0 } \\frac { x _ { t + T } ^ { 2 \\delta } - x ( t + T ) } { x _ { t + T } ^ { \\delta } - x ( t + T ) } = \\frac { ( 2 \\delta ) ^ { k } } { \\delta ^ { k } } = 2 ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 890, + 627, + 928 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here, $x _ { t + T } ^ { \\delta }$ is the result at $\\mathrm { t } { + } \\mathrm { T }$ and move $\\delta$ every step. This shows that the fourth-order method can converge to the exact solution faster than the first-order method when $\\delta 0$ , which means that we can use a bigger iteration interval $\\delta$ to achieve similar global error and a bigger iteration interval means that we can iterate fewer times to get results with high quality. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.3 PSEUDO SECOND-ORDER METHOD ", + "text_level": 1, + "bbox": [ + 178, + 176, + 464, + 190 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We introduce two second-order numerical methods. First is improved Euler method satisfying: ", + "bbox": [ + 173, + 202, + 794, + 217 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c1ba7dd912872baaca2453aa54a576252ff9a919d946c753d9daf8d55b8bbbff.jpg", + "text": "$$\n\\left\\{ \\begin{array} { r l } & { k _ { 1 } = f ( x _ { t } , t ) } \\\\ & { k _ { 2 } = f ( x _ { t } + \\delta k _ { 1 } , t + \\delta ) } \\\\ & { x _ { t + \\delta } = x _ { t } + \\frac { \\delta } { 2 } ( k _ { 1 } + k _ { 2 } ) } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 400, + 223, + 598, + 277 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Second is another linear multi-step method called second-order linear multi-step method satisfying: ", + "bbox": [ + 173, + 287, + 823, + 303 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e396cac90746b9dba1d908cb5d77749c6b44140889c114562c0d0788b2770af2.jpg", + "text": "$$\nx _ { t + \\delta } = x _ { t } + \\frac \\delta 2 ( 3 f _ { t } - f _ { t - \\delta } )\n$$", + "text_format": "latex", + "bbox": [ + 403, + 306, + 594, + 338 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "And the corresponding pseudo improved Euler methods satisfying: ", + "bbox": [ + 173, + 349, + 612, + 364 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a0c5107fc32627c9ab88b02804bdaafdb85aa5aee1e9d2970fed08491231ef9f.jpg", + "text": "$$\n\\left\\{ \\begin{array} { l l } { \\ { e } _ { t } ^ { 1 } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ { \\ { x } _ { t } ^ { 1 } = \\phi ( x _ { t } , { e } _ { t } ^ { 1 } , t , t + \\delta ) } \\\\ { \\ { e } _ { t } ^ { 2 } = \\epsilon _ { \\theta } ( { x } _ { t } ^ { 1 } , t + \\delta ) } \\\\ { \\ { e } _ { t } ^ { \\prime } = \\frac { 1 } { 2 } ( { e } _ { t } ^ { 1 } + { e } _ { t } ^ { 2 } ) } \\\\ { \\ { x } _ { t + \\delta } = \\phi ( { x } _ { t } , { e } _ { t } ^ { \\prime } , t , t + \\delta ) } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 398, + 368, + 599, + 457 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Pseudo second-order linear multi-step method satisfying: ", + "bbox": [ + 173, + 468, + 549, + 483 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/0d51619c6b74fffd24ebd31e1c802e1bfc1f6153633cd4768bdc31a0582e524a.jpg", + "text": "$$\n\\left\\{ \\begin{array} { r l } & { e _ { t } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { e _ { t } ^ { \\prime } = \\frac { 1 } { 2 } ( 3 e _ { t } - e _ { t - \\delta } ) } \\\\ & { x _ { t + \\delta } = \\phi ( x _ { t } , e _ { t } ^ { \\prime } , t , t + \\delta ) } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 398, + 488, + 601, + 541 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Similar to what we do to get F-PNDMs, We combine them to get S-PNDMs. Abbreviate Equation (22) and (23) as ", + "bbox": [ + 173, + 559, + 513, + 602 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Algorithm 3 S-PNDMs ", + "text_level": 1, + "bbox": [ + 531, + 561, + 689, + 577 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/2d44e01738cf7d8a7de4bf66ec503c71e7eca25a14e61186da8a9e0c621e4d1d.jpg", + "text": "$$\n\\begin{array} { r l } & { x _ { t + \\delta } , e _ { t } = P I E ( x _ { t } , \\{ e _ { p } \\} _ { p < t } , t , t + \\delta ) , } \\\\ & { x _ { t + \\delta } , e _ { t } ^ { 1 } = P L M S ^ { \\prime } ( x _ { t } , t , t + \\delta ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 607, + 473, + 647 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1: $x _ { T } \\sim \\mathcal { N } ( 0 , I )$ \n2: for t = T − 1 do \n3: $x _ { t } , e _ { t } = P I E ( x _ { t + 1 } , t + 1 , t )$ \n4: end for \n5: for $t = T - 2 , \\cdots , 1 , 0$ do \n6: $x _ { t } , e _ { t } = P L M S ^ { \\prime } ( x _ { t + 1 } , \\{ e _ { p } \\} _ { p > t } , t + 1 , t )$ \n7: end for \n8: return $x _ { 0 }$ ", + "bbox": [ + 532, + 580, + 825, + 685 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 THE EXISTENCE OF A DERIVATIVE ", + "text_level": 1, + "bbox": [ + 174, + 695, + 459, + 710 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Because $\\bar { \\alpha } _ { t }$ is usually obtained by multiplying a linear variance schedule $\\beta _ { t }$ . So we have ", + "bbox": [ + 171, + 720, + 753, + 736 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ae412bbce2ff3387a7045e0fc900ab3ad1cc7e8fdc973cffd08f2eeb416d3c74.jpg", + "text": "$$\n\\bar { \\alpha } _ { t } = e ^ { a t ^ { 2 } + b t + c } ,\n$$", + "text_format": "latex", + "bbox": [ + 444, + 741, + 553, + 761 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "and $\\bar { \\alpha } _ { 0 } = 1$ , so $c = 0$ . Now we have ", + "bbox": [ + 174, + 767, + 413, + 781 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/6ac5768c4773e356d338aaf1040dfb5cf8218c88ac341fa33d4a59eaebb97139.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\delta 0 } { \\operatorname* { l i m } } \\frac { x _ { t - \\delta } - x _ { t } } { \\delta } } \\\\ & { = \\underset { \\delta 0 } { \\operatorname* { l i m } } \\frac { \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } } { \\delta } ( \\frac { x _ { t } } { \\sqrt { \\bar { \\alpha } _ { t } } ( \\sqrt { \\bar { \\alpha } _ { t - \\delta } } + \\sqrt { \\bar { \\alpha } _ { t } } ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } ( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } ) } ) } \\\\ & { = \\underset { \\delta 0 } { \\operatorname* { l i m } } \\frac { \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } } { \\delta } ( \\frac { x _ { t } } { 2 \\bar { \\alpha } _ { t } } - \\frac { \\epsilon _ { \\theta } ( x _ { t } , t ) } { 2 \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\bar { \\alpha } _ { t } } ) = ( e ^ { a t ^ { 2 } + b t } ) ^ { \\prime } ( \\frac { x _ { t } } { 2 \\bar { \\alpha } _ { t } } - \\frac { \\epsilon _ { \\theta } ( x _ { t } , t ) } { 2 \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\bar { \\alpha } _ { t } } ) } \\\\ & { = ( 2 a t + b ) \\bar { \\alpha } _ { t } ( \\frac { x _ { t } } { 2 \\bar { \\alpha } _ { t } } - \\frac { \\epsilon _ { \\theta } ( x _ { t } , t ) } { 2 \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\bar { \\alpha } _ { t } } ) = \\frac { 1 } { 2 } ( 2 a t + b ) ( x _ { t } - \\frac { \\epsilon _ { \\theta } ( x _ { t } , t ) } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 786, + 781, + 931 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To make $\\begin{array} { r } { \\operatorname* { l i m } _ { \\delta \\to 0 } \\frac { x _ { t - \\delta } - x _ { t } } { \\delta } \\big | _ { t = 0 } } \\end{array}$ is well-defined, $b$ must equal to zero, or $\\begin{array} { r } { ( 2 a t + b ) \\frac { \\epsilon _ { \\theta } ( x _ { t } , t ) } { \\sqrt { 1 - \\bar { \\alpha } _ { t } } } } \\end{array}$ will tend to infinity. This is a strong condition that most variance schedules do not satisfy. In practice, DDPMs can choose the variance schedule very freely. This means that treating DDPMs as ODEs directly is not proper and has theoretical weakness. ", + "bbox": [ + 173, + 101, + 825, + 162 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.5 RELATIONSHIP BETWEEN $t$ , $\\epsilon _ { \\theta }$ AND $x _ { t }$ ", + "text_level": 1, + "bbox": [ + 176, + 179, + 480, + 194 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Relationship between $t$ and $\\epsilon _ { \\theta }$ . In Figure 5, we can see that the denoising process tends to converge, whether in the $\\epsilon _ { \\theta }$ domain or the sample/image domain when the step-index tends to zero. Therefore, we can say that the noise becomes more and more precise when step, namely $t$ , tends to zero. ", + "bbox": [ + 174, + 205, + 825, + 247 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/211a013ff6622187ddb6bed4a28631541d76a3f3c521f36bda044c85bdad9169.jpg", + "image_caption": [ + "Figure 5: The norm $\\delta$ of the difference between two adjacent terms under different steps " + ], + "image_footnote": [], + "bbox": [ + 359, + 263, + 637, + 419 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Relationship between $\\epsilon _ { \\theta }$ and $x _ { t }$ To prove Property 3.1, assume that $x _ { t } = \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon$ , NN is the neural network and $\\epsilon _ { \\theta } = \\mathrm { N N } ( x _ { t } , t )$ . Because we assume that the gradient part is precise, then we have $\\epsilon _ { \\theta } = \\epsilon$ . Then for all $t ^ { \\prime } \\leq t$ , we have: ", + "bbox": [ + 174, + 459, + 825, + 502 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/cf1fb35fa123cfd26a6460ee520f33dcd0e7ba12c57a1e7c4f80abe19932b643.jpg", + "text": "$$\n\\begin{array} { r l } & { x _ { t ^ { \\prime } } = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } \\left( \\frac { x _ { t } - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon _ { \\theta } } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon _ { \\theta } } \\\\ & { ~ = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } \\left( \\frac { \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { ~ = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 506, + 730, + 598 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here, we can find that $x _ { t ^ { \\prime } } = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon$ is also precise, so Property 3.1 is true. ", + "bbox": [ + 173, + 603, + 756, + 619 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.6 ORDER ANALYSIS OF PSEUDO METHOD ", + "text_level": 1, + "bbox": [ + 174, + 635, + 496, + 650 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For the convenience of theoretical analysis, we generalize the problem. Let $\\phi ( x ( t ) , \\epsilon , t , \\delta ) \\ =$ $f ( x ( t ) , t , \\delta ) + g ( t , \\delta ) \\epsilon ( x ( t ) , t )$ and we have the property $f ( x ( t ) , \\dot { } t , 0 ) = g ( t , 0 ) = \\dot { 0 }$ . Then we have: ", + "bbox": [ + 174, + 661, + 823, + 703 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/90c7538ec9fed753ee60114e81ee534a2c065ef047cd65f1161566cb1adfba50.jpg", + "text": "$$\n\\begin{array} { r l } & { x ( 1 ) = x ( 0 ) + \\displaystyle \\sum _ { \\delta \\to 0 } ( y ( t + \\delta ) - y ( t ) ) } \\\\ & { \\quad \\quad = x ( 0 ) + \\displaystyle \\sum _ { \\delta \\to 0 } ( f ( x ( t ) , t , \\delta ) + g ( t , \\delta ) \\epsilon ( x ( t ) , t ) ) } \\\\ & { \\quad \\quad = x ( 0 ) + \\displaystyle \\int _ { 0 } ^ { 1 } \\left( \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) + \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\epsilon ( x ( t ) , t ) \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 705, + 696, + 813 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now, Equation (10) becomes a special case of this more general version and, in this special case, we have: ", + "bbox": [ + 173, + 815, + 823, + 843 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/6c2ebbe1517ee5c532e8d3d00662648cdbdb1748026b7222f6b95e19d142beb2.jpg", + "text": "$$\n\\begin{array} { l } { f ( x ( t ) , t , \\delta ) = \\left( \\displaystyle \\frac { \\sqrt { \\alpha ( t + \\delta ) } } { \\sqrt { \\alpha } } - 1 \\right) x ( t ) } \\\\ { g ( t , \\delta ) = \\sqrt { 1 - \\alpha ( t + \\delta ) } - \\displaystyle \\frac { \\sqrt { ( 1 - \\alpha ( t ) ) \\alpha ( t + \\delta ) } } { \\sqrt { \\alpha ( t ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 845, + 684, + 930 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now, we compute the local error of S-PNDMs. We first compute the theoretical and numerical results of different numerical methods. We have: ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/3e736a1eb4acc494b2861325df614e9585bd20441b8759cbed0a7905e90b7898.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle x ( t ) } } \\\\ { { \\displaystyle x ( t ) + \\delta \\left( \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) + \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\epsilon ( x ( t ) , t ) \\right) + } } \\\\ { { \\displaystyle \\quad \\frac { \\delta ^ { 2 } } { 2 } \\left( \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) + \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\epsilon ( x ( t ) , t ) \\right) ^ { \\prime } + O ( \\delta ^ { 3 } ) } } \\\\ { { \\displaystyle - x ( t ) + \\delta \\left( \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) + \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\epsilon ( x ( t ) , t ) \\right) + O ( \\delta ^ { 3 } ) + } } \\\\ { { \\displaystyle \\quad \\frac { \\delta ^ { 2 } } { 2 } \\left( \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial t } ( x ( t ) , t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\left( \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) + \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\epsilon ( x ( t ) , t ) \\right) \\right) + } } \\\\ { { \\displaystyle \\quad \\frac { \\delta ^ { 2 } } { 2 } \\left( \\frac { \\partial ^ { 2 } g } { \\partial \\delta \\partial t } ( t , 0 ) \\epsilon ( x ( t ) , t ) + \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\epsilon ^ { \\prime } ( x ( t ) , t ) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 147, + 769, + 348 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 364, + 200, + 378 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d451c5c5a7f66ead9f2a4e892a684ed6786b4419d5a9e465a0fc7d963c9bbffa.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { S u s s ~ o u s : } \\quad \\mathcal { A } _ { 1 } ^ { \\mathrm { n o t } } } \\\\ & { = \\nu _ { 1 } ^ { ( n ) } - \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { = \\nu _ { 1 } ^ { ( n ) } - \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { = \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { = \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad ( \\frac { \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } } { \\mu _ { 1 } ^ { ( 1 ) } } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad - \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad ( \\frac { \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } } { \\mu _ { 1 } ^ { ( 1 ) } } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad - \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad ( \\frac { \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } } { \\mu _ { 1 } ^ { ( 1 ) } } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad - \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & { \\quad ( \\frac { \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } } { \\mu _ { 1 } ^ { ( 1 ) } } \\mu _ { 1 } ^ { ( 1 ) } , } \\\\ & \\quad ( \\frac { \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ { ( 1 ) } } { \\mu _ { 1 } ^ { ( 1 ) } } \\mu _ { 1 } ^ { ( 1 ) } ) ^ { 2 } ) ^ { 2 } \\mu _ { 1 } ^ { ( 1 ) } \\mu _ { 1 } ^ { ( 1 ) } + \\nu _ { 1 } ^ { ( n ) } \\mu _ { 1 } ^ ( \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 383, + 743, + 791 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Then we compute the difference between the theoretical and numerical results. We have: ", + "bbox": [ + 171, + 804, + 754, + 820 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/085e967bce097c2bba6045d12c0e8b92bf007299374799d21d6eadcbbeb6a76b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad x ( t + \\delta ) - x _ { \\mathrm { S } \\cdot \\mathrm { P N D M } } ( x + \\delta ) } \\\\ & { = \\frac { \\delta ^ { 2 } } { 2 } \\left( ( \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial t } - \\frac { \\partial ^ { 2 } f } { \\partial \\delta ^ { 2 } } ) ( x ( t ) , t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) \\right) + } \\\\ & { \\quad \\frac { \\delta ^ { 2 } } { 2 } \\left( ( \\frac { \\partial ^ { 2 } g } { \\partial \\delta \\partial t } - \\frac { \\partial ^ { 2 } g } { \\partial \\delta ^ { 2 } } ) ( t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) \\right) \\epsilon ( x ( t ) , t ) + O ( \\delta ^ { 3 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 837, + 753, + 931 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this special case, we compute the derivatives of some items needed in Equation (31). We have: ", + "bbox": [ + 169, + 103, + 808, + 118 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d2d2cbcdd088e6656cb4d594a5d294d912d956b6073218c2e7f01683b35e3232.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\\\ & { \\qquad \\frac { \\partial \\theta _ { \\theta } } { \\partial t } \\theta ( \\theta , \\phi , \\phi ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 112, + 738, + 518 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Now we can compute the final result of Equation (31). We split it into three parts and the values of the first two terms. We have: ", + "bbox": [ + 174, + 666, + 823, + 694 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/cf206a2405b8218f0b7ff48c2f88a9ef8c2be7e33aa6db61ce07251a80cfa06a.jpg", + "text": "$$\n\\begin{array} { l } { { ( \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial t } - \\frac { \\partial ^ { 2 } f } { \\partial \\delta ^ { 2 } } ) ( x ( t ) , t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) } } \\\\ { { = \\left( \\frac { \\alpha ^ { \\prime \\prime } ( t ) } { 2 \\alpha ( t ) } x ( t ) - \\frac { \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 2 \\alpha ( t ) ^ { 2 } } x ( t ) \\right) - \\left( \\frac { \\alpha ^ { \\prime \\prime } ( t ) } { 2 \\alpha ( t ) } x ( t ) + \\frac { - \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) ^ { 2 } } x ( t ) \\right) + \\frac { 1 } { 2 \\alpha ( t ) } \\frac { \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } x ( t ) } } \\\\ { { = 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 693, + 764, + 775 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 780, + 200, + 792 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/b384a472b637fedc0ff2a4b66fe60a45a24e8597bb5bef520412e9e03f6fa110.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle ( \\frac { \\partial ^ { 2 } g } { \\partial \\delta \\partial t } - \\frac { \\partial ^ { 2 } g } { \\partial \\delta ^ { 2 } } ) ( t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) } \\\\ { = \\frac { \\sqrt { 1 - \\alpha ( t ) } \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) ^ { 2 } } + \\frac { \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) \\sqrt { 1 - \\alpha ( t ) } } + \\frac { \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } \\left( \\frac { - \\alpha ^ { \\prime } ( t ) } { 2 \\sqrt { 1 - \\alpha ( ( t ) ) } } - \\frac { \\sqrt { 1 - \\alpha ( t ) } \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } \\right) } \\\\ { = \\frac { \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) ^ { 2 } \\sqrt { 1 - \\alpha ( t ) } } + \\frac { \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } \\left( \\frac { - \\alpha ^ { \\prime } ( t ) } { 2 \\sqrt { 1 - \\alpha ( t ) } \\alpha ( t ) } \\right) } \\\\ { = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 791, + 779, + 928 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Finally, we get the final result of Equation (31): ", + "bbox": [ + 174, + 103, + 485, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/49da3801dea17615bb4fac992473becb8a1fe6ebfd820c88bd4fe160a57e9b12.jpg", + "text": "$$\nx ( t + \\delta ) - x _ { \\mathrm { S - P N D M } } ( x + \\delta ) = O ( \\delta ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 375, + 122, + 620, + 140 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "And the computation of the convergence order of F-PNDMs is similar, and we ignore it here. Therefore, Property 3.2 is true. ", + "bbox": [ + 173, + 143, + 823, + 172 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.7 VARIANCE SCHEDULE ", + "text_level": 1, + "bbox": [ + 176, + 190, + 370, + 204 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "According to Cifar10 (cosine) in Table 2, PNDMs can be used on both linear variance schedule and cosine variance schedule. However, we also notice cosine variance schedule can make FID lower when we use relatively big generation steps, but the effort is limited when the number of steps is small. F-PNDM uses information from four consecutive steps, so the smoothness of the schedule is more important for F-PNDM than DDIM. According to this experiment, our work can be used with works that pay attention to variance schedules to improve the acceleration effect further. However, a variance schedule that fits pseudo numerical methods better remains to be found in further work. ", + "bbox": [ + 174, + 215, + 825, + 314 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.8 TOY EXAMPLE ", + "text_level": 1, + "bbox": [ + 176, + 329, + 320, + 344 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Here, we design a toy example to test our new methods without the influence of neural networks. We randomly generate the initial input $x _ { 1 } = ( m _ { 1 } , m _ { 2 } )$ , $m _ { i } \\sim U ( 0 , 1 )$ and use a simple analytic equations $\\epsilon _ { \\theta } ( x ) = ( \\sin x [ 0 ] , \\cos x [ 1 ] )$ to replace the neural networks in real cases. Let $\\phi$ in Equation (11) is unchanged and $\\bar { \\alpha } _ { t } = \\alpha ( t ) = 1 - t$ , then we get: ", + "bbox": [ + 174, + 356, + 825, + 412 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/de0a509dc540c55f66c0825c694793e78d313e740f1d50ef40cdc8882856f2cd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } ) , t , t - \\delta ) } \\\\ & { = \\frac { \\sqrt { \\bar { \\alpha } _ { t - \\delta } } } { \\sqrt { \\bar { \\alpha } _ { t } } } x _ { t } - \\frac { \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { \\left( 1 - \\bar { \\alpha } _ { t - \\delta } \\right) \\bar { \\alpha } _ { t } } + \\sqrt { \\left( 1 - \\bar { \\alpha } _ { t } \\right) \\bar { \\alpha } _ { t - \\delta } } \\right) } \\epsilon _ { t } } \\\\ & { = \\frac { \\sqrt { 1 - \\left( t - \\delta \\right) } } { \\sqrt { 1 - t } } x _ { t } - \\frac { \\delta } { \\sqrt { 1 - t } \\left( \\sqrt { \\left( t - \\delta \\right) \\left( 1 - t \\right) } + \\sqrt { t ( 1 - \\left( t - \\delta \\right) ) } \\right) } \\epsilon _ { \\theta } ( x _ { t } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 434, + 750, + 537 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Here, we use three different numerical methods to generate $x _ { 0 }$ ", + "bbox": [ + 173, + 541, + 586, + 555 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For DDIM, we have: ", + "bbox": [ + 173, + 561, + 312, + 577 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/68e0ddf0f84f3be3ba5cb1f8825f9ecb50a3047e5abc9a718026fee0d3d66e75.jpg", + "text": "$$\nx _ { t - \\delta } = x _ { t } + \\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } ) , t , t - \\delta )\n$$", + "text_format": "latex", + "bbox": [ + 382, + 575, + 616, + 592 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For FON, according to Equation (10), we have: ", + "bbox": [ + 173, + 594, + 485, + 609 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8c72a4506a485312ef992ab12e0b29a8c9b1a29b0859aaad1261cc9f4c9f5bea.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle e _ { t } ^ { \\prime } = \\bar { \\alpha } ^ { \\prime } ( t ) \\left( \\frac { x _ { t } } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } \\right) } } \\\\ { { \\displaystyle ~ = - \\left( \\frac { x _ { t } } { 2 ( 1 - t ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } ) } { 2 ( 1 - t ) \\sqrt { t } } \\right) } } \\\\ { { \\displaystyle x _ { t - \\delta } = x _ { t } + \\frac { \\delta } { 2 4 } ( 5 5 e _ { t } ^ { \\prime } - 5 9 e _ { t + \\delta } ^ { \\prime } + 3 7 e _ { t + 2 \\delta } ^ { \\prime } - 9 e _ { t + 3 \\delta } ^ { \\prime } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 612, + 678, + 724 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For F-PNDM, we have: ", + "bbox": [ + 173, + 726, + 330, + 739 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/e98b5d5de615dbbac51d6d8c4a87f94e518b7a6175fde5e5eb65433b2ce2fe24.jpg", + "text": "$$\n\\begin{array} { c } { { e ^ { \\prime } = \\displaystyle \\frac { 1 } { 2 4 } \\big ( 5 5 \\epsilon _ { \\theta } ( x _ { t } ) - 5 9 \\epsilon _ { \\theta } ( x _ { t + \\delta } ) + 3 7 \\epsilon _ { \\theta } ( x _ { t + 2 \\delta } ) - 9 \\epsilon _ { \\theta } ( x _ { t + 3 \\delta } ) \\big ) } } \\\\ { { x _ { t - \\delta } = x _ { t } + \\phi ( x _ { t } , e ^ { \\prime } , t , t - \\delta ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 744, + 707, + 794 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Then we draw the corresponding generation curves in Figure 6. We find that the result is similar to the real cases. The main difference here is that FON can correct its results while the real case cannot. The reason is that the gradient is well-defined everywhere, while in real cases, the gradient is meaningful on the high-density region of the data $x _ { t }$ of DDPMs. ", + "bbox": [ + 173, + 796, + 825, + 853 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.9 MORE FID RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 869, + 362, + 883 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Here, We provide our more detailed FID results on Cifar10, CelebA, LSUN-church and LSUNbedroom in Table 3, 4, 5, 6. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/be45ffbce3a2d38e669a4bc00e9042c1345f1381c96712d8462385105dfde8f6.jpg", + "image_caption": [ + "Figure 6: The generation curve of our toy example. " + ], + "image_footnote": [], + "bbox": [ + 169, + 109, + 820, + 358 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/5bcd7fbdadf084d9a45bccacee893eb03e6579bc5e2624eead9bbe85741bb767.jpg", + "table_caption": [ + "Table 3: Cifar10 image generation measured in FID. The upper part uses linear variance schedule and the bottom half uses cosine variance schedule. The first line shows the FID provided by Song et al. (2020a). " + ], + "table_footnote": [], + "table_body": "
steps510202540501001252002505001000
DDIM13.46.844.674.164.04
DDIM*44.518.510.99.617.656.995.525.194.694.524.174.00
FON98.013.17.416.415.505.264.654.544.234.123.843.71
S-PNDM22.811.67.566.795.575.184.344.183.973.913.813.80
F-PNDM13.97.035.004.764.103.953.723.643.603.603.643.70
DDIM*28.714.58.797.836.415.864.924.754.424.303.983.69
S-PNDM18.38.645.775.454.764.463.943.853.693.713.603.38
F-PNDM18.27.054.614.323.853.683.533.463.473.493.443.26
", + "bbox": [ + 173, + 407, + 839, + 555 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/6e8c9df7eb4e85f6d4247922724ae1f4ac9cb63d571b8de0d03111bf45cebb45.jpg", + "table_caption": [ + "Table 4: CelebA image generation measured in FID. All of them use linear variance schedule. " + ], + "table_footnote": [], + "table_body": "
steps510202540501001252002505001000
DDIM17.313.79.176.533.51
DDIM*24.416.913.412.39.998.956.365.744.784.443.753.41
FON60.216.011.610.68.898.136.706.285.455.144.494.17
S-PNDM15.212.29.458.426.505.694.033.723.303.193.012.99
F-PNDM11.37.715.514.753.673.342.812.752.712.712.772.86
", + "bbox": [ + 174, + 640, + 838, + 739 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/db86133464a47a70c8749e6adda06441de30853d4f02c13db526ba06be756b1b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
steps51020254050100125200250
DDIM19.512.510.810.6
DDIM*48.818.811.711.010.110.09.849.839.859.88
S-PNDM20.511.89.209.139.319.499.829.8810.010.0
F-PNDM14.88.699.139.339.699.8910.19.9910.110.1
", + "bbox": [ + 215, + 795, + 782, + 881 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Table 5: LSUN-church image generation measured in FID. All of them use linear variance schedule. ", + "bbox": [ + 174, + 897, + 823, + 912 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/d553eda39c26a00f8f90a6293ac819c1174420053f8091098efd90bc07870dc2.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
steps-51020254050100125200250
DDIM17.08.896.756.62
DDIM*51.316.48.477.416.276.055.976.036.236.32
S-PNDM18.110.26.506.025.745.816.296.446.696.75
F-PNDM12.66.995.685.746.176.446.916.967.036.92
", + "bbox": [ + 215, + 102, + 781, + 188 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Table 6: LSUN-bedroom image generation measured in FID. All of them use linear variance schedule. ", + "bbox": [ + 173, + 203, + 823, + 232 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.10 MORE IMAGE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 258, + 390, + 273 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Here, we show more generated images on Cifar10, CelebA, LSUN-church and LSUN-bedroom in Figure 7, 9, 8, 10, 11, 12. ", + "bbox": [ + 176, + 284, + 825, + 314 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/c4f5582ce5980bcd0e8ac8379e97701ac88de76a8102e86df5aca16d27de35cf.jpg", + "image_caption": [ + "Figure 7: 5, 10, 20, 50, 100, 250, 500-steps generated results using DDIMs, classical numerical methods and PNDMs on Cifar10. " + ], + "image_footnote": [], + "bbox": [ + 245, + 320, + 751, + 496 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/e2e55ffc863d3d86c6415bfe1c8e64af10a0a61e5ee41f0c7f50268bacabf7df.jpg", + "image_caption": [ + "Figure 8: 5, 10, 20, 50, 100, 250, 500-steps generated results using DDIMs, classical numerical methods and PNDMs on CelebA. " + ], + "image_footnote": [], + "bbox": [ + 243, + 545, + 750, + 869 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/8ca89fae4dfa5d0cdbfd05cf769104951e67eaa94405628c9c6e5dcf092ce347.jpg", + "image_caption": [ + "Figure 9: Generated images of PNDMs on Cifar10. " + ], + "image_footnote": [], + "bbox": [ + 271, + 116, + 725, + 465 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/707d628b8fe0c94f27302502b323c2339811ec6f9da5f802750e7f5b6dd536b0.jpg", + "image_caption": [ + "Figure 10: Generated images of PNDMs on CelebA. " + ], + "image_footnote": [], + "bbox": [ + 271, + 529, + 725, + 881 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/0cd7787d65de975dc53a90ae9f8d3dc7ad1ba175cb337e0fc21718a6c69f32fa.jpg", + "image_caption": [ + "Figure 11: 5, 10, 20, 50, 100-steps generated results using DDIMs, classical numerical methods and PNDMs on LSUN-church. " + ], + "image_footnote": [], + "bbox": [ + 207, + 85, + 790, + 906 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/ef1a6784558da2575fe140e396e950f9d691d70145b931cb4fa8ea9ddebd8afd.jpg", + "image_caption": [ + "Figure 12: 5, 10, 20, 50, 100-steps generated results using DDIMs, classical numerical methods and PNDMs on LSUN-bedroom. " + ], + "image_footnote": [], + "bbox": [ + 207, + 82, + 789, + 907 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "A.11 MORE VISUALIZATION RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 454, + 117 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Here, we put more visualization results similar to Figure 4 in Figure 13. ", + "bbox": [ + 174, + 128, + 643, + 145 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/eb24a50a2b3b3413ce7f38719c70a88204c3334e245ab79b82a85e12dd3e3dcb.jpg", + "image_caption": [ + "Figure 13: Visualization results under 5, 10, 20, 25, 40 and 50 steps. " + ], + "image_footnote": [], + "bbox": [ + 171, + 156, + 818, + 657 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "A.12 FID RESULT WITH STANDARD DEVIATION ", + "text_level": 1, + "bbox": [ + 176, + 704, + 519, + 718 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Here, we report the mean and standard deviation of FID results, tested over four sampling runs. ", + "bbox": [ + 174, + 729, + 797, + 746 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/a8708d930bff52812cc0daa24888d8eb08329c784fd57882816a48c288ea7656.jpg", + "table_caption": [], + "table_footnote": [ + "Table 7: Image generation measured in FID on Cifar10. DDIM\\* means a kind of pseudo numerical method and also a retest of DDIM. " + ], + "table_body": "
datasetFID\\ step model1020501002501000
Cifar10 (linear)DDIM*18.50±.0610.86±.086.95±.045.49±.064.52±.024.02±.04
FON13.00±.117.33±.065.24±.054.64±.044.12±.033.73±.03
S-PNDM11.58±.107.53±.075.15±.054.34±.033.93±.023.83±.03
F-PNDM6.12±.075.04±.074.01±.023.75±.043.67±.033.78±.04
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DDPMs have been applied successfully to a variety", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "of applications, including image generation (Ho et al., 2020; Song et al., 2020b), text generation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "(Hoogeboom et al., 2021; Austin et al., 2021), 3D point cloud generation (Luo & Hu, 2021), text-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 547, + 497, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 497, + 560 + ], + "score": 1.0, + "content": "to-speech (Kong et al., 2021; Chen et al., 2020) and image super-resolution (Saharia et al., 2021).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "Unlike Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which require careful", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "hyperparameter tuning according to different model structures and datasets, DDPMs can use similar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "model structures and be trained by a simple denoising objective which makes the models fit the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "score": 1.0, + "content": "noise in the data. 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(2021). Some works focus on changing the inference equation, including Song et al. 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However, DDPMs require hundreds", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 246, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 246, + 469, + 257 + ], + "score": 1.0, + "content": "to thousands of iterations to produce final samples. Several prior works have", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 470, + 270 + ], + "score": 1.0, + "content": "successfully accelerated DDPMs through adjusting the variance schedule (e.g.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "Improved Denoising Diffusion Probabilistic Models) or the denoising equation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "score": 1.0, + "content": "(e.g., Denoising Diffusion Implicit Models (DDIMs)). However, these acceler-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 142, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "ation methods cannot maintain the quality of samples and even introduce new", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "score": 1.0, + "content": "noise at a high speedup rate, which limit their practicability. To accelerate the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 324 + ], + "score": 1.0, + "content": "inference process while keeping the sample quality, we provide a fresh perspec-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 469, + 334 + ], + "score": 1.0, + "content": "tive that DDPMs should be treated as solving differential equations on manifolds.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 332, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 470, + 347 + ], + "score": 1.0, + "content": "Under such a perspective, we propose pseudo numerical methods for diffusion", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 142, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "models (PNDMs). Specifically, we figure out how to solve differential equations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 367 + ], + "score": 1.0, + "content": "on manifolds and show that DDIMs are simple cases of pseudo numerical meth-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 470, + 378 + ], + "score": 1.0, + "content": "ods. We change several classical numerical methods to corresponding pseudo", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 377, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 470, + 389 + ], + "score": 1.0, + "content": "numerical methods and find that the pseudo linear multi-step method is the best", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 399 + ], + "score": 1.0, + "content": "in most situations. According to our experiments, by directly using pre-trained", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "models on Cifar10, CelebA and LSUN, PNDMs can generate higher quality syn-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 469, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 469, + 423 + ], + "score": 1.0, + "content": "thetic images with only 50 steps compared with 1000-step DDIMs (20x speedup),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "score": 1.0, + "content": "significantly outperform DDIMs with 250 steps (by around 0.4 in FID) and have", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 355, + 443 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 355, + 443 + ], + "score": 1.0, + "content": "good generalization on different variance schedules.1", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 15.5, + "bbox_fs": [ + 141, + 222, + 470, + 443 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 206, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 208, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 208, + 482 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Denoising Diffusion Probabilistic Models (DDPMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "is a class of generative models which model the data distribution through an iterative denoising", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "process reversing a multi-step noising process. DDPMs have been applied successfully to a variety", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "of applications, including image generation (Ho et al., 2020; Song et al., 2020b), text generation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "(Hoogeboom et al., 2021; Austin et al., 2021), 3D point cloud generation (Luo & Hu, 2021), text-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 547, + 497, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 497, + 560 + ], + "score": 1.0, + "content": "to-speech (Kong et al., 2021; Chen et al., 2020) and image super-resolution (Saharia et al., 2021).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 491, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "Unlike Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which require careful", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "hyperparameter tuning according to different model structures and datasets, DDPMs can use similar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "model structures and be trained by a simple denoising objective which makes the models fit the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "score": 1.0, + "content": "noise in the data. To generate samples, the iterative denoising process starts from white noise and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "score": 1.0, + "content": "progressively denoises it into the target domain according to the noise predicted by the model at", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "every step. 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(2021). Some works focus on changing the inference equation, including Song et al. (2020a)", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 667, + 506, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "and Song et al. (2020b). Denoising Diffusion Implicit Models (DDIMs) (Song et al., 2020a) relying", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "on a non-Markovian process accelerate the denoising process by taking multiple steps every itera-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "tion. Probability Flows (PFs) (Song et al., 2020b) build a connection between the denoising process", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "and solving ordinary differential equations and use numerical methods of differential equations to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 501, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 501, + 140 + ], + "score": 1.0, + "content": "accelerate the denoising process. Additionally, we introduce more related works in Appendix A.1.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 256, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 257, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 257, + 155 + ], + "score": 1.0, + "content": "However, this direct connection be-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 258, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 258, + 165 + ], + "score": 1.0, + "content": "tween DDPMs and numerical meth-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 258, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 258, + 177 + ], + "score": 1.0, + "content": "ods (e.g., forward Euler method, lin-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 258, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 258, + 189 + ], + "score": 1.0, + "content": "ear multi-step method and Runge-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 257, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 257, + 199 + ], + "score": 1.0, + "content": "Kutta method (Timothy, 2017)) has", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 258, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 258, + 209 + ], + "score": 1.0, + "content": "weaknesses in both speed and effect", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 258, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 258, + 221 + ], + "score": 1.0, + "content": "(see Section 3.1). Some numerical", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 258, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 258, + 232 + ], + "score": 1.0, + "content": "methods are straightforward, like the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 257, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 257, + 243 + ], + "score": 1.0, + "content": "forward Euler method, but they can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 257, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 257, + 254 + ], + "score": 1.0, + "content": "only trade quality for speed. 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Furthermore, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "also notice that numerical methods can introduce noticeable noise at a high speedup rate, which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "makes high-order numerical methods (e.g., Runge-Kutta method) even less effective than DDIMs.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 319, + 358, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 358, + 331 + ], + "score": 1.0, + "content": "This phenomenon is also mentioned in Salimans & Ho (2022).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 347 + ], + "score": 1.0, + "content": "To figure out the reason for the performance degradation in classical numerical methods, we conduct", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 347, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 504, + 357 + ], + "score": 1.0, + "content": "some analyses and find that classical numerical methods may sample data far away from the main", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "distribution area of the data, and the inference equations of DDPMs do not satisfy a necessary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 369, + 401, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 401, + 380 + ], + "score": 1.0, + "content": "condition of numerical methods at the last several steps (see Section 3.2).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "To tackle these problems, we design new numerical methods called pseudo numerical methods for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 396, + 504, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 412, + 409 + ], + "score": 1.0, + "content": "diffusion models (PNDMs) to generate samples along a specific manifold in", + "type": "text" + }, + { + "bbox": [ + 413, + 397, + 426, + 407 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 396, + 504, + 409 + ], + "score": 1.0, + "content": ", which is the high-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "density region of the data. We first compute the corresponding differential equations of diffusion", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "models directly and self-consistently, which builds a theoretical connection between DDPMs and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "numerical methods. Considering that classical numerical methods cannot guarantee to generate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "samples on certain manifolds, we provide brand-new numerical methods called pseudo numerical", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "methods based on our theoretical analyses. 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According to our experiments, our methods have several advantages:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "• Our methods combine the benefits of DDIMs and high-order numerical methods successfully. We", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 113, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 113, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "theoretically prove that our new methods PNDMs are second-order convergent while DDIMs are", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 114, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 114, + 553, + 300, + 567 + ], + "score": 1.0, + "content": "first-order convergent, which makes PNDMs", + "type": "text" + }, + { + "bbox": [ + 301, + 554, + 317, + 564 + ], + "score": 0.52, + "content": "2 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "faster without loss of quality on Cifar10 and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 114, + 565, + 150, + 576 + ], + "spans": [ + { + "bbox": [ + 114, + 565, + 150, + 576 + ], + "score": 1.0, + "content": "CelebA.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "• Our methods can reduce the best FID of pre-trained models with even shorter sampling time. With", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 114, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 114, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "only 250 steps, our new denoising process can reduce the best FID by around 0.4 points Cifar10", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 113, + 600, + 389, + 613 + ], + "spans": [ + { + "bbox": [ + 113, + 600, + 389, + 613 + ], + "score": 1.0, + "content": "and CelebA. We achieve a new SOTA FID score of 2.71 on CelebA.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 613, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 628 + ], + "score": 1.0, + "content": "• Our methods work well with different variance schedules, which means that our methods have a", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 113, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 113, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "good generalization and can be used together with those works introducing better variance sched-", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 114, + 637, + 305, + 649 + ], + "spans": [ + { + "bbox": [ + 114, + 637, + 305, + 649 + ], + "score": 1.0, + "content": "ules to accelerate the denoising process further.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 52.5 + }, + { + "type": "title", + "bbox": [ + 107, + 663, + 200, + 676 + ], + "lines": [ + { + "bbox": [ + 104, + 662, + 201, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 201, + 678 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 58 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "In this section, we introduce some backgrounds. Firstly, we present the classical understanding of", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "DDPMs. Then we provide another understanding based on Song et al. (2020b), which inspires us", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "to use numerical methods to accelerate the denoising process of diffusion models. After that, we", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 720, + 406, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 406, + 734 + ], + "score": 1.0, + "content": "introduce some background on numerical methods used later in this paper.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 60.5 + } + ], + "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 2022", + "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, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "and Song et al. (2020b). Denoising Diffusion Implicit Models (DDIMs) (Song et al., 2020a) relying", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "on a non-Markovian process accelerate the denoising process by taking multiple steps every itera-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "tion. Probability Flows (PFs) (Song et al., 2020b) build a connection between the denoising process", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "and solving ordinary differential equations and use numerical methods of differential equations to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 501, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 501, + 140 + ], + "score": 1.0, + "content": "accelerate the denoising process. 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Some numerical", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 258, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 258, + 232 + ], + "score": 1.0, + "content": "methods are straightforward, like the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 257, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 257, + 243 + ], + "score": 1.0, + "content": "forward Euler method, but they can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 257, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 257, + 254 + ], + "score": 1.0, + "content": "only trade quality for speed. 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Furthermore, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "also notice that numerical methods can introduce noticeable noise at a high speedup rate, which", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "makes high-order numerical methods (e.g., Runge-Kutta method) even less effective than DDIMs.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 319, + 358, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 358, + 331 + ], + "score": 1.0, + "content": "This phenomenon is also mentioned in Salimans & Ho (2022).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 285, + 506, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 347 + ], + "score": 1.0, + "content": "To figure out the reason for the performance degradation in classical numerical methods, we conduct", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 347, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 504, + 357 + ], + "score": 1.0, + "content": "some analyses and find that classical numerical methods may sample data far away from the main", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "distribution area of the data, and the inference equations of DDPMs do not satisfy a necessary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 369, + 401, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 401, + 380 + ], + "score": 1.0, + "content": "condition of numerical methods at the last several steps (see Section 3.2).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 336, + 505, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "To tackle these problems, we design new numerical methods called pseudo numerical methods for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 396, + 504, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 412, + 409 + ], + "score": 1.0, + "content": "diffusion models (PNDMs) to generate samples along a specific manifold in", + "type": "text" + }, + { + "bbox": [ + 413, + 397, + 426, + 407 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 396, + 504, + 409 + ], + "score": 1.0, + "content": ", which is the high-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "density region of the data. We first compute the corresponding differential equations of diffusion", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "models directly and self-consistently, which builds a theoretical connection between DDPMs and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "numerical methods. Considering that classical numerical methods cannot guarantee to generate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "samples on certain manifolds, we provide brand-new numerical methods called pseudo numerical", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "methods based on our theoretical analyses. We also find that DDIMs are simple cases of pseudo", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 462, + 504, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 504, + 474 + ], + "score": 1.0, + "content": "numerical methods, which means that we also provide a new way to understand DDIMs better.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "Furthermore, we find that the pseudo linear multi-step method is the fastest method for diffusion", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 485, + 266, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 266, + 496 + ], + "score": 1.0, + "content": "models under similar generated quality.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 385, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 502, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "Besides, we provide a detailed theoretical analysis of our new theory and give visualization results to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 511, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 504, + 525 + ], + "score": 1.0, + "content": "support our theory intuitively. According to our experiments, our methods have several advantages:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 501, + 504, + 525 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 531, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "• Our methods combine the benefits of DDIMs and high-order numerical methods successfully. 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As a byproduct, we can directly use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "pre-trained models from DDPMs. After establishing this connection, we provide detailed analyses", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "on the weakness of classical numerical methods. To solve the problems in classical numerical meth-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "ods, we dive into the structure of numerical methods by dividing their equations into a gradient part", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 328, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 340 + ], + "score": 1.0, + "content": "and a transfer part and define pseudo numerical methods by introducing nonlinear transfer parts. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "find that DDIMs can be regarded as simple pseudo numerical methods. Then, We explore the pros", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "and cons of different numerical methods and choose the linear multi-step method to make numeri-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "cal methods faster. Finally, we summarize our findings and analyses and safely propose our novel", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "pseudo numerical methods for diffusion models (PNDMs), which combine our proposed transfer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "part and the gradient part of the linear multi-step method. Furthermore, we analyze the convergence", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 497, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 497, + 406 + ], + "score": 1.0, + "content": "order of pseudo numerical methods to demonstrate the effectiveness of our methods theoretically.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 258, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 259, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 259, + 430 + ], + "score": 1.0, + "content": "3.1 FORMULA TRANSFORMATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 452, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 453, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 453, + 452 + ], + "score": 1.0, + "content": "According to Song et al. (2020a), the reverse process of DDPMs and DDIMs satisfies:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 456, + 466, + 485 + ], + "lines": [ + { + "bbox": [ + 145, + 456, + 466, + 485 + ], + "spans": [ + { + "bbox": [ + 145, + 456, + 466, + 485 + ], + "score": 0.92, + "content": "x _ { t - 1 } = \\sqrt { \\bar { \\alpha } _ { t - 1 } } \\left( \\frac { x _ { t } - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon _ { \\theta } ( x _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t - 1 } - \\sigma _ { t } ^ { 2 } } \\epsilon _ { \\theta } ( x _ { t } , t ) + \\sigma _ { t } \\epsilon _ { t } .", + "type": "interline_equation", + "image_path": "f02079c70488e8c098c1e787f0af456fd5d016e5a3de26400fdf3aba98171010.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 145, + 456, + 466, + 465.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 145, + 465.6666666666667, + 466, + 475.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 145, + 475.33333333333337, + 466, + 485.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 131, + 502 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 132, + 491, + 142, + 501 + ], + "score": 0.86, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 489, + 299, + 502 + ], + "score": 1.0, + "content": "controls the ratio of random noise. If", + "type": "text" + }, + { + "bbox": [ + 300, + 491, + 311, + 501 + ], + "score": 0.86, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "equals one, Equation (8) represents the reverse", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 198, + 514 + ], + "score": 1.0, + "content": "process of DDPMs; if", + "type": "text" + }, + { + "bbox": [ + 198, + 502, + 208, + 512 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "equals zero, this equation represents the reverse process of DDIMs. And", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 153, + 524 + ], + "score": 1.0, + "content": "only when", + "type": "text" + }, + { + "bbox": [ + 153, + 513, + 164, + 523 + ], + "score": 0.83, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "equals zero, this equation removes the random item and becomes a discrete form", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "of a certain ODE. Theoretically, the numerical methods that can be used on differential equations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "with random items are limited. And Song et al. (2020b) have done enough research in this case.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "Empirically, Song et al. (2020a) have shown that DDIMs have a better acceleration effect when the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 443, + 568 + ], + "score": 1.0, + "content": "number of total steps is relatively small. 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\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "according to (Song et al., 2020a) and change this equation into a differential form, namely,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 594, + 284, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 140, + 607 + ], + "score": 1.0, + "content": "subtract", + "type": "text" + }, + { + "bbox": [ + 141, + 596, + 151, + 605 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 594, + 284, + 607 + ], + "score": 1.0, + "content": "from both sides of this equation:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 611, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 111, + 611, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 111, + 611, + 505, + 645 + ], + "score": 0.92, + "content": "\\mathfrak { z } _ { t - \\delta } - \\mathfrak { x } _ { t } = \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) \\left( \\frac { \\mathfrak { x } _ { t } } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { \\bar { \\alpha } _ { t - \\delta } } + \\sqrt { \\bar { \\alpha } _ { t } } \\right) } - \\frac { \\epsilon _ { \\theta } ( \\mathfrak { x } _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } \\right) } \\right) .", + "type": "interline_equation", + "image_path": "577c3ec5876a9604c90387e3d534299c1ae1f71d2d18a024141de10a63208af1.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 111, + 611, + 505, + 622.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 111, + 622.3333333333334, + 505, + 633.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 111, + 633.6666666666667, + 505, + 645.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 142, + 666 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 142, + 654, + 148, + 664 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 653, + 285, + 666 + ], + "score": 1.0, + "content": "is a continuous variable from 0 to", + "type": "text" + }, + { + "bbox": [ + 285, + 655, + 289, + 664 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 653, + 504, + 666 + ], + "score": 1.0, + "content": ", we can now compute the derivative of the generation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 125, + 689 + ], + "score": 1.0, + "content": "data", + "type": "text" + }, + { + "bbox": [ + 126, + 673, + 136, + 682 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 671, + 187, + 689 + ], + "score": 1.0, + "content": "and get that", + "type": "text" + }, + { + "bbox": [ + 188, + 665, + 385, + 690 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\underset { \\delta 0 } { \\operatorname* { l i m } } \\frac { x _ { t } - x _ { t - \\delta } } { \\delta } = - \\bar { \\alpha } ^ { \\prime } ( t ) ( \\frac { x ( t ) } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x ( t ) , t ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 670, + 414, + 684 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 414, + 671, + 433, + 684 + ], + "score": 0.91, + "content": "\\bar { \\alpha } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "is the continuous", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 687, + 507, + 705 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 150, + 705 + ], + "score": 1.0, + "content": "version of", + "type": "text" + }, + { + "bbox": [ + 150, + 689, + 184, + 703 + ], + "score": 0.93, + "content": "\\{ \\bar { \\alpha } _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 687, + 271, + 705 + ], + "score": 1.0, + "content": "like the definition of", + "type": "text" + }, + { + "bbox": [ + 271, + 691, + 289, + 702 + ], + "score": 0.9, + "content": "x ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 687, + 463, + 705 + ], + "score": 1.0, + "content": ". 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The reason why these methods achieve higher", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 252, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 252, + 732 + ], + "score": 1.0, + "content": "accuracy can be found in Appendix A.2", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 26, + 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 2022", + "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": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 396, + 97 + ], + "score": 1.0, + "content": "Forward Euler Method For a certain differential equation satisfying", + "type": "text" + }, + { + "bbox": [ + 396, + 81, + 452, + 95 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { d x } { d t } = f ( x , t ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 81, + 505, + 97 + ], + "score": 1.0, + "content": ". The trivial", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 416, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 321, + 107 + ], + "score": 1.0, + "content": "numerical method is forward Euler method satisfying", + "type": "text" + }, + { + "bbox": [ + 322, + 93, + 412, + 106 + ], + "score": 0.91, + "content": "x _ { t + \\delta } = x _ { t } + \\delta f ( x _ { t } ^ { ^ { \\mathrm { a } \\delta } } t )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 92, + 416, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 107 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 110, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "score": 1.0, + "content": "Runge-Kutta Method Runge-Kutta method uses more information at every step, so it can achieve", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 305, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 305, + 134 + ], + "score": 1.0, + "content": "higher accuracy 2. Runge-Kutta method satisfies:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 109, + 505, + 134 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 138, + 425, + 182 + ], + "lines": [ + { + "bbox": [ + 186, + 138, + 425, + 182 + ], + "spans": [ + { + "bbox": [ + 186, + 138, + 425, + 182 + ], + "score": 0.94, + "content": "\\left\\{ \\begin{array} { r l } & { k _ { 1 } = f ( x _ { t } , t ) , \\quad \\quad k _ { 2 } = f ( x _ { t } + \\frac { \\delta } { 2 } k _ { 1 } , t + \\frac { \\delta } { 2 } ) } \\\\ & { k _ { 3 } = f ( x _ { t } + \\frac { \\delta } { 2 } k _ { 2 } , t + \\frac { \\delta } { 2 } ) , \\quad k _ { 4 } = f ( x _ { t } + \\delta k _ { 3 } , t + \\delta ) } \\\\ & { x _ { t + \\delta } = x _ { t } + \\frac { \\delta } { 6 } ( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } ) . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "10be13b41eedb3c348a243bd90aecd0c8d00cb3f3f5da9b53c8d26fb786c835d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 186, + 138, + 425, + 152.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 186, + 152.66666666666666, + 425, + 167.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 167.33333333333331, + 425, + 181.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 192, + 495, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 495, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 495, + 205 + ], + "score": 1.0, + "content": "Linear Multi-Step Method Linear multi-step method is another numerical method and satisfies:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 192, + 495, + 205 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 209, + 447, + 235 + ], + "lines": [ + { + "bbox": [ + 164, + 209, + 447, + 235 + ], + "spans": [ + { + "bbox": [ + 164, + 209, + 447, + 235 + ], + "score": 0.93, + "content": "x _ { t + \\delta } = x _ { t } + \\frac { \\delta } { 2 4 } \\big ( 5 5 f _ { t } - 5 9 f _ { t - \\delta } + 3 7 f _ { t - 2 \\delta } - 9 f _ { t - 3 \\delta } \\big ) , f _ { t } = f ( x _ { t } , t ) .", + "type": "interline_equation", + "image_path": "3f0434bbb9374a4e7e10c11e531cc8211374412b7f94e0c45a66e44af6b5c6de.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 164, + 209, + 447, + 235 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 246, + 348, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 349, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 349, + 261 + ], + "score": 1.0, + "content": "3 PSEUDO NUMERICAL METHOD FOR DDPM", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 272, + 506, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "In this section, we first compute the corresponding differential equations of diffusion models to build", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 283, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 506, + 296 + ], + "score": 1.0, + "content": "a direct connection between DDPMs and numerical methods. As a byproduct, we can directly use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 104, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "pre-trained models from DDPMs. After establishing this connection, we provide detailed analyses", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "on the weakness of classical numerical methods. To solve the problems in classical numerical meth-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "ods, we dive into the structure of numerical methods by dividing their equations into a gradient part", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 328, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 340 + ], + "score": 1.0, + "content": "and a transfer part and define pseudo numerical methods by introducing nonlinear transfer parts. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "find that DDIMs can be regarded as simple pseudo numerical methods. Then, We explore the pros", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "and cons of different numerical methods and choose the linear multi-step method to make numeri-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "cal methods faster. Finally, we summarize our findings and analyses and safely propose our novel", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "pseudo numerical methods for diffusion models (PNDMs), which combine our proposed transfer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "part and the gradient part of the linear multi-step method. Furthermore, we analyze the convergence", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 393, + 497, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 497, + 406 + ], + "score": 1.0, + "content": "order of pseudo numerical methods to demonstrate the effectiveness of our methods theoretically.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 272, + 506, + 406 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 258, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 259, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 259, + 430 + ], + "score": 1.0, + "content": "3.1 FORMULA TRANSFORMATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 452, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 453, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 453, + 452 + ], + "score": 1.0, + "content": "According to Song et al. (2020a), the reverse process of DDPMs and DDIMs satisfies:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 438, + 453, + 452 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 456, + 466, + 485 + ], + "lines": [ + { + "bbox": [ + 145, + 456, + 466, + 485 + ], + "spans": [ + { + "bbox": [ + 145, + 456, + 466, + 485 + ], + "score": 0.92, + "content": "x _ { t - 1 } = \\sqrt { \\bar { \\alpha } _ { t - 1 } } \\left( \\frac { x _ { t } - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon _ { \\theta } ( x _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t - 1 } - \\sigma _ { t } ^ { 2 } } \\epsilon _ { \\theta } ( x _ { t } , t ) + \\sigma _ { t } \\epsilon _ { t } .", + "type": "interline_equation", + "image_path": "f02079c70488e8c098c1e787f0af456fd5d016e5a3de26400fdf3aba98171010.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 145, + 456, + 466, + 465.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 145, + 465.6666666666667, + 466, + 475.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 145, + 475.33333333333337, + 466, + 485.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 131, + 502 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 132, + 491, + 142, + 501 + ], + "score": 0.86, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 489, + 299, + 502 + ], + "score": 1.0, + "content": "controls the ratio of random noise. If", + "type": "text" + }, + { + "bbox": [ + 300, + 491, + 311, + 501 + ], + "score": 0.86, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "equals one, Equation (8) represents the reverse", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 198, + 514 + ], + "score": 1.0, + "content": "process of DDPMs; if", + "type": "text" + }, + { + "bbox": [ + 198, + 502, + 208, + 512 + ], + "score": 0.85, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "equals zero, this equation represents the reverse process of DDIMs. And", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 153, + 524 + ], + "score": 1.0, + "content": "only when", + "type": "text" + }, + { + "bbox": [ + 153, + 513, + 164, + 523 + ], + "score": 0.83, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "equals zero, this equation removes the random item and becomes a discrete form", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "of a certain ODE. Theoretically, the numerical methods that can be used on differential equations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "with random items are limited. And Song et al. (2020b) have done enough research in this case.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "Empirically, Song et al. (2020a) have shown that DDIMs have a better acceleration effect when the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 443, + 568 + ], + "score": 1.0, + "content": "number of total steps is relatively small. Therefore, our work concentrate on the case", + "type": "text" + }, + { + "bbox": [ + 444, + 557, + 454, + 567 + ], + "score": 0.84, + "content": "\\sigma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "equals zero.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 489, + 505, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 378, + 585 + ], + "score": 1.0, + "content": "To find the corresponding ODE of Equation (8), we replace discrete", + "type": "text" + }, + { + "bbox": [ + 378, + 573, + 399, + 583 + ], + "score": 0.78, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "with a continuous version", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 130, + 594 + ], + "score": 0.88, + "content": "t - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "according to (Song et al., 2020a) and change this equation into a differential form, namely,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 594, + 284, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 140, + 607 + ], + "score": 1.0, + "content": "subtract", + "type": "text" + }, + { + "bbox": [ + 141, + 596, + 151, + 605 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 594, + 284, + 607 + ], + "score": 1.0, + "content": "from both sides of this equation:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 572, + 505, + 607 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 611, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 111, + 611, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 111, + 611, + 505, + 645 + ], + "score": 0.92, + "content": "\\mathfrak { z } _ { t - \\delta } - \\mathfrak { x } _ { t } = \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) \\left( \\frac { \\mathfrak { x } _ { t } } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { \\bar { \\alpha } _ { t - \\delta } } + \\sqrt { \\bar { \\alpha } _ { t } } \\right) } - \\frac { \\epsilon _ { \\theta } ( \\mathfrak { x } _ { t } , t ) } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } \\right) } \\right) .", + "type": "interline_equation", + "image_path": "577c3ec5876a9604c90387e3d534299c1ae1f71d2d18a024141de10a63208af1.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 111, + 611, + 505, + 622.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 111, + 622.3333333333334, + 505, + 633.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 111, + 633.6666666666667, + 505, + 645.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 142, + 666 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 142, + 654, + 148, + 664 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 653, + 285, + 666 + ], + "score": 1.0, + "content": "is a continuous variable from 0 to", + "type": "text" + }, + { + "bbox": [ + 285, + 655, + 289, + 664 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 653, + 504, + 666 + ], + "score": 1.0, + "content": ", we can now compute the derivative of the generation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 125, + 689 + ], + "score": 1.0, + "content": "data", + "type": "text" + }, + { + "bbox": [ + 126, + 673, + 136, + 682 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 671, + 187, + 689 + ], + "score": 1.0, + "content": "and get that", + "type": "text" + }, + { + "bbox": [ + 188, + 665, + 385, + 690 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\underset { \\delta 0 } { \\operatorname* { l i m } } \\frac { x _ { t } - x _ { t - \\delta } } { \\delta } = - \\bar { \\alpha } ^ { \\prime } ( t ) ( \\frac { x ( t ) } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x ( t ) , t ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 670, + 414, + 684 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 414, + 671, + 433, + 684 + ], + "score": 0.91, + "content": "\\bar { \\alpha } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "is the continuous", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 687, + 507, + 705 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 150, + 705 + ], + "score": 1.0, + "content": "version of", + "type": "text" + }, + { + "bbox": [ + 150, + 689, + 184, + 703 + ], + "score": 0.93, + "content": "\\{ \\bar { \\alpha } _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 687, + 271, + 705 + ], + "score": 1.0, + "content": "like the definition of", + "type": "text" + }, + { + "bbox": [ + 271, + 691, + 289, + 702 + ], + "score": 0.9, + "content": "x ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 687, + 463, + 705 + ], + "score": 1.0, + "content": ". Therefore, the corresponding ODE when", + "type": "text" + }, + { + "bbox": [ + 463, + 691, + 469, + 700 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 687, + 507, + 705 + ], + "score": 1.0, + "content": "tends to", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 653, + 507, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 200, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 202, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 202, + 96 + ], + "score": 1.0, + "content": "zero of Equation (9) is:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 100, + 399, + 133 + ], + "lines": [ + { + "bbox": [ + 212, + 100, + 399, + 133 + ], + "spans": [ + { + "bbox": [ + 212, + 100, + 399, + 133 + ], + "score": 0.94, + "content": "\\frac { d x } { d t } = - \\bar { \\alpha } ^ { \\prime } ( t ) \\left( \\frac { x ( t ) } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x ( t ) , t ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } \\right) .", + "type": "interline_equation", + "image_path": "b8e28e0672da3f0640c702d9463c85d624cbd79ee3f60d28c45f64be27d3b94d.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 100, + 399, + 116.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 212, + 116.5, + 399, + 133.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 146, + 278, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 279, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 279, + 159 + ], + "score": 1.0, + "content": "3.2 CLASSICAL NUMERICAL METHOD", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "After getting the target ODE, the easiest way to solve it is through classical numerical methods.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 179, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 504, + 190 + ], + "score": 1.0, + "content": "However, We notice that classical numerical methods can introduce noticeable noise at a high", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "speedup rate, making high-order numerical methods (e.g., Runge-Kutta method) even less effec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "tive than DDIMs. This phenomenon is also mentioned in Salimans & Ho (2022). To make better", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 504, + 223 + ], + "score": 1.0, + "content": "use of numerical methods, we analyze the differences between Equation (10) and usual differen-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "tial equations and find two main problems when we directly use numerical methods with diffusion", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 141, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 141, + 245 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 344, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 344, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 288, + 263 + ], + "score": 1.0, + "content": "The first problem is that the neural network", + "type": "text" + }, + { + "bbox": [ + 288, + 252, + 298, + 261 + ], + "score": 0.82, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 249, + 344, + 263 + ], + "score": 1.0, + "content": "and Equa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 260, + 344, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 344, + 273 + ], + "score": 1.0, + "content": "tion (10) are well-defined only in a limited area. Equation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 271, + 345, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 200, + 285 + ], + "score": 1.0, + "content": "(2) shows that the data", + "type": "text" + }, + { + "bbox": [ + 201, + 273, + 212, + 283 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 271, + 345, + 285 + ], + "score": 1.0, + "content": "is generated along a curve close", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 345, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 270, + 295 + ], + "score": 1.0, + "content": "to an arc. According to Figure 2, most of", + "type": "text" + }, + { + "bbox": [ + 271, + 284, + 281, + 294 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 284, + 345, + 295 + ], + "score": 1.0, + "content": "is concentrated", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 345, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 345, + 306 + ], + "score": 1.0, + "content": "in a band with a width of around 0.1, namely the red area", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 345, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 312, + 317 + ], + "score": 1.0, + "content": "in Figure 2. This means that the neural network", + "type": "text" + }, + { + "bbox": [ + 312, + 307, + 322, + 316 + ], + "score": 0.82, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 304, + 345, + 317 + ], + "score": 1.0, + "content": "can-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 345, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 345, + 329 + ], + "score": 1.0, + "content": "not get enough examples to fit the noise successfully away", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 327, + 344, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 210, + 339 + ], + "score": 1.0, + "content": "from this area. Therefore,", + "type": "text" + }, + { + "bbox": [ + 211, + 329, + 221, + 338 + ], + "score": 0.82, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 327, + 344, + 339 + ], + "score": 1.0, + "content": "and Equation (10), which con-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 338, + 345, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 128, + 350 + ], + "score": 1.0, + "content": "tains", + "type": "text" + }, + { + "bbox": [ + 128, + 340, + 138, + 349 + ], + "score": 0.84, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 338, + 345, + 350 + ], + "score": 1.0, + "content": ", are only well-defined in this limited area. How-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 345, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 345, + 362 + ], + "score": 1.0, + "content": "ever, all classical numerical methods generate results along", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 345, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 345, + 372 + ], + "score": 1.0, + "content": "a straight line instead of an arc. The generation process", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 345, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 345, + 382 + ], + "score": 1.0, + "content": "may generate samples away from the well-defined area and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 381, + 344, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 344, + 394 + ], + "score": 1.0, + "content": "then introduce new errors. In Section 4.3 we will give more", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 392, + 251, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 251, + 405 + ], + "score": 1.0, + "content": "visualization results to support this.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5 + }, + { + "type": "image", + "bbox": [ + 353, + 252, + 503, + 365 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 353, + 252, + 503, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 353, + 252, + 503, + 365 + ], + "spans": [ + { + "bbox": [ + 353, + 252, + 503, + 365 + ], + "score": 0.967, + "type": "image", + "image_path": "5d0a4adc964cc9dab17aeddee06c037ae60160c3540a2b0026d4c04a47a7047d.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 353, + 252, + 503, + 308.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 353, + 308.5, + 503, + 365.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 352, + 373, + 504, + 395 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 351, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 351, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "Figure 2: the density distribution of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 351, + 383, + 437, + 396 + ], + "spans": [ + { + "bbox": [ + 351, + 383, + 437, + 396 + ], + "score": 1.0, + "content": "the norm of the data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "The second problem is that Equation (10) is unbounded at most cases. We find that for most linear", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 182, + 433 + ], + "score": 1.0, + "content": "variance schedules", + "type": "text" + }, + { + "bbox": [ + 182, + 421, + 192, + 432 + ], + "score": 0.86, + "content": "\\beta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 421, + 339, + 433 + ], + "score": 1.0, + "content": ", Equation (10) tends to infinity when", + "type": "text" + }, + { + "bbox": [ + 339, + 421, + 344, + 431 + ], + "score": 0.52, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "tends to zero (see Appendix A.4), which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "does not satisfy the condition of numerical methods mentioned in Section 2.3. This is an apparent", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "theoretical weakness that previous works have not explored. On the contrary, in the original DDPMs", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 273, + 466 + ], + "score": 1.0, + "content": "and DDIMs, the prediction of the sample", + "type": "text" + }, + { + "bbox": [ + 273, + 455, + 284, + 464 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 453, + 340, + 466 + ], + "score": 1.0, + "content": "and the noise", + "type": "text" + }, + { + "bbox": [ + 340, + 455, + 351, + 464 + ], + "score": 0.84, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "in the data are more and more precise", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 156, + 477 + ], + "score": 1.0, + "content": "as the index", + "type": "text" + }, + { + "bbox": [ + 156, + 465, + 162, + 474 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "tends to zero (see Appendix A.5). This means original diffusion models do not make", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "a significant error in the last several steps, whereas using numerical methods on Equation (10) does.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 486, + 408, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 408, + 498 + ], + "score": 1.0, + "content": "This explains why DDIMs are better than higher-order numerical methods.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 512, + 329, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 330, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 330, + 525 + ], + "score": 1.0, + "content": "3.3 PSEUDO NUMERICAL METHOD ON MANIFOLD", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "The first problem above shows that we should try to solve our problems on certain manifolds.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 360, + 557 + ], + "score": 1.0, + "content": "Here, target manifolds are the high-density region of the data √", + "type": "text" + }, + { + "bbox": [ + 360, + 546, + 370, + 555 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "of DDPMs, which is defined by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 298, + 567 + ], + "score": 0.77, + "content": "x _ { t } ( x _ { 0 } , \\epsilon ) \\overset { } { = } \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon , \\epsilon \\sim \\bar { \\mathcal { N } } ( 0 , \\bar { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 554, + 506, + 568 + ], + "score": 1.0, + "content": ". Ernst & Gerhard (1996) show several numerical", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "methods to solve differential equations on manifolds that have analytic expressions. Unfortunately,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 482, + 590 + ], + "score": 1.0, + "content": "it’s challenging to use the above expression of manifolds. Because we do not know the target", + "type": "text" + }, + { + "bbox": [ + 482, + 578, + 493, + 588 + ], + "score": 0.84, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 588, + 359, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 258, + 600 + ], + "score": 1.0, + "content": "the reverse process and random items", + "type": "text" + }, + { + "bbox": [ + 258, + 590, + 264, + 598 + ], + "score": 0.73, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 588, + 359, + 600 + ], + "score": 1.0, + "content": "are hard to handle, too.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "In this paper, we design a different way that we make our new equation of denoising process more", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "fits with the equation of original DDIMs to make their results share similar data distribution. Firstly,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "we divide the classical numerical methods into two parts: gradient and transfer parts. The gradient", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "part determines the gradient at each step, while the transfer part generates the result at the next step.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 102, + 644, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 102, + 644, + 402, + 667 + ], + "score": 1.0, + "content": "For example, linear multi-step method can be divided into the gradient part", + "type": "text" + }, + { + "bbox": [ + 403, + 648, + 505, + 662 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ^ { \\prime } = \\frac { \\delta } { 2 4 } ( 5 5 f _ { t } - 5 9 f _ { t - \\delta } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 107, + 660, + 183, + 672 + ], + "score": 0.91, + "content": "3 7 f _ { t - 2 \\delta } - 9 f _ { t - 3 \\delta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 659, + 267, + 673 + ], + "score": 1.0, + "content": "and the transfer part", + "type": "text" + }, + { + "bbox": [ + 267, + 660, + 335, + 672 + ], + "score": 0.93, + "content": "x _ { t + \\delta } = x _ { t } + \\delta f ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". All classical numerical methods have the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 672, + 343, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 343, + 683 + ], + "score": 1.0, + "content": "same linear transfer part, while gradient parts are different.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "score": 1.0, + "content": "We define those numerical methods which use a nonlinear transfer part as pseudo numerical meth-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "ods. And an expected transfer part should have the property that when the result from the gradient", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "part is precise, then the result of the transfer part is as close to the manifold as possible and the error", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 719, + 448, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 448, + 734 + ], + "score": 1.0, + "content": "of this result is as small as possible. We find that Equation (9) satisfies such property.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 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 2022", + "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, + 200, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 202, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 202, + 96 + ], + "score": 1.0, + "content": "zero of Equation (9) is:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 202, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 100, + 399, + 133 + ], + "lines": [ + { + "bbox": [ + 212, + 100, + 399, + 133 + ], + "spans": [ + { + "bbox": [ + 212, + 100, + 399, + 133 + ], + "score": 0.94, + "content": "\\frac { d x } { d t } = - \\bar { \\alpha } ^ { \\prime } ( t ) \\left( \\frac { x ( t ) } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x ( t ) , t ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } \\right) .", + "type": "interline_equation", + "image_path": "b8e28e0672da3f0640c702d9463c85d624cbd79ee3f60d28c45f64be27d3b94d.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 100, + 399, + 116.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 212, + 116.5, + 399, + 133.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 146, + 278, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 279, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 279, + 159 + ], + "score": 1.0, + "content": "3.2 CLASSICAL NUMERICAL METHOD", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "After getting the target ODE, the easiest way to solve it is through classical numerical methods.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 179, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 504, + 190 + ], + "score": 1.0, + "content": "However, We notice that classical numerical methods can introduce noticeable noise at a high", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "speedup rate, making high-order numerical methods (e.g., Runge-Kutta method) even less effec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "tive than DDIMs. This phenomenon is also mentioned in Salimans & Ho (2022). To make better", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 504, + 223 + ], + "score": 1.0, + "content": "use of numerical methods, we analyze the differences between Equation (10) and usual differen-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "tial equations and find two main problems when we directly use numerical methods with diffusion", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 141, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 141, + 245 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 167, + 506, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 344, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 344, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 288, + 263 + ], + "score": 1.0, + "content": "The first problem is that the neural network", + "type": "text" + }, + { + "bbox": [ + 288, + 252, + 298, + 261 + ], + "score": 0.82, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 249, + 344, + 263 + ], + "score": 1.0, + "content": "and Equa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 260, + 344, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 344, + 273 + ], + "score": 1.0, + "content": "tion (10) are well-defined only in a limited area. Equation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 271, + 345, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 200, + 285 + ], + "score": 1.0, + "content": "(2) shows that the data", + "type": "text" + }, + { + "bbox": [ + 201, + 273, + 212, + 283 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 271, + 345, + 285 + ], + "score": 1.0, + "content": "is generated along a curve close", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 284, + 345, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 270, + 295 + ], + "score": 1.0, + "content": "to an arc. According to Figure 2, most of", + "type": "text" + }, + { + "bbox": [ + 271, + 284, + 281, + 294 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 284, + 345, + 295 + ], + "score": 1.0, + "content": "is concentrated", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 345, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 345, + 306 + ], + "score": 1.0, + "content": "in a band with a width of around 0.1, namely the red area", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 345, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 312, + 317 + ], + "score": 1.0, + "content": "in Figure 2. This means that the neural network", + "type": "text" + }, + { + "bbox": [ + 312, + 307, + 322, + 316 + ], + "score": 0.82, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 304, + 345, + 317 + ], + "score": 1.0, + "content": "can-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 345, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 345, + 329 + ], + "score": 1.0, + "content": "not get enough examples to fit the noise successfully away", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 327, + 344, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 210, + 339 + ], + "score": 1.0, + "content": "from this area. Therefore,", + "type": "text" + }, + { + "bbox": [ + 211, + 329, + 221, + 338 + ], + "score": 0.82, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 327, + 344, + 339 + ], + "score": 1.0, + "content": "and Equation (10), which con-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 338, + 345, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 128, + 350 + ], + "score": 1.0, + "content": "tains", + "type": "text" + }, + { + "bbox": [ + 128, + 340, + 138, + 349 + ], + "score": 0.84, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 338, + 345, + 350 + ], + "score": 1.0, + "content": ", are only well-defined in this limited area. How-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 345, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 345, + 362 + ], + "score": 1.0, + "content": "ever, all classical numerical methods generate results along", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 345, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 345, + 372 + ], + "score": 1.0, + "content": "a straight line instead of an arc. The generation process", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 345, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 345, + 382 + ], + "score": 1.0, + "content": "may generate samples away from the well-defined area and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 381, + 344, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 344, + 394 + ], + "score": 1.0, + "content": "then introduce new errors. In Section 4.3 we will give more", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 392, + 251, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 251, + 405 + ], + "score": 1.0, + "content": "visualization results to support this.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 249, + 345, + 405 + ] + }, + { + "type": "image", + "bbox": [ + 353, + 252, + 503, + 365 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 353, + 252, + 503, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 353, + 252, + 503, + 365 + ], + "spans": [ + { + "bbox": [ + 353, + 252, + 503, + 365 + ], + "score": 0.967, + "type": "image", + "image_path": "5d0a4adc964cc9dab17aeddee06c037ae60160c3540a2b0026d4c04a47a7047d.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 353, + 252, + 503, + 308.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 353, + 308.5, + 503, + 365.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 352, + 373, + 504, + 395 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 351, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 351, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "Figure 2: the density distribution of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 351, + 383, + 437, + 396 + ], + "spans": [ + { + "bbox": [ + 351, + 383, + 437, + 396 + ], + "score": 1.0, + "content": "the norm of the data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "The second problem is that Equation (10) is unbounded at most cases. We find that for most linear", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 182, + 433 + ], + "score": 1.0, + "content": "variance schedules", + "type": "text" + }, + { + "bbox": [ + 182, + 421, + 192, + 432 + ], + "score": 0.86, + "content": "\\beta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 421, + 339, + 433 + ], + "score": 1.0, + "content": ", Equation (10) tends to infinity when", + "type": "text" + }, + { + "bbox": [ + 339, + 421, + 344, + 431 + ], + "score": 0.52, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "tends to zero (see Appendix A.4), which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "does not satisfy the condition of numerical methods mentioned in Section 2.3. This is an apparent", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "theoretical weakness that previous works have not explored. On the contrary, in the original DDPMs", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 273, + 466 + ], + "score": 1.0, + "content": "and DDIMs, the prediction of the sample", + "type": "text" + }, + { + "bbox": [ + 273, + 455, + 284, + 464 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 453, + 340, + 466 + ], + "score": 1.0, + "content": "and the noise", + "type": "text" + }, + { + "bbox": [ + 340, + 455, + 351, + 464 + ], + "score": 0.84, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "in the data are more and more precise", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 156, + 477 + ], + "score": 1.0, + "content": "as the index", + "type": "text" + }, + { + "bbox": [ + 156, + 465, + 162, + 474 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "tends to zero (see Appendix A.5). This means original diffusion models do not make", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "a significant error in the last several steps, whereas using numerical methods on Equation (10) does.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 486, + 408, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 408, + 498 + ], + "score": 1.0, + "content": "This explains why DDIMs are better than higher-order numerical methods.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 410, + 505, + 498 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 512, + 329, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 330, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 330, + 525 + ], + "score": 1.0, + "content": "3.3 PSEUDO NUMERICAL METHOD ON MANIFOLD", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "The first problem above shows that we should try to solve our problems on certain manifolds.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 360, + 557 + ], + "score": 1.0, + "content": "Here, target manifolds are the high-density region of the data √", + "type": "text" + }, + { + "bbox": [ + 360, + 546, + 370, + 555 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "of DDPMs, which is defined by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 298, + 567 + ], + "score": 0.77, + "content": "x _ { t } ( x _ { 0 } , \\epsilon ) \\overset { } { = } \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon , \\epsilon \\sim \\bar { \\mathcal { N } } ( 0 , \\bar { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 554, + 506, + 568 + ], + "score": 1.0, + "content": ". Ernst & Gerhard (1996) show several numerical", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "methods to solve differential equations on manifolds that have analytic expressions. Unfortunately,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 482, + 590 + ], + "score": 1.0, + "content": "it’s challenging to use the above expression of manifolds. Because we do not know the target", + "type": "text" + }, + { + "bbox": [ + 482, + 578, + 493, + 588 + ], + "score": 0.84, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 588, + 359, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 258, + 600 + ], + "score": 1.0, + "content": "the reverse process and random items", + "type": "text" + }, + { + "bbox": [ + 258, + 590, + 264, + 598 + ], + "score": 0.73, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 588, + 359, + 600 + ], + "score": 1.0, + "content": "are hard to handle, too.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 533, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "In this paper, we design a different way that we make our new equation of denoising process more", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "fits with the equation of original DDIMs to make their results share similar data distribution. Firstly,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "we divide the classical numerical methods into two parts: gradient and transfer parts. The gradient", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "part determines the gradient at each step, while the transfer part generates the result at the next step.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 102, + 644, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 102, + 644, + 402, + 667 + ], + "score": 1.0, + "content": "For example, linear multi-step method can be divided into the gradient part", + "type": "text" + }, + { + "bbox": [ + 403, + 648, + 505, + 662 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ^ { \\prime } = \\frac { \\delta } { 2 4 } ( 5 5 f _ { t } - 5 9 f _ { t - \\delta } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 107, + 660, + 183, + 672 + ], + "score": 0.91, + "content": "3 7 f _ { t - 2 \\delta } - 9 f _ { t - 3 \\delta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 659, + 267, + 673 + ], + "score": 1.0, + "content": "and the transfer part", + "type": "text" + }, + { + "bbox": [ + 267, + 660, + 335, + 672 + ], + "score": 0.93, + "content": "x _ { t + \\delta } = x _ { t } + \\delta f ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". All classical numerical methods have the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 672, + 343, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 343, + 683 + ], + "score": 1.0, + "content": "same linear transfer part, while gradient parts are different.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47, + "bbox_fs": [ + 102, + 605, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "score": 1.0, + "content": "We define those numerical methods which use a nonlinear transfer part as pseudo numerical meth-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "ods. And an expected transfer part should have the property that when the result from the gradient", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "part is precise, then the result of the transfer part is as close to the manifold as possible and the error", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 719, + 448, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 448, + 734 + ], + "score": 1.0, + "content": "of this result is as small as possible. We find that Equation (9) satisfies such property.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 266, + 96 + ], + "score": 1.0, + "content": "Property 3.1 If ϵ is the precise noise in", + "type": "text" + }, + { + "bbox": [ + 266, + 84, + 277, + 93 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 81, + 349, + 96 + ], + "score": 1.0, + "content": ", then the result of", + "type": "text" + }, + { + "bbox": [ + 349, + 83, + 369, + 94 + ], + "score": 0.88, + "content": "x _ { t - \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "from Equation (9) is also precise.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 409, + 114 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 407, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 407, + 115 + ], + "score": 1.0, + "content": "And we put the proof of this property in Appendix A.5. Therefore, we use:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 126, + 463, + 155 + ], + "lines": [ + { + "bbox": [ + 149, + 126, + 463, + 155 + ], + "spans": [ + { + "bbox": [ + 149, + 126, + 463, + 155 + ], + "score": 0.94, + "content": "\\phi ( x _ { t } , \\epsilon _ { t } , t , t - \\delta ) = \\frac { \\sqrt { \\bar { \\alpha } _ { t - \\delta } } } { \\sqrt { \\bar { \\alpha } _ { t } } } x _ { t } - \\frac { \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) } { \\sqrt { \\bar { \\alpha } _ { t } } ( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } ) } \\epsilon _ { t }", + "type": "interline_equation", + "image_path": "0505272c5972520578859285aff50d4ab192f035bd95522592e4f5b6ce879226.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 149, + 126, + 463, + 135.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 149, + 135.66666666666666, + 463, + 145.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 149, + 145.33333333333331, + 463, + 154.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 504, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 199, + 174 + ], + "score": 1.0, + "content": "as the transfer part and", + "type": "text" + }, + { + "bbox": [ + 199, + 163, + 209, + 172 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 160, + 320, + 174 + ], + "score": 1.0, + "content": "as the gradient part. That if", + "type": "text" + }, + { + "bbox": [ + 320, + 163, + 330, + 172 + ], + "score": 0.86, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 160, + 423, + 174 + ], + "score": 1.0, + "content": "is precise, the result of", + "type": "text" + }, + { + "bbox": [ + 423, + 163, + 443, + 172 + ], + "score": 0.88, + "content": "x _ { t - \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "is also precise,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 177, + 184 + ], + "score": 1.0, + "content": "which means that", + "type": "text" + }, + { + "bbox": [ + 177, + 173, + 187, + 183 + ], + "score": 0.86, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "can determine the direction of the denoising process to generate the final results.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "Therefore, such a choice also satisfies the definition of a gradient part. Now, we have our gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 214, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 124, + 206 + ], + "score": 1.0, + "content": "part", + "type": "text" + }, + { + "bbox": [ + 124, + 195, + 134, + 205 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 194, + 203, + 206 + ], + "score": 1.0, + "content": "and transfer part", + "type": "text" + }, + { + "bbox": [ + 203, + 194, + 210, + 205 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 194, + 214, + 206 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "This combination solves the two problems mentioned above successfully. Firstly, our new transfer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "parts do not introduce new errors. This property also means that it keeps the results at the next step", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "on the target manifold because generating samples away is a kind of error. This shows that we solve", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 339, + 257 + ], + "score": 1.0, + "content": "the first problem. Secondly, we know that the prediction of", + "type": "text" + }, + { + "bbox": [ + 339, + 245, + 349, + 255 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "is more and more precise in the reverse", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "process in the above subsection. And our new transfer part can generate precise results according", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 218, + 279 + ], + "score": 1.0, + "content": "to the precise prediction of", + "type": "text" + }, + { + "bbox": [ + 219, + 267, + 228, + 277 + ], + "score": 0.83, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 266, + 505, + 279 + ], + "score": 1.0, + "content": ". Therefore, our generation results are more and more precise using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 288 + ], + "score": 1.0, + "content": "pseudo numerical methods, while classical numerical methods can introduce obvious error at the last", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "several steps. This shows that we solve the second problem, too. We also find that their combination", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 107, + 298, + 202, + 311 + ], + "score": 0.91, + "content": "\\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } , t ) , t , t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "is just the inference equation used by DDIMs, so DDIMs is a simple case", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "of pseudo numerical methods. Here, we define DDIMs as DDIMs*, emphasizing that it is a pseudo", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 183, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 183, + 331 + ], + "score": 1.0, + "content": "numerical method.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 344, + 203, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 205, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 205, + 358 + ], + "score": 1.0, + "content": "3.4 GRADIENT PART", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "score": 1.0, + "content": "Because we split numerical methods into two parts, we can use the same gradient part from different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "classical numerical methods freely (e.g., linear multi-step method), although we change the transfer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "part of our inference equation. Our theoretical analyses and experiments show that the gradient part", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 412 + ], + "score": 1.0, + "content": "from different classical methods can work well with our new transfer part (see Section 3.6, 4.2). By", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 388, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 388, + 422 + ], + "score": 1.0, + "content": "using the same gradient part of the linear multi-step method, we have:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 428, + 317, + 471 + ], + "lines": [ + { + "bbox": [ + 112, + 428, + 317, + 471 + ], + "spans": [ + { + "bbox": [ + 112, + 428, + 317, + 471 + ], + "score": 0.91, + "content": "\\left\\{ \\begin{array} { l l } & { e _ { t } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { e _ { t } ^ { \\prime } = \\frac { 1 } { 2 4 } ( 5 5 e _ { t } - 5 9 e _ { t - \\delta } + 3 7 e _ { t - 2 \\delta } - 9 e _ { t - 3 \\delta } ) } \\\\ & { x _ { t + \\delta } = \\phi ( x _ { t } , e _ { t } ^ { \\prime } , t , t + \\delta ) . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "1588a7d1247e616e51fcf13ac73292868a0dd72cfb3203e0977ccf70dbeacb20.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 112, + 428, + 317, + 442.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 112, + 442.3333333333333, + 317, + 456.66666666666663 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 112, + 456.66666666666663, + 317, + 470.99999999999994 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 322, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 322, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 322, + 492 + ], + "score": 1.0, + "content": "By using the same gradient part of Runge-Kutta", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 490, + 180, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 180, + 502 + ], + "score": 1.0, + "content": "method, we have:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 328, + 432, + 414, + 443 + ], + "lines": [ + { + "bbox": [ + 327, + 431, + 415, + 446 + ], + "spans": [ + { + "bbox": [ + 327, + 431, + 415, + 446 + ], + "score": 1.0, + "content": "Algorithm 1 DDIMs", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 330, + 446, + 498, + 499 + ], + "lines": [ + { + "bbox": [ + 331, + 447, + 397, + 459 + ], + "spans": [ + { + "bbox": [ + 331, + 447, + 343, + 459 + ], + "score": 1.0, + "content": "1:", + "type": "text" + }, + { + "bbox": [ + 343, + 448, + 397, + 459 + ], + "score": 0.79, + "content": "x _ { T } \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 330, + 456, + 445, + 469 + ], + "spans": [ + { + "bbox": [ + 330, + 456, + 356, + 469 + ], + "score": 1.0, + "content": "2: for", + "type": "text" + }, + { + "bbox": [ + 356, + 459, + 431, + 468 + ], + "score": 0.72, + "content": "t = T - 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Propagating forward four times along a neural", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "network makes the denoising process slower. However, we find that the linear multi-step method can", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 178, + 711 + ], + "score": 1.0, + "content": "reuse the result of", + "type": "text" + }, + { + "bbox": [ + 179, + 700, + 188, + 709 + ], + "score": 0.83, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 699, + 304, + 711 + ], + "score": 1.0, + "content": "four times and only compute", + "type": "text" + }, + { + "bbox": [ + 304, + 700, + 314, + 709 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "once at every step. And theoretical analyses tell", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "us that the Runge-Kutta and linear multi-step method have the same convergence order and similar", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 104, + 721, + 137, + 731 + ], + "spans": [ + { + "bbox": [ + 104, + 721, + 137, + 731 + ], + "score": 1.0, + "content": "results.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 53 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 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 2022", + "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": "text", + "bbox": [ + 107, + 82, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 266, + 96 + ], + "score": 1.0, + "content": "Property 3.1 If ϵ is the precise noise in", + "type": "text" + }, + { + "bbox": [ + 266, + 84, + 277, + 93 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 81, + 349, + 96 + ], + "score": 1.0, + "content": ", then the result of", + "type": "text" + }, + { + "bbox": [ + 349, + 83, + 369, + 94 + ], + "score": 0.88, + "content": "x _ { t - \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "from Equation (9) is also precise.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 505, + 96 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 409, + 114 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 407, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 407, + 115 + ], + "score": 1.0, + "content": "And we put the proof of this property in Appendix A.5. Therefore, we use:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 101, + 407, + 115 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 126, + 463, + 155 + ], + "lines": [ + { + "bbox": [ + 149, + 126, + 463, + 155 + ], + "spans": [ + { + "bbox": [ + 149, + 126, + 463, + 155 + ], + "score": 0.94, + "content": "\\phi ( x _ { t } , \\epsilon _ { t } , t , t - \\delta ) = \\frac { \\sqrt { \\bar { \\alpha } _ { t - \\delta } } } { \\sqrt { \\bar { \\alpha } _ { t } } } x _ { t } - \\frac { \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) } { \\sqrt { \\bar { \\alpha } _ { t } } ( \\sqrt { ( 1 - \\bar { \\alpha } _ { t - \\delta } ) \\bar { \\alpha } _ { t } } + \\sqrt { ( 1 - \\bar { \\alpha } _ { t } ) \\bar { \\alpha } _ { t - \\delta } } ) } \\epsilon _ { t }", + "type": "interline_equation", + "image_path": "0505272c5972520578859285aff50d4ab192f035bd95522592e4f5b6ce879226.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 149, + 126, + 463, + 135.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 149, + 135.66666666666666, + 463, + 145.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 149, + 145.33333333333331, + 463, + 154.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 504, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 199, + 174 + ], + "score": 1.0, + "content": "as the transfer part and", + "type": "text" + }, + { + "bbox": [ + 199, + 163, + 209, + 172 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 160, + 320, + 174 + ], + "score": 1.0, + "content": "as the gradient part. That if", + "type": "text" + }, + { + "bbox": [ + 320, + 163, + 330, + 172 + ], + "score": 0.86, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 160, + 423, + 174 + ], + "score": 1.0, + "content": "is precise, the result of", + "type": "text" + }, + { + "bbox": [ + 423, + 163, + 443, + 172 + ], + "score": 0.88, + "content": "x _ { t - \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "is also precise,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 177, + 184 + ], + "score": 1.0, + "content": "which means that", + "type": "text" + }, + { + "bbox": [ + 177, + 173, + 187, + 183 + ], + "score": 0.86, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "can determine the direction of the denoising process to generate the final results.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "Therefore, such a choice also satisfies the definition of a gradient part. Now, we have our gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 214, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 124, + 206 + ], + "score": 1.0, + "content": "part", + "type": "text" + }, + { + "bbox": [ + 124, + 195, + 134, + 205 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 194, + 203, + 206 + ], + "score": 1.0, + "content": "and transfer part", + "type": "text" + }, + { + "bbox": [ + 203, + 194, + 210, + 205 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 194, + 214, + 206 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 160, + 506, + 206 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "This combination solves the two problems mentioned above successfully. Firstly, our new transfer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "parts do not introduce new errors. This property also means that it keeps the results at the next step", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "on the target manifold because generating samples away is a kind of error. This shows that we solve", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 339, + 257 + ], + "score": 1.0, + "content": "the first problem. Secondly, we know that the prediction of", + "type": "text" + }, + { + "bbox": [ + 339, + 245, + 349, + 255 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "is more and more precise in the reverse", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "process in the above subsection. And our new transfer part can generate precise results according", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 218, + 279 + ], + "score": 1.0, + "content": "to the precise prediction of", + "type": "text" + }, + { + "bbox": [ + 219, + 267, + 228, + 277 + ], + "score": 0.83, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 266, + 505, + 279 + ], + "score": 1.0, + "content": ". Therefore, our generation results are more and more precise using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 288 + ], + "score": 1.0, + "content": "pseudo numerical methods, while classical numerical methods can introduce obvious error at the last", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "several steps. This shows that we solve the second problem, too. We also find that their combination", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 107, + 298, + 202, + 311 + ], + "score": 0.91, + "content": "\\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } , t ) , t , t - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "is just the inference equation used by DDIMs, so DDIMs is a simple case", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "of pseudo numerical methods. Here, we define DDIMs as DDIMs*, emphasizing that it is a pseudo", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 183, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 183, + 331 + ], + "score": 1.0, + "content": "numerical method.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 210, + 506, + 331 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 344, + 203, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 205, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 205, + 358 + ], + "score": 1.0, + "content": "3.4 GRADIENT PART", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "score": 1.0, + "content": "Because we split numerical methods into two parts, we can use the same gradient part from different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "classical numerical methods freely (e.g., linear multi-step method), although we change the transfer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "part of our inference equation. Our theoretical analyses and experiments show that the gradient part", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 412 + ], + "score": 1.0, + "content": "from different classical methods can work well with our new transfer part (see Section 3.6, 4.2). By", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 388, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 388, + 422 + ], + "score": 1.0, + "content": "using the same gradient part of the linear multi-step method, we have:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 365, + 506, + 422 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 428, + 317, + 471 + ], + "lines": [ + { + "bbox": [ + 112, + 428, + 317, + 471 + ], + "spans": [ + { + "bbox": [ + 112, + 428, + 317, + 471 + ], + "score": 0.91, + "content": "\\left\\{ \\begin{array} { l l } & { e _ { t } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { e _ { t } ^ { \\prime } = \\frac { 1 } { 2 4 } ( 5 5 e _ { t } - 5 9 e _ { t - \\delta } + 3 7 e _ { t - 2 \\delta } - 9 e _ { t - 3 \\delta } ) } \\\\ & { x _ { t + \\delta } = \\phi ( x _ { t } , e _ { t } ^ { \\prime } , t , t + \\delta ) . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "1588a7d1247e616e51fcf13ac73292868a0dd72cfb3203e0977ccf70dbeacb20.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 112, + 428, + 317, + 442.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 112, + 442.3333333333333, + 317, + 456.66666666666663 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 112, + 456.66666666666663, + 317, + 470.99999999999994 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 322, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 322, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 322, + 492 + ], + "score": 1.0, + "content": "By using the same gradient part of Runge-Kutta", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 490, + 180, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 180, + 502 + ], + "score": 1.0, + "content": "method, we have:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 479, + 322, + 502 + ] + }, + { + "type": "title", + "bbox": [ + 328, + 432, + 414, + 443 + ], + "lines": [ + { + "bbox": [ + 327, + 431, + 415, + 446 + ], + "spans": [ + { + "bbox": [ + 327, + 431, + 415, + 446 + ], + "score": 1.0, + "content": "Algorithm 1 DDIMs", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "index", + "bbox": [ + 330, + 446, + 498, + 499 + ], + "lines": [ + { + "bbox": [ + 331, + 447, + 397, + 459 + ], + "spans": [ + { + "bbox": [ + 331, + 447, + 343, + 459 + ], + "score": 1.0, + "content": "1:", + "type": "text" + }, + { + "bbox": [ + 343, + 448, + 397, + 459 + ], + "score": 0.79, + "content": "x _ { T } \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 330, + 456, + 445, + 469 + ], + "spans": [ + { + "bbox": [ + 330, + 456, + 356, + 469 + ], + "score": 1.0, + "content": "2: for", + "type": "text" + }, + { + "bbox": [ + 356, + 459, + 431, + 468 + ], + "score": 0.72, + "content": "t = T - 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According to Song", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "et al. (2020a), the algorithm of the original method satisfies Algorithm 1. And our new algorithm of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "DNPMs uses the pseudo linear multi-step and pseudo Runge-Kutta method, which satisfies Algo-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "rithm 2. Here, we cannot use linear multi-step initially because the linear multi-step method cannot", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "start automatically, which needs at least three previous steps’ information to generate results. So", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "we use the Runge-Kutta method to compute the first three steps’ results and then use the linear", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 281, + 290, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 290, + 295 + ], + "score": 1.0, + "content": "multi-step method to calculate the remaining.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "We also use the gradient parts of two second-order numerical methods to get another pseudo nu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "merical method. We introduce the details of this method in Appendix A.3. We call it S-PNDMs,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "because its gradient part uses information from two steps at every step. Similarly, we also call our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 331, + 474, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 474, + 344 + ], + "score": 1.0, + "content": "first PNDMs F-PNDMs, which use data from four steps, when we need to distinguish them.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 231, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 233, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 233, + 370 + ], + "score": 1.0, + "content": "3.6 CONVERGENCE ORDER", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 389 + ], + "score": 1.0, + "content": "Change the transfer part of numerical methods may introduce unknown error. To determine the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "influence of our new transfer part theoretically, we compute the local and global error between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 243, + 412 + ], + "score": 1.0, + "content": "theoretical result of Equation (10)", + "type": "text" + }, + { + "bbox": [ + 243, + 399, + 276, + 411 + ], + "score": 0.91, + "content": "x ( t + \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 398, + 416, + 412 + ], + "score": 1.0, + "content": "and our new methods, we find that", + "type": "text" + }, + { + "bbox": [ + 416, + 399, + 505, + 411 + ], + "score": 0.91, + "content": "x ( t + \\delta ) - x _ { \\mathrm { D D I M } } ( x +", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 410, + 174, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 154, + 422 + ], + "score": 0.93, + "content": "\\delta ) = O ( \\delta ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 410, + 174, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 420, + 384, + 435 + ], + "lines": [ + { + "bbox": [ + 226, + 420, + 384, + 435 + ], + "spans": [ + { + "bbox": [ + 226, + 420, + 384, + 435 + ], + "score": 0.9, + "content": "x ( t + \\delta ) - x _ { \\mathrm { S / F - P N D M } } ( x + \\delta ) = O ( \\delta ^ { 3 } ) .", + "type": "interline_equation", + "image_path": "e92c1130506a0fe0d87ac0b98b06802b20ec41a0334893283ac0931d2035665c.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 226, + 420, + 384, + 435 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 104, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 356, + 451 + ], + "score": 1.0, + "content": "If the target ODE satisfies Lipschitz condition and local error", + "type": "text" + }, + { + "bbox": [ + 356, + 437, + 415, + 449 + ], + "score": 0.9, + "content": "e _ { \\mathrm { l o c a l } } = O ( \\delta ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 437, + 477, + 451 + ], + "score": 1.0, + "content": ", then there are", + "type": "text" + }, + { + "bbox": [ + 478, + 438, + 487, + 448 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 447, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 114, + 459 + ], + "score": 0.76, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 447, + 218, + 462 + ], + "score": 1.0, + "content": "such that the global error", + "type": "text" + }, + { + "bbox": [ + 218, + 450, + 243, + 461 + ], + "score": 0.9, + "content": "e _ { \\mathrm { g l o b a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 447, + 278, + 462 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 278, + 448, + 465, + 461 + ], + "score": 0.89, + "content": "e _ { \\mathrm { g l o b a l } } \\leq C \\delta ^ { k } ( 1 + e ^ { h } + e ^ { 2 h } + \\dot { \\cdot } \\dot { \\cdot } \\dot { \\cdot } ) \\leq C ^ { \\prime } \\delta ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 447, + 507, + 462 + ], + "score": 1.0, + "content": ". And we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "have that the convergence order is equal to the order of the global error. The detailed proof can be", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 470, + 372, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 372, + 484 + ], + "score": 1.0, + "content": "found in Appendix A.6. Therefore, we get the following property:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 105, + 492, + 464, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 466, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 466, + 505 + ], + "score": 1.0, + "content": "Property 3.2 S/F-PNDMs have third-order local error and are second-order convergent.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 194, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 196, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 196, + 534 + ], + "score": 1.0, + "content": "4 EXPERIMENT", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 107, + 544, + 160, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 161, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 161, + 557 + ], + "score": 1.0, + "content": "4.1 SETUP", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 463, + 578 + ], + "score": 1.0, + "content": "We conduct unconditional image generation experiments on four datasets: Cifar10", + "type": "text" + }, + { + "bbox": [ + 463, + 565, + 502, + 576 + ], + "score": 0.83, + "content": "( 3 2 \\times 3 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 253, + 588 + ], + "score": 1.0, + "content": "(Krizhevsky et al., 2009), CelebA", + "type": "text" + }, + { + "bbox": [ + 253, + 576, + 292, + 587 + ], + "score": 0.85, + "content": "( 6 4 \\times 6 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 576, + 435, + 588 + ], + "score": 1.0, + "content": "(Liu et al., 2015), LSUN-church", + "type": "text" + }, + { + "bbox": [ + 436, + 577, + 484, + 587 + ], + "score": 0.87, + "content": "( 2 5 6 \\times 2 5 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 174, + 599 + ], + "score": 1.0, + "content": "LSUN-bedroom", + "type": "text" + }, + { + "bbox": [ + 175, + 587, + 222, + 598 + ], + "score": 0.87, + "content": "( 2 5 6 \\times 2 5 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 588, + 506, + 599 + ], + "score": 1.0, + "content": ") (Yu et al., 2016). According to the analysis in Section 3.1, we can use", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "pre-trained models from prior works in our experiments. The pre-trained models for Cifar10, LSUN-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "church and LSUN-bedroom are taken from Ho et al. (2020) and the pre-trained model for CelebA is", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "taken from Song et al. (2020a). In these models, the number of total steps N is 1000 and the variance", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 631, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 643 + ], + "score": 1.0, + "content": "schedule is linear variance schedule. And we also use a pre-trained model for Cifar10, which uses a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "cosine variance schedule from improved denoising diffusion probabilistic models (iDDPMs (Nichol", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 652, + 189, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 189, + 665 + ], + "score": 1.0, + "content": "& Dhariwal, 2021)).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44 + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 283, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 285, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 285, + 690 + ], + "score": 1.0, + "content": "4.2 SAMPLE EFFICIENCY AND QUALITY", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "To analyze the acceleration effect, we test Fenchel Inception Distance (FID (Heusel et al., 2018))", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "on different datasets under different steps and different numerical methods, including DDIMs,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "S-PNDMs, F-PNDMs and classical fourth-order numerical methods (FONs) (e.g., Runge-Kutta", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 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, + 290, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 290, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 290, + 94 + ], + "score": 1.0, + "content": "Therefore, we use the gradient part of the lin-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 291, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 291, + 105 + ], + "score": 1.0, + "content": "ear multi-step method and our new transfer", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 290, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 290, + 115 + ], + "score": 1.0, + "content": "part as our main pseudo numerical methods", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 291, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 291, + 127 + ], + "score": 1.0, + "content": "for diffusion models (PNDMs). In Table 1, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 291, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 291, + 137 + ], + "score": 1.0, + "content": "show the relationship between different nu-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 291, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 291, + 148 + ], + "score": 1.0, + "content": "merical methods. Here, we can see PNDMs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 290, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 290, + 159 + ], + "score": 1.0, + "content": "combine the benefits of higher-order classical", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 291, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 291, + 171 + ], + "score": 1.0, + "content": "numerical methods (in the gradient part) and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 225, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 225, + 183 + ], + "score": 1.0, + "content": "DDIMs (in the transfer part).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 291, + 183 + ] + }, + { + "type": "table", + "bbox": [ + 298, + 84, + 501, + 137 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 298, + 84, + 501, + 137 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 298, + 84, + 501, + 137 + ], + "spans": [ + { + "bbox": [ + 298, + 84, + 501, + 137 + ], + "score": 0.974, + "html": "
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", + "type": "table", + "image_path": "e2e05a3016a399de590fee2dda6265c4358cabe0f92622bba3cbeb6b75e1bca5.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 298, + 84, + 501, + 97.25 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 298, + 97.25, + 501, + 110.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 298, + 110.5, + 501, + 123.75 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 298, + 123.75, + 501, + 137.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 298, + 145, + 503, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 298, + 145, + 504, + 157 + ], + "spans": [ + { + "bbox": [ + 298, + 145, + 504, + 157 + ], + "score": 1.0, + "content": "Table 1: The relationship between different numer-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 298, + 157, + 353, + 167 + ], + "spans": [ + { + "bbox": [ + 298, + 157, + 353, + 167 + ], + "score": 1.0, + "content": "ical methods.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + } + ], + "index": 12.0 + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 185, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 187, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 187, + 208 + ], + "score": 1.0, + "content": "3.5 ALGORITHM", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 230 + ], + "score": 1.0, + "content": "We can provide our whole algorithm of the denoising process of DDIMs now. According to Song", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "et al. (2020a), the algorithm of the original method satisfies Algorithm 1. And our new algorithm of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "DNPMs uses the pseudo linear multi-step and pseudo Runge-Kutta method, which satisfies Algo-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "rithm 2. Here, we cannot use linear multi-step initially because the linear multi-step method cannot", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "start automatically, which needs at least three previous steps’ information to generate results. So", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "we use the Runge-Kutta method to compute the first three steps’ results and then use the linear", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 281, + 290, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 290, + 295 + ], + "score": 1.0, + "content": "multi-step method to calculate the remaining.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 215, + 506, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "We also use the gradient parts of two second-order numerical methods to get another pseudo nu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "merical method. We introduce the details of this method in Appendix A.3. We call it S-PNDMs,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "because its gradient part uses information from two steps at every step. Similarly, we also call our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 331, + 474, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 474, + 344 + ], + "score": 1.0, + "content": "first PNDMs F-PNDMs, which use data from four steps, when we need to distinguish them.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 298, + 505, + 344 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 231, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 233, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 233, + 370 + ], + "score": 1.0, + "content": "3.6 CONVERGENCE ORDER", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 389 + ], + "score": 1.0, + "content": "Change the transfer part of numerical methods may introduce unknown error. To determine the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "influence of our new transfer part theoretically, we compute the local and global error between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 243, + 412 + ], + "score": 1.0, + "content": "theoretical result of Equation (10)", + "type": "text" + }, + { + "bbox": [ + 243, + 399, + 276, + 411 + ], + "score": 0.91, + "content": "x ( t + \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 398, + 416, + 412 + ], + "score": 1.0, + "content": "and our new methods, we find that", + "type": "text" + }, + { + "bbox": [ + 416, + 399, + 505, + 411 + ], + "score": 0.91, + "content": "x ( t + \\delta ) - x _ { \\mathrm { D D I M } } ( x +", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 410, + 174, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 154, + 422 + ], + "score": 0.93, + "content": "\\delta ) = O ( \\delta ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 410, + 174, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 377, + 506, + 423 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 420, + 384, + 435 + ], + "lines": [ + { + "bbox": [ + 226, + 420, + 384, + 435 + ], + "spans": [ + { + "bbox": [ + 226, + 420, + 384, + 435 + ], + "score": 0.9, + "content": "x ( t + \\delta ) - x _ { \\mathrm { S / F - P N D M } } ( x + \\delta ) = O ( \\delta ^ { 3 } ) .", + "type": "interline_equation", + "image_path": "e92c1130506a0fe0d87ac0b98b06802b20ec41a0334893283ac0931d2035665c.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 226, + 420, + 384, + 435 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 104, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 356, + 451 + ], + "score": 1.0, + "content": "If the target ODE satisfies Lipschitz condition and local error", + "type": "text" + }, + { + "bbox": [ + 356, + 437, + 415, + 449 + ], + "score": 0.9, + "content": "e _ { \\mathrm { l o c a l } } = O ( \\delta ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 437, + 477, + 451 + ], + "score": 1.0, + "content": ", then there are", + "type": "text" + }, + { + "bbox": [ + 478, + 438, + 487, + 448 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 447, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 114, + 459 + ], + "score": 0.76, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 447, + 218, + 462 + ], + "score": 1.0, + "content": "such that the global error", + "type": "text" + }, + { + "bbox": [ + 218, + 450, + 243, + 461 + ], + "score": 0.9, + "content": "e _ { \\mathrm { g l o b a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 447, + 278, + 462 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 278, + 448, + 465, + 461 + ], + "score": 0.89, + "content": "e _ { \\mathrm { g l o b a l } } \\leq C \\delta ^ { k } ( 1 + e ^ { h } + e ^ { 2 h } + \\dot { \\cdot } \\dot { \\cdot } \\dot { \\cdot } ) \\leq C ^ { \\prime } \\delta ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 447, + 507, + 462 + ], + "score": 1.0, + "content": ". And we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "have that the convergence order is equal to the order of the global error. The detailed proof can be", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 470, + 372, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 372, + 484 + ], + "score": 1.0, + "content": "found in Appendix A.6. Therefore, we get the following property:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 437, + 507, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 492, + 464, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 466, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 466, + 505 + ], + "score": 1.0, + "content": "Property 3.2 S/F-PNDMs have third-order local error and are second-order convergent.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 491, + 466, + 505 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 194, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 196, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 196, + 534 + ], + "score": 1.0, + "content": "4 EXPERIMENT", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 107, + 544, + 160, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 161, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 161, + 557 + ], + "score": 1.0, + "content": "4.1 SETUP", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 463, + 578 + ], + "score": 1.0, + "content": "We conduct unconditional image generation experiments on four datasets: Cifar10", + "type": "text" + }, + { + "bbox": [ + 463, + 565, + 502, + 576 + ], + "score": 0.83, + "content": "( 3 2 \\times 3 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 253, + 588 + ], + "score": 1.0, + "content": "(Krizhevsky et al., 2009), CelebA", + "type": "text" + }, + { + "bbox": [ + 253, + 576, + 292, + 587 + ], + "score": 0.85, + "content": "( 6 4 \\times 6 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 576, + 435, + 588 + ], + "score": 1.0, + "content": "(Liu et al., 2015), LSUN-church", + "type": "text" + }, + { + "bbox": [ + 436, + 577, + 484, + 587 + ], + "score": 0.87, + "content": "( 2 5 6 \\times 2 5 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 174, + 599 + ], + "score": 1.0, + "content": "LSUN-bedroom", + "type": "text" + }, + { + "bbox": [ + 175, + 587, + 222, + 598 + ], + "score": 0.87, + "content": "( 2 5 6 \\times 2 5 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 588, + 506, + 599 + ], + "score": 1.0, + "content": ") (Yu et al., 2016). According to the analysis in Section 3.1, we can use", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "pre-trained models from prior works in our experiments. The pre-trained models for Cifar10, LSUN-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "church and LSUN-bedroom are taken from Ho et al. (2020) and the pre-trained model for CelebA is", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "taken from Song et al. (2020a). In these models, the number of total steps N is 1000 and the variance", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 631, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 643 + ], + "score": 1.0, + "content": "schedule is linear variance schedule. And we also use a pre-trained model for Cifar10, which uses a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "cosine variance schedule from improved denoising diffusion probabilistic models (iDDPMs (Nichol", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 652, + 189, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 189, + 665 + ], + "score": 1.0, + "content": "& Dhariwal, 2021)).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 565, + 506, + 665 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 283, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 285, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 285, + 690 + ], + "score": 1.0, + "content": "4.2 SAMPLE EFFICIENCY AND QUALITY", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "To analyze the acceleration effect, we test Fenchel Inception Distance (FID (Heusel et al., 2018))", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "on different datasets under different steps and different numerical methods, including DDIMs,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "S-PNDMs, F-PNDMs and classical fourth-order numerical methods (FONs) (e.g., Runge-Kutta", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 82, + 396, + 264 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 82, + 396, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 82, + 396, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 396, + 264 + ], + "score": 0.981, + "html": "
datasetFID step model1020501002501000time
Cifar10DDIM PF13.46.84 13.84.67 3.894.16 3.693.714.04 3.72
Cifar10 (linear)DDIM*18.510.96.995.524.524.000.337
FON13.17.415.264.654.123.710.390
S-PNDM11.67.565.184.343.913.800.344
Cifar10 (cosine)F-PNDM DDIM7.03 14.55.00 8.793.95 5.863.72 4.923.60 4.303.70 3.690.391 0.505
S-PNDM8.645.774.463.943.713.380.517
F-PNDM7.054.613.683.533.493.260.595
CelebADDIM17.313.79.176.533.51
CelebA (linear)DDIM*16.913.48.956.364.443.411.237
FON16.011.68.136.705.144.171.431
S-PNDM12.29.455.694.033.192.991.258
F-PNDM7.715.513.342.812.712.861.433
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PFs use black", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 396, + 108, + 503, + 120 + ], + "spans": [ + { + "bbox": [ + 396, + 108, + 503, + 120 + ], + "score": 1.0, + "content": "box ODE solvers and we use", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 397, + 118, + 503, + 129 + ], + "spans": [ + { + "bbox": [ + 397, + 118, + 503, + 129 + ], + "score": 1.0, + "content": "the number of score func-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 397, + 128, + 503, + 140 + ], + "spans": [ + { + "bbox": [ + 397, + 128, + 503, + 140 + ], + "score": 1.0, + "content": "tion evaluations as the step", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 397, + 138, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 397, + 138, + 504, + 149 + ], + "score": 1.0, + "content": "of PFs. DDIM* is a retest", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 397, + 148, + 503, + 159 + ], + "spans": [ + { + "bbox": [ + 397, + 148, + 503, + 159 + ], + "score": 1.0, + "content": "of DDIM. The bold results", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 397, + 158, + 503, + 169 + ], + "spans": [ + { + "bbox": [ + 397, + 158, + 503, + 169 + ], + "score": 1.0, + "content": "mean the best ones using the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 396, + 169, + 503, + 178 + ], + "spans": [ + { + "bbox": [ + 396, + 169, + 503, + 178 + ], + "score": 1.0, + "content": "same pretrained model. We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 396, + 178, + 503, + 189 + ], + "spans": [ + { + "bbox": [ + 396, + 178, + 503, + 189 + ], + "score": 1.0, + "content": "use the 50-step, 512 batch", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 396, + 188, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 396, + 188, + 504, + 199 + ], + "score": 1.0, + "content": "size experiment on an RTX-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 397, + 198, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 397, + 198, + 504, + 210 + ], + "score": 1.0, + "content": "3090 to test the computa-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 397, + 208, + 503, + 219 + ], + "spans": [ + { + "bbox": [ + 397, + 208, + 503, + 219 + ], + "score": 1.0, + "content": "tional cost and the column", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 396, + 218, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 396, + 218, + 504, + 230 + ], + "score": 1.0, + "content": "time is the average compu-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 397, + 228, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 397, + 228, + 504, + 240 + ], + "score": 1.0, + "content": "tational cost per step in sec-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 397, + 239, + 503, + 248 + ], + "spans": [ + { + "bbox": [ + 397, + 239, + 503, + 248 + ], + "score": 1.0, + "content": "onds. And we put the results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 396, + 246, + 503, + 260 + ], + "spans": [ + { + "bbox": [ + 396, + 246, + 503, + 260 + ], + "score": 1.0, + "content": "of standard deviation in Ap-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 396, + 258, + 445, + 268 + ], + "spans": [ + { + "bbox": [ + 396, + 258, + 445, + 268 + ], + "score": 1.0, + "content": "pendix A.12", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 504, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 504, + 292 + ], + "score": 1.0, + "content": "method and linear multi-step method). On Cifar10 and CelebA, we first provide the results of previ-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "ous works DDIMs. Then, we use the same pre-trained models to test numerical methods mentioned", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "in this paper and put the results in Cifar10 / CelebA (linear). We also use models from iDDPMs to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "test nonlinear variance schedules and put the results in Cifar10 (cosine). Song et al. (2020b) do not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "provide detailed FID results of probability flows (PFs) under different steps, so we retest the results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 335, + 272, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 272, + 347 + ], + "score": 1.0, + "content": "using its pretrained models by ourselves.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "score": 1.0, + "content": "Efficiency Our two baselines are DDIM and PF. DDIM is a simple case of pseudo numerical meth-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "ods, and PF is a case of classical numerical methods. However, PF uses a much bigger model than", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "DDIM and uses some tricks to improve the sample quality. To ensure the experiment’s fairness, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "use fourth-order numerical methods on Equation (10) and the model from DDIM. In Table 2, we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "find that the performance of FON is limited when the number of steps is small. By contrast, our new", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "methods, including S-PNDM and F-PNDM, can improve the generated results regardless of whether", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "the number of steps used is large or small. 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Be-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 478, + 345, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 345, + 491 + ], + "score": 1.0, + "content": "cause F-PNDM uses the pseudo Runge-Kutta method to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 489, + 345, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 345, + 501 + ], + "score": 1.0, + "content": "generate the first three steps, it is slower than other methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 501, + 345, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 345, + 512 + ], + "score": 1.0, + "content": "at the first several steps. Therefore, S-PNDM can achieve", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 511, + 345, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 345, + 523 + ], + "score": 1.0, + "content": "the best FID initially, then F-PNDM becomes the best and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 522, + 227, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 227, + 535 + ], + "score": 1.0, + "content": "the acceleration is significant.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 343, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 345, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 345, + 552 + ], + "score": 1.0, + "content": "Quality When the number of steps is relatively big, the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 550, + 345, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 345, + 561 + ], + "score": 1.0, + "content": "results of FON become more and more similar to that of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 561, + 345, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 345, + 573 + ], + "score": 1.0, + "content": "pseudo numerical methods. This is because all the meth-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 572, + 345, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 345, + 584 + ], + "score": 1.0, + "content": "ods are solving Equation (10), and their convergent results", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 583, + 345, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 345, + 595 + ], + "score": 1.0, + "content": "should be the same. 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This phenomenon is more pronounced", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 660, + 320, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 320, + 672 + ], + "score": 1.0, + "content": "when we test our methods on LSUN (see Table 5, 6).", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 60.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "According to Cifar10 (cosine), the cosine variance schedule can lower FID using a relatively large", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "number of steps. More analyses about variance schedule can be found in Appendix A.7. 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datasetFID step model1020501002501000time
Cifar10DDIM PF13.46.84 13.84.67 3.894.16 3.693.714.04 3.72
Cifar10 (linear)DDIM*18.510.96.995.524.524.000.337
FON13.17.415.264.654.123.710.390
S-PNDM11.67.565.184.343.913.800.344
Cifar10 (cosine)F-PNDM DDIM7.03 14.55.00 8.793.95 5.863.72 4.923.60 4.303.70 3.690.391 0.505
S-PNDM8.645.774.463.943.713.380.517
F-PNDM7.054.613.683.533.493.260.595
CelebADDIM17.313.79.176.533.51
CelebA (linear)DDIM*16.913.48.956.364.443.411.237
FON16.011.68.136.705.144.171.431
S-PNDM12.29.455.694.033.192.991.258
F-PNDM7.715.513.342.812.712.861.433
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On Cifar10 and CelebA, we first provide the results of previ-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "ous works DDIMs. Then, we use the same pre-trained models to test numerical methods mentioned", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "in this paper and put the results in Cifar10 / CelebA (linear). We also use models from iDDPMs to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "test nonlinear variance schedules and put the results in Cifar10 (cosine). Song et al. (2020b) do not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "provide detailed FID results of probability flows (PFs) under different steps, so we retest the results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 335, + 272, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 272, + 347 + ], + "score": 1.0, + "content": "using its pretrained models by ourselves.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 280, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "score": 1.0, + "content": "Efficiency Our two baselines are DDIM and PF. DDIM is a simple case of pseudo numerical meth-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "ods, and PF is a case of classical numerical methods. However, PF uses a much bigger model than", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "DDIM and uses some tricks to improve the sample quality. To ensure the experiment’s fairness, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "use fourth-order numerical methods on Equation (10) and the model from DDIM. In Table 2, we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "find that the performance of FON is limited when the number of steps is small. By contrast, our new", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "methods, including S-PNDM and F-PNDM, can improve the generated results regardless of whether", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "the number of steps used is large or small. According to Cifar10 / CelebA (linear), F-PNDM can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 462, + 442 + ], + "score": 1.0, + "content": "achieve lower FID than 1000 steps DDIM using only 50 steps, making diffusion models", + "type": "text" + }, + { + "bbox": [ + 462, + 429, + 479, + 439 + ], + "score": 0.58, + "content": "2 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "faster", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 438, + 199, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 199, + 453 + ], + "score": 1.0, + "content": "without losing quality.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 352, + 506, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 344, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 344, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 344, + 468 + ], + "score": 1.0, + "content": "We draw a line chart of computation cost with FID accord-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 467, + 344, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 344, + 479 + ], + "score": 1.0, + "content": "ing to the results of Cifar10 (linear) above in Figure 3. Be-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 478, + 345, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 345, + 491 + ], + "score": 1.0, + "content": "cause F-PNDM uses the pseudo Runge-Kutta method to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 489, + 345, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 345, + 501 + ], + "score": 1.0, + "content": "generate the first three steps, it is slower than other methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 501, + 345, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 345, + 512 + ], + "score": 1.0, + "content": "at the first several steps. Therefore, S-PNDM can achieve", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 511, + 345, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 345, + 523 + ], + "score": 1.0, + "content": "the best FID initially, then F-PNDM becomes the best and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 522, + 227, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 227, + 535 + ], + "score": 1.0, + "content": "the acceleration is significant.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 456, + 345, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 343, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 345, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 345, + 552 + ], + "score": 1.0, + "content": "Quality When the number of steps is relatively big, the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 550, + 345, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 345, + 561 + ], + "score": 1.0, + "content": "results of FON become more and more similar to that of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 561, + 345, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 345, + 573 + ], + "score": 1.0, + "content": "pseudo numerical methods. This is because all the meth-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 572, + 345, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 345, + 584 + ], + "score": 1.0, + "content": "ods are solving Equation (10), and their convergent results", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 583, + 345, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 345, + 595 + ], + "score": 1.0, + "content": "should be the same. However, pseudo numerical methods", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 592, + 344, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 344, + 607 + ], + "score": 1.0, + "content": "still work better using a large number of steps empirically.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 604, + 345, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 345, + 617 + ], + "score": 1.0, + "content": "F-PNDM can improve the best FID around 0.4 using pre-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 615, + 344, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 344, + 627 + ], + "score": 1.0, + "content": "trained models and achieves a new SOTA FID score of 2.71", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "on CelebA, which shows that our work can not only accelerate diffusion models but also improve", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "the sample quality topline. We also notice that the FID results of F-PNDM converge after more than", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "250 steps. The FID results will fluctuate around a value then. This phenomenon is more pronounced", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 660, + 320, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 320, + 672 + ], + "score": 1.0, + "content": "when we test our methods on LSUN (see Table 5, 6).", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 538, + 345, + 627 + ] + }, + { + "type": "image", + "bbox": [ + 352, + 446, + 502, + 561 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 352, + 446, + 502, + 561 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 352, + 446, + 502, + 561 + ], + "spans": [ + { + "bbox": [ + 352, + 446, + 502, + 561 + ], + "score": 0.968, + "type": "image", + "image_path": "988a0e4aa3b27ecdc0d0e15c4f94b23487b35295b934829b8f68160586957c92.jpg" + } + ] + } + ], + "index": 44.5, + "virtual_lines": [ + { + "bbox": [ + 352, + 446, + 502, + 503.5 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 352, + 503.5, + 502, + 561.0 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 352, + 568, + 504, + 624 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 351, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 351, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "Figure 3: The FID results under dif-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 351, + 580, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 351, + 580, + 505, + 590 + ], + "score": 1.0, + "content": "ferent computation costs and different", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 351, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 351, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "numerical methods on Cifar10. The", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 352, + 602, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 352, + 602, + 505, + 613 + ], + "score": 1.0, + "content": "unit of time is the computational cost", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 351, + 612, + 489, + 624 + ], + "spans": [ + { + "bbox": [ + 351, + 612, + 489, + 624 + ], + "score": 1.0, + "content": "of 1-step DDIM, which is 0.337s.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 56 + } + ], + "index": 50.25 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 671 + ], + "lines": [], + "index": 60.5, + "bbox_fs": [ + 105, + 626, + 505, + 672 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "According to Cifar10 (cosine), the cosine variance schedule can lower FID using a relatively large", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "number of steps. More analyses about variance schedule can be found in Appendix A.7. What’s", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "more, we test our methods on other datasets and provide our FID results in Appendix A.9 and", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "image results in Appendix A.10. We can draw similar conclusions on our methods’ acceleration and", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 720, + 388, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 388, + 734 + ], + "score": 1.0, + "content": "sampling quality, regardless of the datasets and the size of the images.", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 65, + "bbox_fs": [ + 105, + 676, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 73, + 505, + 268 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 73, + 505, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 268 + ], + "score": 0.973, + "type": "image", + "image_path": "65efd211782730face9f7f6f648ca37c2a522e6bc77cdc1991a885b3333e5f3f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 73, + 505, + 138.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 138.0, + 505, + 203.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 203.0, + 505, + 268.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 276, + 505, + 321 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "Figure 4: The upper part shows the change of norm with the number of steps using different methods", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "and different steps. The lower part shows the generation curves of two points using different methods", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 254, + 311 + ], + "score": 1.0, + "content": "and different steps. DDIM-n means", + "type": "text" + }, + { + "bbox": [ + 255, + 300, + 261, + 309 + ], + "score": 0.29, + "content": "\\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "-step DDIM method. Experiments in this subsection all use", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 309, + 426, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 426, + 321 + ], + "score": 1.0, + "content": "the Cifar10 dataset and we use the 1000-step DDIM’s result as our target result.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 237, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 239, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 239, + 343 + ], + "score": 1.0, + "content": "4.3 SAMPLE ON MANIFOLDS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "Here, we design visualization experiments to show the effort of our new methods and support our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "analyses. Because it is hard to visualize high-dimensional data, we use the change of a global char-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "acteristic norm and a local characteristic pixel to show the change of the data under different steps.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 383, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 289, + 398 + ], + "score": 1.0, + "content": "For pixel, we randomly choose two positions", + "type": "text" + }, + { + "bbox": [ + 289, + 384, + 313, + 396 + ], + "score": 0.91, + "content": "p ^ { 1 } , p ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 383, + 429, + 398 + ], + "score": 1.0, + "content": ". Then for a series of images", + "type": "text" + }, + { + "bbox": [ + 429, + 386, + 504, + 396 + ], + "score": 0.86, + "content": "x _ { T } , x _ { T - k } , \\cdot \\cdot \\cdot , x _ { 0 }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 395, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 287, + 408 + ], + "score": 1.0, + "content": "derived from the reverse process, we denote", + "type": "text" + }, + { + "bbox": [ + 288, + 396, + 299, + 407 + ], + "score": 0.89, + "content": "y _ { t } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 395, + 362, + 408 + ], + "score": 1.0, + "content": "as the value of", + "type": "text" + }, + { + "bbox": [ + 363, + 397, + 373, + 407 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 395, + 420, + 408 + ], + "score": 1.0, + "content": "at position", + "type": "text" + }, + { + "bbox": [ + 420, + 395, + 431, + 407 + ], + "score": 0.89, + "content": "p ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 395, + 507, + 408 + ], + "score": 1.0, + "content": ". Then we draw a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 405, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 142, + 421 + ], + "score": 1.0, + "content": "polyline", + "type": "text" + }, + { + "bbox": [ + 142, + 406, + 199, + 419 + ], + "score": 0.91, + "content": "( y _ { t } ^ { 1 } , y _ { t } ^ { 2 } ) _ { t = T , \\cdots }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 405, + 211, + 421 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 212, + 406, + 224, + 417 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 405, + 506, + 421 + ], + "score": 1.0, + "content": ". For norm, we first count the distribution of the norm of the training", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "score": 1.0, + "content": "datasets under different steps and use this to make a heat map as the background. After that, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 428, + 484, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 484, + 442 + ], + "score": 1.0, + "content": "draw the norm of our generated results using different methods and steps above this heat map.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "In Figure 4, we can see that the FON may run far away from the high-density area of the data, which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "explains why FON may introduce noticeable noise. However, PNDM can avoid this problem and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "appropriately fit the target result. More visualization results supporting our analysis can be found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "in Appendix A.11. What’s more, we design a toy example to test our new methods without the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "influence of neural networks and get similar conclusions as to the real cases above. We put the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 241, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 241, + 513 + ], + "score": 1.0, + "content": "detailed results in Appendix A.8.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 529, + 190, + 542 + ], + "lines": [ + { + "bbox": [ + 104, + 529, + 192, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 192, + 545 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "In this paper, we provided DNPMs, a new numerical method suitable for solving the corresponding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 566, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 581 + ], + "score": 1.0, + "content": "ODEs of DDPMs. DNPMs can generate high-quality images using fewer steps without loss of qual-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "ity successfully. Based on the idea of this work, further improvement can be explored in our future", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "works: 1) find a better variance schedule for DNPMs: although we tested DNPMs on linear variance", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "schedule and cosine variance schedule in this work, there might be another variance schedule more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "suitable for our proposed numerical methods. 2) Find higher-order convergent pseudo numerical", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "methods: we analyzed the convergence order of S/F-DNPMs, which are both second-order conver-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "gent. However, F-DNPMs achieve better FID than S-DNPMs in most cases. We think this is because", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "the result between our transfer part and target ODE has a higher-order error, which limits the con-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "vergence order of F-DNPMs. This error from the change of the transfer part is theoretical but does", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "not influence the quality of images according to the property of Equation (11). However, making", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "score": 1.0, + "content": "the transfer part higher-order convergent and finding the influence of such change is still exciting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "and needs more research. 3) Extend PNDMs to more general applications: when we proved the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "convergence order of DNPMs, we found that other kinds of transfer parts can keep the convergence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "order unchanged too, which means that pseudo numerical methods can be used on more general", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 721, + 423, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 423, + 733 + ], + "score": 1.0, + "content": "applications, like certain neural ODEs (Chen et al., 2019; Dupont et al., 2019).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 30.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "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": "image", + "bbox": [ + 106, + 73, + 505, + 268 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 73, + 505, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 268 + ], + "score": 0.973, + "type": "image", + "image_path": "65efd211782730face9f7f6f648ca37c2a522e6bc77cdc1991a885b3333e5f3f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 73, + 505, + 138.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 138.0, + 505, + 203.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 203.0, + 505, + 268.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 276, + 505, + 321 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "Figure 4: The upper part shows the change of norm with the number of steps using different methods", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "and different steps. The lower part shows the generation curves of two points using different methods", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 254, + 311 + ], + "score": 1.0, + "content": "and different steps. DDIM-n means", + "type": "text" + }, + { + "bbox": [ + 255, + 300, + 261, + 309 + ], + "score": 0.29, + "content": "\\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "-step DDIM method. Experiments in this subsection all use", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 309, + 426, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 426, + 321 + ], + "score": 1.0, + "content": "the Cifar10 dataset and we use the 1000-step DDIM’s result as our target result.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 237, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 239, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 239, + 343 + ], + "score": 1.0, + "content": "4.3 SAMPLE ON MANIFOLDS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "Here, we design visualization experiments to show the effort of our new methods and support our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "analyses. Because it is hard to visualize high-dimensional data, we use the change of a global char-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "acteristic norm and a local characteristic pixel to show the change of the data under different steps.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 383, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 289, + 398 + ], + "score": 1.0, + "content": "For pixel, we randomly choose two positions", + "type": "text" + }, + { + "bbox": [ + 289, + 384, + 313, + 396 + ], + "score": 0.91, + "content": "p ^ { 1 } , p ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 383, + 429, + 398 + ], + "score": 1.0, + "content": ". Then for a series of images", + "type": "text" + }, + { + "bbox": [ + 429, + 386, + 504, + 396 + ], + "score": 0.86, + "content": "x _ { T } , x _ { T - k } , \\cdot \\cdot \\cdot , x _ { 0 }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 395, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 287, + 408 + ], + "score": 1.0, + "content": "derived from the reverse process, we denote", + "type": "text" + }, + { + "bbox": [ + 288, + 396, + 299, + 407 + ], + "score": 0.89, + "content": "y _ { t } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 395, + 362, + 408 + ], + "score": 1.0, + "content": "as the value of", + "type": "text" + }, + { + "bbox": [ + 363, + 397, + 373, + 407 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 395, + 420, + 408 + ], + "score": 1.0, + "content": "at position", + "type": "text" + }, + { + "bbox": [ + 420, + 395, + 431, + 407 + ], + "score": 0.89, + "content": "p ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 395, + 507, + 408 + ], + "score": 1.0, + "content": ". Then we draw a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 405, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 142, + 421 + ], + "score": 1.0, + "content": "polyline", + "type": "text" + }, + { + "bbox": [ + 142, + 406, + 199, + 419 + ], + "score": 0.91, + "content": "( y _ { t } ^ { 1 } , y _ { t } ^ { 2 } ) _ { t = T , \\cdots }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 405, + 211, + 421 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 212, + 406, + 224, + 417 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 405, + 506, + 421 + ], + "score": 1.0, + "content": ". For norm, we first count the distribution of the norm of the training", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "score": 1.0, + "content": "datasets under different steps and use this to make a heat map as the background. After that, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 428, + 484, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 484, + 442 + ], + "score": 1.0, + "content": "draw the norm of our generated results using different methods and steps above this heat map.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 351, + 507, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "In Figure 4, we can see that the FON may run far away from the high-density area of the data, which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "explains why FON may introduce noticeable noise. However, PNDM can avoid this problem and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "appropriately fit the target result. More visualization results supporting our analysis can be found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "in Appendix A.11. What’s more, we design a toy example to test our new methods without the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "influence of neural networks and get similar conclusions as to the real cases above. We put the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 241, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 241, + 513 + ], + "score": 1.0, + "content": "detailed results in Appendix A.8.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 444, + 505, + 513 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 529, + 190, + 542 + ], + "lines": [ + { + "bbox": [ + 104, + 529, + 192, + 545 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 192, + 545 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "In this paper, we provided DNPMs, a new numerical method suitable for solving the corresponding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 566, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 581 + ], + "score": 1.0, + "content": "ODEs of DDPMs. DNPMs can generate high-quality images using fewer steps without loss of qual-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "ity successfully. Based on the idea of this work, further improvement can be explored in our future", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "works: 1) find a better variance schedule for DNPMs: although we tested DNPMs on linear variance", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "schedule and cosine variance schedule in this work, there might be another variance schedule more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "suitable for our proposed numerical methods. 2) Find higher-order convergent pseudo numerical", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "methods: we analyzed the convergence order of S/F-DNPMs, which are both second-order conver-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "gent. However, F-DNPMs achieve better FID than S-DNPMs in most cases. We think this is because", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "the result between our transfer part and target ODE has a higher-order error, which limits the con-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "vergence order of F-DNPMs. This error from the change of the transfer part is theoretical but does", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "not influence the quality of images according to the property of Equation (11). 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(2020b) com-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "bine DDPMs and NCSNs under the framework of neural differential equations (Chen et al., 2019;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "Dupont et al., 2019). Therefore, numerical methods widely used in neural differential equations can", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "also be applied to accelerate DDPMs. Our work successfully combines the advantages of Song et al.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "(2020a) and Song et al. (2020b). We use a transfer part from DDIMs and use different gradient parts", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "from different numerical methods. Although we and Song et al. (2020b) solve certain differential", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "equations derived from DDPMs, we use different target different equations and different numerical", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 243, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 243, + 105 + ], + "score": 1.0, + "content": "methods, which get better results.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 632, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "equations derived from DDPMs, we use different target different equations and different numerical", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 243, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 243, + 105 + ], + "score": 1.0, + "content": "methods, which get better results.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "The application of DDPMs is not limited to unconditional image generation. Some works apply", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "DDPMs to various types of data successfully, including text-to-speech (Chen et al., 2020; Lam", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "et al., 2021), singing voice (Liu et al., 2021), 3D Point Cloud (Luo & Hu, 2021), text generation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "(Austin et al., 2021). Additionally, DDPMs can also be used to generate conditional samples, too", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 259, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 259, + 167 + ], + "score": 1.0, + "content": "(Jeong et al., 2021; Choi et al., 2021).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 178, + 283, + 190 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 283, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 283, + 191 + ], + "score": 1.0, + "content": "A.2 CONVERGENT ORDER OF METHOD", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 199, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 199, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 506, + 211 + ], + "score": 1.0, + "content": "We use the forward Euler method and linear multi-step method to show what is the order of a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 197, + 223 + ], + "score": 1.0, + "content": "method. Assume that", + "type": "text" + }, + { + "bbox": [ + 197, + 212, + 207, + 221 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 209, + 353, + 223 + ], + "score": 1.0, + "content": "is precise and compute the error at", + "type": "text" + }, + { + "bbox": [ + 353, + 212, + 373, + 222 + ], + "score": 0.89, + "content": "x _ { t + \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 209, + 506, + 223 + ], + "score": 1.0, + "content": ". For forward Euler method, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 132, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 132, + 234 + ], + "score": 1.0, + "content": "have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 230, + 430, + 300 + ], + "lines": [ + { + "bbox": [ + 180, + 230, + 430, + 300 + ], + "spans": [ + { + "bbox": [ + 180, + 230, + 430, + 300 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle e _ { t , \\delta } = \\boldsymbol { x } ( t + \\delta ) - \\boldsymbol { x } _ { t , \\delta } } \\\\ { \\displaystyle = \\bigg ( \\boldsymbol { x } ( t ) + \\delta f ( \\boldsymbol { x } ( t ) , t ) + \\frac { \\delta ^ { 2 } } { 2 } f ^ { \\prime \\prime } ( c ) \\bigg ) - ( \\boldsymbol { x } ( t ) + \\delta f ( \\boldsymbol { x } _ { t } , t ) ) } \\\\ { \\displaystyle = \\frac { \\delta ^ { 2 } } { 2 } f ^ { \\prime \\prime } ( c ) \\leq \\frac { \\delta ^ { 2 } } { 2 } M } \\end{array}", + "type": "interline_equation", + "image_path": "83ad2575f9858c2190d2151954f85be69eb29b6b8608e859266b38b3deaf8035.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 180, + 230, + 430, + 253.33333333333334 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 180, + 253.33333333333334, + 430, + 276.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 180, + 276.6666666666667, + 430, + 300.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 300, + 504, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 197, + 313 + ], + "score": 1.0, + "content": "Here, we assume that", + "type": "text" + }, + { + "bbox": [ + 198, + 300, + 209, + 312 + ], + "score": 0.9, + "content": "f ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 299, + 282, + 313 + ], + "score": 1.0, + "content": "is continuous, so", + "type": "text" + }, + { + "bbox": [ + 283, + 300, + 295, + 312 + ], + "score": 0.89, + "content": "f ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 299, + 411, + 313 + ], + "score": 1.0, + "content": "is bounded in a close area.", + "type": "text" + }, + { + "bbox": [ + 411, + 300, + 447, + 312 + ], + "score": 0.93, + "content": "x ( t + \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "is the precise", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 311, + 320, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 148, + 323 + ], + "score": 1.0, + "content": "result and", + "type": "text" + }, + { + "bbox": [ + 148, + 313, + 165, + 324 + ], + "score": 0.87, + "content": "x _ { t , \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 311, + 278, + 323 + ], + "score": 1.0, + "content": "is the numerical result from", + "type": "text" + }, + { + "bbox": [ + 279, + 313, + 283, + 321 + ], + "score": 0.8, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 311, + 295, + 323 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 295, + 312, + 317, + 322 + ], + "score": 0.9, + "content": "t + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 311, + 320, + 323 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 263, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 263, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 263, + 341 + ], + "score": 1.0, + "content": "For linear multi-step method, we have:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 340, + 483, + 421 + ], + "lines": [ + { + "bbox": [ + 111, + 340, + 483, + 421 + ], + "spans": [ + { + "bbox": [ + 111, + 340, + 483, + 421 + ], + "score": 0.9, + "content": "\\begin{array} { l } { \\displaystyle e _ { t , \\delta } = x ( t + \\delta ) - x _ { t , \\delta } } \\\\ { \\displaystyle = \\Bigg ( x ( t ) + \\frac { \\delta } { 1 ! } f ( x ( t ) , t ) + \\frac { \\delta ^ { 2 } } { 2 ! } f ^ { \\prime } ( x ( t ) , t ) + \\cdots + \\frac { \\delta ^ { 4 } } { 4 ! } f ^ { ( 3 ) } ( x ( t ) , t ) + O \\left( \\delta ^ { 5 } \\right) \\Bigg ) } \\\\ { \\displaystyle \\quad - \\left( x ( t ) + b _ { 1 } \\delta f ( x ( t ) , t ) + \\sum _ { s = 2 } ^ { 4 } b _ { s } \\delta \\left( \\sum _ { k = 0 } ^ { 4 } \\frac { ( - ( s - 1 ) \\delta ) ^ { k } } { ( k ) ! } f ^ { ( k - 1 ) } ( x ( t ) , t ) + O \\left( \\delta ^ { 5 } \\right) \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "a1924594e76b54f7037b6af8d94f6021242cd86fd9603ce9fbcc17d7a2ab0a8e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 111, + 340, + 483, + 367.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 111, + 367.0, + 483, + 394.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 111, + 394.0, + 483, + 421.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 429, + 449 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 425, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 131, + 453 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 436, + 150, + 449 + ], + "score": 0.92, + "content": "\\{ b _ { s } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 432, + 180, + 453 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + }, + { + "bbox": [ + 181, + 435, + 235, + 450 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { s = 1 } ^ { 4 } b _ { s } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 432, + 363, + 453 + ], + "score": 1.0, + "content": "and the following equations for", + "type": "text" + }, + { + "bbox": [ + 364, + 437, + 425, + 449 + ], + "score": 0.87, + "content": "j \\in \\{ 1 , \\cdots , 3 \\}", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 450, + 389, + 476 + ], + "lines": [ + { + "bbox": [ + 221, + 450, + 389, + 476 + ], + "spans": [ + { + "bbox": [ + 221, + 450, + 389, + 476 + ], + "score": 0.87, + "content": "( - 1 ) ^ { j } b _ { 2 } + ( - 2 ) ^ { j } b _ { 3 } + ( - 3 ) ^ { j } b _ { 4 } = \\frac { 1 } { j + 1 } .", + "type": "interline_equation", + "image_path": "55885cd55f64a66acd21027b81db647a019bf5552d6c51d5d2788c357bfa80c2.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 221, + 450, + 389, + 476 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 504, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 185, + 494 + ], + "score": 1.0, + "content": "We call the error at", + "type": "text" + }, + { + "bbox": [ + 185, + 483, + 201, + 493 + ], + "score": 0.88, + "content": "x _ { t , \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 481, + 311, + 494 + ], + "score": 1.0, + "content": "local error, and the error at", + "type": "text" + }, + { + "bbox": [ + 312, + 483, + 340, + 493 + ], + "score": 0.88, + "content": "x _ { t + M \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 482, + 356, + 492 + ], + "score": 0.36, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "is big enough but finite) global error.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 293, + 506 + ], + "score": 1.0, + "content": "Assume the local error of our method has order", + "type": "text" + }, + { + "bbox": [ + 294, + 493, + 311, + 503 + ], + "score": 0.78, + "content": "_ { \\mathrm { k + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 491, + 506, + 506 + ], + "score": 1.0, + "content": "and the target ODE satisfies Lipschitz condition,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 503, + 130, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 130, + 515 + ], + "score": 1.0, + "content": "then:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 512, + 461, + 627 + ], + "lines": [ + { + "bbox": [ + 150, + 512, + 461, + 627 + ], + "spans": [ + { + "bbox": [ + 150, + 512, + 461, + 627 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { x ( t + M \\delta ) - x _ { t , M \\delta } \\leq \\vert x ( t + M \\delta ) - x _ { t + ( M - 1 ) \\delta , \\delta } \\vert + \\vert x _ { t + ( M - 1 ) \\delta , \\delta } - x _ { t , M \\delta } \\vert } \\\\ & { \\qquad \\leq e _ { t + ( M - 1 ) \\delta , \\delta } + e ^ { L \\delta } \\vert x ( t + ( M - 1 ) \\delta ) - x _ { t , ( M - 1 ) \\delta } \\vert } \\\\ & { \\qquad \\leq e _ { t + ( M - 1 ) \\delta , \\delta } + e ^ { L \\delta } e _ { t + ( M - 2 ) \\delta , \\delta } + \\cdots } \\\\ & { \\qquad \\leq C \\delta ^ { k + 1 } \\left( 1 + e ^ { L h } + \\cdots + e ^ { ( i - 1 ) L h } \\right) } \\\\ & { \\qquad = C h ^ { k + 1 } \\frac { e ^ { i L \\delta } - 1 } { e ^ { L \\delta } - 1 } \\leq C \\delta ^ { k + 1 } \\frac { e ^ { i L \\delta } - 1 } { L \\delta } } \\\\ & { \\qquad = O ( \\delta ^ { k } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "bbdf21d2145f10416c60b2785734376cc8056e32d32ef105bcfbf840e614051d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 150, + 512, + 461, + 550.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 150, + 550.3333333333334, + 461, + 588.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 150, + 588.6666666666667, + 461, + 627.0000000000001 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "Therefore, the global error will be one order lower than the local error. From Equation (15), we can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 283, + 655 + ], + "score": 1.0, + "content": "see the forward Euler method has local error", + "type": "text" + }, + { + "bbox": [ + 283, + 643, + 309, + 655 + ], + "score": 0.92, + "content": "O ( \\delta ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 643, + 375, + 655 + ], + "score": 1.0, + "content": "and global error", + "type": "text" + }, + { + "bbox": [ + 375, + 644, + 396, + 655 + ], + "score": 0.89, + "content": "O ( \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 643, + 505, + 655 + ], + "score": 1.0, + "content": ", so we call it the first-order", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 381, + 667 + ], + "score": 1.0, + "content": "numerical method. And the linear multi-step method has local error", + "type": "text" + }, + { + "bbox": [ + 381, + 654, + 407, + 666 + ], + "score": 0.92, + "content": "{ \\dot { O } } ( \\delta ^ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 653, + 475, + 667 + ], + "score": 1.0, + "content": "and global error", + "type": "text" + }, + { + "bbox": [ + 475, + 654, + 501, + 666 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\delta ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 653, + 505, + 667 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 666, + 305, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 305, + 676 + ], + "score": 1.0, + "content": "and we call it the fourth-order numerical method.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 682, + 503, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 504, + 695 + ], + "score": 1.0, + "content": "In addition, assuming that a numerical method has kth-order global error, we can compute the con-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 693, + 275, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 275, + 705 + ], + "score": 1.0, + "content": "vergence speed of this numerical method:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 705, + 384, + 735 + ], + "lines": [ + { + "bbox": [ + 227, + 705, + 384, + 735 + ], + "spans": [ + { + "bbox": [ + 227, + 705, + 384, + 735 + ], + "score": 0.94, + "content": "\\operatorname* { l i m } _ { \\delta 0 } \\frac { x _ { t + T } ^ { 2 \\delta } - x ( t + T ) } { x _ { t + T } ^ { \\delta } - x ( t + T ) } = \\frac { ( 2 \\delta ) ^ { k } } { \\delta ^ { k } } = 2 ^ { k } .", + "type": "interline_equation", + "image_path": "685bb3df44c9b75e9f21adc907ec8f60ecdc68db6a506c09864fa43b7327c4b7.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 227, + 705, + 384, + 720.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 227, + 720.0, + 384, + 735.0 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 11, + "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 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "The application of DDPMs is not limited to unconditional image generation. Some works apply", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "DDPMs to various types of data successfully, including text-to-speech (Chen et al., 2020; Lam", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "et al., 2021), singing voice (Liu et al., 2021), 3D Point Cloud (Luo & Hu, 2021), text generation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "(Austin et al., 2021). Additionally, DDPMs can also be used to generate conditional samples, too", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 259, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 259, + 167 + ], + "score": 1.0, + "content": "(Jeong et al., 2021; Choi et al., 2021).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 110, + 505, + 167 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 178, + 283, + 190 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 283, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 283, + 191 + ], + "score": 1.0, + "content": "A.2 CONVERGENT ORDER OF METHOD", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 199, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 199, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 506, + 211 + ], + "score": 1.0, + "content": "We use the forward Euler method and linear multi-step method to show what is the order of a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 197, + 223 + ], + "score": 1.0, + "content": "method. Assume that", + "type": "text" + }, + { + "bbox": [ + 197, + 212, + 207, + 221 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 209, + 353, + 223 + ], + "score": 1.0, + "content": "is precise and compute the error at", + "type": "text" + }, + { + "bbox": [ + 353, + 212, + 373, + 222 + ], + "score": 0.89, + "content": "x _ { t + \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 209, + 506, + 223 + ], + "score": 1.0, + "content": ". For forward Euler method, we", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 132, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 132, + 234 + ], + "score": 1.0, + "content": "have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 199, + 506, + 234 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 230, + 430, + 300 + ], + "lines": [ + { + "bbox": [ + 180, + 230, + 430, + 300 + ], + "spans": [ + { + "bbox": [ + 180, + 230, + 430, + 300 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle e _ { t , \\delta } = \\boldsymbol { x } ( t + \\delta ) - \\boldsymbol { x } _ { t , \\delta } } \\\\ { \\displaystyle = \\bigg ( \\boldsymbol { x } ( t ) + \\delta f ( \\boldsymbol { x } ( t ) , t ) + \\frac { \\delta ^ { 2 } } { 2 } f ^ { \\prime \\prime } ( c ) \\bigg ) - ( \\boldsymbol { x } ( t ) + \\delta f ( \\boldsymbol { x } _ { t } , t ) ) } \\\\ { \\displaystyle = \\frac { \\delta ^ { 2 } } { 2 } f ^ { \\prime \\prime } ( c ) \\leq \\frac { \\delta ^ { 2 } } { 2 } M } \\end{array}", + "type": "interline_equation", + "image_path": "83ad2575f9858c2190d2151954f85be69eb29b6b8608e859266b38b3deaf8035.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 180, + 230, + 430, + 253.33333333333334 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 180, + 253.33333333333334, + 430, + 276.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 180, + 276.6666666666667, + 430, + 300.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 300, + 504, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 197, + 313 + ], + "score": 1.0, + "content": "Here, we assume that", + "type": "text" + }, + { + "bbox": [ + 198, + 300, + 209, + 312 + ], + "score": 0.9, + "content": "f ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 299, + 282, + 313 + ], + "score": 1.0, + "content": "is continuous, so", + "type": "text" + }, + { + "bbox": [ + 283, + 300, + 295, + 312 + ], + "score": 0.89, + "content": "f ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 299, + 411, + 313 + ], + "score": 1.0, + "content": "is bounded in a close area.", + "type": "text" + }, + { + "bbox": [ + 411, + 300, + 447, + 312 + ], + "score": 0.93, + "content": "x ( t + \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "is the precise", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 311, + 320, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 148, + 323 + ], + "score": 1.0, + "content": "result and", + "type": "text" + }, + { + "bbox": [ + 148, + 313, + 165, + 324 + ], + "score": 0.87, + "content": "x _ { t , \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 311, + 278, + 323 + ], + "score": 1.0, + "content": "is the numerical result from", + "type": "text" + }, + { + "bbox": [ + 279, + 313, + 283, + 321 + ], + "score": 0.8, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 311, + 295, + 323 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 295, + 312, + 317, + 322 + ], + "score": 0.9, + "content": "t + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 311, + 320, + 323 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 299, + 505, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 263, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 263, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 263, + 341 + ], + "score": 1.0, + "content": "For linear multi-step method, we have:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 328, + 263, + 341 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 340, + 483, + 421 + ], + "lines": [ + { + "bbox": [ + 111, + 340, + 483, + 421 + ], + "spans": [ + { + "bbox": [ + 111, + 340, + 483, + 421 + ], + "score": 0.9, + "content": "\\begin{array} { l } { \\displaystyle e _ { t , \\delta } = x ( t + \\delta ) - x _ { t , \\delta } } \\\\ { \\displaystyle = \\Bigg ( x ( t ) + \\frac { \\delta } { 1 ! } f ( x ( t ) , t ) + \\frac { \\delta ^ { 2 } } { 2 ! } f ^ { \\prime } ( x ( t ) , t ) + \\cdots + \\frac { \\delta ^ { 4 } } { 4 ! } f ^ { ( 3 ) } ( x ( t ) , t ) + O \\left( \\delta ^ { 5 } \\right) \\Bigg ) } \\\\ { \\displaystyle \\quad - \\left( x ( t ) + b _ { 1 } \\delta f ( x ( t ) , t ) + \\sum _ { s = 2 } ^ { 4 } b _ { s } \\delta \\left( \\sum _ { k = 0 } ^ { 4 } \\frac { ( - ( s - 1 ) \\delta ) ^ { k } } { ( k ) ! } f ^ { ( k - 1 ) } ( x ( t ) , t ) + O \\left( \\delta ^ { 5 } \\right) \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "a1924594e76b54f7037b6af8d94f6021242cd86fd9603ce9fbcc17d7a2ab0a8e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 111, + 340, + 483, + 367.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 111, + 367.0, + 483, + 394.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 111, + 394.0, + 483, + 421.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 429, + 449 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 425, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 131, + 453 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 436, + 150, + 449 + ], + "score": 0.92, + "content": "\\{ b _ { s } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 432, + 180, + 453 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + }, + { + "bbox": [ + 181, + 435, + 235, + 450 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { s = 1 } ^ { 4 } b _ { s } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 432, + 363, + 453 + ], + "score": 1.0, + "content": "and the following equations for", + "type": "text" + }, + { + "bbox": [ + 364, + 437, + 425, + 449 + ], + "score": 0.87, + "content": "j \\in \\{ 1 , \\cdots , 3 \\}", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 432, + 425, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 450, + 389, + 476 + ], + "lines": [ + { + "bbox": [ + 221, + 450, + 389, + 476 + ], + "spans": [ + { + "bbox": [ + 221, + 450, + 389, + 476 + ], + "score": 0.87, + "content": "( - 1 ) ^ { j } b _ { 2 } + ( - 2 ) ^ { j } b _ { 3 } + ( - 3 ) ^ { j } b _ { 4 } = \\frac { 1 } { j + 1 } .", + "type": "interline_equation", + "image_path": "55885cd55f64a66acd21027b81db647a019bf5552d6c51d5d2788c357bfa80c2.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 221, + 450, + 389, + 476 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 504, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 185, + 494 + ], + "score": 1.0, + "content": "We call the error at", + "type": "text" + }, + { + "bbox": [ + 185, + 483, + 201, + 493 + ], + "score": 0.88, + "content": "x _ { t , \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 481, + 311, + 494 + ], + "score": 1.0, + "content": "local error, and the error at", + "type": "text" + }, + { + "bbox": [ + 312, + 483, + 340, + 493 + ], + "score": 0.88, + "content": "x _ { t + M \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 482, + 356, + 492 + ], + "score": 0.36, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "is big enough but finite) global error.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 293, + 506 + ], + "score": 1.0, + "content": "Assume the local error of our method has order", + "type": "text" + }, + { + "bbox": [ + 294, + 493, + 311, + 503 + ], + "score": 0.78, + "content": "_ { \\mathrm { k + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 491, + 506, + 506 + ], + "score": 1.0, + "content": "and the target ODE satisfies Lipschitz condition,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 503, + 130, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 130, + 515 + ], + "score": 1.0, + "content": "then:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 481, + 506, + 515 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 512, + 461, + 627 + ], + "lines": [ + { + "bbox": [ + 150, + 512, + 461, + 627 + ], + "spans": [ + { + "bbox": [ + 150, + 512, + 461, + 627 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { x ( t + M \\delta ) - x _ { t , M \\delta } \\leq \\vert x ( t + M \\delta ) - x _ { t + ( M - 1 ) \\delta , \\delta } \\vert + \\vert x _ { t + ( M - 1 ) \\delta , \\delta } - x _ { t , M \\delta } \\vert } \\\\ & { \\qquad \\leq e _ { t + ( M - 1 ) \\delta , \\delta } + e ^ { L \\delta } \\vert x ( t + ( M - 1 ) \\delta ) - x _ { t , ( M - 1 ) \\delta } \\vert } \\\\ & { \\qquad \\leq e _ { t + ( M - 1 ) \\delta , \\delta } + e ^ { L \\delta } e _ { t + ( M - 2 ) \\delta , \\delta } + \\cdots } \\\\ & { \\qquad \\leq C \\delta ^ { k + 1 } \\left( 1 + e ^ { L h } + \\cdots + e ^ { ( i - 1 ) L h } \\right) } \\\\ & { \\qquad = C h ^ { k + 1 } \\frac { e ^ { i L \\delta } - 1 } { e ^ { L \\delta } - 1 } \\leq C \\delta ^ { k + 1 } \\frac { e ^ { i L \\delta } - 1 } { L \\delta } } \\\\ & { \\qquad = O ( \\delta ^ { k } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "bbdf21d2145f10416c60b2785734376cc8056e32d32ef105bcfbf840e614051d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 150, + 512, + 461, + 550.3333333333334 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 150, + 550.3333333333334, + 461, + 588.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 150, + 588.6666666666667, + 461, + 627.0000000000001 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "Therefore, the global error will be one order lower than the local error. From Equation (15), we can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 283, + 655 + ], + "score": 1.0, + "content": "see the forward Euler method has local error", + "type": "text" + }, + { + "bbox": [ + 283, + 643, + 309, + 655 + ], + "score": 0.92, + "content": "O ( \\delta ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 643, + 375, + 655 + ], + "score": 1.0, + "content": "and global error", + "type": "text" + }, + { + "bbox": [ + 375, + 644, + 396, + 655 + ], + "score": 0.89, + "content": "O ( \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 643, + 505, + 655 + ], + "score": 1.0, + "content": ", so we call it the first-order", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 381, + 667 + ], + "score": 1.0, + "content": "numerical method. And the linear multi-step method has local error", + "type": "text" + }, + { + "bbox": [ + 381, + 654, + 407, + 666 + ], + "score": 0.92, + "content": "{ \\dot { O } } ( \\delta ^ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 653, + 475, + 667 + ], + "score": 1.0, + "content": "and global error", + "type": "text" + }, + { + "bbox": [ + 475, + 654, + 501, + 666 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\delta ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 653, + 505, + 667 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 666, + 305, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 305, + 676 + ], + "score": 1.0, + "content": "and we call it the fourth-order numerical method.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 631, + 506, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 682, + 503, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 504, + 695 + ], + "score": 1.0, + "content": "In addition, assuming that a numerical method has kth-order global error, we can compute the con-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 693, + 275, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 275, + 705 + ], + "score": 1.0, + "content": "vergence speed of this numerical method:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 681, + 504, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 705, + 384, + 735 + ], + "lines": [ + { + "bbox": [ + 227, + 705, + 384, + 735 + ], + "spans": [ + { + "bbox": [ + 227, + 705, + 384, + 735 + ], + "score": 0.94, + "content": "\\operatorname* { l i m } _ { \\delta 0 } \\frac { x _ { t + T } ^ { 2 \\delta } - x ( t + T ) } { x _ { t + T } ^ { \\delta } - x ( t + T ) } = \\frac { ( 2 \\delta ) ^ { k } } { \\delta ^ { k } } = 2 ^ { k } .", + "type": "interline_equation", + "image_path": "685bb3df44c9b75e9f21adc907ec8f60ecdc68db6a506c09864fa43b7327c4b7.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 227, + 705, + 384, + 720.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 227, + 720.0, + 384, + 735.0 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 131, + 97 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 81, + 153, + 96 + ], + "score": 0.93, + "content": "x _ { t + T } ^ { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 80, + 212, + 97 + ], + "score": 1.0, + "content": "is the result at", + "type": "text" + }, + { + "bbox": [ + 213, + 83, + 230, + 93 + ], + "score": 0.81, + "content": "\\mathrm { t } { + } \\mathrm { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 80, + 271, + 97 + ], + "score": 1.0, + "content": "and move", + "type": "text" + }, + { + "bbox": [ + 272, + 83, + 278, + 92 + ], + "score": 0.74, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 80, + 507, + 97 + ], + "score": 1.0, + "content": "every step. This shows that the fourth-order method can", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 387, + 106 + ], + "score": 1.0, + "content": "converge to the exact solution faster than the first-order method when", + "type": "text" + }, + { + "bbox": [ + 387, + 94, + 414, + 104 + ], + "score": 0.9, + "content": "\\delta 0", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 94, + 505, + 106 + ], + "score": 1.0, + "content": ", which means that we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 246, + 117 + ], + "score": 1.0, + "content": "can use a bigger iteration interval", + "type": "text" + }, + { + "bbox": [ + 247, + 105, + 253, + 114 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "to achieve similar global error and a bigger iteration interval", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 385, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 385, + 128 + ], + "score": 1.0, + "content": "means that we can iterate fewer times to get results with high quality.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 109, + 140, + 284, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 284, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 284, + 152 + ], + "score": 1.0, + "content": "A.3 PSEUDO SECOND-ORDER METHOD", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 486, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 487, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 487, + 175 + ], + "score": 1.0, + "content": "We introduce two second-order numerical methods. First is improved Euler method satisfying:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 177, + 366, + 220 + ], + "lines": [ + { + "bbox": [ + 245, + 177, + 366, + 220 + ], + "spans": [ + { + "bbox": [ + 245, + 177, + 366, + 220 + ], + "score": 0.93, + "content": "\\left\\{ \\begin{array} { r l } & { k _ { 1 } = f ( x _ { t } , t ) } \\\\ & { k _ { 2 } = f ( x _ { t } + \\delta k _ { 1 } , t + \\delta ) } \\\\ & { x _ { t + \\delta } = x _ { t } + \\frac { \\delta } { 2 } ( k _ { 1 } + k _ { 2 } ) } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "c1ba7dd912872baaca2453aa54a576252ff9a919d946c753d9daf8d55b8bbbff.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 177, + 366, + 198.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 245, + 198.5, + 366, + 220.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 228, + 504, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 243 + ], + "score": 1.0, + "content": "Second is another linear multi-step method called second-order linear multi-step method satisfying:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 243, + 364, + 268 + ], + "lines": [ + { + "bbox": [ + 247, + 243, + 364, + 268 + ], + "spans": [ + { + "bbox": [ + 247, + 243, + 364, + 268 + ], + "score": 0.93, + "content": "x _ { t + \\delta } = x _ { t } + \\frac \\delta 2 ( 3 f _ { t } - f _ { t - \\delta } )", + "type": "interline_equation", + "image_path": "e396cac90746b9dba1d908cb5d77749c6b44140889c114562c0d0788b2770af2.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 247, + 243, + 364, + 268 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 375, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 376, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 376, + 292 + ], + "score": 1.0, + "content": "And the corresponding pseudo improved Euler methods satisfying:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 292, + 367, + 362 + ], + "lines": [ + { + "bbox": [ + 244, + 292, + 367, + 362 + ], + "spans": [ + { + "bbox": [ + 244, + 292, + 367, + 362 + ], + "score": 0.94, + "content": "\\left\\{ \\begin{array} { l l } { \\ { e } _ { t } ^ { 1 } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ { \\ { x } _ { t } ^ { 1 } = \\phi ( x _ { t } , { e } _ { t } ^ { 1 } , t , t + \\delta ) } \\\\ { \\ { e } _ { t } ^ { 2 } = \\epsilon _ { \\theta } ( { x } _ { t } ^ { 1 } , t + \\delta ) } \\\\ { \\ { e } _ { t } ^ { \\prime } = \\frac { 1 } { 2 } ( { e } _ { t } ^ { 1 } + { e } _ { t } ^ { 2 } ) } \\\\ { \\ { x } _ { t + \\delta } = \\phi ( { x } _ { t } , { e } _ { t } ^ { \\prime } , t , t + \\delta ) } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a0c5107fc32627c9ab88b02804bdaafdb85aa5aee1e9d2970fed08491231ef9f.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 292, + 367, + 327.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 244, + 327.0, + 367, + 362.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 336, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 368, + 337, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 337, + 386 + ], + "score": 1.0, + "content": "Pseudo second-order linear multi-step method satisfying:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 387, + 368, + 429 + ], + "lines": [ + { + "bbox": [ + 244, + 387, + 368, + 429 + ], + "spans": [ + { + "bbox": [ + 244, + 387, + 368, + 429 + ], + "score": 0.9, + "content": "\\left\\{ \\begin{array} { r l } & { e _ { t } = \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { e _ { t } ^ { \\prime } = \\frac { 1 } { 2 } ( 3 e _ { t } - e _ { t - \\delta } ) } \\\\ & { x _ { t + \\delta } = \\phi ( x _ { t } , e _ { t } ^ { \\prime } , t , t + \\delta ) } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "0d51619c6b74fffd24ebd31e1c802e1bfc1f6153633cd4768bdc31a0582e524a.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 387, + 368, + 408.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 244, + 408.0, + 368, + 429.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 314, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 315, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 315, + 455 + ], + "score": 1.0, + "content": "Similar to what we do to get F-PNDMs, We com-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 453, + 315, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 315, + 466 + ], + "score": 1.0, + "content": "bine them to get S-PNDMs. Abbreviate Equation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 465, + 173, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 173, + 478 + ], + "score": 1.0, + "content": "(22) and (23) as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 325, + 445, + 422, + 457 + ], + "lines": [ + { + "bbox": [ + 324, + 444, + 423, + 459 + ], + "spans": [ + { + "bbox": [ + 324, + 444, + 423, + 459 + ], + "score": 1.0, + "content": "Algorithm 3 S-PNDMs", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 481, + 290, + 513 + ], + "lines": [ + { + "bbox": [ + 131, + 481, + 290, + 513 + ], + "spans": [ + { + "bbox": [ + 131, + 481, + 290, + 513 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { x _ { t + \\delta } , e _ { t } = P I E ( x _ { t } , \\{ e _ { p } \\} _ { p < t } , t , t + \\delta ) , } \\\\ & { x _ { t + \\delta } , e _ { t } ^ { 1 } = P L M S ^ { \\prime } ( x _ { t } , t , t + \\delta ) . } \\end{array}", + "type": "interline_equation", + "image_path": "2d44e01738cf7d8a7de4bf66ec503c71e7eca25a14e61186da8a9e0c621e4d1d.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 131, + 481, + 290, + 497.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 131, + 497.0, + 290, + 513.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 326, + 460, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 327, + 460, + 393, + 473 + ], + "spans": [ + { + "bbox": [ + 327, + 460, + 339, + 473 + ], + "score": 1.0, + "content": "1:", + "type": "text" + }, + { + "bbox": [ + 339, + 461, + 393, + 471 + ], + "score": 0.4, + "content": "x _ { T } \\sim \\mathcal { N } ( 0 , I )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 327, + 470, + 405, + 481 + ], + "spans": [ + { + "bbox": [ + 327, + 470, + 405, + 481 + ], + "score": 1.0, + "content": "2: for t = T − 1 do", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 327, + 480, + 458, + 492 + ], + "spans": [ + { + "bbox": [ + 327, + 480, + 349, + 492 + ], + "score": 1.0, + "content": "3:", + "type": "text" + }, + { + "bbox": [ + 349, + 482, + 458, + 492 + ], + "score": 0.8, + "content": "x _ { t } , e _ { t } = P I E ( x _ { t + 1 } , t + 1 , t )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 327, + 489, + 370, + 502 + ], + "spans": [ + { + "bbox": [ + 327, + 489, + 370, + 502 + ], + "score": 1.0, + "content": "4: end for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 327, + 500, + 441, + 511 + ], + "spans": [ + { + "bbox": [ + 327, + 500, + 353, + 511 + ], + "score": 1.0, + "content": "5: for", + "type": "text" + }, + { + "bbox": [ + 353, + 501, + 429, + 511 + ], + "score": 0.78, + "content": "t = T - 2 , \\cdots , 1 , 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 500, + 441, + 511 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 327, + 511, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 327, + 511, + 338, + 521 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 350, + 511, + 504, + 522 + ], + "score": 0.84, + "content": "x _ { t } , e _ { t } = P L M S ^ { \\prime } ( x _ { t + 1 } , \\{ e _ { p } \\} _ { p > t } , t + 1 , t )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 326, + 519, + 370, + 531 + ], + "spans": [ + { + "bbox": [ + 326, + 519, + 370, + 531 + ], + "score": 1.0, + "content": "7: end for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 326, + 529, + 379, + 543 + ], + "spans": [ + { + "bbox": [ + 326, + 529, + 368, + 543 + ], + "score": 1.0, + "content": "8: return", + "type": "text" + }, + { + "bbox": [ + 369, + 533, + 379, + 541 + ], + "score": 0.27, + "content": "x _ { 0 }", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 551, + 281, + 563 + ], + "lines": [ + { + "bbox": [ + 106, + 551, + 282, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 282, + 563 + ], + "score": 1.0, + "content": "A.4 THE EXISTENCE OF A DERIVATIVE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 105, + 571, + 461, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 462, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 142, + 585 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 142, + 573, + 153, + 583 + ], + "score": 0.88, + "content": "\\bar { \\alpha } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 571, + 398, + 585 + ], + "score": 1.0, + "content": "is usually obtained by multiplying a linear variance schedule", + "type": "text" + }, + { + "bbox": [ + 398, + 572, + 409, + 583 + ], + "score": 0.86, + "content": "\\beta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 571, + 462, + 585 + ], + "score": 1.0, + "content": ". So we have", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 272, + 587, + 339, + 603 + ], + "lines": [ + { + "bbox": [ + 272, + 587, + 339, + 603 + ], + "spans": [ + { + "bbox": [ + 272, + 587, + 339, + 603 + ], + "score": 0.91, + "content": "\\bar { \\alpha } _ { t } = e ^ { a t ^ { 2 } + b t + c } ,", + "type": "interline_equation", + "image_path": "ae412bbce2ff3387a7045e0fc900ab3ad1cc7e8fdc973cffd08f2eeb416d3c74.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 272, + 587, + 339, + 603 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 253, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 254, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 123, + 620 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 609, + 154, + 619 + ], + "score": 0.92, + "content": "\\bar { \\alpha } _ { 0 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 607, + 169, + 620 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 169, + 609, + 192, + 618 + ], + "score": 0.88, + "content": "c = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 607, + 254, + 620 + ], + "score": 1.0, + "content": ". 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This shows that the fourth-order method can", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 387, + 106 + ], + "score": 1.0, + "content": "converge to the exact solution faster than the first-order method when", + "type": "text" + }, + { + "bbox": [ + 387, + 94, + 414, + 104 + ], + "score": 0.9, + "content": "\\delta 0", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 94, + 505, + 106 + ], + "score": 1.0, + "content": ", which means that we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 246, + 117 + ], + "score": 1.0, + "content": "can use a bigger iteration interval", + "type": "text" + }, + { + "bbox": [ + 247, + 105, + 253, + 114 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "to achieve similar global error and a bigger iteration interval", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 385, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 385, + 128 + ], + "score": 1.0, + "content": "means that we can iterate fewer times to get results with high quality.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 80, + 507, + 128 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 140, + 284, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 284, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 284, + 152 + ], + "score": 1.0, + "content": "A.3 PSEUDO SECOND-ORDER METHOD", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 486, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 487, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 487, + 175 + ], + "score": 1.0, + "content": "We introduce two second-order numerical methods. 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Let", + "type": "text" + }, + { + "bbox": [ + 433, + 524, + 504, + 536 + ], + "score": 0.89, + "content": "\\phi ( x ( t ) , \\epsilon , t , \\delta ) \\ =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 232, + 547 + ], + "score": 0.92, + "content": "f ( x ( t ) , t , \\delta ) + g ( t , \\delta ) \\epsilon ( x ( t ) , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 534, + 343, + 548 + ], + "score": 1.0, + "content": "and we have the property", + "type": "text" + }, + { + "bbox": [ + 344, + 535, + 459, + 547 + ], + "score": 0.93, + "content": "f ( x ( t ) , \\dot { } t , 0 ) = g ( t , 0 ) = \\dot { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 534, + 506, + 548 + ], + "score": 1.0, + "content": ". 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In Figure 5, we can see that the denoising process tends to converge,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 165, + 186 + ], + "score": 1.0, + "content": "whether in the", + "type": "text" + }, + { + "bbox": [ + 165, + 175, + 175, + 185 + ], + "score": 0.85, + "content": "\\epsilon _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "domain or the sample/image domain when the step-index tends to zero. 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Because we assume that the gradient part is precise, then", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 386, + 289, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 141, + 398 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 142, + 388, + 169, + 397 + ], + "score": 0.9, + "content": "\\epsilon _ { \\theta } = \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 386, + 222, + 398 + ], + "score": 1.0, + "content": ". Then for all", + "type": "text" + }, + { + "bbox": [ + 222, + 387, + 247, + 397 + ], + "score": 0.9, + "content": "t ^ { \\prime } \\leq t", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 386, + 289, + 398 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 364, + 505, + 398 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 401, + 447, + 474 + ], + "lines": [ + { + "bbox": [ + 163, + 401, + 447, + 474 + ], + "spans": [ + { + "bbox": [ + 163, + 401, + 447, + 474 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { x _ { t ^ { \\prime } } = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } \\left( \\frac { x _ { t } - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon _ { \\theta } } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon _ { \\theta } } \\\\ & { ~ = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } \\left( \\frac { \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon - \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon } { \\sqrt { \\bar { \\alpha } _ { t } } } \\right) + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon _ { \\theta } ( x _ { t } , t ) } \\\\ & { ~ = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon . } \\end{array}", + "type": "interline_equation", + "image_path": "cf1fb35fa123cfd26a6460ee520f33dcd0e7ba12c57a1e7c4f80abe19932b643.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 163, + 401, + 447, + 425.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 163, + 425.3333333333333, + 447, + 449.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 163, + 449.66666666666663, + 447, + 473.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 463, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 464, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 196, + 492 + ], + "score": 1.0, + "content": "Here, we can find that", + "type": "text" + }, + { + "bbox": [ + 197, + 478, + 308, + 491 + ], + "score": 0.93, + "content": "x _ { t ^ { \\prime } } = \\sqrt { \\bar { \\alpha } _ { t ^ { \\prime } } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t ^ { \\prime } } } \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 478, + 464, + 492 + ], + "score": 1.0, + "content": "is also precise, so Property 3.1 is true.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 478, + 464, + 492 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 503, + 304, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 304, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 304, + 516 + ], + "score": 1.0, + "content": "A.6 ORDER ANALYSIS OF PSEUDO METHOD", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 504, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 433, + 537 + ], + "score": 1.0, + "content": "For the convenience of theoretical analysis, we generalize the problem. Let", + "type": "text" + }, + { + "bbox": [ + 433, + 524, + 504, + 536 + ], + "score": 0.89, + "content": "\\phi ( x ( t ) , \\epsilon , t , \\delta ) \\ =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 232, + 547 + ], + "score": 0.92, + "content": "f ( x ( t ) , t , \\delta ) + g ( t , \\delta ) \\epsilon ( x ( t ) , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 534, + 343, + 548 + ], + "score": 1.0, + "content": "and we have the property", + "type": "text" + }, + { + "bbox": [ + 344, + 535, + 459, + 547 + ], + "score": 0.93, + "content": "f ( x ( t ) , \\dot { } t , 0 ) = g ( t , 0 ) = \\dot { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 534, + 506, + 548 + ], + "score": 1.0, + "content": ". 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We split it into three parts and the values of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 224, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 224, + 551 + ], + "score": 1.0, + "content": "the first two terms. 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We split it into three parts and the values of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 224, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 224, + 551 + ], + "score": 1.0, + "content": "the first two terms. We have:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 528, + 506, + 551 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 549, + 468, + 614 + ], + "lines": [ + { + "bbox": [ + 126, + 549, + 468, + 614 + ], + "spans": [ + { + "bbox": [ + 126, + 549, + 468, + 614 + ], + "score": 0.87, + "content": "\\begin{array} { l } { { ( \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial t } - \\frac { \\partial ^ { 2 } f } { \\partial \\delta ^ { 2 } } ) ( x ( t ) , t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\frac { \\partial f } { \\partial \\delta } ( x ( t ) , t , 0 ) } } \\\\ { { = \\left( \\frac { \\alpha ^ { \\prime \\prime } ( t ) } { 2 \\alpha ( t ) } x ( t ) - \\frac { \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 2 \\alpha ( t ) ^ { 2 } } x ( t ) \\right) - \\left( \\frac { \\alpha ^ { \\prime \\prime } ( t ) } { 2 \\alpha ( t ) } x ( t ) + \\frac { - \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) ^ { 2 } } x ( t ) \\right) + \\frac { 1 } { 2 \\alpha ( t ) } \\frac { \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } x ( t ) } } \\\\ { { = 0 } } \\end{array}", + "type": "interline_equation", + "image_path": "cf206a2405b8218f0b7ff48c2f88a9ef8c2be7e33aa6db61ce07251a80cfa06a.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 126, + 549, + 468, + 570.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 126, + 570.6666666666666, + 468, + 592.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 126, + 592.3333333333333, + 468, + 613.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 618, + 123, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 123, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 123, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 618, + 123, + 628 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 627, + 477, + 735 + ], + "lines": [ + { + "bbox": [ + 116, + 627, + 477, + 735 + ], + "spans": [ + { + "bbox": [ + 116, + 627, + 477, + 735 + ], + "score": 0.92, + "content": "\\begin{array} { l } { \\displaystyle ( \\frac { \\partial ^ { 2 } g } { \\partial \\delta \\partial t } - \\frac { \\partial ^ { 2 } g } { \\partial \\delta ^ { 2 } } ) ( t , 0 ) + \\frac { \\partial ^ { 2 } f } { \\partial \\delta \\partial x } ( x ( t ) , t , 0 ) \\frac { \\partial g } { \\partial \\delta } ( t , 0 ) } \\\\ { = \\frac { \\sqrt { 1 - \\alpha ( t ) } \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) ^ { 2 } } + \\frac { \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) \\sqrt { 1 - \\alpha ( t ) } } + \\frac { \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } \\left( \\frac { - \\alpha ^ { \\prime } ( t ) } { 2 \\sqrt { 1 - \\alpha ( ( t ) ) } } - \\frac { \\sqrt { 1 - \\alpha ( t ) } \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } \\right) } \\\\ { = \\frac { \\alpha ^ { \\prime } ( t ) ^ { 2 } } { 4 \\alpha ( t ) ^ { 2 } \\sqrt { 1 - \\alpha ( t ) } } + \\frac { \\alpha ^ { \\prime } ( t ) } { 2 \\alpha ( t ) } \\left( \\frac { - \\alpha ^ { \\prime } ( t ) } { 2 \\sqrt { 1 - \\alpha ( t ) } \\alpha ( t ) } \\right) } \\\\ { = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "b384a472b637fedc0ff2a4b66fe60a45a24e8597bb5bef520412e9e03f6fa110.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 116, + 627, + 477, + 663.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 116, + 663.0, + 477, + 699.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 116, + 699.0, + 477, + 735.0 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 297, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 299, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 299, + 97 + ], + "score": 1.0, + "content": "Finally, we get the final result of Equation (31):", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 97, + 380, + 111 + ], + "lines": [ + { + "bbox": [ + 230, + 97, + 380, + 111 + ], + "spans": [ + { + "bbox": [ + 230, + 97, + 380, + 111 + ], + "score": 0.92, + "content": "x ( t + \\delta ) - x _ { \\mathrm { S - P N D M } } ( x + \\delta ) = O ( \\delta ^ { 3 } )", + "type": "interline_equation", + "image_path": "49da3801dea17615bb4fac992473becb8a1fe6ebfd820c88bd4fe160a57e9b12.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 230, + 97, + 380, + 111 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "And the computation of the convergence order of F-PNDMs is similar, and we ignore it here. There-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 209, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 209, + 138 + ], + "score": 1.0, + "content": "fore, Property 3.2 is true.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 151, + 227, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 229, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 229, + 163 + ], + "score": 1.0, + "content": "A.7 VARIANCE SCHEDULE", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "score": 1.0, + "content": "According to Cifar10 (cosine) in Table 2, PNDMs can be used on both linear variance schedule and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "score": 1.0, + "content": "cosine variance schedule. However, we also notice cosine variance schedule can make FID lower", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 193, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 506, + 205 + ], + "score": 1.0, + "content": "when we use relatively big generation steps, but the effort is limited when the number of steps is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 205, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 216 + ], + "score": 1.0, + "content": "small. F-PNDM uses information from four consecutive steps, so the smoothness of the schedule is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "more important for F-PNDM than DDIM. According to this experiment, our work can be used with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "works that pay attention to variance schedules to improve the acceleration effect further. However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 501, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 501, + 248 + ], + "score": 1.0, + "content": "a variance schedule that fits pseudo numerical methods better remains to be found in further work.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 261, + 196, + 273 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 198, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 198, + 274 + ], + "score": 1.0, + "content": "A.8 TOY EXAMPLE", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "Here, we design a toy example to test our new methods without the influence of neural networks.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 266, + 306 + ], + "score": 1.0, + "content": "We randomly generate the initial input", + "type": "text" + }, + { + "bbox": [ + 267, + 293, + 332, + 305 + ], + "score": 0.88, + "content": "x _ { 1 } = ( m _ { 1 } , m _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 292, + 339, + 306 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 339, + 293, + 397, + 305 + ], + "score": 0.92, + "content": "m _ { i } \\sim U ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "and use a simple analytic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 304, + 147, + 317 + ], + "score": 1.0, + "content": "equations", + "type": "text" + }, + { + "bbox": [ + 147, + 304, + 256, + 316 + ], + "score": 0.91, + "content": "\\epsilon _ { \\theta } ( x ) = ( \\sin x [ 0 ] , \\cos x [ 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 304, + 448, + 317 + ], + "score": 1.0, + "content": "to replace the neural networks in real cases. Let", + "type": "text" + }, + { + "bbox": [ + 448, + 305, + 455, + 316 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "in Equation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 315, + 329, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 197, + 329 + ], + "score": 1.0, + "content": "(11) is unchanged and", + "type": "text" + }, + { + "bbox": [ + 198, + 316, + 273, + 327 + ], + "score": 0.9, + "content": "\\bar { \\alpha } _ { t } = \\alpha ( t ) = 1 - t", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 315, + 329, + 329 + ], + "score": 1.0, + "content": ", then we get:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 344, + 459, + 426 + ], + "lines": [ + { + "bbox": [ + 151, + 344, + 459, + 426 + ], + "spans": [ + { + "bbox": [ + 151, + 344, + 459, + 426 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } ) , t , t - \\delta ) } \\\\ & { = \\frac { \\sqrt { \\bar { \\alpha } _ { t - \\delta } } } { \\sqrt { \\bar { \\alpha } _ { t } } } x _ { t } - \\frac { \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { \\left( 1 - \\bar { \\alpha } _ { t - \\delta } \\right) \\bar { \\alpha } _ { t } } + \\sqrt { \\left( 1 - \\bar { \\alpha } _ { t } \\right) \\bar { \\alpha } _ { t - \\delta } } \\right) } \\epsilon _ { t } } \\\\ & { = \\frac { \\sqrt { 1 - \\left( t - \\delta \\right) } } { \\sqrt { 1 - t } } x _ { t } - \\frac { \\delta } { \\sqrt { 1 - t } \\left( \\sqrt { \\left( t - \\delta \\right) \\left( 1 - t \\right) } + \\sqrt { t ( 1 - \\left( t - \\delta \\right) ) } \\right) } \\epsilon _ { \\theta } ( x _ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "de0a509dc540c55f66c0825c694793e78d313e740f1d50ef40cdc8882856f2cd.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 151, + 344, + 459, + 371.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 151, + 371.3333333333333, + 459, + 398.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 151, + 398.66666666666663, + 459, + 425.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 359, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 356, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 344, + 443 + ], + "score": 1.0, + "content": "Here, we use three different numerical methods to generate", + "type": "text" + }, + { + "bbox": [ + 345, + 431, + 356, + 441 + ], + "score": 0.86, + "content": "x _ { 0 }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 191, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 192, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 192, + 459 + ], + "score": 1.0, + "content": "For DDIM, we have:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 456, + 377, + 469 + ], + "lines": [ + { + "bbox": [ + 234, + 456, + 377, + 469 + ], + "spans": [ + { + "bbox": [ + 234, + 456, + 377, + 469 + ], + "score": 0.9, + "content": "x _ { t - \\delta } = x _ { t } + \\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } ) , t , t - \\delta )", + "type": "interline_equation", + "image_path": "68e0ddf0f84f3be3ba5cb1f8825f9ecb50a3047e5abc9a718026fee0d3d66e75.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 234, + 456, + 377, + 469 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 297, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 297, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 297, + 484 + ], + "score": 1.0, + "content": "For FON, according to Equation (10), we have:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 485, + 415, + 574 + ], + "lines": [ + { + "bbox": [ + 198, + 485, + 415, + 574 + ], + "spans": [ + { + "bbox": [ + 198, + 485, + 415, + 574 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle e _ { t } ^ { \\prime } = \\bar { \\alpha } ^ { \\prime } ( t ) \\left( \\frac { x _ { t } } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } \\right) } } \\\\ { { \\displaystyle ~ = - \\left( \\frac { x _ { t } } { 2 ( 1 - t ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } ) } { 2 ( 1 - t ) \\sqrt { t } } \\right) } } \\\\ { { \\displaystyle x _ { t - \\delta } = x _ { t } + \\frac { \\delta } { 2 4 } ( 5 5 e _ { t } ^ { \\prime } - 5 9 e _ { t + \\delta } ^ { \\prime } + 3 7 e _ { t + 2 \\delta } ^ { \\prime } - 9 e _ { t + 3 \\delta } ^ { \\prime } ) } } \\end{array}", + "type": "interline_equation", + "image_path": "8c72a4506a485312ef992ab12e0b29a8c9b1a29b0859aaad1261cc9f4c9f5bea.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 485, + 415, + 499.8333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 198, + 499.8333333333333, + 415, + 514.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 198, + 514.6666666666666, + 415, + 529.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 198, + 529.5, + 415, + 544.3333333333334 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 198, + 544.3333333333334, + 415, + 559.1666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 198, + 559.1666666666667, + 415, + 574.0000000000001 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 575, + 202, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 203, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 203, + 588 + ], + "score": 1.0, + "content": "For F-PNDM, we have:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 590, + 433, + 629 + ], + "lines": [ + { + "bbox": [ + 178, + 590, + 433, + 629 + ], + "spans": [ + { + "bbox": [ + 178, + 590, + 433, + 629 + ], + "score": 0.92, + "content": "\\begin{array} { c } { { e ^ { \\prime } = \\displaystyle \\frac { 1 } { 2 4 } \\big ( 5 5 \\epsilon _ { \\theta } ( x _ { t } ) - 5 9 \\epsilon _ { \\theta } ( x _ { t + \\delta } ) + 3 7 \\epsilon _ { \\theta } ( x _ { t + 2 \\delta } ) - 9 \\epsilon _ { \\theta } ( x _ { t + 3 \\delta } ) \\big ) } } \\\\ { { x _ { t - \\delta } = x _ { t } + \\phi ( x _ { t } , e ^ { \\prime } , t , t - \\delta ) } } \\end{array}", + "type": "interline_equation", + "image_path": "e98b5d5de615dbbac51d6d8c4a87f94e518b7a6175fde5e5eb65433b2ce2fe24.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 178, + 590, + 433, + 603.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 178, + 603.0, + 433, + 616.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 178, + 616.0, + 433, + 629.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "Then we draw the corresponding generation curves in Figure 6. We find that the result is similar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "to the real cases. The main difference here is that FON can correct its results while the real case", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "cannot. The reason is that the gradient is well-defined everywhere, while in real cases, the gradient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 664, + 376, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 315, + 677 + ], + "score": 1.0, + "content": "is meaningful on the high-density region of the data", + "type": "text" + }, + { + "bbox": [ + 315, + 666, + 326, + 675 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 664, + 376, + 677 + ], + "score": 1.0, + "content": "of DDPMs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 222, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 223, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 223, + 702 + ], + "score": 1.0, + "content": "A.9 MORE FID RESULTS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Here, We provide our more detailed FID results on Cifar10, CelebA, LSUN-church and LSUN-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 219, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 219, + 732 + ], + "score": 1.0, + "content": "bedroom in Table 3, 4, 5, 6.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "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": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 297, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 299, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 299, + 97 + ], + "score": 1.0, + "content": "Finally, we get the final result of Equation (31):", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 299, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 97, + 380, + 111 + ], + "lines": [ + { + "bbox": [ + 230, + 97, + 380, + 111 + ], + "spans": [ + { + "bbox": [ + 230, + 97, + 380, + 111 + ], + "score": 0.92, + "content": "x ( t + \\delta ) - x _ { \\mathrm { S - P N D M } } ( x + \\delta ) = O ( \\delta ^ { 3 } )", + "type": "interline_equation", + "image_path": "49da3801dea17615bb4fac992473becb8a1fe6ebfd820c88bd4fe160a57e9b12.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 230, + 97, + 380, + 111 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 114, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "And the computation of the convergence order of F-PNDMs is similar, and we ignore it here. There-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 209, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 209, + 138 + ], + "score": 1.0, + "content": "fore, Property 3.2 is true.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 114, + 505, + 138 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 151, + 227, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 229, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 229, + 163 + ], + "score": 1.0, + "content": "A.7 VARIANCE SCHEDULE", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "score": 1.0, + "content": "According to Cifar10 (cosine) in Table 2, PNDMs can be used on both linear variance schedule and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 194 + ], + "score": 1.0, + "content": "cosine variance schedule. However, we also notice cosine variance schedule can make FID lower", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 193, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 506, + 205 + ], + "score": 1.0, + "content": "when we use relatively big generation steps, but the effort is limited when the number of steps is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 205, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 216 + ], + "score": 1.0, + "content": "small. F-PNDM uses information from four consecutive steps, so the smoothness of the schedule is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "more important for F-PNDM than DDIM. According to this experiment, our work can be used with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "works that pay attention to variance schedules to improve the acceleration effect further. However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 501, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 501, + 248 + ], + "score": 1.0, + "content": "a variance schedule that fits pseudo numerical methods better remains to be found in further work.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 172, + 506, + 248 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 261, + 196, + 273 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 198, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 198, + 274 + ], + "score": 1.0, + "content": "A.8 TOY EXAMPLE", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "Here, we design a toy example to test our new methods without the influence of neural networks.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 266, + 306 + ], + "score": 1.0, + "content": "We randomly generate the initial input", + "type": "text" + }, + { + "bbox": [ + 267, + 293, + 332, + 305 + ], + "score": 0.88, + "content": "x _ { 1 } = ( m _ { 1 } , m _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 292, + 339, + 306 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 339, + 293, + 397, + 305 + ], + "score": 0.92, + "content": "m _ { i } \\sim U ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "and use a simple analytic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 304, + 147, + 317 + ], + "score": 1.0, + "content": "equations", + "type": "text" + }, + { + "bbox": [ + 147, + 304, + 256, + 316 + ], + "score": 0.91, + "content": "\\epsilon _ { \\theta } ( x ) = ( \\sin x [ 0 ] , \\cos x [ 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 304, + 448, + 317 + ], + "score": 1.0, + "content": "to replace the neural networks in real cases. Let", + "type": "text" + }, + { + "bbox": [ + 448, + 305, + 455, + 316 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "in Equation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 315, + 329, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 197, + 329 + ], + "score": 1.0, + "content": "(11) is unchanged and", + "type": "text" + }, + { + "bbox": [ + 198, + 316, + 273, + 327 + ], + "score": 0.9, + "content": "\\bar { \\alpha } _ { t } = \\alpha ( t ) = 1 - t", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 315, + 329, + 329 + ], + "score": 1.0, + "content": ", then we get:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 282, + 506, + 329 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 344, + 459, + 426 + ], + "lines": [ + { + "bbox": [ + 151, + 344, + 459, + 426 + ], + "spans": [ + { + "bbox": [ + 151, + 344, + 459, + 426 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } ) , t , t - \\delta ) } \\\\ & { = \\frac { \\sqrt { \\bar { \\alpha } _ { t - \\delta } } } { \\sqrt { \\bar { \\alpha } _ { t } } } x _ { t } - \\frac { \\left( \\bar { \\alpha } _ { t - \\delta } - \\bar { \\alpha } _ { t } \\right) } { \\sqrt { \\bar { \\alpha } _ { t } } \\left( \\sqrt { \\left( 1 - \\bar { \\alpha } _ { t - \\delta } \\right) \\bar { \\alpha } _ { t } } + \\sqrt { \\left( 1 - \\bar { \\alpha } _ { t } \\right) \\bar { \\alpha } _ { t - \\delta } } \\right) } \\epsilon _ { t } } \\\\ & { = \\frac { \\sqrt { 1 - \\left( t - \\delta \\right) } } { \\sqrt { 1 - t } } x _ { t } - \\frac { \\delta } { \\sqrt { 1 - t } \\left( \\sqrt { \\left( t - \\delta \\right) \\left( 1 - t \\right) } + \\sqrt { t ( 1 - \\left( t - \\delta \\right) ) } \\right) } \\epsilon _ { \\theta } ( x _ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "de0a509dc540c55f66c0825c694793e78d313e740f1d50ef40cdc8882856f2cd.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 151, + 344, + 459, + 371.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 151, + 371.3333333333333, + 459, + 398.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 151, + 398.66666666666663, + 459, + 425.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 429, + 359, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 356, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 344, + 443 + ], + "score": 1.0, + "content": "Here, we use three different numerical methods to generate", + "type": "text" + }, + { + "bbox": [ + 345, + 431, + 356, + 441 + ], + "score": 0.86, + "content": "x _ { 0 }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 426, + 356, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 191, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 192, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 192, + 459 + ], + "score": 1.0, + "content": "For DDIM, we have:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 444, + 192, + 459 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 456, + 377, + 469 + ], + "lines": [ + { + "bbox": [ + 234, + 456, + 377, + 469 + ], + "spans": [ + { + "bbox": [ + 234, + 456, + 377, + 469 + ], + "score": 0.9, + "content": "x _ { t - \\delta } = x _ { t } + \\phi ( x _ { t } , \\epsilon _ { \\theta } ( x _ { t } ) , t , t - \\delta )", + "type": "interline_equation", + "image_path": "68e0ddf0f84f3be3ba5cb1f8825f9ecb50a3047e5abc9a718026fee0d3d66e75.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 234, + 456, + 377, + 469 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 297, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 297, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 297, + 484 + ], + "score": 1.0, + "content": "For FON, according to Equation (10), we have:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 470, + 297, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 485, + 415, + 574 + ], + "lines": [ + { + "bbox": [ + 198, + 485, + 415, + 574 + ], + "spans": [ + { + "bbox": [ + 198, + 485, + 415, + 574 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle e _ { t } ^ { \\prime } = \\bar { \\alpha } ^ { \\prime } ( t ) \\left( \\frac { x _ { t } } { 2 \\bar { \\alpha } ( t ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } ) } { 2 \\bar { \\alpha } ( t ) \\sqrt { 1 - \\bar { \\alpha } ( t ) } } \\right) } } \\\\ { { \\displaystyle ~ = - \\left( \\frac { x _ { t } } { 2 ( 1 - t ) } - \\frac { \\epsilon _ { \\theta } ( x _ { t } ) } { 2 ( 1 - t ) \\sqrt { t } } \\right) } } \\\\ { { \\displaystyle x _ { t - \\delta } = x _ { t } + \\frac { \\delta } { 2 4 } ( 5 5 e _ { t } ^ { \\prime } - 5 9 e _ { t + \\delta } ^ { \\prime } + 3 7 e _ { t + 2 \\delta } ^ { \\prime } - 9 e _ { t + 3 \\delta } ^ { \\prime } ) } } \\end{array}", + "type": "interline_equation", + "image_path": "8c72a4506a485312ef992ab12e0b29a8c9b1a29b0859aaad1261cc9f4c9f5bea.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 485, + 415, + 499.8333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 198, + 499.8333333333333, + 415, + 514.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 198, + 514.6666666666666, + 415, + 529.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 198, + 529.5, + 415, + 544.3333333333334 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 198, + 544.3333333333334, + 415, + 559.1666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 198, + 559.1666666666667, + 415, + 574.0000000000001 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 575, + 202, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 203, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 203, + 588 + ], + "score": 1.0, + "content": "For F-PNDM, we have:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 573, + 203, + 588 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 590, + 433, + 629 + ], + "lines": [ + { + "bbox": [ + 178, + 590, + 433, + 629 + ], + "spans": [ + { + "bbox": [ + 178, + 590, + 433, + 629 + ], + "score": 0.92, + "content": "\\begin{array} { c } { { e ^ { \\prime } = \\displaystyle \\frac { 1 } { 2 4 } \\big ( 5 5 \\epsilon _ { \\theta } ( x _ { t } ) - 5 9 \\epsilon _ { \\theta } ( x _ { t + \\delta } ) + 3 7 \\epsilon _ { \\theta } ( x _ { t + 2 \\delta } ) - 9 \\epsilon _ { \\theta } ( x _ { t + 3 \\delta } ) \\big ) } } \\\\ { { x _ { t - \\delta } = x _ { t } + \\phi ( x _ { t } , e ^ { \\prime } , t , t - \\delta ) } } \\end{array}", + "type": "interline_equation", + "image_path": "e98b5d5de615dbbac51d6d8c4a87f94e518b7a6175fde5e5eb65433b2ce2fe24.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 178, + 590, + 433, + 603.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 178, + 603.0, + 433, + 616.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 178, + 616.0, + 433, + 629.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "Then we draw the corresponding generation curves in Figure 6. We find that the result is similar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "to the real cases. The main difference here is that FON can correct its results while the real case", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "cannot. The reason is that the gradient is well-defined everywhere, while in real cases, the gradient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 664, + 376, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 315, + 677 + ], + "score": 1.0, + "content": "is meaningful on the high-density region of the data", + "type": "text" + }, + { + "bbox": [ + 315, + 666, + 326, + 675 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 664, + 376, + 677 + ], + "score": 1.0, + "content": "of DDPMs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 631, + 506, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 222, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 223, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 223, + 702 + ], + "score": 1.0, + "content": "A.9 MORE FID RESULTS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Here, We provide our more detailed FID results on Cifar10, CelebA, LSUN-church and LSUN-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 219, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 219, + 732 + ], + "score": 1.0, + "content": "bedroom in Table 3, 4, 5, 6.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 87, + 502, + 284 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 87, + 502, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 87, + 502, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 87, + 502, + 284 + ], + "score": 0.972, + "type": "image", + "image_path": "be45ffbce3a2d38e669a4bc00e9042c1345f1381c96712d8462385105dfde8f6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 87, + 502, + 152.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 152.66666666666669, + 502, + 218.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 218.33333333333337, + 502, + 284.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 201, + 291, + 409, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 201, + 289, + 409, + 306 + ], + "spans": [ + { + "bbox": [ + 201, + 289, + 409, + 306 + ], + "score": 1.0, + "content": "Figure 6: The generation curve of our toy example.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "table", + "bbox": [ + 106, + 323, + 514, + 440 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 323, + 514, + 440 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 323, + 514, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 514, + 440 + ], + "score": 0.975, + "html": "
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datasetFID\\ step model1020501002501000
Cifar10 (linear)DDIM*18.50±.0610.86±.086.95±.045.49±.064.52±.024.02±.04
FON13.00±.117.33±.065.24±.054.64±.044.12±.033.73±.03
S-PNDM11.58±.107.53±.075.15±.054.34±.033.93±.023.83±.03
F-PNDM6.12±.075.04±.074.01±.023.75±.043.67±.033.78±.04
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datasetFID\\ step model1020501002501000
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1: xT ~ N(0,I)2:fort=T-i,T-2,T-3do
3: 4: end forXt,et =PRK(xt+1,t+1,t)
5:1for t=T-4,.,1,0do
6:Xt,et =PLMS(xt+1,{ep}p>t,t+1,t)
7: end for
8:return xo
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datasetFID step model1020501002501000time
Cifar10DDIM PF13.46.84 13.84.67 3.894.16 3.693.714.04 3.72
Cifar10 (linear)DDIM*18.510.96.995.524.524.000.337
FON13.17.415.264.654.123.710.390
S-PNDM11.67.565.184.343.913.800.344
Cifar10 (cosine)F-PNDM DDIM7.03 14.55.00 8.793.95 5.863.72 4.923.60 4.303.70 3.690.391 0.505
S-PNDM8.645.774.463.943.713.380.517
F-PNDM7.054.613.683.533.493.260.595
CelebADDIM17.313.79.176.533.51
CelebA (linear)DDIM*16.913.48.956.364.443.411.237
FON16.011.68.136.705.144.171.431
S-PNDM12.29.455.694.033.192.991.258
F-PNDM7.715.513.342.812.712.861.433
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1: xT ~ N(0,I)2:fort=T-ido 3:Xt,et = PIE(xt+1,t+1,t)
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datasetFID\\ step model1020501002501000
Cifar10 (linear)DDIM*18.50±.0610.86±.086.95±.045.49±.064.52±.024.02±.04
FON13.00±.117.33±.065.24±.054.64±.044.12±.033.73±.03
S-PNDM11.58±.107.53±.075.15±.054.34±.033.93±.023.83±.03
F-PNDM6.12±.075.04±.074.01±.023.75±.043.67±.033.78±.04
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+1,351 @@ +# DETECTING ADVERSARIAL EXAMPLES IS (NEARLY) AS HARD AS CLASSIFYING THEM + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Making classifiers robust to adversarial examples is challenging. Thus, many defenses tackle the seemingly easier task of detecting perturbed inputs. + +We show a barrier towards this goal. We prove a general hardness reduction between detection and classification of adversarial examples: given a robust detector for attacks at distance $\epsilon$ (in some metric), we show how to build a similarly robust (but inefficient) classifier for attacks at distance $\epsilon / 2$ —and vice-versa. + +Our reduction is computationally inefficient, and thus cannot be used to build practical classifiers. Instead, it is a useful sanity check to test whether empirical detection results imply something much stronger than the authors presumably anticipated. + +To illustrate, we revisit 14 empirical detector defenses published over the past years. For $1 2 / 1 4$ defenses, we show that the claimed detection results imply an inefficient classifier with robustness far beyond the state-of-the-art— thus casting some doubts on the results’ validity. + +Finally, we show that our reduction applies in both directions: a robust classifier for attacks at distance $\epsilon / 2$ implies an inefficient robust detector at distance . Thus, we argue that robust classification and robust detection should be regarded as (near)-equivalent problems, if we disregard their computational complexity. + +# 1 INTRODUCTION + +Building models that are robust to adversarial examples (Szegedy et al., 2014; Biggio et al., 2013) is a major challenge and open-problem in machine learning. Due to the inherent difficulty in building robust classifiers, researchers have attempted to build techniques to at least detect adversarial examples, a weaker task that is largely considered easier than robust classification (Xu et al., 2018; Pang et al., 2021; Sheikholeslami et al., 2021). + +Yet, evaluating the robustness of empirical detector defenses is challenging. This is in part due to a lack of strong evaluation guidelines and benchmarks—akin to those developed for robust classifiers (Carlini et al., 2019; Croce et al., 2020)—as well as to a lack of long-standing comparative baselines such as adversarial training (Madry et al., 2018). + +To illustrate, consider the following (fictitious) claims about two defenses against adversarial examples on CIFAR-10: + +• defense A is a classifier that achieves robust accuracy of $9 0 \%$ under $\ell _ { \infty }$ -perturbations bounded by $\epsilon = { } ^ { 4 } / 2 5 5$ ; +• defense B also has a “rejection” option, and achieves robust accuracy of $9 0 \%$ under $\ell _ { \infty }$ - perturbations bounded by $\epsilon = { \bf 8 } / 2 5 5$ (we say that defense B is robust for some example if it classifies that example correctly, and either rejects/detects or correctly classifies all perturbed examples at distance $\epsilon .$ .) + +Which of these two (empirical) claims are you more likely to believe to be correct? + +Defense A claims much higher robustness than the current best result achieved with adversarial training (Madry et al., 2018; Rebuffi et al., 2021), the only empirical defense against adversarial examples that has stood the test of time. Indeed, the state-of-the-art $\ell _ { \infty }$ robustness for $\epsilon = { ^ 4 } / { 2 5 } 5$ on CIFAR-10 (without external data) is $\approx 7 9 \%$ (Rebuffi et al., 2021). Thus, the claim of defense A would likely be met with some initial skepticism and heightened scrutiny, as could be expected for such a claimed breakthrough result. + +The claim of defense B is harder to assess, due to a lack of long-standing baselines for robust detectors (many detection defenses have been shown to be broken (Carlini & Wagner, 2017; Tramer\` et al., 2020)). On one hand, detection of adversarial examples has largely been considered to be an easier task than classification $\mathrm { { X u } }$ et al., 2018; Pang et al., 2021; Sheikholeslami et al., 2021). On the other hand, defense B claims robustness to perturbations that are twice as large as defense A ${ \bf \zeta } ( \epsilon = 8 / 2 5 5$ vs. $\epsilon = 4 / 2 5 5$ ). + +In this paper, we show that the claims of defenses A and B are, in fact, equivalent! (up to computational efficiency.) + +We prove a general hardness reduction between classification and detection of adversarial examples. Given a detector defense that achieves robust risk $\alpha$ for attacks at distance $\epsilon$ (under any metric), we show how to build an explicit but inefficient classifier that achieves robust risk $\alpha$ for classifying attacks at distance $\epsilon / 2$ . The reverse implication also holds: a classifier robust at distance $\epsilon / 2$ implies an explicit but inefficient robust detector at distance $\epsilon$ . + +To the authors knowledge, there is no known way of leveraging computational inefficiency to build more robust models. We should thus be as “surprised” by the claim made by defense B as by the claim made by defense A. + +Our reduction provides a way of assessing the plausibility of new robust detection claims, by contrasting them with results from the more mature literature on robust classification. To illustrate, we revisit 14 published detection defenses across three datasets, and show that in 12/14 cases the defense’s robust detection claims would imply an inefficient classifier with robustness far superior to the current state-of-the-art. Yet, none of these detection papers make the claim that their techniques should imply such a breakthrough in robust classification. + +Using our reduction, it is obvious that many detection defenses are claiming much stronger robustness than we believe feasible with current techniques. And indeed, many of these defenses were later shown to have overestimated their robustness (Carlini & Wagner, 2017; Tramer et al., 2020). \` + +Remarkably, we find that for certified defenses, the state-of-the-art results for provable robust classification and detection perfectly match the results implied by our reduction. For example, Sheikholeslami et al. (2021) recently proposed a certified detector on CIFAR-10 with provable robust error that is within $3 \%$ of the provable error of the inefficient detector obtained by combining our result with the state-of-the-art robust classifier of Zhang et al. (2020a). + +In summary, we prove that giving classifiers access to a detection option does not help robustness (or at least, not much). Our work provides, to our knowledge, the first example of a hardness reduction between different approaches for robust machine learning. As in the case of computational complexity, we believe that such reductions can be useful for identifying research questions or areas that are unlikely to bear fruit (bar a significant breakthrough)—so that the majority of the community’s efforts can be redirected elsewhere. + +On a technical level, our reduction exposes a natural connection between robustness and error correcting codes, which may be of independent interest. + +# 2 HARDNESS REDUCTIONS BETWEEN ROBUST CLASSIFIERS AND DETECTORS + +In this section, we prove our main result: a reduction between robust detectors and robust classifiers, and vice-versa. We first introduce some useful notation and define the (robust) risk of classifiers with and without a detection option. + +# 2.1 PRELIMINARIES + +We consider a classification task with a distribution $\mathcal { D }$ over examples $x \in \mathbb { R } ^ { d }$ with labels $y \in$ $[ C ]$ . A classifier is a function $f : \mathbb { R } ^ { d } [ C ]$ . A detector is a classifier with an extra “rejection” + +or ”detection” option $\perp$ , that indicates the absence of a classification. We assume for simplicity that classifiers and detectors are deterministic. Our results can easily be extended to randomized functions as well. The binary indicator function $\mathbb { 1 } _ { \{ A \} }$ is 1 if and only if the predicate $A$ is true. + +We first define a classifier’s risk, i.e., its classification error on unperturbed samples. + +Definition 1 (Risk). Let $f : \mathbb { R } ^ { d } [ C ] \cup \{ \bot \}$ be a classifier (optionally with a detection output $\perp$ ). The risk of $f$ is the expected rate at which $f$ fails to correctly classify a sample: + +$$ +R ( f ) : = \operatorname* { \mathbb { E } } _ { ( x , y ) \sim \mathcal { D } } \left[ \mathbb { 1 } _ { \{ f ( x ) \neq y \} } \right] +$$ + +Note that for a detector, rejecting an unperturbed example sampled from the distribution $\mathcal { D }$ is counted as an error. + +For classifiers without a rejection option, we define the robust risk as the risk on worst-case adversarial examples (Madry et al., 2018). Given an input $x$ sampled from $\mathcal { D }$ , an adversarial example $\hat { x }$ is constrained to being within distance $d ( x , { \hat { x } } ) \leq { \bar { \epsilon } }$ from $x$ , where $d$ is some distance measure. + +Definition 2 (Robust risk). Let $f : \mathbb { R } ^ { d } [ C ]$ be a classifier. The robust risk at distance $\epsilon$ is: + +$$ +R _ { a d \nu } ^ { \epsilon } ( f ) : = \operatorname* { \mathbb { E } } _ { ( x , y ) \sim \mathcal { D } } \left[ \operatorname* { m a x } _ { d ( x , \hat { x } ) \leq \epsilon } \mathbb { 1 } _ { \{ f ( \hat { x } ) \neq y \} } \right] +$$ + +Thus, a sample $( x , y )$ is robustly classified if and only if every point within distance $\epsilon$ of $x$ (including $x$ itself) is correctly classified as $y$ . + +For a detector (a classifier with an extra detection/rejection output), we analogously define the robust risk with detection. The classifier is now allowed to reject adversarial examples. + +Definition 3 (Robust risk with detection). Let $f : \mathbb { R } ^ { d } [ C ] \cup \{ \bot \}$ be a classifier with an extra detection output $\perp$ . The robust risk with detection at distance $\epsilon$ is: + +$$ +R _ { a d \nu - d e t } ^ { \epsilon } ( f ) : = \underset { ( x , y ) \sim \mathcal { D } } { \mathbb { E } } \left[ \operatorname* { m a x } _ { d ( x , \hat { x } ) \leq \epsilon } \mathbb { 1 } _ { \left\{ f ( x ) \neq y \vee f ( \hat { x } ) \notin \left\{ y , \perp \right\} \right\} } \right] +$$ + +That is, a detector defense $f$ is robust on a natural input $x$ if and only if the defense classifies the natural input $x$ correctly, and the defense either rejects or correctly classifies every perturbed input $\hat { x }$ within distance $\epsilon$ from $x$ . The requirement that the defense correctly classify natural examples eliminates pathological defenses that reject all inputs. + +# 2.2 ROBUST DETECTION IMPLIES INEFFICIENT ROBUST CLASSIFICATION + +We are now ready to introduce our main result, a reduction from a robust detector for adversarial examples at distance $\epsilon$ , to an inefficient robust classifier at distance $\epsilon / 2$ . We later prove that this reduction also holds in the reverse direction, thereby demonstrating the equivalence between robust detection and classification—up to computational hardness. + +Theorem 4 $\epsilon$ -robust detection implies inefficient $\epsilon / 2$ -robust classification). Let $d ( \cdot , \cdot )$ be an arbitrary metric. Let $f$ be a detector that achieves risk $R ( f ) = \alpha _ { \mathrm { { } } }$ , and robust risk with detection $R _ { a d \nu - d e t } ^ { \epsilon } ( f ) = \beta$ . Then, we can construct an explicit (but inefficient) classifier $g$ that achieves risk $R ( g ) \leq \alpha$ and robust risk $R _ { a d \nu } ^ { \epsilon / 2 } ( g ) \le \beta$ . + +The classifier $g$ is constructed as follows on input $x$ : + +• Run the detector model $y f ( x )$ . If the input is not rejected, i.e., $y \ne \perp$ , then output the label y that was predicted by the detector. + +• Otherwise, find an input $x ^ { \prime }$ within distance $\epsilon / 2$ of $x$ that is not rejected, i.e., $d ( x , x ^ { \prime } ) \leq \epsilon / 2$ and $f ( x ^ { \prime } ) \neq \bot$ . If such an input $x ^ { \prime }$ exists, output the label $y \gets f ( x ^ { \prime } )$ . Else, output a uniformly random label $y \in [ C ]$ . + +An intuitive illustration for our construction, and for the proof of the theorem (see below) is in Figure 1. + +![](images/1703fa013d6a1ee6b00378a016c2a93fc0e3073c9d6734ac06a5bcc5e5e8f741.jpg) +Figure 1: Illustration of the construction of a robust classifier from a robust detector in Theorem 4. The outer blue circle represents all inputs at distance at most $\epsilon$ from the input $x$ . For a detector $f$ , the areas in green correspond to correctly classified inputs, and ratcheted gray areas correspond to rejected inputs. The detector $f$ is thus robust on $x$ up to distance $\epsilon$ . The classifier $g$ classifies a perturbed input $\hat { x }$ , at distance $\epsilon / 2$ from $x$ , by finding any input within distance $\epsilon / 2$ from $\hat { x }$ (the red dashed circle) that is not rejected by $f$ . Such an input necessarily exists and is correctly labeled by $f$ . The classifier $g$ is thus robust on $x$ up to distance $\epsilon / 2$ . + +Our construction can be viewed as an analog of minimum distance decoding in coding theory. We can view a clean data point sampled from $\mathcal { D }$ as a codeword, and an adversarial example $\hat { x }$ as a noisy message with a certain number of errors (where the error magnitude is measured using an arbitrary metric on $\mathbb { R } ^ { d }$ rather than the Hamming distance that is typically used for error correcting codes). A standard result in coding theory states that if a code can detect $\alpha$ errors, then it can correct $\alpha / 2$ errors. This result follows from a “ball-packing” argument: if $\alpha$ errors can be detected, then any two valid codewords must be at least at distance $\alpha$ from each other, and therefore $\alpha / 2$ errors can be corrected via minimum distance decoding. + +Proof of Theorem 4. First, note that the natural accuracy of our constructed classifier $g$ is at least as high as that of the detector $f$ , since $g$ always mimics the output of $f$ whenever $f$ does not reject an input sampled from $\mathcal { D }$ . Thus, $R ( g ) \dot { \leq } R ( f ) = \alpha$ . + +Now, for the sake of contradiction, consider an input $( x , y ) \sim \mathcal { D }$ for which the constructed classifier $g$ is not robust at distance $\epsilon / 2$ . By construction, this means that there exists some input $\hat { x }$ at distance $\epsilon / 2$ from $x$ such that $\hat { x }$ is misclassified, i.e., $g ( \hat { x } ) = \hat { y } \neq y$ . We will show that the detector $f$ is not robust with detection for $x$ either (for attacks at distance up to $\epsilon$ ). + +By definition of the classifier $g$ , if $g ( \hat { x } ) = \hat { y } \neq y$ then either: + +• The detector $f$ also misclassifies $\hat { x }$ , i.e., $f ( \hat { x } ) = \hat { y }$ . So $f$ is not robust with detection for $x$ at distance $\epsilon$ . +• There exists an input $x ^ { \prime }$ within distance $\epsilon / 2$ of $x$ , such that the detector $f$ misclassifies $x ^ { \prime }$ , i.e. $\boldsymbol { f } ( \boldsymbol { x } ^ { \prime } ) = \boldsymbol { \hat { y } }$ . Note that by the triangular inequality, $d ( x , x ^ { \prime } ) \leq d ( x , \hat { x } ) + d ( \hat { x } , x ^ { \prime } ) \leq \epsilon / 2 + \epsilon / 2 = \epsilon ,$ and thus $f$ is not robust with detection for $x$ at distance $\epsilon$ . +• The detector $f$ rejects all inputs $x ^ { \prime }$ within distance $\epsilon / 2$ of $x$ (and thus $g$ has output $\hat { y }$ by sampling a label at random). Since $d ( x , { \hat { x } } ) \leq \epsilon / 2$ , this implies that the detector also rejects the clean input $x$ , i.e., $f ( x ) = \bot$ , and thus $f$ is not robust with detection for $x$ . + +In summary, whenever the constructed classifier $g$ fails to robustly classify an input $x$ up to distance $\epsilon / 2$ , the detector $f$ also fails to robustly classify $x$ with detection up to distance $\epsilon$ . Taking expectations over the entire distribution $\mathcal { D }$ concludes the proof. □ + +Note that the classifier $g$ constructed in Theorem 4 is computationally inefficient. Indeed, the second step of the defense consists in finding a non-rejected input within some metric ball. If the original detector $f$ is a non-convex function (e.g., a deep neural network), then this step consists in solving an intractable non-convex optimization problem. Our reduction is thus typically not suitable for building a practical robust classifier. Instead, it demonstrates the existence of an inefficient but explicit robust classifier. We discuss the implications of this result more thoroughly in Section 3. + +A corollary to our reduction is that many “information theoretic” results about robust classifiers can +be directly extended to robust detectors. For example, Tsipras et al. (2019) prove that there exists a +formal tradeoff between a classifier’s clean accuracy and robust accuracy for certain natural tasks. +Since their result applies to any classifier (including inefficient ones), combining their result with our +reduction implies that a similar accuracy-robustness tradeoff exists for detectors. More precisely, +Tsipras et al. (2019) show that for certain classification tasks and suitable choices of parameters +$\alpha , \beta , \epsilon$ , any classifier $g$ which achieves risk $R ( g ) \leq \alpha$ must have robust risk at least $R _ { \mathrm { a d v } } ^ { \bar { \epsilon } } ( g ) \geq \beta$ inst k at $\ell _ { \infty }$ -t s boumust d by o hav $\epsilon$ . By our reduction, this implies t robust risk with detection at least $f$ withainst $R ( f ) \leq \alpha$ $R _ { \mathrm { a d v - d e t } } ^ { \epsilon / 2 } ( f ) \geq \beta$ +$\ell _ { \infty }$ $\epsilon / 2$ + +Similar arguments can be applied to show, for instance, that the increased data complexity of robust generalization from Schmidt et al. (2018), or the tradeoff between robustness to multiple perturbation types from Tramer & Boneh (2019), also apply to robust detectors. \` + +Our reduction does not apply for “computational” hardness results that have been shown for robust classification. For example, Garg et al. (2020) and Bubeck et al. (2018) show (“unnatural”) distributions where learning a robust classifier is computationally hard—under standard cryptographic assumptions. We cannot use Theorem 4 to conclude that learning a robust detector is hard for these distributions, since the existence of such a detector would only imply an inefficient robust classifier which does not contradict the results of Garg et al. (2020) or Bubeck et al. (2018). + +# 2.3 ROBUST CLASSIFICATION IMPLIES INEFFICIENT ROBUST DETECTION + +A similar argument as in Theorem 4 can be used in the opposite direction, to show that a robust classifier at distance $\epsilon / 2$ implies an inefficient robust detector at distance $\epsilon$ . Taken together, Theorem 4 and Theorem 5 show that robust detection and classification are equivalent, up to a factor 2 in the norm bound and up to computational constraints. + +Theorem 5 $( \epsilon / 2$ robust-classification implies inefficient $\epsilon$ -robust detection). Let $d ( \cdot , \cdot )$ be an arbi +trary metric. Let g be a defense than explicit (but inefficient) defense achieves robust rithat achieves risk $R _ { a d \nu } ^ { \epsilon / 2 } ( f ) = \beta$ . Then, we can constructrobust risk with detection $f$ $R ( f ) \le \beta$ +$R _ { a d \nu - d e t } ^ { \epsilon } ( f ) \leq \beta$ . + +The defense $f$ is constructed as follows on input $x$ : + +• Run the classifier $y g ( x )$ . + +• Find a perturbed input $x ^ { \prime }$ withing distance $\epsilon / 2$ of $x$ that is classified differently, i.e., $d ( x , x ^ { \prime } ) \leq \epsilon / 2$ and $g ( x ^ { \prime } ) \neq y$ . If such an input $x ^ { \prime }$ exists, reject the input and output ⊥. Else, output the class $y$ . + +We provide the proof of Theorem 5 in Appendix A. + +A main distinction between Theorem 4 and Theorem 5 is that the construction in Theorem 4 preserves clean accuracy, but the construction in Theorem 5 does not. That is, the constructed robust detector in Theorem 5 has clean accuracy that is equal to the robust classifier’s robust accuracy. + +The construction in Theorem 5 can be efficiently (but approximately) instantiated by a certifiably robust classifier (Wong & Kolter, 2018; Raghunathan et al., 2018). These defenses can certify that a classifier’s output is constant for all points within some distance $\epsilon$ of the input. For an adversarial example $\hat { x }$ for $g$ , the certification always fails and thus the constructed detector $f$ will reject $\hat { x }$ . If $g$ is robust and the certification succeeds, the detector $f$ copies the output of $g$ . However, a certified defense may fail to certify a robust input (a false negative), and thus the detector $f$ may reject more inputs than with the “optimal” construction in Theorem 5. This reduction from a certified classifier to a detector is implicit in (Wong & Kolter, 2018, Section 3.1). + +# 3 WHAT ARE DETECTION DEFENSES CLAIMING? + +We now survey 14 detection defenses, and consider the robust classification performance that these defenses implicitly claim (via Theorem 4). As we will see, in 12/14 cases, the defenses’ detection results imply an inefficient classifier with far better robust accuracy than the state-of-the-art. + +Before presenting our experimental setup and the explicit results from the reduction, we first discuss how we believe these results should be interpreted. + +Interpreting our reduction. Suppose that some detector defense claims a robust accuracy that implies—via our reduction—an inefficient classifier with much higher robustness that the state-ofthe-art (e.g., the defense A described in the introduction of this paper). + +A first possible interpretation of our reduction is that this robust detector implies the existence of a robust classifier. This interpretation is rather weak however, since it is typically presumed that robust classification is possible, and that human perception is one concrete example of a robust classifier. The mere existence of a robust classifier is thus typically already assumed to be true. + +Our reduction yields a stronger result. It provides an explicit construction of an (inefficient) robust classifier from a robust detector. The question then is whether we should expect the construction of inefficient robust classifiers to be easier than the construction of efficient ones. That is, do we expect that we can leverage computational inefficiency to build more robust classifiers that the current state-of-the-art? + +We do not know of a positive answer to this question, and there is evidence to suggest that the answer may be negative.1 For example, the work of Schmidt et al. (2018) proves that for a synthetic classification task between Gaussian distributions, building more robust classifiers requires additional data regardless of the amount of computation power. Their results are corroborated by current state-ofthe-art robust classifiers based on adversarial training (Madry et al., 2018), which do not appear to be limited by computational constraints. On CIFAR-10 for example, adversarial training achieves $100 \%$ robust training accuracy (Schmidt et al., 2018). Thus, it is unclear how computational inefficiency could be leveraged to build more robust classifiers using existing techniques. + +Candidate approaches could be to train much larger models (e.g., with an exponential number of parameters), or to perform an exhaustive architecture search to find more robust models. Yet, note that the robust classifier constructed in our reduction only uses its unbounded computational power at inference time. That is, the classifier that is built in Theorem 4 uses a trained detector model as a subroutine (which is presumed to be efficient), and then solves a non-convex optimization problem at inference time. The classifier built in our reduction is thus presumably weaker than a robust classifier that can be trained with unbounded computational power. + +To summarize, when a detector defense claims a certain robust accuracy, this implies the existence of a concretely instantiatable robust classifier with an inefficient inference procedure. If this inefficient classifier is much more robust than the current state-of-the-art, this does not necessarily mean that the defense’s claims is wrong. But given how challenging robust classification is proving to be, we have reason to be skeptical of such a major breakthrough (even for inefficient classifiers). To compound this, many proposed detection defenses are quite simple, and reject adversarial inputs based on some standard statistical test over a neural network’s features. It would thus be particularly surprising if such simple techniques could yield robust classifiers, given that “simple” approaches to adversarial robustness (denoising, compression, randomness, etc.) are ineffective (He et al., 2017). + +As a result, it is not too surprising that a number of the detector defenses that we survey have already been broken by stronger attacks (Carlini & Wagner, 2017; Tramer et al., 2020). Our reduction would \` have already suggested that such a break was likely to happen. + +Experimental setup. We choose 14 detector defenses from the literature (see Table 1). Our selection of these defenses was partially motivated by a pragmatic consideration on the easiness of translating the defenses’ claims into a bound on the robust risk with detection $R _ { \mathrm { a d v - d e t } } ^ { \epsilon }$ . Indeed, some defenses simply report a single AUC score for the detector’s performance, from which we cannot derive a useful bound on the robust risk. We thus focus on defenses that either directly report a robust error akin to Definition 3, or that provide concrete pairs of false-positive and false-negative rates (e.g., a full ROC curve). In the latter case, we compute a “best-effort” bound on the robust risk with detection2 as: + +$$ +R _ { \mathrm { a d v - d e t } } ^ { \epsilon } ( f ) \leq \mathrm { F P R } + \mathrm { F N R } + R ( f ) , +$$ + +where FPR and FNR are the detector’s false-positive and false-negative rates for a fixed detection threshold, and $R ( f )$ is the defense’s standard risk (i.e., the test error on natural examples). + +The above union bound in Equation (4) is quite pessimistic, as we may over-count examples that lead to multiple sources of errors (e.g., a natural input that is misclassified and erroneously detected). The true robustness claim made by these detector defenses might thus be stronger than what we obtain from our bound. We encourage future detection papers to report their adversarial risk with detection, $R _ { \mathrm { a d v - d e t } } ^ { \epsilon }$ , to facilitate direct comparisons with robust classifiers using our reduction. + +The 14 detector defenses use three datasets: MNIST, CIFAR-10 and ImageNet, and consider adversarial examples under the $\ell _ { \infty }$ or $\ell _ { 2 }$ norms. Given a claim of robust detection at distance , we contrast it to a state-of-the-art robust classification result for distance $\epsilon / 2$ : + +• On MNIST with $\ell _ { \infty }$ attacks, we use the adversarially-trained TRADES classifier (Zhang et al., 2019) and measure robust error with the Square attack (Andriushchenko et al., 2020). +• On MNIST with $\ell _ { 2 }$ attacks, we use the adversarially-trained classifier from Tramer & \` Boneh (2019) and measure robust error with PGD (Madry et al., 2018). +• On CIFAR-10, for both $\ell _ { \infty }$ and $\ell _ { 2 }$ attacks we use the adversarially-trained classifier of Rebuffi et al. (2021) (trained without external data), and attack it using the APGD-CE attack from AutoAttack (Croce & Hein, 2020). +• For ImageNet, for both $\ell _ { \infty }$ and $\ell _ { 2 }$ attacks we use adversarially-trained classifiers and PGD attacks from Engstrom et al. (2019). + +We also consider two certified defenses for $\ell _ { \infty }$ attacks on CIFAR-10: the robust classifier of Zhang et al. (2020a), and a recent certified detector of Sheikholeslami et al. (2021). + +Results. As we can see from Table 1, most defenses claim a detection performance that implies a far greater robust accuracy than our current best robust classifiers. To illustrate with a concrete example, the CIFAR-10 detector of Miller et al. (2019) claims to achieve robust accuracy with detection of $7 5 \%$ for $\ell _ { 2 }$ attacks with $\epsilon = 2 . 9$ . Using Theorem 4, this implies an inefficient classifier with robust accuracy of $7 5 \%$ for $\ell _ { 2 }$ attacks with $\epsilon = { } ^ { 2 . 9 } / 2 = 1 . 4 5$ . Yet, the current state-of-the-art robust accuracy for such a perturbation budget is only $3 0 \%$ (Rebuffi et al., 2021). If this detector defense’s robustness claim were correct, it would imply a remarkable breakthrough in robust classification. + +Why do many of these defenses claim robust accuracies that appear “too good to be true”? A primary reason is that the vast majority of the above detector defenses do not consider evaluations against adaptive attacks (Carlini et al., 2019; Athalye et al., 2018; Tramer et al., 2020). That is, \` these defenses show that they can detect some fixed attacks, and thereafter conclude that the detector is robust against all attacks. As in the case of robust classifiers, such an evaluation is clearly insufficient! Some defenses do evaluate against adaptive adversaries, but fail to build a sufficiently strong attack to reliably approximate the worst-case robust risk. Because of the lack of a strong comparative baseline, it is not always immediately clear that these results are overly strong. + +Table 1: For each detector defense, we compute a (best-effort) bound on the claimed robust risk with detection $R _ { \mathrm { a d v - d e t } } ^ { \epsilon }$ using Equation (4), and report the complement (the robust accuracy with detection), $1 - R _ { \mathrm { a d v - d e t } } ^ { \epsilon }$ . For each detector’s robustness claim (at distance $\epsilon$ ), we report the stateof-the-art robust classification accuracy for attacks at distance $\epsilon / 2$ , denoted $1 - R _ { \mathrm { a d v } } ^ { \epsilon / 2 }$ . Detection defense claims that imply a higher robust classification accuracy than the current state-of-the-art are highlighted in red. + +
DatasetDefenseNormE1-Radv-det1- R
MNISTGrosse et al. (2017)l80.5≥ 98%94%
Ma et al. (2018)l24.2≥ 99%72%
Raghuram et al. (2021)l28.9≥ 74%0%
CIFAR-10Yin et al. (2020)l21.7≥ 90%66%
Feinman et al. (2017)l22.7≥ 43%36%
Miller et al. (2019)l22.9≥ 75%30%
Raghuram et al. (2021)l4.0≥ 56%10%
Ma & Liu (2019)l84/255≥ 96%85%
Roth et al. (2019)l88/255≥ 66%79%
Lee et al. (2018)lo20/255≥ 81%59%
Li et al. (2019)lo26/255≥ 80%44%
ImageNetXu et al. (2018)l1.0≥ 67%54%
Ma & Liu (2019)l82/255≥ 68%55%
Jha et al. (2019)lo2/255≥ 30%55%
Hendrycks & Gimpel (2017)l810/255≥ 76%30%
Yu et al. (2019)l826/255M 7%5%
+ +For example, the recent work of Raghuram et al. (2021, ICML Long Talk) builds a detector on MNIST with a FNR of $\leq 5 \%$ at a FPR of $\leq 2 0 \%$ , for adaptive $\ell _ { 2 }$ attacks bounded by $\epsilon = 8 . 9$ . Yet, this perturbation bound is much larger than the average distance between an MNIST image and the nearest image from a different class! Thus, an attack within this perturbation bound can trivially reduce the detector’s accuracy to chance. On CIFAR-10, the same detector achieves $9 5 \%$ clean accuracy, and a FNR of $\leq 1 9 \%$ at a FPR of $\leq 2 0 \%$ for adaptive $\ell _ { 2 }$ attacks bounded by $\epsilon = 4$ . Using Equation (4), this yields a bound on the robust accuracy with detection of $1 - R _ { \mathrm { a d v - d e t } } ^ { \epsilon } ( f ) \geq$ $1 - \bar { ( 5 \% + 1 9 \% + 2 0 \% ) } = 5 6 \%$ . In contrast, the best robust classifier we are aware of for $\ell _ { 2 }$ attacks bounded by $\epsilon = 2$ achieves robust accuracy of only $1 0 \%$ (Rebuffi et al., 2021). In summary, the adaptive attack considered in this detector defense’s evaluation is highly unlikely to be good approximation of a worst-case attack, and this defense can likely be broken by stronger attacks. + +Certifiably robust detection and classification. In Table 2, we look at the robust accuracy with detection, and standard robust accuracy achieved by certified defenses (for which the claimed robustness numbers are necessarily mathematically correct). + +We note that our reduction is not as meaningful in the case of certified defenses, since it is highly plausible that computational inefficiency can be leveraged to build better certified classifiers. Indeed, given any robust classifier (e.g., an adversarially trained model), the classifier’s robustness can always be certified inefficiently (by enumerating over all points within an $\epsilon$ -ball). Thus, the existence of an inefficient classifier with higher certified robustness than the state-of-the-art is to be expected. + +Nevertheless, we find that existing results for certified classifiers and detectors perfectly match what is implied by our reduction (up to $\pm 2 \%$ error). For example, Zhang et al. (2020a) follow a long line of results on robust classifiers and achieve $3 9 \%$ robust accuracy on CIFAR-10 for perturbations of $\ell _ { \infty }$ -norm below $^ { 4 } / 2 5 5$ . Together with Theorem 5, this implies an inefficient detector with $3 9 \%$ robust detection accuracy for perturbations of $\ell _ { \infty }$ -norm below $^ { 8 } / 2 5 5$ . The recent work of Sheikholeslami et al. (2021) nearly matches that bound $3 7 \%$ robust accuracy with detection), with a defense that has the advantage of being concretely efficient. + +Table 2: Certified robust accuracy $1 - R _ { \mathrm { a d v } } ^ { \epsilon / 2 }$ for the defense of Zhang et al. (2020a), and certified robust accuracy with detection $1 - R _ { \mathrm { a d v - d e t } } ^ { \epsilon }$ dv for the defense of Sheikholeslami et al. (2021). + +
E1-Radv-det
8/25537%39%
16/25532%33%
+ +These results give additional credence to our thesis: with current techniques, robust classification is indeed approximately twice as hard (in terms of the perturbation bounds covered) than robust detection. + +Extensions and open problems. The main open problem raised by our work is of course whether it could be possible to show an efficient reduction between classification and detection of adversarial examples, but this seems implausible (at least with our minimum distance decoding approach). + +Another interesting question is whether a similar reduction can be shown for robustness to less “structured” perturbations than $\ell _ { p }$ balls and other metric spaces. For example, there has been a line of research on defending against adversarial patches (Brown et al., 2017), using empirical (Hayes, 2018; Naseer et al., 2019; Chou et al., 2020) and certifiable techniques (Chiang et al., 2020; Zhang et al., 2020b; Xiang et al., 2021). To use our result, we would have to define some metric to measure the size of an adversarial patch’s perturbation. Yet, the size of a patch is typically defined by the number of contiguously perturbed pixels, which does not define a metric (in particular, it does not satisfy the triangular inequality which our reduction relies on). + +Finally, similar hardness reductions might exist between other candidate approaches for building robust classifiers. For example, the question of whether (test-time) randomness can be leveraged to build more robust models is also intriguing .Empirical defenses that use randomness can be notoriously hard to evaluate (Athalye et al., 2018; Tramer et al., 2020), so a reduction similar to ours \` might be useful in showing that we should not expect such approaches to bare fruit. + +# 4 CONCLUSION + +We have shown formal reductions between robust classification with, and without, a detection option. Our results show that significant progress on one of these two tasks implies similar progress on the other—unless computational inefficiency can somehow be leveraged to build more robust models. This raises the question on whether we should spend our efforts on studying both of these tasks, or focus our efforts on a single one. + +On one hand, the two tasks represent different ways of tackling a common goal, and working on either task might result in new techniques or ideas that apply to the other task as well. On the other hand, our reductions show that unless we make progress on both tasks, work on one of the tasks can merely aim to match the robustness of our inefficient constructions, whilst improving their computational complexity. + +We believe our reduction will serve as a useful sanity-check when assessing the claims of future detector defenses. Detector defenses’ robustness evaluations have received less stringent scrutiny than robust classifiers over the past years, perhaps in part due to a lack of strong comparative baselines. Instead of having to wait until some detector defense’s claims pass the test-of-time, we show that detection results can be directly contrasted against long-standing results for robust classification. + +When applying this approach to past detector defenses, we find that many make robustness claims that imply significant breakthroughs in robust classification. We believe our reduction could have been useful in highlighting the suspiciously strong claims made by many of these defenses—before they were explicitly broken by stronger attacks. + +Ethics statement. Our paper demonstrates a fundamental barrier towards detecting against adversarial examples, under the assumption that our current techniques are insufficient to achieve strong (inefficient) robust classification. We do not however explicitly break any existing defenses (our results merely strongly suggest that many existing detector defenses’ claims are suspiciously high). Our paper therefore cannot lead to any explicit harms, but aims to further our understanding of the hardness of robust classification and detection. + +Reproducibility statement. Our paper’s contribution is mainly of theoretical nature. Section 2 is self-contained and clearly states our assumptions, results and proofs (except for the proof of Theorem 5 in Appendix A). The experiments in Section 3 use only public datasets and pre-trained models, with clearly indicates hyper-parameters for all attacks that we evaluate. + +# REFERENCES + +Maksym Andriushchenko, Francesco Croce, Nicolas Flammarion, and Matthias Hein. Square attack: a query-efficient black-box adversarial attack via random search. In European Conference on Computer Vision, 2020. + +Anish Athalye, Nicholas Carlini, and David Wagner. 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In International Conference on Learning Representations, 2020a. + +Zhanyuan Zhang, Benson Yuan, Michael McCoyd, and David Wagner. Clipped bagnet: Defending against sticker attacks with clipped bag-of-features. In 2020 IEEE Security and Privacy Workshops (SPW), pp. 55–61. IEEE, 2020b. + +# A PROOF OF THEOREM 5. + +We recall Theorem 5: + +Theorem 5 $( \epsilon / 2$ robust-classification implies inefficient $\epsilon$ -robust detection). Let $d ( \cdot , \cdot )$ be an arbi +trary metric. Let g be a defense than explicit (but inefficient) defense achieves robust rithat achieves risk $R _ { a d \nu } ^ { \epsilon / 2 } ( f ) = \beta$ . Then, we can constructrobust risk with detection $f$ $R ( f ) \leq \beta$ +$R _ { a d \nu - d e t } ^ { \epsilon } ( f ) \leq \beta$ . + +The defense $f$ is constructed as follows on input $x$ : + +• Run the classifier $y g ( x )$ . + +• Find a perturbed input $x ^ { \prime }$ withing distance $\epsilon / 2$ of $x$ that is classified differently, i.e., $\bar { d } ( x , x ^ { \prime } ) \bar { \ } \leq \ \epsilon / 2$ and $g ( x ^ { \prime } ) \neq y$ . If such an input $x ^ { \prime }$ exists, reject the input and output ⊥. Else, output the class $y$ . + +Proof of Theorem 5. Note that for any input $( x , y )$ for which the classifier $g$ is robust at distance $\epsilon / 2$ , no input $x ^ { \prime }$ above exists and so $f ( x ) = y$ . Thus, the risk of $f$ is at most the robust risk of $g$ , so $R ( f ) \leq \beta$ . + +Now, consider an input $( x , y ) \sim \mathcal { D }$ for which $f$ is not robust with detection at distance $\epsilon$ . That is, either $f ( x ) \neq y$ , or there exists an input $\hat { x }$ at distance $d ( x , { \hat { x } } ) \leq \epsilon$ such that $f ( \hat { x } ) = \hat { y } \not \in \{ y , \bot \}$ . We will show that the defense $g$ is not robust for $x$ either (for attacks at distance up to $\epsilon / 2$ .) + +If $f ( x ) \neq y$ , then by the same argument as above it cannot be the case that $g$ is robust at distance $\epsilon / 2$ for $x$ . + +So let us consider the case where $f ( \hat { x } ) = \hat { y } \not \in \{ y , \bot \}$ . By the definition of $f$ , this means that for all $x ^ { \prime }$ at distance at most $\epsilon / 2$ from $\hat { x }$ , we have $g ( x ^ { \prime } ) = \hat { y }$ . But, note that there exists a point $x ^ { * }$ that is at distance at most $\epsilon / 2$ from both $\hat { x }$ and $x$ . 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To evaluate the capabilities of our model, we introduce two", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 414, + 454, + 428 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 454, + 428 + ], + "score": 1.0, + "content": "new benchmark datasets with explicit focus on long-term temporal dynamics.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26.5, + "bbox_fs": [ + 141, + 272, + 471, + 428 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 190, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 192, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 192, + 459 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "Videos are data that change over time, with complex patterns of camera viewpoint, motion, deforma-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "tion and occlusion. In certain respects, videos are unbounded — they may last arbitrarily long and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "there is no limit to the amount of new content that may become visible over time. Yet videos that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "depict the real world must also remain consistent with physical laws that dictate which changes over", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "time are feasible. For example, the camera may only move through 3D space along a smooth path,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "objects cannot morph between each other, and time cannot go backward. Generating long videos thus", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 533, + 491, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 491, + 546 + ], + "score": 1.0, + "content": "requires the ability to produce endless new content while maintaining appropriate consistencies.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 466, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "In this work, we focus on generating long videos with rich dynamics and new content that arises over", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "time. While existing video generation models can produce “infinite” videos, the type and amount of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "change along the time axis is highly limited. For example, a synthesized infinite video of a person", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "talking will only include small motions of the mouth and head. Moreover, common video generation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "datasets often contain short clips with little new content over time, which may inadvertently bias the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "score": 1.0, + "content": "design choices toward training on short segments or pairs of frames, forcing content in videos to stay", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 614, + 367, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 367, + 627 + ], + "score": 1.0, + "content": "fixed, or using architectures with small temporal receptive fields.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 549, + 506, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "We make the time axis a first-class citizen for video generation. To this end, we introduce two new", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 640, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 655 + ], + "score": 1.0, + "content": "datasets that contain motion, changing camera viewpoints, and entrances/exits of objects and scenery", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "score": 1.0, + "content": "over time. We learn long-term consistencies by training on long videos and design a temporal latent", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "score": 1.0, + "content": "representation that enables modeling complex temporal changes. Figure 1 illustrates the rich motion", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 674, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 506, + 687 + ], + "score": 1.0, + "content": "and scenery changes that our model is capable of generating. See our webpage1 for video results,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 686, + 242, + 697 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 242, + 697 + ], + "score": 1.0, + "content": "code, data and pretrained models.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 630, + 506, + 697 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 70, + 504, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 70, + 504, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 504, + 232 + ], + "score": 0.974, + "type": "image", + "image_path": "562863548ff907597e11c6cbb57974aa93aa276baa0e5f95dac2fcddfba84a77.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 70, + 504, + 124.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 124.0, + 504, + 178.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 178.0, + 504, + 232.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 237, + 505, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "Figure 1: We aim to generate videos that accurately portray motion, changing camera viewpoint,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 249, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 261 + ], + "score": 1.0, + "content": "and new content that arises over time. Top: Our horseback riding dataset exhibits these types of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "changes as the horse moves forward in the environment. Middle: StyleGAN-V, a state-of-the-art", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "video generation baseline, is incapable of generating new content over time; the horse fails to move", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "forward past the obstacle, the scene does not change, and the video morphs back and forth within a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "short window of motion. Bottom: Our novel video generation model prioritizes the time axis and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "generates realistic motion and scenery changes over long durations. The same videos can be viewed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 314, + 229, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 229, + 327 + ], + "score": 1.0, + "content": "on the supplemental webpage.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "Our main contribution is a hierarchical generator architecture that employs a vast temporal receptive", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "field and a novel temporal embedding. We employ a multi-resolution strategy, where we first generate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "videos at low resolution and then refine them using a separate super-resolution network. Naively", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "training on long videos at high spatial resolution is prohibitively expensive, but we find that the main", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "aspects of a video persist at a low spatial resolution. This observation allows us to train with long", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "videos at low resolution and short videos at high resolution, enabling us to prioritize the time axis", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "and ensure that long-term changes are accurately portrayed. The low-resolution and super-resolution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "networks are trained independently with an RGB bottleneck in between. This modular design", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "allows iterating on each network independently and leveraging the same super-resolution network for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 445, + 279, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 279, + 457 + ], + "score": 1.0, + "content": "different low-resolution network ablations.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 108, + 461, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "We compare our results to several recent video generative models and demonstrate state-of-the-art", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "performance in producing long videos with realistic motion and changes in content. Code, new", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 484, + 402, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 402, + 496 + ], + "score": 1.0, + "content": "datasets, and pre-trained models on these datasets will be made available.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 510, + 182, + 523 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 185, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 185, + 526 + ], + "score": 1.0, + "content": "2 Prior work", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "Video generation is a challenging problem with a long history. The classic early works, Video", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "Textures [50] and Dynamic Textures [10], model videos as textures by analogy with image textures.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 555, + 507, + 571 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 507, + 571 + ], + "score": 1.0, + "content": "That is, they explicitly assume the content to be stationary over time, e.g., fire burning, smoke rising,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "foliage falling, pendulum swinging, etc., and use non-parametric [50] or parametric [10] approaches", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "to model that stationary distribution. Although subsequent video synthesis works have dropped the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "“texture” moniker, much of the limitations remain similar — short training videos and models which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "produce little or no new objects entering the frame during the video. Below we summarize some of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 611, + 283, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 283, + 624 + ], + "score": 1.0, + "content": "the more recent efforts on video generation.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "Unconditional video generation. Many video generation works are based on GANs [14], including", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "early models that output fixed-length videos [1, 47, 60] and approaches that use recurrent networks", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "to produce a sequence of latent codes used to generate frames [9, 12, 55, 56]. MoCoGAN [56]", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "explicitly disentangles “motion” from “content” and keeps the latter fixed over the entire generated", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "video. StyleGAN-V [52] is a recent state-of-the-art model we use as a primary baseline. Similar", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "to MoCoGAN, StyleGAN-V employs a global latent code that controls content of an entire video.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "MoCoGAN-HD [55], which we also compare with, and StyleVideoGAN [12] attempt to generate", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "videos by navigating the latent space of a pretrained StyleGAN2 model [29], but struggle to produce", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 70, + 504, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 70, + 504, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 504, + 232 + ], + "score": 0.974, + "type": "image", + "image_path": "562863548ff907597e11c6cbb57974aa93aa276baa0e5f95dac2fcddfba84a77.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 70, + 504, + 124.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 124.0, + 504, + 178.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 178.0, + 504, + 232.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 237, + 505, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "Figure 1: We aim to generate videos that accurately portray motion, changing camera viewpoint,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 249, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 261 + ], + "score": 1.0, + "content": "and new content that arises over time. Top: Our horseback riding dataset exhibits these types of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "changes as the horse moves forward in the environment. Middle: StyleGAN-V, a state-of-the-art", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "video generation baseline, is incapable of generating new content over time; the horse fails to move", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "forward past the obstacle, the scene does not change, and the video morphs back and forth within a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "short window of motion. Bottom: Our novel video generation model prioritizes the time axis and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "generates realistic motion and scenery changes over long durations. The same videos can be viewed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 314, + 229, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 229, + 327 + ], + "score": 1.0, + "content": "on the supplemental webpage.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "Our main contribution is a hierarchical generator architecture that employs a vast temporal receptive", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "field and a novel temporal embedding. We employ a multi-resolution strategy, where we first generate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "videos at low resolution and then refine them using a separate super-resolution network. Naively", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "training on long videos at high spatial resolution is prohibitively expensive, but we find that the main", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "aspects of a video persist at a low spatial resolution. This observation allows us to train with long", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "videos at low resolution and short videos at high resolution, enabling us to prioritize the time axis", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "and ensure that long-term changes are accurately portrayed. The low-resolution and super-resolution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "networks are trained independently with an RGB bottleneck in between. This modular design", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "allows iterating on each network independently and leveraging the same super-resolution network for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 445, + 279, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 279, + 457 + ], + "score": 1.0, + "content": "different low-resolution network ablations.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 347, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 461, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "We compare our results to several recent video generative models and demonstrate state-of-the-art", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "performance in producing long videos with realistic motion and changes in content. Code, new", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 484, + 402, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 402, + 496 + ], + "score": 1.0, + "content": "datasets, and pre-trained models on these datasets will be made available.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 461, + 505, + 496 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 510, + 182, + 523 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 185, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 185, + 526 + ], + "score": 1.0, + "content": "2 Prior work", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "Video generation is a challenging problem with a long history. The classic early works, Video", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "Textures [50] and Dynamic Textures [10], model videos as textures by analogy with image textures.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 555, + 507, + 571 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 507, + 571 + ], + "score": 1.0, + "content": "That is, they explicitly assume the content to be stationary over time, e.g., fire burning, smoke rising,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "foliage falling, pendulum swinging, etc., and use non-parametric [50] or parametric [10] approaches", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "to model that stationary distribution. Although subsequent video synthesis works have dropped the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "“texture” moniker, much of the limitations remain similar — short training videos and models which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "produce little or no new objects entering the frame during the video. Below we summarize some of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 611, + 283, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 283, + 624 + ], + "score": 1.0, + "content": "the more recent efforts on video generation.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 534, + 507, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "Unconditional video generation. Many video generation works are based on GANs [14], including", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "early models that output fixed-length videos [1, 47, 60] and approaches that use recurrent networks", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "to produce a sequence of latent codes used to generate frames [9, 12, 55, 56]. MoCoGAN [56]", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "explicitly disentangles “motion” from “content” and keeps the latter fixed over the entire generated", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "video. StyleGAN-V [52] is a recent state-of-the-art model we use as a primary baseline. Similar", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "to MoCoGAN, StyleGAN-V employs a global latent code that controls content of an entire video.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "MoCoGAN-HD [55], which we also compare with, and StyleVideoGAN [12] attempt to generate", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "videos by navigating the latent space of a pretrained StyleGAN2 model [29], but struggle to produce", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "realistic motion. Unlike previous StyleGAN-based [28] video models, we prioritize the time axis in", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "our generator through a new temporal latent representation, temporal upsampling, and spatiotemporal", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "modulated convolutions. We also compare with DIGAN [66] that employs an implicit representation", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 252, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 252, + 118 + ], + "score": 1.0, + "content": "to generate the video pixel by pixel.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 633, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "realistic motion. Unlike previous StyleGAN-based [28] video models, we prioritize the time axis in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "our generator through a new temporal latent representation, temporal upsampling, and spatiotemporal", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "modulated convolutions. We also compare with DIGAN [66] that employs an implicit representation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 252, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 252, + 118 + ], + "score": 1.0, + "content": "to generate the video pixel by pixel.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "Transformers are another class of models used for video generation [13, 42, 61, 65]. We com-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "score": 1.0, + "content": "pare with TATS [13] that generates long unconditional videos with transformers, improving upon", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "VideoGPT [65]. Both TATS and VideoGPT employ a GPT-like autoregressive transformer [4] that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 167 + ], + "score": 1.0, + "content": "represents videos as sequences of tokens. However, the resulting videos tend to accumulate error", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "over time and often diverge or change too rapidly. The models are also expensive to train and deploy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "due to their autoregressive nature over time and space. In concurrent work, promising results in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 188, + 468, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 468, + 200 + ], + "score": 1.0, + "content": "generating diverse videos have also been demonstrated using diffusion-based models [20].", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "Conditional video prediction. A separate line of research focuses on predicting future video frames", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "conditioned on one or more real video frames [3, 23, 34, 36, 39, 41] or past frames accompanied by an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "action label [6, 15, 30, 31]. Some video prediction methods focus specifically on generating infinite", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "scenery by conditioning on camera trajectory [37, 44] and/or explicitly predicting depth [2, 37] to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "then simulate a virtual camera flying through a 3D scene. Our goal, on the other hand, is to support", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 268, + 480, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 480, + 281 + ], + "score": 1.0, + "content": "camera movement as well as moving objects by having the scene structure emerge implicitly.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 307 + ], + "score": 1.0, + "content": "Multi-resolution training. Training at multiple scales is a common strategy for image generation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "models [7, 25, 43, 46, 58]. Transformer-based video generators also employ a related two-phase", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "setup [65, 13]. Saito et al. [48] subsample frames at higher resolutions in their video generator", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "architecture to improve efficiency. A similar idea is also used in SlowFast [11] networks where", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "different network pathways are used for high and low frame rate video streams. Acharya et al. [1]", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 362 + ], + "score": 1.0, + "content": "propose a multi-scale GAN for video generation that increases both spatial resolution and sequence", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "length during training to produce a fixed-length video. In contrast, our multi-resolution approach is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 368, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 384 + ], + "score": 1.0, + "content": "designed to enable generating arbitrarily long videos with rich long-term dynamics by leveraging", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 285, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 285, + 393 + ], + "score": 1.0, + "content": "training of long sequences at low resolution.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 189, + 423 + ], + "lines": [ + { + "bbox": [ + 104, + 408, + 190, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 190, + 426 + ], + "score": 1.0, + "content": "3 Our method", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "Modeling the long-term temporal behavior observed in real videos presents us with two main", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "challenges. 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Second, we must ensure that the networks themselves are capable of operating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 494 + ], + "score": 1.0, + "content": "over long time scales; if, e.g., the receptive field of the generator spans only 8 adjacent frames, any", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 474, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 474, + 503 + ], + "score": 1.0, + "content": "two frames taken more than 8 frames apart will necessarily be uncorrelated with each other.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 507, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 507, + 521 + ], + "score": 1.0, + "content": "Figure 2a shows the overall design of our generator. We seed the generation process with a variable-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 518, + 507, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 507, + 532 + ], + "score": 1.0, + "content": "length stream of temporal noise, consisting of 8 scalar components per frame drawn from i.i.d.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Gaussian distribution. 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Specifically,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "the low-resolution generator is designed to be fully convolutional over time, so the duration and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "time offset of the generated video can be controlled by shifting and reshaping the temporal noise,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "respectively. The super-resolution network, on the other hand, operates on a frame-by-frame basis. 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The combination of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "fully-convolutional and per-frame processing enables us to generate arbitrary frames in arbitrary", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 677, + 440, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 440, + 689 + ], + "score": 1.0, + "content": "order, which is highly desirable for, e.g., interactive editing and real-time playback.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 702, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 117, + 699, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 699, + 505, + 714 + ], + "score": 1.0, + "content": "2We handle datasets with non-square aspect ratio by shrinking all intermediate data accordingly. With", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 450, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 142, + 721 + ], + "score": 0.84, + "content": "2 5 6 \\times 1 4 4", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 711, + 382, + 722 + ], + "score": 1.0, + "content": "target resolution, for example, the low-resolution frames will have", + "type": "text" + }, + { + "bbox": [ + 382, + 712, + 409, + 721 + ], + "score": 0.87, + "content": "6 4 \\times 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 711, + 450, + 722 + ], + "score": 1.0, + "content": "resolution.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 505, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 122, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "Transformers are another class of models used for video generation [13, 42, 61, 65]. We com-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "score": 1.0, + "content": "pare with TATS [13] that generates long unconditional videos with transformers, improving upon", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "VideoGPT [65]. Both TATS and VideoGPT employ a GPT-like autoregressive transformer [4] that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 167 + ], + "score": 1.0, + "content": "represents videos as sequences of tokens. However, the resulting videos tend to accumulate error", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "over time and often diverge or change too rapidly. The models are also expensive to train and deploy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "due to their autoregressive nature over time and space. In concurrent work, promising results in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 188, + 468, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 468, + 200 + ], + "score": 1.0, + "content": "generating diverse videos have also been demonstrated using diffusion-based models [20].", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 121, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "Conditional video prediction. A separate line of research focuses on predicting future video frames", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "conditioned on one or more real video frames [3, 23, 34, 36, 39, 41] or past frames accompanied by an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "action label [6, 15, 30, 31]. Some video prediction methods focus specifically on generating infinite", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "scenery by conditioning on camera trajectory [37, 44] and/or explicitly predicting depth [2, 37] to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 269 + ], + "score": 1.0, + "content": "then simulate a virtual camera flying through a 3D scene. Our goal, on the other hand, is to support", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 268, + 480, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 480, + 281 + ], + "score": 1.0, + "content": "camera movement as well as moving objects by having the scene structure emerge implicitly.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 212, + 506, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 307 + ], + "score": 1.0, + "content": "Multi-resolution training. Training at multiple scales is a common strategy for image generation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "models [7, 25, 43, 46, 58]. Transformer-based video generators also employ a related two-phase", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "setup [65, 13]. Saito et al. [48] subsample frames at higher resolutions in their video generator", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "architecture to improve efficiency. A similar idea is also used in SlowFast [11] networks where", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "different network pathways are used for high and low frame rate video streams. Acharya et al. 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In contrast, our multi-resolution approach is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 368, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 384 + ], + "score": 1.0, + "content": "designed to enable generating arbitrarily long videos with rich long-term dynamics by leveraging", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 285, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 285, + 393 + ], + "score": 1.0, + "content": "training of long sequences at low resolution.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 292, + 506, + 393 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 189, + 423 + ], + "lines": [ + { + "bbox": [ + 104, + 408, + 190, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 190, + 426 + ], + "score": 1.0, + "content": "3 Our method", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "Modeling the long-term temporal behavior observed in real videos presents us with two main", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "challenges. 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Second, we must ensure that the networks themselves are capable of operating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 494 + ], + "score": 1.0, + "content": "over long time scales; if, e.g., the receptive field of the generator spans only 8 adjacent frames, any", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 474, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 474, + 503 + ], + "score": 1.0, + "content": "two frames taken more than 8 frames apart will necessarily be uncorrelated with each other.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 436, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 507, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 507, + 521 + ], + "score": 1.0, + "content": "Figure 2a shows the overall design of our generator. We seed the generation process with a variable-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 518, + 507, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 507, + 532 + ], + "score": 1.0, + "content": "length stream of temporal noise, consisting of 8 scalar components per frame drawn from i.i.d.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Gaussian distribution. The temporal noise is first processed by a low-resolution generator to obtain", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 231, + 553 + ], + "score": 1.0, + "content": "a sequence of RGB frames at", + "type": "text" + }, + { + "bbox": [ + 231, + 540, + 247, + 551 + ], + "score": 0.9, + "content": "6 4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "resolution that are then refined by a separate super-resolution", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 261, + 564 + ], + "score": 1.0, + "content": "network to produce the final frames at", + "type": "text" + }, + { + "bbox": [ + 261, + 551, + 282, + 562 + ], + "score": 0.87, + "content": "2 5 6 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "resolution.2 The role of the low-resolution generator is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "to model major aspects of the motion and scene composition, which necessitates strong expressive", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "power and a large receptive field over time, whereas the super-resolution network is responsible for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 585, + 366, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 366, + 596 + ], + "score": 1.0, + "content": "the more fine-grained task of hallucinating the remaining details.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 507, + 507, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 688 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "Our two-stage design provides maximum flexibility in terms of generating long videos. Specifically,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "the low-resolution generator is designed to be fully convolutional over time, so the duration and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "time offset of the generated video can be controlled by shifting and reshaping the temporal noise,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "respectively. The super-resolution network, on the other hand, operates on a frame-by-frame basis. It", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "receives a short sequence of 9 consecutive low-resolution frames and outputs a single high-resolution", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "frame; each output frame is processed independently using a sliding window. 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(a) To achieve long temporal receptive field and high spatial", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "resolution, we split our generator into two components: a low-resolution generator, responsible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "for modeling major aspects of the motion and scene composition, and a super-resolution network,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "responsible for hallucinating fine details. (b) The low-resolution generator (Section 3.1) employs a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 411, + 328 + ], + "score": 1.0, + "content": "wide temporal receptive field and is trained with sequences of 128 frames at", + "type": "text" + }, + { + "bbox": [ + 411, + 316, + 427, + 326 + ], + "score": 0.87, + "content": "6 4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "resolution. (c) The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "super-resolution network (Section 3.2) is conditioned on short sequences of low-resolution frames", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 338, + 385, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 319, + 350 + ], + "score": 1.0, + "content": "and trained to produce their plausible counterparts at", + "type": "text" + }, + { + "bbox": [ + 319, + 338, + 340, + 348 + ], + "score": 0.86, + "content": "2 5 6 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 338, + 385, + 350 + ], + "score": 1.0, + "content": "resolution.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "The low-resolution and super-resolution networks are modular with an RGB bottleneck in between.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "This greatly simplifies experimentation, since the networks are trained independently and can be used", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "in different combinations during inference. We will first describe the training and architecture of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 410, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 421 + ], + "score": 1.0, + "content": "low-resolution generator in Section 3.1 and then discuss the super-resolution network in Section 3.2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 440, + 239, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 241, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 241, + 455 + ], + "score": 1.0, + "content": "3.1 Low-resolution generator", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "Figure 2b shows our training setup for the low-resolution generator. In each iteration, we provide", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "the generator with a fresh set of temporal noise to produce sequences of 128 frames (4.3 seconds at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 483, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 499 + ], + "score": 1.0, + "content": "30 fps). To train the discriminator, we sample corresponding sequences from the training data by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 496, + 429, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 429, + 507 + ], + "score": 1.0, + "content": "choosing a random video and a random interval of 128 frames within that video.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "We have observed that training with long sequences tends to exacerbate the issue of overfitting [26].", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "As the sequence length increases, we suspect that it becomes harder for the generator to simultaneously", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "score": 1.0, + "content": "model temporal dynamics at multiple time scales, but at the same time, easier for the discriminator", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "to spot any mistakes. In practice, we have found strong discriminator augmentation [26, 69] to be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "necessary in order to stabilize the training. We employ DiffAug [69] using the same transformation for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 384, + 580 + ], + "score": 1.0, + "content": "each frame in a sequence, as well as fractional time stretching between", + "type": "text" + }, + { + "bbox": [ + 384, + 566, + 399, + 579 + ], + "score": 0.91, + "content": "\\bar { \\frac { 1 } { 2 } } \\bar { \\times }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 567, + 416, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 416, + 567, + 430, + 578 + ], + "score": 0.84, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "; see Appendix C.1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 577, + 151, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 151, + 589 + ], + "score": 1.0, + "content": "for details.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "Architecture. Figure 3 illustrates the architecture of our low-resolution generator. Our main goal is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "to make the time axis a first-class citizen, including careful design of a temporal latent representation,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "temporal style modulation, spatiotemporal convolutions, and temporal upsamples. 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(a) To achieve long temporal receptive field and high spatial", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "resolution, we split our generator into two components: a low-resolution generator, responsible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "for modeling major aspects of the motion and scene composition, and a super-resolution network,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "responsible for hallucinating fine details. (b) The low-resolution generator (Section 3.1) employs a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 411, + 328 + ], + "score": 1.0, + "content": "wide temporal receptive field and is trained with sequences of 128 frames at", + "type": "text" + }, + { + "bbox": [ + 411, + 316, + 427, + 326 + ], + "score": 0.87, + "content": "6 4 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "resolution. (c) The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "super-resolution network (Section 3.2) is conditioned on short sequences of low-resolution frames", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 338, + 385, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 319, + 350 + ], + "score": 1.0, + "content": "and trained to produce their plausible counterparts at", + "type": "text" + }, + { + "bbox": [ + 319, + 338, + 340, + 348 + ], + "score": 0.86, + "content": "2 5 6 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 338, + 385, + 350 + ], + "score": 1.0, + "content": "resolution.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "The low-resolution and super-resolution networks are modular with an RGB bottleneck in between.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "This greatly simplifies experimentation, since the networks are trained independently and can be used", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "in different combinations during inference. We will first describe the training and architecture of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 410, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 421 + ], + "score": 1.0, + "content": "low-resolution generator in Section 3.1 and then discuss the super-resolution network in Section 3.2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 376, + 506, + 421 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 440, + 239, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 241, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 241, + 455 + ], + "score": 1.0, + "content": "3.1 Low-resolution generator", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "Figure 2b shows our training setup for the low-resolution generator. In each iteration, we provide", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "the generator with a fresh set of temporal noise to produce sequences of 128 frames (4.3 seconds at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 483, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 499 + ], + "score": 1.0, + "content": "30 fps). To train the discriminator, we sample corresponding sequences from the training data by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 496, + 429, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 429, + 507 + ], + "score": 1.0, + "content": "choosing a random video and a random interval of 128 frames within that video.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 463, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "We have observed that training with long sequences tends to exacerbate the issue of overfitting [26].", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "As the sequence length increases, we suspect that it becomes harder for the generator to simultaneously", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 546 + ], + "score": 1.0, + "content": "model temporal dynamics at multiple time scales, but at the same time, easier for the discriminator", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "to spot any mistakes. In practice, we have found strong discriminator augmentation [26, 69] to be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "necessary in order to stabilize the training. We employ DiffAug [69] using the same transformation for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 384, + 580 + ], + "score": 1.0, + "content": "each frame in a sequence, as well as fractional time stretching between", + "type": "text" + }, + { + "bbox": [ + 384, + 566, + 399, + 579 + ], + "score": 0.91, + "content": "\\bar { \\frac { 1 } { 2 } } \\bar { \\times }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 567, + 416, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 416, + 567, + 430, + 578 + ], + "score": 0.84, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "; see Appendix C.1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 577, + 151, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 151, + 589 + ], + "score": 1.0, + "content": "for details.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 510, + 506, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "Architecture. Figure 3 illustrates the architecture of our low-resolution generator. Our main goal is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "to make the time axis a first-class citizen, including careful design of a temporal latent representation,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "temporal style modulation, spatiotemporal convolutions, and temporal upsamples. Through these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "mechanisms, our generator spans a vast temporal receptive field (5k frames), allowing it to represent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 651, + 287, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 287, + 663 + ], + "score": 1.0, + "content": "temporal correlations at multiple time scales.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 607, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "We employ a style-based design, similar to Karras et al. [29, 27], that maps the input temporal noise", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 261, + 691 + ], + "score": 1.0, + "content": "into a sequence of intermediate latents", + "type": "text" + }, + { + "bbox": [ + 262, + 679, + 283, + 690 + ], + "score": 0.91, + "content": "\\{ w _ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "used to modulate the behavior of each layer in the main", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "synthesis path. Each intermediate latent is associated with a specific frame, but it can significantly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "influence the scene composition and temporal behavior of several frames through hierarchical 3D", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 711, + 275, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 275, + 723 + ], + "score": 1.0, + "content": "convolutions that appear in the main path.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 667, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 72, + 503, + 297 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 72, + 503, + 297 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 72, + 503, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 503, + 297 + ], + "score": 0.972, + "type": "image", + "image_path": "076266e0bcaa70f10582dc49f85bdf9ecb33cf99075855164135f9dc4acbcfee.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 72, + 503, + 147.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 147.0, + 503, + 222.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 222.0, + 503, + 297.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 303, + 506, + 380 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 371, + 315 + ], + "score": 1.0, + "content": "Figure 3: Low-resolution generator architecture, illustrated for", + "type": "text" + }, + { + "bbox": [ + 372, + 303, + 399, + 313 + ], + "score": 0.86, + "content": "6 4 { \\times } 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "output. Left: The input", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 371, + 327 + ], + "score": 1.0, + "content": "temporal noise is mapped to a sequence of intermediate latents", + "type": "text" + }, + { + "bbox": [ + 371, + 314, + 393, + 326 + ], + "score": 0.91, + "content": "\\{ w _ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "that modulate the interme-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 323, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 506, + 338 + ], + "score": 1.0, + "content": "diate activations of the main synthesis path. Top right: To facilitate the modeling of long-term", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "dependencies, we enrich the temporal noise by passing it through a series of lowpass filters whose", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "temporal footprints range all the way from 100 to 5000 frames. Bottom right: The main synthesis", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "path consists of spatiotemporal (ST) and spatial (S) blocks that gradually increase the resolution over", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 369, + 171, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 171, + 381 + ], + "score": 1.0, + "content": "time and space.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "In order to reap the full benefits of the style-based design, it is crucial for the intermediate latents", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "to capture long-term temporal correlations, such as weather changes or persistent objects. To this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "end, we adopt a scheme where we first enrich the input temporal noise using a series of temporal", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "lowpass filters and then pass it through a fully-connected mapping network on a frame-by-frame", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "basis. The goal of the lowpass filtering is to provide the mapping network with sufficient long-term", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "context across a wide range of different time scales. 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It then gradually increases the temporal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "and spatial resolutions through a series of processing blocks, illustrated in Figure 3 (bottom right),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "focusing first on the time dimension (ST) and then the spatial dimensions (S). The first four blocks", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "have 512 channels, followed by two blocks with 256, two with 128 and two with 64 channels. 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We introduce first-person datasets of (a)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "mountain biking and (b) horseback riding videos that contain complex motion and new content over", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 175, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 505, + 187 + ], + "score": 1.0, + "content": "time. 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We feel that replacing this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "simple network with a more advanced model from the video super-resolution literature [16, 24, 49, 54]", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 261, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 261, + 462 + ], + "score": 1.0, + "content": "is a promising avenue for future work.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 405, + 506, + 462 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 491, + 169, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 490, + 171, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 171, + 506 + ], + "score": 1.0, + "content": "4 Datasets", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "Most of the existing video datasets introduce little or no new content over time. For example, talking", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "head datasets [8, 45, 62, 63] show the same person for the duration of each video. UCF101 [53]", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "portrays diverse human actions, but the videos are short and contain limited camera motion and little", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 559, + 305, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 305, + 570 + ], + "score": 1.0, + "content": "or no new objects that enter the videos over time.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 524, + 506, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "To best evaluate our model, we introduce two new video datasets of first-person mountain biking and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "horseback riding (Figure 4a,b) that exhibit complex changes over time. Our new datasets include", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "subject motion of the horse or biker, a first-person camera viewpoint that moves through space, and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "new scenery and objects over time. The videos are available in high definition and were manually", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "trimmed to remove problematic segments, scene cuts, text overlays, obstructed views, etc. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "mountain biking dataset has 1202 videos with a median duration of 330 frames at 30 fps, and the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "horseback dataset has 66 videos with a median duration of 6504 frames also at 30fps. We have", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "permission from the content owners to publicly release the datasets for research purposes. We believe", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 662, + 385, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 385, + 674 + ], + "score": 1.0, + "content": "our new datasets will serve as important benchmarks for future work.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 574, + 506, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "We also evaluate our model on the ACID dataset [38] (Figure 4c) that contains significant camera", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "motion but lacks other types of motion, as well as the commonly used SkyTimelapse dataset [67]", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "(Figure 4d) that exhibits new content over time as the clouds pass by, but the videos are relatively", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 711, + 286, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 286, + 723 + ], + "score": 1.0, + "content": "homogeneous and the camera remains fixed.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 677, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 76, + 505, + 179 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 76, + 505, + 179 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 76, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 76, + 505, + 179 + ], + "score": 0.969, + "type": "image", + "image_path": "51a7018ce7ee7153bed2227caa13b3dc9ccca39dc48ab201bd0e52c57e0b9491.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 76, + 505, + 110.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 110.33333333333334, + 505, + 144.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 144.66666666666669, + 505, + 179.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 185, + 505, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 197 + ], + "score": 1.0, + "content": "Figure 5: Color similarity (Eq. 1) of real and generated videos as a function of frame separation,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 195, + 502, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 502, + 208 + ], + "score": 1.0, + "content": "reported as the mean (solid lines) and standard deviation (shaded regions) over 1000 random clips.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 106, + 227, + 163, + 241 + ], + "lines": [ + { + "bbox": [ + 104, + 226, + 165, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 165, + 243 + ], + "score": 1.0, + "content": "5 Results", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 506, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "We evaluate our model through qualitative examination of the generated videos (Section 5.1), ana-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "lyzing color change over time (Section 5.2), computing the FVD metric (Section 5.3), and ablating", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "the key design choices (Section 5.4). We compare with StyleGAN-V [52] on all datasets. Mountain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 387, + 298 + ], + "score": 1.0, + "content": "biking, horseback riding and ACID [37] datasets contain videos with a", + "type": "text" + }, + { + "bbox": [ + 387, + 286, + 409, + 296 + ], + "score": 0.86, + "content": "1 6 { \\times } 9", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "widescreen aspect ratio.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 152, + 309 + ], + "score": 1.0, + "content": "We train at", + "type": "text" + }, + { + "bbox": [ + 152, + 297, + 189, + 307 + ], + "score": 0.82, + "content": "2 5 6 { \\times } 1 4 4", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "resolution on these datasets to preserve the aspect ratio. Since StyleGAN-V is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "based on StyleGAN2 [29], we can easily extend it to support non-square aspect ratios by masking", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 419, + 332 + ], + "score": 1.0, + "content": "real and generated frames during training. We found it necessary to increase the", + "type": "text" + }, + { + "bbox": [ + 420, + 318, + 441, + 330 + ], + "score": 0.35, + "content": "{ \\mathrm { R 1 } } \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "hyperparameter", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 328, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 119, + 343 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 330, + 136, + 340 + ], + "score": 0.85, + "content": "1 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 328, + 506, + 343 + ], + "score": 1.0, + "content": "to produce good results with StyleGAN-V on our new datasets that exhibit complex changes", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "over time. We compare with MoCoGAN-HD [56], TATS [13] and DIGAN [66] using pre-trained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 349, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 266, + 364 + ], + "score": 1.0, + "content": "models for the SkyTimelapse dataset at", + "type": "text" + }, + { + "bbox": [ + 266, + 351, + 286, + 361 + ], + "score": 0.87, + "content": "1 2 8 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 349, + 506, + 364 + ], + "score": 1.0, + "content": "resolution. For these comparisons, we train a separate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 305, + 374 + ], + "score": 1.0, + "content": "super-resolution network to output the frames at", + "type": "text" + }, + { + "bbox": [ + 306, + 361, + 326, + 372 + ], + "score": 0.86, + "content": "1 2 8 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "resolution, but use the same low-resolution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 372, + 255, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 182, + 385 + ], + "score": 1.0, + "content": "generator as in the", + "type": "text" + }, + { + "bbox": [ + 182, + 372, + 203, + 383 + ], + "score": 0.87, + "content": "2 5 6 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 373, + 255, + 385 + ], + "score": 1.0, + "content": "comparison.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 208, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 209, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 209, + 411 + ], + "score": 1.0, + "content": "5.1 Qualitative results", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "The major qualitative difference in results is that our model generates realistic new content over time,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "whereas StyleGAN-V continually repeats the same content. The effect is best observed by watching", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "videos on the supplemental webpage and is additionally illustrated in Figure 1. Scenery changes", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "over time in real videos and our results as the horse moves forward through space. However, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "videos generated by StyleGAN-V tend to morph back to the same scene at regular intervals. Similar", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "repeated content from StyleGAN-V is apparent on all datasets. For example, results on the webpage", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "for the SkyTimelapse dataset show that clouds generated by StyleGAN-V repeatedly move back and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "forth. MoCoGAN-HD and TATS suffer from unrealistic rapid changes over time that diverge, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "DIGAN results contain periodic patterns visible in both space and time. Our model is capable of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 284, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 284, + 528 + ], + "score": 1.0, + "content": "generating a constant stream of new clouds.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "As a further validation of our observations, we conducted a preliminary user study on Amazon", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "Mechanical Turk. We created 50 pairs of videos for each of the 4 datasets. Each pair contained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "a random video generated by StyleGAN-V and one generated by our method, and we asked the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "participants which of them exhibited more realistic motion in a forced-choice response. Each pair", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 577, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 313, + 588 + ], + "score": 1.0, + "content": "was shown to 10 participants, resulting in a total of", + "type": "text" + }, + { + "bbox": [ + 314, + 577, + 351, + 587 + ], + "score": 0.88, + "content": "5 0 { \\times } 4 { \\times } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 577, + 506, + 588 + ], + "score": 1.0, + "content": "responses. Our method was preferred", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 587, + 411, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 126, + 599 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 126, + 587, + 146, + 598 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 588, + 411, + 599 + ], + "score": 1.0, + "content": "of the time for every dataset. Please see Appendix A.1 for details.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 612, + 272, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 274, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 274, + 627 + ], + "score": 1.0, + "content": "5.2 Analyzing color change over time", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "To gain insight into how well different methods produce new content at appropriate rates, we analyze", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "how the overall color scheme changes as a function of time. 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BikingHorsebackACIDSky 2562
FVD128FVD16FVD128FVD16FVD128FVD16FVD128FVD16
StyleGAN-V533.3353.7427.0319.2112.491.5151.248.4
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The curves", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "trend downward over time for real videos as content and scenery gradually change. StyleGAN-V", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "and DIGAN are biased toward colors changing too slowly — both of these models include a global", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "latent code that is fixed over the entire video. On the other extreme, MoCoGAN-HD and TATS are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 346 + ], + "score": 1.0, + "content": "biased toward colors changing too quickly. These models use recurrent and autoregressive networks,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 358 + ], + "score": 1.0, + "content": "respectively, both of which suffer from accumulating errors. 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In Appendix A.3", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "we also consider LPIPS [68] perceptual distance instead of color similarly and observe the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 394, + 192, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 192, + 407 + ], + "score": 1.0, + "content": "trends in most cases.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 373, + 505, + 407 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 441, + 255, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 256, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 256, + 456 + ], + "score": 1.0, + "content": "5.3 Fréchet video distance (FVD)", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "score": 1.0, + "content": "The commonly used Fréchet video distance (FVD) [57] attempts to measure similarity between real", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "and generated video distributions. 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Note that even though we ablate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "aspects of the low-resolution generator, we still compute FVD using the final high-resolution videos", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 430, + 278, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 278, + 441 + ], + "score": 1.0, + "content": "produced by the super-resolution network.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 363, + 506, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "Footprint of the temporal lowpass filters. Our temporal latent representation serves a vital role", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "in expanding the receptive field of our generator, modeling patterns over different time scales, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "enabling the generation of new content over time. While we primarily leverage long training videos", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "to learn long-term consistencies from data, the size of our temporal lowpass filters plays a role in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "encouraging the low-resolution mapping network to learn correlations at appropriate time scales.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "Table 2b demonstrates the negative impact of using inappropriately sized filters. We find that our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "model performs well with the same filter configuration for all datasets, although it is possible that the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 525, + 306, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 306, + 540 + ], + "score": 1.0, + "content": "ideal settings may vary slightly between datasets.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 449, + 506, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "Effectiveness of the super-resolution network. Figure 6a,b shows examples of low-resolution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "frames generated by our model along with the corresponding high-resolution frames produced by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "our super-resolution network; we find that the super-resolution network generally performs well. To", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 579, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 591 + ], + "score": 1.0, + "content": "ensure that the quality of our results is not disproportionately limited by the super-resolution network,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "we further measure FVD when providing the super-resolution network with real low-resolution", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "videos as input in Figure 6c. Indeed, FVD greatly improves in this case, which indicates that there", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 611, + 465, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 465, + 624 + ], + "score": 1.0, + "content": "are still significant gains to be realized by further improving the low-resolution generator.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 546, + 506, + 624 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 185, + 647 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 187, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 187, + 650 + ], + "score": 1.0, + "content": "5.5 Failure cases", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "Separate low- and super-resolution networks makes the problem computationally feasible, but it may", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "somewhat compromise the quality of the final high-resolution frames. We observed that “swirly”", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "artifacts are most prominent in the super-resolution output and not in the low-resolution output. Our", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "model also struggles with long-term consistency of small details (e.g., distant jumps in generated", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "horseback riding videos) that begin to appear before quickly fading out. We believe these issues are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "due to limitations of our super-resolution network, and that improving the super-resolution network", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "would benefit the model in this regard. Another failure case we observed is difficulty preserving 3D", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "consistency for scenes with very little motion, such as in the ACID dataset. In cases where there is", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 377, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 377, + 107 + ], + "score": 1.0, + "content": "little motion, one may consider using an explicit 3D representation.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 656, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "would benefit the model in this regard. Another failure case we observed is difficulty preserving 3D", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "consistency for scenes with very little motion, such as in the ACID dataset. In cases where there is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 377, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 377, + 107 + ], + "score": 1.0, + "content": "little motion, one may consider using an explicit 3D representation.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 118, + 187, + 132 + ], + "lines": [ + { + "bbox": [ + 104, + 115, + 190, + 134 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 190, + 134 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 506, + 155 + ], + "score": 1.0, + "content": "Video generation has historically focused on relatively short clips with little new content over time.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "We consider longer videos with complex temporal changes, and uncover several open questions and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "video generation practices worth reassessing — the temporal latent representation and generator", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "architecture, the training sequence length and recipes for using long videos, and the right evaluation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 237, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 237, + 199 + ], + "score": 1.0, + "content": "metrics for long-term dynamics.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "We have shown that representations over many time scales serve as useful building blocks for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "modeling complex motions and the introduction of new content over time. We feel that the form of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "the latent space most suitable for video remains an open, almost philosophical question, leaving a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "large design space to explore. For example, what is the right latent representation to model persistent", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "objects that exit from a video and re-enter later in the video while maintaining a consistent identity?", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "The benefits we find from training on longer sequences open up further questions. Would video", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "generation benefit from even longer training sequences? Currently we train using segments of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 500, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 500, + 298 + ], + "score": 1.0, + "content": "adjacent frames, but it might be beneficial to use larger frame spacings to cover longer time spans.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Quantitative evaluation of the results continues to be challenging. As we observed, FVD goes only a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "part of the way, being essentially blind to repetitive, even very implausible results. Our tests with how", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "the colors and LPIPS distance change as a function of time partially bridge this gap, but we feel that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "this area deserves a thorough, targeted investigation of its own. We hope our work encourages further", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 494, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 494, + 357 + ], + "score": 1.0, + "content": "research into video generation that focuses on more complex and longer-term changes over time.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "Negative societal impacts Our work falls within data-driven generative modeling, which, as a field,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "has well known potential for misuse with increasing quality improvements. The training of video", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "generators is even more intensive computationally than training still image generators, increasing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 492, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 492, + 414 + ], + "score": 1.0, + "content": "energy usage. Our project consumed 300MWh on an in-house cluster of V100 and A100 GPUs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "Acknowledgements We thank William Peebles, Samuli Laine, Axel Sauer and David Luebke", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "for helpful discussion and feedback; Ivan Skorokhodov for providing additional results and in-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "sight into the StyleGAN-V baseline; Tero Kuosmanen for maintaining compute infrastructure;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "Elisa Wallace Eventing (https://www.youtube.com/c/WallaceEventing) and Brian Kennedy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "(https://www.youtube.com/c/bkxc) for videos used to make the horseback riding and mountain", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "biking datasets. Tim Brooks is supported by the National Science Foundation Graduate Research", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 276, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 276, + 504 + ], + "score": 1.0, + "content": "Fellowship under Grant No. 2020306087.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 163, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 165, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 165, + 535 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 111, + 542, + 507, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 110, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "[1] Dinesh Acharya, Zhiwu Huang, Danda Pani Paudel, and Luc Van Gool. 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ICLR, 2018.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 110, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 110, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "[4] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 127, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 127, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 128, + 657, + 353, + 668 + ], + "spans": [ + { + "bbox": [ + 128, + 657, + 353, + 668 + ], + "score": 1.0, + "content": "few-shot learners. Proc. 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Recurrent environ-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 127, + 711, + 284, + 723 + ], + "spans": [ + { + "bbox": [ + 127, + 711, + 284, + 723 + ], + "score": 1.0, + "content": "ment simulators. In Proc. ICLR, 2017.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 73, + 505, + 107 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 118, + 187, + 132 + ], + "lines": [ + { + "bbox": [ + 104, + 115, + 190, + 134 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 190, + 134 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 506, + 155 + ], + "score": 1.0, + "content": "Video generation has historically focused on relatively short clips with little new content over time.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "We consider longer videos with complex temporal changes, and uncover several open questions and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "video generation practices worth reassessing — the temporal latent representation and generator", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "architecture, the training sequence length and recipes for using long videos, and the right evaluation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 237, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 237, + 199 + ], + "score": 1.0, + "content": "metrics for long-term dynamics.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 142, + 506, + 199 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "We have shown that representations over many time scales serve as useful building blocks for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "modeling complex motions and the introduction of new content over time. We feel that the form of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "the latent space most suitable for video remains an open, almost philosophical question, leaving a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "large design space to explore. 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Would video", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "generation benefit from even longer training sequences? Currently we train using segments of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 500, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 500, + 298 + ], + "score": 1.0, + "content": "adjacent frames, but it might be beneficial to use larger frame spacings to cover longer time spans.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 261, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Quantitative evaluation of the results continues to be challenging. As we observed, FVD goes only a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "part of the way, being essentially blind to repetitive, even very implausible results. Our tests with how", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "the colors and LPIPS distance change as a function of time partially bridge this gap, but we feel that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "this area deserves a thorough, targeted investigation of its own. We hope our work encourages further", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 494, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 494, + 357 + ], + "score": 1.0, + "content": "research into video generation that focuses on more complex and longer-term changes over time.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 301, + 506, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "Negative societal impacts Our work falls within data-driven generative modeling, which, as a field,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "has well known potential for misuse with increasing quality improvements. The training of video", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "generators is even more intensive computationally than training still image generators, increasing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 492, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 492, + 414 + ], + "score": 1.0, + "content": "energy usage. Our project consumed 300MWh on an in-house cluster of V100 and A100 GPUs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 369, + 506, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "Acknowledgements We thank William Peebles, Samuli Laine, Axel Sauer and David Luebke", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "for helpful discussion and feedback; Ivan Skorokhodov for providing additional results and in-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "sight into the StyleGAN-V baseline; Tero Kuosmanen for maintaining compute infrastructure;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "Elisa Wallace Eventing (https://www.youtube.com/c/WallaceEventing) and Brian Kennedy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "(https://www.youtube.com/c/bkxc) for videos used to make the horseback riding and mountain", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "biking datasets. Tim Brooks is supported by the National Science Foundation Graduate Research", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 276, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 276, + 504 + ], + "score": 1.0, + "content": "Fellowship under Grant No. 2020306087.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 426, + 506, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 163, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 165, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 165, + 535 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "list", + "bbox": [ + 111, + 542, + 507, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 110, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "[1] Dinesh Acharya, Zhiwu Huang, Danda Pani Paudel, and Luc Van Gool. 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However, despite being a general technique, recent works reveal that applying binarization in various practical scenarios, including multiple tasks, architectures, and hardware, is not trivial. Moreover, common challenges, such as severe degradation in accuracy and limited efficiency gains, suggest that specific attributes of binarization are not thoroughly studied and adequately understood. To comprehensively understand binarization methods, we present BiBench, a carefully engineered benchmark with in-depth analysis for network binarization. We first inspect the requirements of binarization in the actual production setting. Then for the sake of fairness and systematic, we define the evaluation tracks and metrics. We also perform a comprehensive evaluation with a rich collection of milestone binarization algorithms. Our benchmark results show that binarization still faces severe accuracy challenges, and newer state-ofthe-art binarization algorithms bring diminishing improvements, even at the expense of efficiency. Moreover, the actual deployment of certain binarization operations reveals a surprisingly large deviation from their theoretical consumption. Finally, based on our benchmark results and analysis, we suggest establishing a paradigm for accurate and efficient binarization among existing techniques. We hope BiBench paves the way toward more extensive adoption of network binarization and serves as a fundamental work for future research. ", + "bbox": [ + 233, + 267, + 764, + 558 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 585, + 336, + 602 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since the rising of modern deep learning, the contradiction between ever-increasing model size and limited deployment resources has persisted. For this reason, compression technologies are crucial for practical deep learning and have been widely studied, including model quantization (Gong et al., 2014; Wu et al., 2016; Vanhoucke et al., 2011; Gupta et al., 2015), network pruning (Han et al., 2015; 2016; He et al., 2017), knowledge distillation (Hinton et al., 2015; Xu et al., 2018; Chen et al., 2018; Yim et al., 2017; Zagoruyko & Komodakis, 2017), lightweight architecture design (Howard et al., 2017; Sandler et al., 2018; Zhang et al., 2018b; Ma et al., 2018), and low-rank decomposition (Denton et al., 2014; Lebedev et al., 2015; Jaderberg et al., 2014; Lebedev & Lempitsky, 2016). ", + "bbox": [ + 174, + 617, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As a compression approach that extremely reduces the bit-width to 1-bit, network binarization is regarded as the most aggressive quantization technology (Rusci et al., 2020; Choukroun et al., 2019; Qin et al., 2022; Shang et al., 2022b; Zhang et al., 2022b; Bethge et al., 2020; 2019; Martinez et al., 2019; Helwegen et al., 2019). The binarized models leverage the most compact 1-bit parameters, which take little storage and memory and accelerate the inference by efficient bitwise operations. Compared to other compression technologies like network pruning and architecture design, network binarization enjoys stronger topological generics since it only applies to parameters. Therefore, in academic research, network binarization is widely studied as an independent compression technique instead of the 1-bit specialization of quantization (Gong et al., 2019; Gholami et al., 2021). It is impressive that State-of-The-Art (SoTA) binarization algorithms push binarized models to fullprecision performance on large-scale tasks (Deng et al., 2009; Liu et al., 2020). ", + "bbox": [ + 174, + 736, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, existing network binarization is still far from practical. We point out that two worrisome trends are emerging from accuracy and efficiency perspectives in current binarization research: ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/7e5f1ccd33fd55f9a3e7fe5c51344d6ecce734ec1eebddb4b080b106c9a53994.jpg", + "image_caption": [ + "Figure 1: Evaluation tracks of BiBench. Our benchmark evaluates binarization algorithms on the most comprehensive evaluation tracks, including “Learning Task”, “Neural Architecture”, “Corruption Robustness”, “Training Consumption”, “Theoretical Complexity”, and “Hardware Inference”. " + ], + "image_footnote": [], + "bbox": [ + 199, + 59, + 802, + 219 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Trend-1. Accuracy comparison converging to limited scope. In recent binarization research, several image classification tasks, e.g., CIFAR-10 and ImageNet, are becoming standard options for comparing accuracy. The typical selection of evaluation tasks helps the clear and fair comparison of accuracy performance among different binarization algorithms. However, since most binarization algorithm studies are engineered for learning tasks with image modality inputs, the presented insights and conclusions are rarely verified in a broader range of other modalities and tasks. The monotonic tasks also hinder the comprehensive evaluation from an architectural perspective. Besides, data noise like corruption is a common problem on low-cost edge devices and is widely studied in compression (Lin et al., 2018; Rakin et al., 2021), whereas few advanced binarization algorithms consider the robustness of binarized models. ", + "bbox": [ + 173, + 286, + 825, + 425 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Trend-2. Efficiency analysis remaining at the theoretical level. Network binarization is widely recognized for its significant storage and computation savings. For example, theoretical savings are up to $3 2 \\times$ and $6 4 \\times$ for convolutions, respectively (Rastegari et al., 2016; Bai et al., 2021). However, since lacking support from hardware libraries, the models compressed by binarization algorithms can hardly be evaluated on real-world edge hardware, leaving their efficiency claims lacking experimental evidence. In addition, the training efficiency of the binarization algorithm is usually neglected in current research, which causes several negative phenomena in training a binary network, such as the increasing demand for computation resources and time consumption, being sensitive to hyperparameters, and requiring detailed tuning in optimization, etc. ", + "bbox": [ + 173, + 433, + 825, + 558 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we present BiBench, a network Binarization Benchmark to evaluate binarization algorithms comprehensively from accuracy and efficiency perspectives (Table 1). Based on BiBench, we benchmark 8 representative binarization algorithms on 9 deep learning datasets, 13 different neural architectures, 2 deployment libraries, 14 hardware chips, and various hyperparameter settings. It costs us about 4 GPU years of computation time to build our BiBench, devoted to promoting comprehensive evaluation for network binarization from the perspectives of accuracy and efficiency. Furthermore, we analyze the benchmark results in depth and reveal insights along evaluation tracks, and give suggestions for designing practical binarization algorithms. ", + "bbox": [ + 173, + 564, + 825, + 676 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 695, + 326, + 712 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 NETWORK BINARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 727, + 397, + 741 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Binarization compresses weights ${ \\pmb w } \\in \\mathbb { R } ^ { c _ { \\mathrm { i n } } \\times c _ { \\mathrm { o u t } } \\times k \\times k }$ and activations $\\pmb { a } ~ \\in ~ \\mathbb { R } ^ { c _ { \\mathrm { i n } } \\times w \\times h }$ to 1-bit in computationally dense convolution, where $c _ { \\mathrm { i n } }$ , $k$ , $c _ { \\mathrm { o u t } }$ , $w$ , and $h$ denote the input channel, kernel size, output channel, input width, and input height. The computation can be expressed as ", + "bbox": [ + 173, + 751, + 825, + 795 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/53553f047315625d2eccd3e25aed34e791b6d2495f8d6ddcb9edaf766e4a4cad.jpg", + "text": "$$\n\\pmb { o } = \\alpha \\operatorname { p o p c o u n t } \\left( \\operatorname { x n o r } \\left( \\operatorname { s i g n } ( \\pmb { a } ) , \\operatorname { s i g n } ( \\pmb { w } ) \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 796, + 645, + 814 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $^ o$ denotes the outputs and $\\alpha \\in \\mathbb { R } ^ { c _ { \\mathrm { o u t } } }$ denotes the optional scaling factor calculated as $\\alpha =$ $\\frac { \\| w \\| } { n }$ (Courbariaux et al., 2016b; Rastegari et al., 2016), xnor and popcount are bitwise instructions defined as (Arm, 2020; AMD, 2022). Though enjoying extreme compression and acceleration, severely limited representation causes the degradation of binarized networks. Therefore, various algorithms emerge constantly to improve accuracy (Yuan & Agaian, 2021). ", + "bbox": [ + 174, + 815, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The vast majority of existing binarization algorithms focus on improving the binarized operators. As shown in Eq. (1), the fundamental difference between binarized and full-precision networks is the former applies binarized operators, which also affect binarized models’ optimization properties directly and determine their hardware efficiency (Alizadeh et al., 2019; Geiger & Team, 2020). In addition, the improvements of binarized operators are widely flexible across neural architectures and learning tasks (Wang et al., 2020b; Qin et al., 2022; Zhao et al., 2022), which fully exerts the generalizability of the bit-width compression. We thereby consider 8 generic binarization algorithms in our BiBench (Courbariaux et al., 2016b; Rastegari et al., 2016; Zhou et al., 2016b; Liu et al., 2018b; Bulat et al., 2019; Liu et al., 2020; Xu et al., 2021b;a). And techniques of these binarization algorithms focus on binarized operator improvement and can be categorized into several types broadly, i.e., scaling factors, parameter redistribution, and gradient approximation. Note that the techniques requiring specified local structures or training pipelines are excluded for a fair comparison, such as the bi-real shortcut (Liu et al., 2018a) and duplicate activation (Liu et al., 2020). For more details of binarization algorithms for our BiBench, please see Appendix. A. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/a740e48144c22b080e074dfa7ade3d703cac28a746265c89a3ddf6f2f0826275.jpg", + "table_caption": [ + "Table 1: Comparison between BiBench and existing binarization works along evaluation tracks. " + ], + "table_footnote": [ + "1 “ $\\dot { \\cdot } \\sqrt { \\dot { \\bar { \\mathbf { \\rho } } } }$ and “ $\\times$ ” indicates the track is considered in the original paper of the binarization algorithm, while “\\*” indicates only being studied in other related studies. “s”, $\" \\tau \"$ ”, or “ $\\cdot _ { g }$ ” indicates “scaling factor”, “parameter redistribution”, or “gradient approximation” techniques proposed in this work, respectively. " + ], + "table_body": "
AlgorithmTechniqueAccurate BinarizationEfficient Binarization
STg#Task#Arch.Corruption RobustnessTraining ConsumptionTheoretical ComplexityHardware Inference
BNN (Courbariaux et al., 2016b)××√33*√√
XNOR (Rastegari et al., 2016)√××23*√√
DoReFa (Zhou et al., 2016b)√××22*×>>>>×
Bi-Real (Liu et al., 2018b)××√12×××
XNOR++ (Bulat et al., 2019)√××12××××
ReActNet (Liu et al., 2020)×√×12××√×
ReCU (Xu et al., 2021b)×√√24××××
FDA (Xu et al., 2021a)××√16××××
Our Benchmark (BiBench)√√√913√√√√
", + "bbox": [ + 174, + 88, + 830, + 257 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 327, + 825, + 493 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 CHALLENGES FOR BINARIZATION ", + "text_level": 1, + "bbox": [ + 178, + 511, + 449, + 525 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Since about 2015, network binarization has attracted great interest in different research fields, including but not limited to vision, language understanding, etc. However, there are still various challenges in the production and deployment process of network binarization in actual practice. The goal in production is to train accurate binarized networks with controllable resources. Recent works have revealed the capability of binarization algorithms evaluated on image classification is not always appliable to new learning tasks and neural architectures (Qin et al., 2020a; Wang et al., 2020b; Qin et al., 2021; Liu et al., 2022). And to achieve higher accuracy, some binarization algorithms require several times of training resources compared with the training of full-precision networks. Ideal binarized networks should be hardware-friendly and robust when deployed to edge devices. Unlike well-supported multi-bit (2-8 bit) quantization, most mainstream inference libraries have not yet supported the deployment of binarization on hardware (NVIDIA, 2022; HUAWEI, 2022; Qualcomm, 2022), which also causes the theoretical efficiency of existing binarization algorithms cannot align their performance on actual hardware. Moreover, the data collected by low-cost devices in natural edge scenarios is not always clean and high-quality, which affects the robustness of binarized models severely (Lin et al., 2018; Ye et al., 2019; Cygert & Czyzewski ˙ , 2021). However, recent binarization algorithms rarely consider corruption robustness when designed. ", + "bbox": [ + 174, + 536, + 825, + 758 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 BIBENCH: TRACKS AND METRICS FOR BINARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 779, + 660, + 795 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we present BiBench towards accurate and efficient network binarization. As Figure 1 shows, we build 6 evaluation tracks and corresponding metrics upon the practical challenges in production and deployment of binarization. For all tracks, higher metrics mean better performance. ", + "bbox": [ + 176, + 810, + 823, + 853 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 TOWARDS ACCURATE BINARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 869, + 472, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The evaluation tracks for accurate network binarization in our BiBench are “Learning Task”, “Neural Architecture” (for production), and “Corruption Robustness” (for deployment). ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "$\\textcircled{1}$ Learning Task. We selected 9 learning tasks for 4 different data modalities to comprehensively evaluate network binarization algorithms. For the widely-evaluated 2D visual modality, in addition to image classification tasks on CIFAR-10 (Krizhevsky et al., 2014) and ImageNet (Krizhevsky et al., 2012), we also include object detection tasks on PASCAL VOC (Hoiem et al., 2009) and COCO (Lin et al., 2014) across all algorithms. For 3D visual modality, we evaluate the algorithm on ModelNet40 classification (Wu et al., 2015) and ShapeNet segmentation (Chang et al., 2015) tasks of 3D point clouds. For textual modality, the natural language understanding tasks in GLUE benchmark (Wang et al., 2018) are applied for evaluation. For speech modality, we evaluate algorithms on Speech Commands KWS task (Warden, 2018). The details of tasks and datasets are in Appendix. B. ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then we build the evaluation metric for this track. For a particular binarization algorithm, we take the accuracy of full-precision models as baselines and calculate the mean relative accuracy for all architectures on each task. Then we calculate the Overall Metric (OM) of the task track as the quadratic mean of all tasks (Curtis & Marshall, 2000). The equation of evaluation metric is ", + "bbox": [ + 174, + 236, + 825, + 291 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/89e5216f01e36140a7d8c2efe2d5593b59430895e96720180b2f7cff11a02451.jpg", + "text": "$$\n\\mathrm { O M } _ { \\mathrm { t a s k } } = \\sqrt { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { t a s k } _ { i } } ^ { b i } } { A _ { \\mathrm { t a s k } _ { i } } } \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 383, + 299, + 612, + 349 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $A _ { \\mathrm { t a s k } _ { i } } ^ { b i }$ and $A _ { \\mathrm { t a s k } _ { i } }$ denote the accuracy results of the binarized and full-precision models on $i$ -th task, $N$ is the number of tasks, and $\\mathbb { E } ( \\cdot )$ denote taking the mean value. The quadratic mean form is uniformly applied in BiBench to unify all metrics, which prevents metrics from being unduly influenced by certain bad items and thus can measure the overall performance on each track. ", + "bbox": [ + 174, + 356, + 825, + 415 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$\\textcircled{2}$ Neural Architecture. We choose diverse neural architectures with mainstream CNN-based, transformer-based, and MLP-based architectures to evaluate the generalizability of binarization algorithms from the neural architecture perspective. We adopt standard ResNet-18/20/34 (He et al., 2016) and VGG (Simonyan & Zisserman, 2015) to evaluate CNN architectures, and the FasterRCNN (Ren et al., 2015) and SSD300 (Liu et al., 2016) frameworks are applied as detectors. We binarize BERT-Tiny4/Tiny6/Base (Kenton & Toutanova, 2019) with the bi-attention mechanism for convergence (Qin et al., 2021) to evaluate transformer-based architectures. And we evaluate PointNetvanilla and PointNet (Qi et al., 2017) with EMA aggregator (Qin et al., 2020a), FSMN (Zhang et al., 2015), and Deep-FSMN (Zhang et al., 2018a) as typical MLP-based architectures for their linear unit composition. The details of these architectures are presented in Appendix. C. ", + "bbox": [ + 173, + 421, + 825, + 561 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Similar to the overall metric for learning task track, we build the metric for neural architecture: ", + "bbox": [ + 173, + 568, + 795, + 583 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c97b82b78d7f43bdf4ffe09febdf4ea251812857f1da88031980f22553303410.jpg", + "text": "$$\n\\mathrm { O M } _ { \\mathrm { a r c h } } = \\sqrt { \\frac { 1 } { 3 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { C N N } } ^ { b i } } { A _ { \\mathrm { C N N } } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { T r a n s f o r m e r } } ^ { b i } } { A _ { \\mathrm { T r a n s f o r m e r } } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { M L P } } ^ { b i } } { A _ { \\mathrm { M L P } } } \\right) \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 264, + 589, + 733, + 632 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$\\textcircled{3}$ Corruption Robustness. The corruption robustness of binarization on deployment is critical to deal with bad cases like perceptual device damage, a common problem with low-cost equipment in real-world implementations. We consider the robustness of binarized models to corruption of 2D visual data and evaluate algorithms on the CIFAR10-C (Hendrycks & Dietterich, 2018) benchmark. ", + "bbox": [ + 173, + 645, + 825, + 702 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Therefore, we evaluate binarization algorithms’ performance on the corrupted data compared with the normal data using the corruption generalization gap (Zhang et al., 2022a): ", + "bbox": [ + 174, + 708, + 821, + 738 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4ddaabe321d3718c23d6fb945e70615308b2a72b3e41dbbaff1e41fb59ad6862.jpg", + "text": "$$\n\\begin{array} { r } { G _ { \\mathrm { t a s k } _ { i } } = A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { n o r m } } - A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { c o r r } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 415, + 744, + 581, + 763 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { c o r r } }$ and $A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { n o r m } }$ denote the accuracy results under all architectures on $i$ -th corruption task and corresponding normal task, respectively. And the overall metric on this track is calculated by ", + "bbox": [ + 173, + 768, + 825, + 797 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8547a0e7eb2fa8cb76c456b327a283eece2ff3f000cd22422758e62be3832e2a.jpg", + "text": "$$\n\\mathrm { O M } _ { \\mathrm { r o b u s t } } = \\sqrt { \\frac { 1 } { C } \\sum _ { i = 1 } ^ { C } \\mathbb { E } ^ { 2 } \\left( \\frac { G _ { \\mathrm { t a s k } _ { i } } } { G _ { \\mathrm { t a s k } _ { i } } ^ { b i } } \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 804, + 619, + 854 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 TOWARDS EFFICIENT BINARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 869, + 470, + 883 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As for the efficiency of network binarization, we evaluate “Training Consumption” for production, “Theoretical Complexity” and “Hardware Inference” for deployment. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$\\textcircled{4}$ Training Consumption. We consider the occupied training resource and the hyperparameter sensitivity of binarization algorithms, which affect the consumption of one training and the whole tuning process, respectively. For each algorithm, we train its binarized networks with various hyperparameter settings, including different learning rates, learning rate schedulers, optimizers, and even random seeds, to evaluate whether the binarization algorithm is easy to tune to an optimal state. We align the epochs for binarized and full-precision networks and compare their consumption and time. ", + "bbox": [ + 173, + 102, + 825, + 188 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The evaluation metric for the training consumption track is related to the training time and hyperparameter sensitivity. For a specific binarization algorithm, we have ", + "bbox": [ + 171, + 194, + 821, + 223 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a59e99a0fc5c5320a409a3bed69687db0c631fa5722fc945630d567b15d02224.jpg", + "text": "$$\n\\mathrm { O M } _ { \\mathrm { t r a i n } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { T _ { \\mathrm { t r a i n } } } { T _ { \\mathrm { t r a i n } } ^ { b i } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { \\mathrm { s t d } ( A _ { \\mathrm { h y p e r } } ) } { \\mathrm { s t d } ( A _ { \\mathrm { h y p e r } } ^ { b i } ) } \\right) \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 228, + 679, + 279 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\pmb { T } _ { \\mathrm { t r a i n } }$ denotes the set of time in once training, $A _ { \\mathrm { h y p e r } }$ is the set of results with different hyperparameter settings, and $\\operatorname { s t d } ( \\cdot )$ is taking standard deviation values. ", + "bbox": [ + 174, + 281, + 820, + 310 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\textcircled{5}$ Theoretical Complexity. When evaluating theoretical complexity, we calculate the compression and speedup ratio before and after binarization on architectures such as ResNet18. ", + "bbox": [ + 173, + 316, + 823, + 345 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The evaluation metric relates to model size saving (MB) and computational floating-point operations (FLOPs) at inference. For binarized parameters, the storage occupation of binarized parameters is 1/32 as their 32-bit floating-point counterparts (Rastegari et al., 2016). For binarized operations, the multiplication between a 1-bit number (weight) and a 1-bit number (activation) approximately takes $1 ^ { * } 1 / 6 4$ FLOPs for a CPU with the instruction size of 64 bits (Zhou et al., 2016b; Liu et al., 2018b; Li et al., 2019). The compression ratio $\\mathbf { \\nabla } _ { \\mathbf { r } _ { c } }$ and speedup ratio $\\mathbf { \\Delta } _ { r _ { s } }$ are ", + "bbox": [ + 174, + 351, + 825, + 436 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/786f54912324a3ca4ffee190faaad61ca39aa062b5d9ae7974b3c2c93f31060c.jpg", + "text": "$$\nr _ { c } = \\frac { | M | _ { \\ell 0 } } { \\frac { 1 } { 3 2 } \\left( | M | _ { \\ell 0 } - | \\hat { M } | _ { \\ell 0 } \\right) + | \\hat { M } | _ { \\ell 0 } } , \\quad r _ { s } = \\frac { \\mathrm { F L O P s } _ { M } } { \\frac { 1 } { 6 4 } \\left( \\mathrm { F L O P s } _ { M } - \\mathrm { F L O P s } _ { \\hat { M } } \\right) + \\mathrm { F L O P s } _ { \\hat { M } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 192, + 440, + 785, + 484 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $M$ and $\\hat { M }$ are the amount of full-precision parameters in the original and binarized models, respectively, and $\\mathrm { F L O P s } _ { M }$ and $\\mathrm { F L O P s } _ { \\hat { M } }$ denote the computation related to these parameters, respectively. And the overall metric for theoretical complexity is ", + "bbox": [ + 174, + 491, + 825, + 534 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a46db1217a1899ecf08ad4d0f099d7f3085ee76e09d55211d9f21042bce8c953.jpg", + "text": "$$\n\\mathrm { O M } _ { \\mathrm { c o m p } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } ( \\pmb { r } _ { c } ) + \\mathbb { E } ^ { 2 } ( \\pmb { r } _ { s } ) \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 539, + 619, + 574 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\textcircled{6}$ Hardware Inference. Since the limited support of binarization in hardware deployment, just two inference libraries, Larq’s Compute Engine (Geiger & Team, 2020) and JD’s daBNN (Zhang et al., 2019) can deploy and evaluate the binarized models on ARM hardware in practice. Regarding target hardware devices, we mainly focus on ARM CPU inference as this is the mainstream hardware type for edge scenarios, including HUAWEI Kirin, Qualcomm Snapdragon, Apple M1, MediaTek Dimensity, and Raspberry Pi. We put the hardware details in Appendix. D. ", + "bbox": [ + 174, + 584, + 825, + 670 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "And for a certain binarization algorithm, we take the saving of storage and inference time under different inference libraries and hardware as evaluation metrics: ", + "bbox": [ + 176, + 675, + 823, + 704 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/12e4456d6a4543908bcee133ea7d612dd9c7617a209192aad00ceaba442fa6ed.jpg", + "text": "$$\n\\mathrm { O M } _ { \\mathrm { i n f e r } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { T _ { \\mathrm { i n f e r } } } { T _ { \\mathrm { i n f e r } } ^ { b i } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { S _ { \\mathrm { i n f e r } } } { S _ { \\mathrm { i n f e r } } ^ { b i } } \\right) \\right) } ,\n$$", + "text_format": "latex", + "bbox": [ + 338, + 707, + 656, + 750 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbf { \\mathit { T } } _ { \\mathrm { i n f e r } }$ and $S _ { \\mathrm { i n f e r } }$ denote the inference time and storage on different devices, respectively. ", + "bbox": [ + 171, + 752, + 784, + 768 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 BIBENCH IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 787, + 444, + 804 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This section shows the implementation details and training and inference pipelines of our BiBench. ", + "bbox": [ + 171, + 818, + 821, + 834 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Implementation details. We implement BiBench with PyTorch (Paszke et al., 2019) packages. The definitions of the binarized operators are entirely independent of corresponding single files. And the corresponding operator in the original model can be flexibly replaced by the binarized one while evaluating different tasks and architectures. When deployed, we export well-trained binarized models of a binarization algorithm to the Open Neural Network Exchange (ONNX) format (developers, 2021) and then feed them to the inference libraries in BiBench (if it applies to this algorithm). ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/957df513c54ecce033d1650890432983b9be71cfc3dec180f93ab96706dadf57.jpg", + "table_caption": [ + "Table 2: Accuracy benchmark for network binarization. Blue: best in a row. Red: worst in a row. " + ], + "table_footnote": [], + "table_body": "
TrackMetricBinarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCUFDA
Learning Task (%)CIFAR1094.5494.7395.0395.6194.5295.9296.7294.66
ImageNet75.8177.2476.6178.3875.0178.6477.9878.15
VOC0776.9774.6176.3580.0774.4181.3881.6579.02
COC01777.9475.3778.3181.6279.4183.8285.6682.35
ModelNet4054.1993.8693.7493.2385.2092.4195.0794.38
ShapeNet48.9673.6270.7968.1341.1668.5140.6571.16
GLUE49.7559.6366.6069.4249.3367.6450.6670.61
SpeechCom.75.0376.9376.6482.4268.6581.8676.9877.90
OMtask70.8278.9779.8281.6372.8981.8177.9681.49
Neural Architecture (%)CNNs72.9083.7483.8685.0278.9586.2083.5086.34
Transformers49.7559.6366.6069.4249.3367.6450.6670.61
MLPs64.6185.4085.1987.8376.9287.1386.0286.14
OMarch63.1577.1679.0181.1669.7280.8275.1481.36
RobustnessCIFAR10-C95.26100.9781.4396.5692.6994.01103.2998.35
Corruption (%)OMcorr95.26100.9781.4396.5692.6994.01103.2998.35
", + "bbox": [ + 176, + 98, + 823, + 313 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training and inference pipelines. Hype-parameters: We train the binarized networks with the same number of epochs as their full-precision counterparts. Inspired by results in Section 5.2.1, we use the Adam optimizer for all binarized models for better convergence, the initial learning rates are $1 e - 3$ as default (or $0 . 1 \\times$ of the default learning rate), and the learning rate scheduler is CosineAnnealingLR (Loshchilov & Hutter, 2017). Architecture: In BiBench, we thoroughly follow the original architectures of full-precision models and binarize their convolution, linear, and multiplication units with the binarization algorithms. Hardtanh is uniformly used as the activation function to avoid the all-one feature. Pretrains: We adopt finetuning for all binarization algorithms, and for each of them, we initialize all binarized models by the same pre-trained model for specific neural architectures and learning tasks to eliminate the inconsistency at initialization. ", + "bbox": [ + 174, + 337, + 825, + 476 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 BIBENCH EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 400, + 511 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This section shows our evaluation results and analysis in BiBench. The main accuracy results are in Table 2 and the efficiency results are in Table 3. More detailed results are in Appendix. E. ", + "bbox": [ + 173, + 526, + 823, + 554 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 ACCURACY ANALYSIS FOR BINARIZATION ", + "text_level": 1, + "bbox": [ + 174, + 571, + 509, + 585 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For the accuracy results of network binarization, we present the evaluation results in Table 2 for each accuracy-related track using the metrics defined in Section 3.1. ", + "bbox": [ + 176, + 598, + 823, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1.1 LEARNING TASK TRACK ", + "text_level": 1, + "bbox": [ + 176, + 640, + 400, + 655 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We present the evaluation results of binarization on various tasks. Besides the overall metrics ${ \\mathrm { O M } } _ { \\mathrm { t a s k } }$ we also present the relative accuracy of binarized networks compared to full-precision ones. ", + "bbox": [ + 173, + 665, + 821, + 694 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Accuracy retention is still the most rigorous challenge for network binarization. With fully unified training pipelines and neural architectures, there is a large gap between the performance of binarized and full-precision networks on most learning tasks. For example, the results on large-scale ImageNet and COCO tasks are usually less than $80 \\%$ of their full-precision counterparts. Moreover, the marginal effect of advanced binarization algorithms is starting to appear, e.g., on the ImageNet, the SoTA algorithm ReCU is less $3 \\%$ higher than vanilla BNN. ", + "bbox": [ + 174, + 700, + 825, + 784 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The binarization algorithms’ performances under different data modes are significantly different. When comparing various tasks, an interesting phenomenon is that the binarized networks suffer a huge accuracy drop in language understanding GLUE benchmark, but it can almost approach full-precision performance on the ModelNet40 point cloud classification task. Similar phenomena suggest that the direct transfer of binarization studies’ insights across different tasks is non-trivial. ", + "bbox": [ + 174, + 791, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As for the overall performance, both ReCU and ReActeNet show high accuracy across different learning tasks. Surprisingly, although ReCU wins the championship on most four individual tasks, ReActNet stands out in the overall metric comparison finally. They both apply reparameterization in the forward propagation and gradient approximation in the backward propagation. ", + "bbox": [ + 176, + 867, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/f3bb50439f99bebe5dac08a4feac681ce36e9aff12a4f7aa45ba6d628e1495ac.jpg", + "table_caption": [ + "Table 3: Efficiency benchmark for network binarization. Blue: best in a row. Red: worst in a row. " + ], + "table_footnote": [], + "table_body": "
TrackMetricBinarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCUFDA
Training Consumption (%)Sensitivity27.28175.53113.40144.5928.66146.0653.3336.62
Time Cost82.1971.4376.9245.8068.1845.4558.2520.62
OMtrain61.23134.0096.89107.2552.30108.1655.8429.72
Theoretical Complexity (x）Speedup12.6012.2612.3712.3712.2612.2612.3712.37
Compression13.2713.2013.2013.2013.1613.2013.2013.20
OMcomp12.9412.7412.7912.7912.7112.7412.7912.79
Hardware Deployment (x）Speedup5.45False5.455.45False4.895.45 5.45
Compression15.62False15.6215.62False15.5215.6215.62
OMinfer11.70False11.7011.70False11.5111.7011.70
", + "bbox": [ + 176, + 88, + 821, + 237 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.2 NEURAL ARCHITECTURE TRACK ", + "text_level": 1, + "bbox": [ + 176, + 265, + 457, + 279 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Binarization exhibits a clear advantage on CNN-based and MLP-based architectures over transformer-based ones. Since being widely studied, the advanced binarization algorithms can achieve about $78 \\% - 8 6 \\%$ of the full-precision accuracy in CNNs, and the binarized networks with MLP architectures even approach the full-precision performance (e.g., Bi-Real Net $8 7 . 8 3 \\%$ ). In contrast, the transformer-based one suffers from extremely significant performance degradation when binarized, and none of the algorithms achieves an overall accuracy metric higher than $70 \\%$ . Compared to CNNs and MLPs, the results show that transformer-based architectures constructed by unique attention mechanisms require specific binarization designs instead of direct binarizing. ", + "bbox": [ + 174, + 290, + 825, + 401 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The overall winner on the architecture track is the FDA algorithm. In this track, FDA has achieved the best results in both CNN and Transformer. The evaluation of these two tracks proves that these binarization algorithms, which apply statistical channel-wise scaling factor and custom gradient approximation like FDA and ReActNet, have the advantage of stability to a certain degree. ", + "bbox": [ + 176, + 409, + 825, + 463 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.3 CORRUPTION ROBUSTNESS TRACK ", + "text_level": 1, + "bbox": [ + 178, + 481, + 472, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The binarized network can approach full-precision level robustness for corruption. Surprisingly, binarized networks show robustness close to full-precision counterparts in corruption evaluation. Evaluation results on the CIFAR10-C dataset show that the binarized network performs close to the full-precision network in the typical 2D image corruption. ReCU and XNOR-Net even outperform their full-precision counterparts. If corruption robustness requirements are the same, the binarized version network requires little additional designs or supervision for robustness. Thus, binarized networks usually show comparable robustness for corruption against full-precision counterparts, which can be seen as a general property of binarized networks rather than specific algorithms. ", + "bbox": [ + 174, + 507, + 823, + 618 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 EFFICIENCY ANALYSIS FOR BINARIZATION ", + "text_level": 1, + "bbox": [ + 174, + 637, + 513, + 651 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As for efficiency, we discuss and analysis the metrics of training consumption, theoretical complexity, and hardware inference tracks below. ", + "bbox": [ + 178, + 665, + 821, + 691 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2.1 TRAINING CONSUMPTION TRACK ", + "text_level": 1, + "bbox": [ + 176, + 710, + 462, + 724 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We comprehensively investigate the training cost of binarization algorithms on ResNet18 of CIFAR10 and present the sensitivity and training time results for different binarization algorithms in Table 3 and Figure 3, respectively. ", + "bbox": [ + 176, + 736, + 823, + 777 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "“Binarization $\\nVDash$ Sensitivity”: existing techniques can stabilize binarization-aware training. An existing common intuition is that the training of binarized networks is usually more sensitive to the training settings than full-precision networks, caused by the representation limits and gradient approximation errors brought by the high degree of discretization. However, we find that the hyperparameter sensitivities of existing binarization algorithms are polarized. Some of them are even more hyperparameter-stable than the training of full-precision networks, while others fluctuate hugely. The reason for this problem is the difference in the techniques applied by the binarized operators of these algorithms. The training-stable binarization algorithms often have the following commonalities: (1) Channel-wise floating-point scaling factors based on learning or statistics; (2) Soft gradient approximation to reduce gradient error. These hyperparameter-stable binarization algorithms may not certainly outperform other algorithms but can simplify the tuning process in production and obtain reliable accuracy in one training. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ce936e7dd0211ceaee8c117b81d59f14136795fbe8fdbfb74f2225ad2e3a2987.jpg", + "image_caption": [ + "Figure 2: Comparisons of accuracy under different training settings. " + ], + "image_footnote": [], + "bbox": [ + 174, + 83, + 820, + 215 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 263, + 823, + 291 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The preference for hyperparameter settings is evident for the training of binarization. From the statistical results in Figure 2, training with Adam optimizer, $1 \\times$ (identical to full-precision network) learning rate and CosineAnnealingLR scheduler is more stable than that with other settings. Inspired by this, we adopt this setting in evaluating binarization as part of the standard training pipelines. ", + "bbox": [ + 173, + 297, + 825, + 354 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The soft gradient approximation in binarization brings a significant training time increase. Comparing the time consumed by each binarization algorithm, we found that the training time of algorithms using the custom gradient approximation techniques such as Bi-Real and ReActNet increased significantly, and the metric of FDA is even as worse as $2 0 . 6 2 \\%$ , which means that the training time he spent is close to $5 \\times$ the full-precision network training. ", + "bbox": [ + 174, + 361, + 825, + 431 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2.2 THEORETICAL COMPLEXITY TRACK ", + "text_level": 1, + "bbox": [ + 176, + 446, + 480, + 462 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "There is no obvious difference in the theoretical complexity among binarization algorithms. The leading cause of the difference in compression rate is the definition of the static scaling factor of each model, e.g., BNN does not apply any factors and enjoys the most compression. For theoretical acceleration, the main difference comes from two aspects. First, the static scaling factor reduction also improves the theoretical speedup. Second, real-time re-scaling and mean-shifting for activation bring additional computation, such as ReActNet, which harms $0 . 1 1 \\times$ speedup. In general, the theoretical complexity of each method is similar, and the overall metrics are in the range of [12.71, 12.94]. These results show binarization algorithms should have similar inference efficiency. ", + "bbox": [ + 173, + 472, + 825, + 583 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2.3 HARDWARE INFERENCE TRACK ", + "text_level": 1, + "bbox": [ + 176, + 601, + 447, + 614 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Compared to other tracks, the hardware inference track brings us some insights for binarization in a real-world deployment. ", + "bbox": [ + 174, + 626, + 482, + 667 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Limited inference libraries lead to an almost fixed paradigm of binarization deployment. After investigating existing open-source inference libraries, we find that few inference libraries support the deployment of binarization algorithms on hardware. And there are only Larq (Geiger & Team, 2020) and daBNN (Zhang et al., 2019) with complete deployment pipelines and mainly support deployment on ARM devices. We first evaluate the ", + "bbox": [ + 174, + 674, + 483, + 811 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/3408fa6d58833db1c58435c092835f1ee3f43ab3bb707ced4395a9717a3ac371.jpg", + "image_caption": [ + "Figure 3: The lower the chip’s computing power, the higher the inference speedup of deployed binarized models. " + ], + "image_footnote": [], + "bbox": [ + 503, + 589, + 818, + 768 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "deployment capability of the two inference libraries in Table 4. Both inference libraries support a channel-wise scaling factor in floating-point form and force it to fuse into the BN layer (BN must follow every convolution of the binarized model). Neither supports dynamic activation of activations statistics nor re-scaling in inference. The only difference is that Larq further supports mean-shifting activation with a fixed bias. Constrained by the inference libraries, the practical deployment of binarization algorithms is limited. The scale factor shape of $\\mathrm { X N O R + + }$ led to its failed deployment, and XNOR also failed because of the activation re-scaling technique. The vast majority of binarization methods have almost identical inference performance, and the mean-shifting operation of ReAct", + "bbox": [ + 174, + 813, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/c1f4fb8c553c6f5cbc2907135a698477335be8ee76067c5d3b91ada3dcfd3e07.jpg", + "table_caption": [ + "Table 4: Deployment capability of different inference libraries on real hardware. " + ], + "table_footnote": [], + "table_body": "
Infer. Lib.Providers Granularitys FormFlod BNAct. Re-scalingAct.Mean-shifting
LarqLarqChannel-wiseFP32√×√
daBNNJDChannel-wiseFP32√××
AlgorithmDeployables GranularitysFormFlod BNAct. Re-scalingAct. Mean-shifting
BNN√N/AN/AN/A××
XNOR×Channel-wiseFP32>>>×√>>√×
DoReFa√Channel-wiseFP32××
Bi-Real√Channel-wiseFP32××
XNOR++×Spatial-wiseFP32××
ReActNetChannel-wiseFP32×√
ReCU√V√Channel-wiseFP32××
FDAChannel-wiseFP32××
", + "bbox": [ + 176, + 88, + 820, + 248 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Net on activation slightly affects the efficiency, i.e., binarized models must satisfy fixed deployment paradigms and have almost identical efficiency performance. ", + "bbox": [ + 176, + 273, + 821, + 303 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Born for the edge: the lower the chip’s computing power, the higher the binarization speedup. After deploying and evaluating binarized models across dozen chips, we compare the average speedup of the binarization algorithm on each chip. A counterintuitive finding is that the higher the chip capability, the worse the speedup of binarization on the chip (Figure 3). Further observation showed that the contradiction is mainly because higher-performance chips have more acceleration brought by multi-threading when running floating-point models. Thus the speedup of binarized models is relatively slow in these chips. The scenarios where network binarization technology comes into play better are edge chips with low performance and cost, and the extreme compression and acceleration of binarization are making deployment of advanced neural networks on edge possible. ", + "bbox": [ + 174, + 309, + 825, + 434 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 SUGGESTIONS FOR BINARIZATION ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 452, + 545, + 465 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Based on the above evaluation and analysis, we attempt to summarize a paradigm for accurate and efficient network binarization among existing techniques: (1) Soft quantization approximation is an undisputed must-have technique. This binarization technique does not affect hardware inference efficiency and is adopted by all winning binarization algorithms on accuracy tracks, including BiReal, ReActNet, and ReCU. (2) Channel-wise scaling factors are the only option available for practical binarization. The results of the learning task and neural architecture tracks demonstrate the advantage of floating-point scaling factors, and analysis of efficiency tracks further limits it to the channel-wise form. (3) Mean-shifting the input with a fixed bias is an optional helpful operation. Our results show that this technique in ReActNet effectively improves accuracy and consumes almost no extra computation, but not all inference libraries support it. ", + "bbox": [ + 173, + 477, + 825, + 616 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have to stress that although the benchmarking on evaluation tracks leads us to several ground rules towards accurate and efficient binarization, none of the binarization techniques or algorithms work well across all scenarios so far. In the future, binarization research should focus on breaking the mutual restrictions between production and deployment, and the binarization algorithms should consider deployability and efficiency. The inference libraries are also expected to support more advanced binarized operators. ", + "bbox": [ + 174, + 623, + 825, + 707 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 727, + 310, + 742 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we present BiBench, a versatile and comprehensive benchmark that delves into the most fundamental questions of model binarization. BiBench covers 8 network binarization algorithms, 9 deep learning datasets (including one corruption benchmark), 13 different neural architectures, 2 deployment libraries, 14 real-world hardware, and various hyperparameter settings. Moreover, BiBench proposes evaluation tracks specifically designed to measure critical aspects such as accuracy under multiple conditions and efficiency when deployed on actual hardware. More importantly, by collating experiment results and analysis, BiBench hopes to establish an empirically optimized paradigm with several critical considerations for designing accurate and efficient binarization methods. We hope BiBench can facilitate a fair comparison of algorithms through a systematic investigation with metrics that reflect the fundamental requirements and serve as a foundation for applying model binarization in broader and more practical scenarios. ", + "bbox": [ + 174, + 758, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES \nMilad Alizadeh, Javier Fernandez-Marqu ´ es, Nicholas D. Lane, and Yarin Gal. A systematic study ´ of binary neural networks’ optimisation. In ICLR, 2019. \nAMD. 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", + "bbox": [ + 171, + 738, + 823, + 781 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "APPENDIX FOR BIBENCH ", + "text_level": 1, + "bbox": [ + 176, + 99, + 480, + 121 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A DETAILS OF BINARIZATION ALGORITHM ", + "bbox": [ + 176, + 141, + 549, + 157 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "General Binarization: ", + "text_level": 1, + "bbox": [ + 174, + 174, + 326, + 188 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In previous studies, quantization schemes with lower bit-widths were regarded as more aggressive schemes (Rusci et al., 2020; Choukroun et al., 2019; Qin et al., 2022), because lower bit-widths usually lead to higher compression and speed-up but result in more Severe accuracy loss. With the lowest bit-width among all quantization approaches, 1-bit quantization (binarization) is regarded as the most aggressive quantization technique (Qin et al., 2022), facing severe challenges in terms of accuracy but enjoying the highest compression and speedup ratios. ", + "bbox": [ + 174, + 195, + 825, + 279 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Training. During the training of a general binarized model, the sign function is usually applied in forward propagation, and STE or other gradient approximations is applied in backward propagation to make the binarized model trainable. Since the parameters are quantized to binary, network binarization approaches usually use a simple sign function as the quantizer instead of directly sharing the quantizer with multi-bit (2-8 bit) quantization (Gong et al., 2019; Gholami et al., 2021). Specifically, as Gong et al. (2019) describes, for multi-bit uniform quantization, given the bit width b and the floating-point activation/weight $\\mathbf { X }$ following in the range $( \\mathbb { 1 } , \\mathbb { u } )$ , the complete quantizationdequantization process of uniform quantization can be defined as ", + "bbox": [ + 174, + 285, + 825, + 397 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f608efe04308633cd3e241dcfd55ba3b903571d112c30a75223e80a8076b7201.jpg", + "text": "$$\nQ _ { U } ( { \\pmb x } ) = \\mathrm { r o u n d } \\left( \\frac { { \\pmb x } } { \\Delta } \\right) \\Delta ,\n$$", + "text_format": "latex", + "bbox": [ + 411, + 404, + 586, + 433 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the original range $( l , u )$ is divided into $2 ^ { b } - 1$ intervals P $\\mathrm { i } , \\ i \\ \\in \\ ( 0 , 1 , \\cdot \\cdot \\cdot , 2 ^ { b } - 1 )$ , and $\\begin{array} { r } { \\Delta = \\frac { u - l } { 2 ^ { b } - 1 } } \\end{array}$ is the interval length. When $b = 1$ , the $Q _ { U } ( { \\pmb x } )$ equals the sign function, and the binary function is expressed as ", + "bbox": [ + 174, + 439, + 823, + 484 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/63290e52204f3f7cc802e2c51574aa9b8a2ac119d8986f228a96b792d36fa59a.jpg", + "text": "$$\nQ _ { B } ( { \\pmb x } ) = \\mathrm { s i g n } ( { \\pmb x } ) .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 486, + 563, + 502 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore, binarization can be regarded as the 1-bit specialization of quantization. ", + "bbox": [ + 174, + 506, + 709, + 522 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Deployment. For real-world hardware deployment, every 32 binarized parameters are packed together using 32-bit instructions and computed simultaneously, which is the main principle for acceleration. For compression of binary algorithms. Instructions including XNOR (or combine EOR and NOT) and popcount enable binarized networks to deploy on real-world hardware. XNOR (exclusiveXOR) gate, a combination of an XOR gate followed by an inverter. XOR (also known as EOR) is a pervasive instruction that has long existed in assembly instructions for all target platforms. The popcount instruction means Population Count per byte. This instruction counts the number of bits with one value in each vector element in the source register, places the result into a vector, and writes the vector to the destination register (Arm, 2020). This instruction is applied to accelerate the inference of binarized networks (Hubara et al., 2016; Rastegari et al., 2016) and is widely supported by various hardware, e.g., the definitions of popcount in ARM and $\\mathbf { \\boldsymbol { x } } 8 6$ are in (Arm, 2020) and (AMD, 2022), respectively. ", + "bbox": [ + 173, + 527, + 825, + 694 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Comparison with other compression techniques. Most existing network compression technologies aim to reduce the size and computation of full-precision models. Specifically, knowledge distillation supervises compact small (student) models by intermediate features and/or soft outputs of the large (teacher) model (Hinton et al., 2015; Xu et al., 2018; Chen et al., 2018; Yim et al., 2017; Zagoruyko & Komodakis, 2017); model pruning (Han et al., 2015; 2016; He et al., 2017) and lowrank decomposition (Denton et al., 2014; Lebedev et al., 2015; Jaderberg et al., 2014; Lebedev & Lempitsky, 2016) reduce network parameters and computation by pruning and low-rank approximation; compact model design directly designs a compact model (Howard et al., 2017; Sandler et al., 2018; Zhang et al., 2018b; Ma et al., 2018). Although these compression technologies can effectively reduce the number of parameters, the compressed model still uses 32-bit floating-point numbers, which leaves room for further compression using model quantization/binarization technologies. Compared with multi-bit (2-8 bit) model quantization compressing parameters to integers (Gong et al., 2014; Wu et al., 2016; Vanhoucke et al., 2011; Gupta et al., 2015), binarization usually directly applies the sign function to compress the model to a more compact 1-bit (Rusci et al., 2020; Choukroun et al., 2019; Qin et al., 2022; Shang et al., 2022b; Qin et al., 2020b). Moreover, due to the application of binary parameters, bitwise operations (XNOR and popcount) can be applied to inference during actual deployment instead of integer multiply-add operations in 2-8 bit model quantization. Therefore, binarization is considered to take advantage of the hardware and can achieve more speedup than multi-bit quantization. ", + "bbox": [ + 174, + 702, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/1a246d91b4dd3137405f58021b75e6a4c991485deaed42e550aca3eb12169b3e.jpg", + "table_caption": [ + "Table 5: The considered binarization algorithms and our final selections in BiBench. Bold means that the algorithm has an advantage in that column. " + ], + "table_footnote": [], + "table_body": "
AlgorithmYear ConferenceCitation (to 2022/11/08) Techniques SourceOperatorOpen Specified Structure / Training-pipeline
BNN (Courbariaux et al., 2016a)2016NeurIPS1846YesYesNo
XNOR-Net (Rastegari et al., 2016) 2016ECCV4313YesYesNo
DoReFa (Zhou et al., 2016a)2016ArXiv174YesYesNo
Bi-Real Net (Liu et al., 2018a)2018ECCV371YesYesOptional
CI-BCNN (Wang et al., 2019)2019CVPR77YesYesYes
XNOR-Net++ (Bulat et al., 2019)2019BMVC118YesYesNo
RBNN (Lin et al., 2020)2020NeurIPS62YesYesNo
ReActNet (Liu et al., 2020)2020ECCV151YesYesOptional
Si-BNN (Wang et al., 2020a)2020AAAI24YesNoNo
ProxyBNN (He et al., 2020)2020ECCV24YesNoYes
FDA (Xu et al., 2021a)2021NeurIPS11YesYesNo
ReCU (Xu et al., 2021b)2021ICCV15YesYesNo
LCR-BNN (Shang et al., 2022a)2022ECCV0YesYesYes
", + "bbox": [ + 173, + 128, + 833, + 333 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 366, + 823, + 409 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Selection Rules: ", + "text_level": 1, + "bbox": [ + 174, + 415, + 284, + 429 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "First of all, we state that we obey some general rules for selecting binarization algorithms, i.e., the selected binarization algorithms should improve the binarized operator since it is the fundamental difference between binarized and full-precision models (as discussed in Section 2.1). And we thus also exclude the algorithms and techniques requiring specified local structures or training pipelines for a fair comparison. ", + "bbox": [ + 174, + 436, + 825, + 506 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Then, we explain in detail the choice of binarization algorithms and why they are representative. When we built the BiBench, we considered various binarization algorithms with improved operator techniques in binarization research, and now we list them in detail in Table 5. We consider from the following perspectives, the purposes are making the selected binarization algorithms representative and should complete all evaluations in BiBench fairly: Operator Techniques (Yes/No), Year, Conference, Citation (to 2022/11/08), Open source (Yes/No), and Specified Structure / Training-pipeline (Yes/No/Optional). ", + "bbox": [ + 174, + 512, + 825, + 611 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(1) We analyze the techniques proposed in these works. Following the general rules we mentioned, all considered binarization algorithms should have significant contributions to the improvement of the binarization operator (Operator Techniques: Yes) and should not include techniques that are bound to specific architectures and training pipelines to complete well all the evaluations of the learning task, neural architecture, and training consumption tracks in BiBench (Specified Structure / Training-pipeline: No/Optional, Optional means the techniques are included but can be decoupled with binarized operator totally). ", + "bbox": [ + 173, + 617, + 825, + 715 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(2) We also consider the impact and reproducibility of these works. We prioritized the selection of works with more than 100 citations, which means they are more discussed and compared in binarization research and thus have higher impacts. Works in 2021 and later are regarded as the SoTA binarization algorithms and also prioritized. Furthermore, we hope the selected works have official open-source implementations for reproducibility. ", + "bbox": [ + 174, + 722, + 825, + 792 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Based on the above selections, eight binarization algorithms, i.e., BNN, XNOR-Net, DoReFa-Net, Bi-Real Net, XNOR-Net $^ { + + }$ , ReActNet, FDA, and ReCU, stand out and are fully evaluated by our BiBench. ", + "bbox": [ + 173, + 799, + 825, + 840 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "BNN (Courbariaux et al., 2016b): During the training process, BNN uses the straight-through estimator (STE) to calculate gradient $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { g } } _ { \\mathbf { \\lambda } } }$ which takes into account the saturation effect: ", + "bbox": [ + 171, + 848, + 823, + 877 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/725f9f0ccfb8c74ba66eee499e4b65efa904a0434b3d598e9d0594f92ad1c766.jpg", + "text": "$$\n\\operatorname { s i g n } ( x ) = { \\left\\{ \\begin{array} { l l } { + 1 , } & { { \\mathrm { i f ~ } } x \\geq 0 } \\\\ { - 1 , } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. } \\qquad g _ { x } = { \\left\\{ \\begin{array} { l l } { g _ { b } , } & { { \\mathrm { i f ~ } } x \\in ( - 1 , 1 ) } \\\\ { 0 , } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 284, + 892, + 712, + 928 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "And during inference, the computation process is expressed as ", + "bbox": [ + 174, + 103, + 583, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/1749fa204ca8e3ef4643f55ebf025558b2a1db07141da839777e810cafab2d1b.jpg", + "text": "$$\n\\pmb { o } = \\mathrm { s i g n } ( \\pmb { a } ) \\circledast \\mathrm { s i g n } ( \\pmb { w } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 123, + 581, + 141 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\circledast$ indicates a convolutional operation using XNOR and bitcount operations. ", + "bbox": [ + 173, + 145, + 715, + 161 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "XNOR-Net (Rastegari et al., 2016): XNOR-Net obtains the channel-wise scaling factors $\\begin{array} { r } { \\alpha = \\frac { \\| \\pmb { w } \\| } { | \\pmb { w } | } } \\end{array}$ for the weight and $\\kappa$ contains scaling factors $\\beta$ for all sub-tensors in activation $^ { a }$ . We can approximate the convolution between activation $^ { a }$ and weight $\\pmb { w }$ mainly using binary operations: ", + "bbox": [ + 173, + 169, + 825, + 215 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/e2d0a82830b3e8de1b6d16389fec5349b475c3cb19244bc08e8bf1917df7c999.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { o } = ( \\mathrm { s i g n } ( \\pmb { a } ) \\circledast \\mathrm { s i g n } ( \\pmb { w } ) ) \\odot \\pmb { K } \\pmb { \\alpha } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 220, + 612, + 238 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\pmb { w } \\in \\mathbb { R } ^ { c \\times w \\times h }$ and $\\pmb { a } \\in \\mathbb { R } ^ { c \\times w _ { \\mathrm { i n } } \\times h _ { \\mathrm { i n } } }$ denote the weight and input tensor, respectively. And the STE is also applied in the backward propagation of the training process. ", + "bbox": [ + 171, + 242, + 825, + 272 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "DoReFa-Net (Zhou et al., 2016b): DoReFa-Net applies the following function for 1-bit weight and activation: ", + "bbox": [ + 169, + 277, + 825, + 305 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/90235f84918f57ed1f8904f44fab039f1170cc21d714bed64338b92e61f0934a.jpg", + "text": "$$\n\\pmb { o } = ( \\mathrm { s i g n } ( \\pmb { a } ) \\circledast \\mathrm { s i g n } ( \\pmb { w } ) ) \\odot \\pmb { \\alpha } .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 305, + 604, + 321 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "And the STE is also applied in the backward propagation with the full-precision gradient. ", + "bbox": [ + 173, + 324, + 759, + 339 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Bi-Real Net (Liu et al., 2018b): Bi-Real Net proposes a piece-wise polynomial function as the gradient approximation function: ", + "bbox": [ + 173, + 344, + 823, + 375 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/e185de3002509e3ef99d027cd32174639fcc9ea5c54b5f21c94032228271e355.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { b i r e a l } \\left( a \\right) = \\left\\{ \\begin{array} { l l } { - 1 } & { \\mathrm { i f ~ } a < - 1 } \\\\ { 2 a + a ^ { 2 } } & { \\mathrm { i f ~ } - 1 \\leqslant a < 0 } \\\\ { 2 a - a ^ { 2 } } & { \\mathrm { i f ~ } 0 \\leqslant a < 1 } \\\\ { 1 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. , \\quad \\frac { \\partial \\mathrm { b i r e a l } \\left( a \\right) } { \\partial a } = \\left\\{ \\begin{array} { l l } { 2 + 2 a } & { \\mathrm { i f ~ } - 1 \\leqslant a < 0 } \\\\ { 2 - 2 a } & { \\mathrm { i f ~ } 0 \\leqslant a < 1 } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 378, + 810, + 439 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "And the forward propagation of Bi-Real Net is the same as Eq. (15). ", + "bbox": [ + 173, + 450, + 620, + 465 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "XNOR- $\\mathbf { N e t } { + } { + }$ (Bulat et al., 2019): XNOR-Net+ $^ +$ proposes to re-formulate Eq. (14) as: ", + "bbox": [ + 171, + 472, + 746, + 487 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/9fbfff87cfed923921e482390133f9476c8ad5c6bfa0af91a034e4b371e0d42c.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { o } = ( \\mathrm { s i g n } ( \\pmb { a } ) \\circledast \\mathrm { s i g n } ( \\pmb { w } ) ) \\odot \\mathbf { \\vec { r } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 492, + 602, + 510 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "and we adopt the $\\mathbf { \\delta T }$ as the following form in experiments (achieve the best performance in the original paper): ", + "bbox": [ + 176, + 513, + 823, + 542 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/015ae25284f9aa2640ae6831c2c370369f51d541540a86a60b202d17c1424ef7.jpg", + "text": "$$\n\\Gamma = \\alpha \\otimes \\beta \\otimes \\gamma , \\quad \\alpha \\in \\mathbb { R } ^ { o } , \\beta \\in \\mathbb { R } ^ { h _ { \\mathrm { o u t } } } , \\gamma \\in \\mathbb { R } ^ { w _ { \\mathrm { o u t } } } ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 541, + 666, + 559 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\alpha , \\beta$ , and $\\gamma$ are learnable during training. ", + "bbox": [ + 174, + 561, + 490, + 577 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ReActNet (Liu et al., 2020): ReActNet defines an RSign as a binarization function with channelwise learnable thresholds: ", + "bbox": [ + 176, + 582, + 823, + 611 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2ec2f18d5b674c992b38ffb74e207d00f6170d6c4c6717b6d98d129c9c482399.jpg", + "text": "$$\n\\pmb { x } = \\mathrm { r s i g n } \\left( \\pmb { x } \\right) = \\left\\{ \\begin{array} { l l } { + 1 , } & { \\mathrm { ~ i f ~ } \\pmb { x } > \\pmb { \\alpha } } \\\\ { - 1 , } & { \\mathrm { ~ i f ~ } \\pmb { x } \\le \\pmb { \\alpha } } \\end{array} . \\right.\n$$", + "text_format": "latex", + "bbox": [ + 369, + 613, + 625, + 648 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $_ { \\pmb { \\alpha } }$ is a learnable coefficient controlling the threshold. And the forward propagation is ", + "bbox": [ + 169, + 652, + 771, + 667 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/7e7c157a7c2e40b7aab5badabc5b2f8f99ff781c4f7afc863e6c65f0e1314308.jpg", + "text": "$$\n\\pmb { o } = ( \\mathrm { r s i g n } ( \\pmb { a } ) \\circledast \\mathrm { s i g n } ( \\pmb { w } ) ) \\odot \\pmb { \\alpha } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 672, + 607, + 690 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ReCU (Xu et al., 2021b): As described in their paper, ReCU is formulated as ", + "bbox": [ + 173, + 702, + 681, + 718 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0d2dbc8f6a0491c59e89c027583a83dcd2a3c6c5dd3a7bbe0d7975a3505b4080.jpg", + "text": "$$\n\\mathrm { r e c u } ( { \\pmb w } ) = \\mathrm { m a x } \\left( \\mathrm { m i n } \\left( { \\pmb w } , { \\pmb Q } _ { ( \\tau ) } \\right) , { \\pmb Q } _ { ( 1 - \\tau ) } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 722, + 643, + 742 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $Q _ { ( \\tau ) }$ and $Q _ { ( 1 - \\tau ) }$ denote the $\\tau$ quantile and $1 - \\tau$ quantile of $\\pmb { w }$ , respectively. And other implementations also strictly follow the original paper and official code. ", + "bbox": [ + 174, + 746, + 825, + 775 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "FDA $\\mathrm { { X u } }$ et al., 2021a): FDA computes the gradients of $^ o$ in the backward propagation as: ", + "bbox": [ + 171, + 780, + 766, + 796 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/4eb656320571a324d9f2248873949614111c74e36e47d94f5ea3531b82846dff.jpg", + "text": "$$\n\\frac { \\partial \\ell } { \\partial \\mathbf { t } } = \\frac { \\partial \\ell } { \\partial \\pmb { \\sigma } } \\pmb { w } _ { 2 } ^ { \\top } \\odot \\left( \\left( \\mathbf { t } \\pmb { w } _ { 1 } \\right) \\geq 0 \\right) \\pmb { w } _ { 1 } ^ { \\top } + \\frac { \\partial \\ell } { \\partial \\pmb { \\sigma } } \\pmb { \\eta } ^ { \\prime } ( \\mathbf { t } ) + \\frac { \\partial \\ell } { \\partial \\pmb { \\sigma } } \\odot \\frac { 4 \\omega } { \\pi } \\sum _ { i = 0 } ^ { n } \\cos ( ( 2 i + 1 ) \\omega \\mathbf { t } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 235, + 799, + 761, + 842 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\frac { \\partial \\ell } { \\partial o }$ is the gradient from the upper layers, $\\odot$ represents element-wise multiplication, and $\\frac { \\partial \\ell } { \\partial \\mathbf { t } }$ is the partial gradient on $\\mathbf { t }$ that backward propagates to the former layer. And ${ \\pmb w } _ { 1 }$ and ${ \\pmb w } _ { 2 }$ are weights in the original models and the noise adaptation modules respectively. FDA updates them as ", + "bbox": [ + 173, + 848, + 823, + 891 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/292322b9b417d464af07235227d701c71d24fdd9b80888f873d55bbb9acaf557.jpg", + "text": "$$\n\\frac { \\partial \\ell } { \\partial w _ { 1 } } = \\mathbf { t } ^ { \\top } \\frac { \\partial \\ell } { \\partial o } \\pmb { w } _ { 2 } ^ { \\top } \\odot \\left( \\left( \\mathbf { t } w _ { 1 } \\right) \\geq 0 \\right) , \\qquad \\frac { \\partial \\ell } { \\partial \\pmb { w } _ { 2 } } = \\sigma \\left( \\mathbf { t } w _ { 1 } \\right) ^ { \\top } \\frac { \\partial \\ell } { \\partial o } .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 895, + 710, + 929 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B DETAILS OF LEARNING TASKS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 464, + 117 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Selection Rules: ", + "text_level": 1, + "bbox": [ + 174, + 137, + 284, + 152 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To comprehensively evaluate the performance of the binarization algorithm in various learning tasks, we should select various representative tasks. First, representative perception modalities are selected in our deep learning, including (2D/3D) vision, text, and speech. Research on these modalities has progressed rapidly and has a broad impact, so we choose specific tasks and datasets in these modalities. Specifically, (1) on the 2D vision modality, we choose the basic image classification task and object detection task (one of the most popular downstream tasks), the former including CIFAR10 and ImageNet datasets, the latter including Pascal VOC and COCO datasets. These datasets ImageNet and COCO are both more challenging large datasets, while CIFAR10 and Pascal VOC are more basic. For other modalities, binarization is still challenging even with the underlying tasks and datasets in the field, since there are few related binarization studies: (2) In the 3D vision modality, the basic point cloud classification ModelNet40 dataset is selected to evaluate the binarization performance, which is regarded as one of the most fundamental tasks in 3D point cloud research and is widely studied. (3) In the text modality, the General Language Understanding Evaluation (GLUE) benchmark is usually recognized as the most popular dataset, including nine sentence- or sentence-pair language understanding tasks. (4) In the speech modality, the keyword spotting task was chosen as the base task, specifically the Google Speech Commands classification dataset. ", + "bbox": [ + 174, + 160, + 825, + 381 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Based on the above reasons and rules, we have selected a series of challenging and representative tasks for BiBench to evaluate binarization comprehensively and have obtained a series of reliable and informative conclusions and experiences. ", + "bbox": [ + 176, + 388, + 821, + 430 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "CIFAR10 (Krizhevsky et al., 2014): The CIFAR-10 dataset (Canadian Institute For Advanced Research) is a collection of images commonly used to train machine learning and computer vision algorithms. This dataset is widely used for image classification tasks. There are 60,000 color images, each of which measures $3 2 \\mathrm { x } 3 2 $ pixels. All images are categorized into 10 different classes: airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks. Each class has 6000 images, where 5000 are for training and 1000 are for testing. The evaluation metric of the CIFAR-10 dataset is accuracy, defined as: ", + "bbox": [ + 174, + 436, + 825, + 535 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/2fe8fbe0b023f4ac47e418eb1035aa1366e2646912e7a75a7a8afe1c5edf9233.jpg", + "text": "$$\nA c c u r a c y = \\frac { T P + T N } { T P + T N + F P + F N } ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 541, + 630, + 574 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $T P$ (True Positive) means cases correctly identified as positive, TN (True Negative) means cases correctly identified as negative, $F P$ (False Positive) means cases incorrectly identified as positive and $F N$ (False Negative) means cases incorrectly identified as negative. To estimate the accuracy, we should calculate the proportion of $T P$ and $T N$ in all evaluated cases. ", + "bbox": [ + 174, + 585, + 823, + 642 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ImageNet (Krizhevsky et al., 2012): ImageNet is a dataset of over 15 million labeled high-resolution images belonging to roughly 22,000 categories. The images are collected from the web and labeled by human labelers using a crowd-sourced image labeling service called Amazon Mechanical Turk. As part of the Pascal Visual Object Challenge, ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) was established in 2010. There are approximately 1.2 million training images, 50,000 validation images, and 150,000 testing images in total in ILSVRC. ILSVRC uses a subset of ImageNet, with about 1000 images in each of the 1000 categories. ", + "bbox": [ + 173, + 647, + 825, + 747 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ImageNet also uses accuracy to evaluate the predicted results, which is defined above. ", + "bbox": [ + 176, + 752, + 733, + 767 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Pascal VOC07 (Hoiem et al., 2009): The PASCAL Visual Object Classes 2007 (VOC07) dataset contains 20 object categories including vehicles, households, animals, and other: airplane, bicycle, boat, bus, car, motorbike, train, bottle, chair, dining table, potted plant, sofa, TV/monitor, bird, cat, cow, dog, horse, sheep, and person. As a benchmark for object detection, semantic segmentation, and object classification, this dataset contains pixel-level segmentation annotations, bounding box annotations, and object class annotations. The VOC07 dataset uses mean average precision $( m A P )$ to evaluate results, which is defined as: ", + "bbox": [ + 173, + 773, + 825, + 872 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/b80704c22c6690c3a5f84571a294d6a5c40e1c1b760d66c552669879f08d71cb.jpg", + "text": "$$\nm A P = \\frac { 1 } { n } \\sum _ { k = 1 } ^ { k = n } A P _ { k }\n$$", + "text_format": "latex", + "bbox": [ + 429, + 885, + 568, + 928 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $A P _ { k }$ denotes the average precision of the $\\mathbf { k }$ -th category, which calculates the area under the precision-recall curve: ", + "bbox": [ + 171, + 103, + 823, + 131 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/b819ae4ac38d94b82ae06eacf984d94e70e60aeb53cfa443fe605c68c911110f.jpg", + "text": "$$\nA P _ { k } = \\int _ { 0 } ^ { 1 } p _ { k } ( r ) d r .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 130, + 568, + 165 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Especially for VOC07, we apply 11-point interpolated $A P$ , which divides the recall value to $\\{ \\bar { 0 . 0 } , 0 . 1 , \\bar { . . . } , 1 . 0 \\}$ and then computes the average of maximum precision value for these 11 recall values as: ", + "bbox": [ + 174, + 167, + 825, + 209 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/9a6e5ea0c338e3e2f7f6dd6b79d7e5a6aa9632c1a62749ae97e3b3694b2b3d71.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\cal A } { \\cal P } = \\frac { 1 } { 1 1 } \\sum _ { r \\in \\{ 0 . 0 , \\ldots , 1 . 0 \\} \\atop { \\displaystyle = \\frac { 1 } { 1 1 } \\sum _ { r \\in \\{ 0 . 0 , \\ldots , 1 . 0 \\} } p _ { \\mathrm { i n t e r p } } r . } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 395, + 213, + 602, + 296 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The maximum precision value equals to the right of its recall level: ", + "bbox": [ + 173, + 300, + 612, + 315 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/96fbe3731c33bad1f541e582d7c26bb5ec36b56bb056ee5786abdbfc9e6d4c2b.jpg", + "text": "$$\np _ { \\mathrm { i n t e r p } } r = \\operatorname* { m a x } _ { \\tilde { r } \\geq r } p ( \\tilde { r } ) .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 321, + 566, + 347 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "COCO17 (Lin et al., 2014): The MS COCO (Microsoft Common Objects in Context) dataset is a large-scale object detection, segmentation, key-point detection, and captioning dataset. The dataset consists of 328K images. According to community feedback, in the 2017 release, the training/validation split was changed from 83K/41K to 118K/5K. And the images and annotations are the same. The 2017 test set is a subset of 41K images from the 2015 test set. Additionally, 123K images are included in the unannotated dataset. The COCO17 dataset also uses mean average precision $( m A P )$ as defined above PASCAL VOC07 uses, which is defined as above. ", + "bbox": [ + 173, + 359, + 825, + 458 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ModelNet40 (Wu et al., 2015): The ModelNet40 dataset contains point clouds of synthetic objects. As the most widely used benchmark for point cloud analysis, ModelNet40 is popular due to the diversity of categories, clean shapes, and well-constructed dataset. In the original ModelNet40, 12,311 CAD-generated meshes are divided into 40 categories, where 9,843 are for training, and 2,468 are for testing. The point cloud data points are sampled by a uniform sampling method from mesh surfaces and then scaled into a unit sphere by moving to the origin. The ModelNet40 dataset also uses accuracy as the metric, which has been defined above in CIFAR10. ", + "bbox": [ + 173, + 464, + 825, + 563 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ShapeNet (Chang et al., 2015): ShapeNet is a large-scale repository for 3D CAD models developed by researchers from Stanford University, Princeton University, and the Toyota Technological Institute in Chicago, USA. ", + "bbox": [ + 174, + 569, + 825, + 611 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Using WordNet hypernym-hyponym relationships, the repository contains over 300M models, with 220,000 classified into 3,135 classes. There are 31,693 meshes in the ShapeNet Parts subset, divided into 16 categories of objects (i.e., tables, chairs, planes, etc.). Each shape contains 2-5 parts (with 50 part classes in total). ", + "bbox": [ + 173, + 617, + 825, + 674 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "GLUE (Wang et al., 2018): General Language Understanding Evaluation (GLUE) benchmark is a collection of nine natural language understanding tasks, including single-sentence tasks CoLA and SST-2, similarity and paraphrasing tasks MRPC, STS-B and QQP, and natural language inference tasks MNLI, QNLI, RTE, and WNLI. Among them, SST-2, MRPC, QQP, MNLI, QNLI, RTE, and WNLI use accuracy as the metric, which is defined in CIFAR10. CoLA is measured by Matthews Correlation Coefficient (MCC), which is better in binary classification since the number of positive and negative samples are extremely unbalanced: ", + "bbox": [ + 173, + 680, + 825, + 779 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/2a7ce8b6624db916f6b750712bd7326c4c522c216da79986d04cbd14bc2fac79.jpg", + "text": "$$\nM C C = { \\frac { T P \\times T N - F P \\times F N } { \\sqrt { ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) } } } .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 785, + 715, + 821 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "And STS-B is measured by Pearson/Spearman Correlation Coefficient: ", + "bbox": [ + 174, + 827, + 638, + 842 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/37ebe74077efb44a27ad97e6ac4c7742a93617e9a3d47ddb0292a69f0a80ba1e.jpg", + "text": "$$\nr _ { P e a r s o n } = \\frac { 1 } { n - 1 } \\sum _ { i = 1 } ^ { n } \\left( \\frac { X _ { i } - \\bar { X } } { s _ { X } } \\right) \\left( \\frac { Y _ { i } - \\bar { Y } } { s _ { Y } } \\right) , r _ { S p e a r m a n } = 1 - \\frac { 6 \\sum _ { i } d _ { i } ^ { 2 } } { n ( n ^ { 2 } - 1 ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 253, + 848, + 743, + 890 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $n$ is the number of observations, $s _ { X }$ and $s _ { Y }$ indicate the sum of squares of $X$ and $Y$ respectively, and $d _ { i }$ is the difference between the ranks of corresponding variables. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "SpeechCom. (Warden, 2018): As part of its training and evaluation process, SpeechCom provides a collection of audio recordings containing spoken words. Its primary goal is to provide a way to build and test small models that detect a single word that belongs to a set of ten target words. Models should detect as few false positives as possible from background noise or unrelated speech while providing as few false positives as possible. The accuracy metric for SpeechCom is also the same as CIFAR10. ", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CIFAR10-C (Hendrycks & Dietterich, 2018): CIFAR10-C is a dataset generated by adding 15 common corruptions and 4 extra corruptions to the test images in the Cifar10 dataset. It benchmarks the frailty of classifiers under corruption, including noise, blur, weather, and digital influence. And each type of corruption has five levels of severity, resulting in 75 distinct corruptions. We report the accuracy of the classifiers under each level of severity and each corruption. Meanwhile, we use the $E _ { S e t t i n g s } ^ { N e t w o r k }$ d relative corruption error as metrics. Denote the error rate of Network under Settings as. The classifier’s aggregate performance across the five severities of the corruption types.ruption Errors of a certain type of Corruption is computed with the formula: ", + "bbox": [ + 174, + 193, + 825, + 308 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/3a3acb2bee88d6a8517ed0ce149f7383132e5eec470929935f1437e92593170e.jpg", + "text": "$$\nC E _ { C o r r u p t i o n } ^ { N e t w o r k } = \\sum _ { s = 1 } ^ { 5 } E _ { s , C o r r u p t i o n } ^ { N e t w o r k } / \\sum _ { s = 1 } ^ { 5 } E _ { s , C o r r u p t i o n } ^ { A l e x N e t } .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 314, + 658, + 357 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To make Corruption Errors comparable across corruption types, the difficulty is usually adjusted by dividing by AlexNet’s errors. ", + "bbox": [ + 173, + 362, + 823, + 391 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C DETAILS OF NEURAL ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 174, + 411, + 534, + 428 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "ResNet (He et al., 2016): Residual Networks, or ResNets, learn residual functions concerning the layer inputs instead of learning unreferenced functions. Instead of making stacked layers directly fit a desired underlying mapping, residual nets let these layers fit a residual mapping. There is empirical evidence that these networks are easier to optimize and can achieve higher accuracy with considerably increased depth. ", + "bbox": [ + 174, + 441, + 825, + 512 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "VGG (Simonyan & Zisserman, 2015): VGG is a classical convolutional neural network architecture. It is proposed by an analysis of how to increase the depth of such networks. It is characterized by its simplicity: the network utilizes small $3 \\times 3$ filters, and the only other components are pooling layers and a fully connected layer. ", + "bbox": [ + 174, + 518, + 823, + 575 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "MobileNetV2 (Sandler et al., 2018): MobileNetV2 is a convolutional neural network architecture that performs well on mobile devices. This model has an inverted residual structure with residual connections between the bottleneck layers. The intermediate expansion layer employs lightweight depthwise convolutions to filter features as a source of nonlinearity. In MobileNetV2, the architecture begins with an initial layer of 32 convolution filters, followed by 19 residual bottleneck layers. ", + "bbox": [ + 174, + 582, + 823, + 652 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Faster-RCNN (Ren et al., 2015): Faster R-CNN is an object detection model that improves Fast R-CNN by utilizing a region proposal network (RPN) with the CNN model. The RPN shares fullimage convolutional features with the detection network, enabling nearly cost-free region proposals. A fully convolutional network is used to predict the bounds and objectness scores of objects at each position simultaneously. RPNs use end-to-end training to produce region proposals of high quality and instruct the unified network where to search. Sharing their convolutional features allows RPN and Fast R-CNN to be combined into a single network. Faster R-CNN consists of two modules. The first module is a deep, fully convolutional network that proposes regions, and the second is the detector that uses the proposals for giving the final prediction boxes. ", + "bbox": [ + 174, + 659, + 825, + 785 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "SSD (Liu et al., 2016): SSD is a single-stage object detection method that discretizes the output space of bounding boxes into a set of default boxes over different aspect ratios and scales per feature map location. During prediction, each default box is adjusted to match better the shape of the object based on its scores for each object category. In addition, the network automatically handles objects of different sizes by combining predictions from multiple feature maps with different resolutions. ", + "bbox": [ + 174, + 790, + 823, + 861 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "BERT (Kenton & Toutanova, 2019): BERT, or Bidirectional Encoder Representations from Transformers, improves upon standard Transformers by removing the unidirectionality constraint using a masked language model (MLM) pre-training objective. By masking some tokens from the input, the masked language model attempts to estimate the original vocabulary id of the masked word based solely on its context. An MLM objective differs from a left-to-right language model in that it enables the representation to integrate the left and right contexts, which facilitates pre-training a deep bidirectional Transformer. Additionally, BERT uses a next-sentence prediction task that pre-trains text-pair representations along with the masked language model. Note that we replace the direct binarized attention with a bi-attention mechanism to prevent the model from completely crashing (Qin et al., 2021). ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 188 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "PointNet (Qi et al., 2017): PointNet is a unified architecture for applications ranging from object classification and part segmentation to scene semantic parsing. The architecture directly receives point clouds as input and outputs either class labels for the entire input or point segment/part labels. PointNet-Vanilla is a variant of PointNet, which drops off the T-Net module. And for all PointNet models, we apply the EMA-Max (Qin et al., 2020a) as the aggregator, because directly following the max pooling aggregator will cause the binarized PointNets to fail to converge. ", + "bbox": [ + 173, + 194, + 825, + 277 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "FSMN (Zhang et al., 2015): Feedforward sequential memory networks or FSMN is a novel neural network structure to model long-term dependency in time series without using recurrent feedback. It is a standard fully connected feedforward neural network containing some learnable memory blocks. As a short-term memory mechanism, the memory blocks encode long context information using a tapped-delay line structure. ", + "bbox": [ + 174, + 285, + 825, + 354 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Deep-FSMN (Zhang et al., 2018a): The Deep-FSMN architecture is an improved feedforward sequential memory network (FSMN) with skip connections between memory blocks in adjacent layers. By utilizing skip connections, information can be transferred across layers, and thus the gradient vanishing problem can be avoided when building very deep structures. ", + "bbox": [ + 174, + 361, + 823, + 417 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "D DETAILS OF HARDWARE ", + "text_level": 1, + "bbox": [ + 176, + 438, + 413, + 453 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Hisilicon Kirin (Hisilicon, 2022): Kirin is a series of ARM-based systems-on-a-chip (SoCs) produced by HiSilicon. Their products include Kirin 970, Kirin 980, Kirin 985, etc. ", + "bbox": [ + 169, + 468, + 823, + 496 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "MediaTek Dimensity (MediaTek, 2022): Dimensity is a series of ARM-based systems-on-a-chip (SoCs) produced by MediaTek. Their products include Dimensity 820, Dimensity 8100, Dimensity 9000, etc. ", + "bbox": [ + 173, + 502, + 821, + 545 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Qualcomm Snapdragon (Singh & Jain, 2014): Snapdragon is a family of mobile systems-ona-chip (SoC) processor architecture provided by Qualcomm. The original Snapdragon chip, the Scorpio, was similar to the ARM Cortex-A8 core based upon the ARMv7 instruction set, but it was enhanced by the use of SIMD operations, which provided higher performance. Qualcomm Snapdragon processors are based on the Krait architecture. They are equipped with an integrated LTE modem, providing seamless connectivity across 2G and 3G LTE networks. ", + "bbox": [ + 174, + 551, + 825, + 636 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Raspberrypi (Wikipedia, 2022b): Raspberry Pi is a series of small single-board computers (SBCs) developed in the United Kingdom by the Raspberry Pi Foundation in association with Broadcom. Raspberry Pi was originally designed to promote the teaching of basic computer science in schools and in developing countries. As a result of its low cost, modularity, and open design, it is used in many applications, including weather monitoring, and is sold outside the intended market. It is typically used by computer hobbyists and electronic enthusiasts due to the adoption of HDMI and USB standards. ", + "bbox": [ + 174, + 642, + 825, + 739 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Apple M1 (Wikipedia, 2022a): Apple M1 is a series of ARM-based systems-on-a-chip (SoCs) designed by Apple Inc. As a central processing unit (CPU) and graphics processing unit (GPU) for Macintosh desktops and notebooks, as well as iPad Pro and iPad Air tablets. In November 2020, Apple introduced the M1 chip, followed by the professional-oriented M1 Pro and M1 Max chips in 2021. Apple launched the M1 Ultra in 2022, which combines two M1 Max chips in a single package. The M1 Max is a higher-performance version of the M1 Pro, with larger GPU cores and memory bandwidth. ", + "bbox": [ + 174, + 747, + 825, + 844 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E FULL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 864, + 333, + 881 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 6-7 shows the accuracy of different binarization algorithms on 2D and 3D vision tasks, including CIFAR10, ImageNet, PASCAL VOC07, COCO14 for 2D vision tasks and ModelNet40 for ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/e2163393fab98d0d1e8f1722b5c951e5e97387628314dea4d3c4f829d4261b96.jpg", + "table_caption": [ + "Table 6: Accuracy on 2D and 3D Vision Tasks. " + ], + "table_footnote": [], + "table_body": "
TaskArch.FP32Binarization Algorithm
BNNXNOR DoReFa Bi-Real XNOR++ ReActNet ReCU FDA
CIFAR10ResNet2091.9985.3185.5385.1885.5685.4186.1886.4286.38
ResNet1894.8289.6991.491.5591.2090.0491.5592.7990.42
ResNet3495.3490.8289.5890.9592.5090.5992.6993.6489.59
VGG-Small93.8089.6689.6589.6690.2589.3490.2790.8489.48
ImageNetResNet1869.9052.9953.9953.5554.7952.4354.9754.5154.63
VOC07Faster-RCNN76.0658.5456.7558.0760.9056.6061.9062.1060.10
SSD30077.349.0933.7230.7031.909.4138.419.8043.68
COC014Faster-RCNN27.2021.2020.5021.3022.2021.6022.8023.3022.40
ModelNet40PointNetvanilla86.8085.1383.4785.2185.3785.6685.1385.2185.49
PointNet88.209.0880.7578.7777.7163.2576.5081.1279.62
", + "bbox": [ + 173, + 119, + 826, + 510 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "3D vision task. And for each task, we cover several representative model architectures and binarize them with the binarization algorithms mentioned above. ", + "bbox": [ + 176, + 534, + 821, + 563 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We also evaluate binarization algorithms on language and speech tasks, for which we test TinyBERT (4 layers and 6 layers) on GLUE Benchmark and FSMN and D-FSMN on the SpeechCommand dataset. Results are listed in Table 8. ", + "bbox": [ + 174, + 569, + 825, + 611 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To demonstrate the robustness corruption of binarized algorithms, we show the results on the CIFAR10-C benchmark, which is used to benchmark the robustness to common perturbations in Table 9 and Table 10. It includes 15 kinds of noise, blur, weather, and digital corruption, each with five levels of severity. ", + "bbox": [ + 174, + 618, + 825, + 674 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The sensitivity of hyperparameters while training is shown in Table 11-12. For each binarization algorithm, we use SGD or Adam optimizer, $1 \\times$ or $0 . 1 \\times$ of the original learning rate, cosine or step learning scheduler, and 200 training epochs. Each case is tested five times to show the training stability. We also calculate the mean and standard deviation (std) of accuracy. The best accuracy and the lowest std for each binarization algorithm are bolded. ", + "bbox": [ + 174, + 681, + 825, + 751 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We conduct comprehensive deployment and inference on various kinds of hardware, including the Kirin series (970, 980, 985, 990, and 9000E), Dimensity series (820 and 9000), Snapdragon series $( 8 5 5 +$ , 870 and 888), Raspberrypi ( $3 \\mathrm { B } +$ and 4B) and Apple M1 series (M1 and M1 Max). Limited to the support of frameworks, we can only test BNN and ReAct with Larq compute engine and only BNN with daBNN. We convert models to enable the actual inference on real hardware, including ResNet18/34 and VGG-Small on Larq, and only ResNet18/34 on daBNN. And we test 1, 2, 4, and 8 threads for each hardware and additionally test 16 threads for Apple Silicons on Larq. And daBNN only supports single-thread inference. Results are showcased in Table 13-16. ", + "bbox": [ + 174, + 757, + 825, + 869 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/11e69bfda258fa7d3f1ad2d8f7411e88a6ef058e1ae28f88049d7e5cc4dedf83.jpg", + "table_caption": [ + "Table 7: Accuracy on ShapeNet dataset. " + ], + "table_footnote": [], + "table_body": "
TaskArch.CategoryFP32Binarization Algorithm
BNN XNOR DoReFa Bi-Real XNOR++ ReActNet ReCU FDA
Airplane83.737.5 74.1467.267.6130.3666.1231.6165.34
Bag79.644.2 4955.3447.1137.4450.2838.5848.62
Cap92.344.3 73.3251.2161.4140.3756.7340.1356
Car76.824.3 55.2752.2449.3924.0749.1123.9258.5
Chair90.961.6 85.6283.9683.641.8983.8341.583.27
Earphone 70.238.530.9734.9435.2426.336.7223.01 34.46
Guitar91.132.969.1767.965.9923.4564.1828.3878.69
Knife85.74378.1676.1675.5337.6275.0138.8177.07
ShapeNet PointNet Lamp8251.26968.756049.4566.1348.41 67.45
Laptop95.549.493.2992.9392.7941.8992.9342.28 93.66
Motorbike 64.416.319.0418.8818.6913.1818.5911.2620.38
Mug93.649.164.3253.5652.0147.5852.5146.83 53.48
Pistol80.825.562.2959.1551.4326.9653.7927.81 62.61
Rocket54.426.930.9527.9226.6122.3826.0119.32 23.08
Skateboard 70.741.245.750.1543.7828.6343.7426.71 45.81
Table81.451.373.6872.6969.7245.7469.5645.2173.45
Overall80.81875 39.8260.8758.3156.3133.5856.5833.3658.68
", + "bbox": [ + 184, + 185, + 813, + 859 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/6eaeca15af95641e1d6989ee85b39dc4232a7ebbfc0be8b71f8b4dbf76d920f0.jpg", + "table_caption": [ + "Table 8: Accuracy on Language and Speech Tasks. " + ], + "table_footnote": [], + "table_body": "
TaskArch.FP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNet ReCUFDA
BERT-Tiny4L82.8136.9041.2052.3155.0937.2755.5238.5559.41
MNLI-mBERT-Tiny6L84.7637.0151.1763.0966.8137.9866.4737.9568.46
BERT-Base84.8835.4541.4060.6762.4735.4560.2235.4563.49
BERT-Tiny4L83.0836.5441.5553.0155.5736.0755.8937.6259.76
MNLI-mmBERT-Tiny6L84.4236.4750.9263.8766.8238.1167.6436.9168.98
BERT-Base85.4535.2241.1860.9663.1735.2261.1935.2263.72
BERT-Tiny4L90.4766.1973.6975.7977.3864.9776.9267.3278.92
QQPBERT-Tiny6L85.9863.1878.9080.9382.4263.1982.9563.383.19
BERT-Base91.5163.1871.9377.0780.0163.1881.1663.1883.26
BERT-Tiny4L87.4651.7160.5961.1561.9252.7962.6753.9962.29
QNLIBERT-Tiny6L90.7952.2262.7566.8869.7251.8470.2751.3272.72
BERT-Base92.1451.860.2970.7870.1454.0769.4451.8772.43
GLUEBERT-Tiny4L92.4352.9879.9382.4584.0654.0184.1754.2486.12
SST-2BERT-Tiny6L90.2558.1484.7486.2387.7369.3887.9552.4087.72
BERT-Base93.2352.2978.7886.0186.3553.3284.452.4087.93
BERT-Tiny4L49.616.557.2212.6916.86014.716.2517.80
CoLABERT-Tiny6L54.172.5712.5715.9717.94015.242.2422.21
BERT-Base59.714.6304.7415.9504.630.404.63
BERT-Tiny4L86.354.3118.0518.7422.657.4522.738.2027.56
STS-BBERT-Tiny6L89.791.0414.7222.3124.595.7023.408.2237.15
BERT-Base90.066.9412.1918.2620.764.998.736.5910.14
BERT-Tiny4L85.5068.3071.7471.9971.7468.3071.7471.2571.49
MRPCBERT-Tiny6L87.7168.3070.7671.7471.4968.3071.7469.0471.74
BERT-Base86.2468.3068.370.0270.2768.3071.2568.3069.04
BERT-Tiny4L65.3456.3153.4356.3155.5954.1557.7661.0159.20
RTEBERT-Tiny6L68.9556.3154.5154.5158.1249.0953.4358.8454.87
BERT-Base72.2053.4357.0455.2354.5154.8754.5155.2355.23
FSMN94.8956.4556.4568.6573.6075.0473.8056.45
Speech CommandsD-FSMN97.5188.3292.0378.9285.1156.7783.8092.1156.45 93.91
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NoiseFP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCU
Origin94.8289.6991.4091.55 91.2090.0491.55 92.7990.42
gaussian_noise-178.2374.2276.0074.9774.9575.0775.15 78.25 61.3277.36
gaussian_noise-256.7256.7362.4455.9458.3357.5255.9760.48
gaussian_noise-336.9342.6947.5839.56 43.4740.7937.99 43.3244.26
gaussian_noise-431.0338.3541.4333.24 36.6534.6831.47 35.9137.30
gaussian_noise-525.5434.0536.2228.6631.7830.1325.49 30.1932.09
ipulse_nosie-182.5484.5786.9484.6887.3085.7286.89 88.7385.8
ipulse_nosie-270.1277.1380.7477.3581.1479.6280.25 82.8080.3
ipulse_nosie-359.8870.5875.0169.2074.5971.8272.16 76.0572.6
ipulse_nosie-440.5954.4259.3949.4856.6652.6149.79 58.4456.45
ipulse_nosie-526.0339.8641.5432.7237.2835.1228.42 38.2639.98
shot_noise-185.7581.5181.3181.8481.5880.6681.88 83.9882.42
shot_noise-276.6172.0474.0272.2172.8173.0372.32 76.7075.1
shot_noise-352.2153.9057.0850.6654.5953.7651.31 57.2256.56
shot_noise-444.1347.5851.2943.5948.3646.6444.21 48.7848.91
shot_noise-532.7339.9340.7933.8038.5036.4731.79 36.4637.8
speckle_noise-186.3081.2981.9480.9380.7781.1482.17 84.1782.62
speckle_noise-271.9468.0770.1467.569.2269.3568.26 72.9471.70
speckle_noise-364.4762.1264.1360.2463.4462.5061.14 66.8964.27
speckle_noise-449.8151.9353.7747.9352.7550.5948.39 54.1352.40
speckle_noise-538.7044.2543.6038.6543.1642.0937.78 42.1342.57
gaussian_blur-194.1789.0390.589.3390.5689.0091.05 92.1689.33
gaussian_blur-287.0478.381.9878.8180.4277.7581.20 84.8078.93
gaussian_blur-375.1567.7468.2767.6768.1664.5467.42 73.6266.29
gaussian_blur-459.555.1753.6355.7454.0852.4452.72 60.3253.37
gaussian_blur-536.0337.3133.9637.5037.5436.7734.08 39.2234.93
defocus_blur-194.288.7391.0689.190.3288.9190.92 91.9889.58
defocus_blur-292.7585.9788.9986.5988.3185.5887.91 90.4787.01
defocus_blur-387.3879.0282.4378.8880.7177.5880.88 84.8579.52
defocus_blur-476.9969.1371.0268.2968.3365.9668.42 74.4068.22
defocus_blur-552.0948.8551.9948.8249.1748.4546.92 55.7048.27
glass_blur-154.9356.5751.7257.9456.7857.2956.27 58.8258.92
glass-blur-256.3757.9353.4660.4259.2159.3258.03 60.2560.56
glass_blur-359.2161.4356.9864.1161.7262.4160.3962.84 63.32
glass_blur-445.6546.5042.7248.4847.1947.8346.8849.09 49.23
glass_blur-549.1949.5246.4052.0649.8350.0249.0851.14 51.82
otion_blur-189.4081.5783.2782.0083.1181.4884.1982.83
otion_blur-281.9571.5274.7173.3872.4870.9974.3577.75 74.09
otion_blur-372.4861.8766.2163.8663.3961.5763.8568.31 64.36
otion_blur-472.7962.4066.1863.9462.8462.0364.54 67.8864.13
otion_blur-563.9154.1457.9856.0755.9054.3555.60 59.7156.65
zoo_blur-187.3678.2581.3178.5679.4577.2080.22 83.6978.55
zoo_blur-283.89 77.7374.84 69.0077.73 70.9875.3975.8872.8375.74 80.46 68.34 74.3374.72 68.21
zoo_blur-3 zoo_blur-4
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NoiseFP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCU
fog-194.0488.1790.8988.84 89.9188.5190.84 92.0889.43
fog-293.0384.5888.8585.48 87.2684.7788 89.8786.76
fog-390.6978.0785.280.0783.3278.9483.77 86.8282.78
fog-486.7269.5678.9272.27 75.8971.0177.96 81.5675.62
fog-568.649.0453.952.33 52.8849.6857.67 62.2955.18
frost-189.9783.6684.784.0785.7683.6485.51 87.7584.85
frost-284.4277.8878.9777.4778.9677.2679.14 81.479.16
frost-374.8567.6769.367.14 68.7666.0369.58 72.5470.14
frost-473.3265.9367.1465.97 68.5265.3767.93 71.4469.41
frost-562.1355.0256.6755.62 56.7754.2657.69 61.1159.88
snow-189.2684.5885.4484.59 86.4385.0785.95 87.8285.67
snow-278.9672.0173.3773.14 73.4272.4373.65 78.0573.84
snow-382.8575.676.8475.74 76.9575.7477.76 79.9875.95
snow-480.2970.8472.5671.93 72.3971.2572.97 76.1272.16
snow-574.9463.8566.9465.9 65.7164.2466.29 70.3366.53
contrast-193.8287.2389.9687.93 89.6887.8890.16 91.5388.84
contrast-290.5376.028477.2782.176.382.8 86.0281.37
contrast-385.8463.9777.165.7772.7764.1874.62 79.3071.89
contrast-475.0844.0762.3747.35 55.5744.9459.87 65.7255.14
contrast-529.362025.6720.18 22.1821.0425.28 25.6624.04
elastic_transfor-189.9782.9784.583.3884.7482.4884.42 86.5483.25
elastic_transfor-289.4382.1284.7982.4484.0781.9784.61 86.2083.15
elastic_transfor-385.5277.5680.7177.9279.117779.53 82.2778.35
elastic_transfor-479.4873.7574.8373.9273.4172.7773.98 77.5374.17
elastic_transfor-575.0270.9770.2271.3671.0371.6371.65 75.3171.49
jpeg_copression-187.3683.2883.9384.0783.8383.7784.3 85.6583.63
jpeg_copression-281.6880.0979.6679.7780.0380.2980.59 81.6679.8
jpeg_copression-379.9878.5578.2178.3278.2778.9978.51 79.9478.44
jpeg_copression-477.1777.1275.7877.4477.0477.4677.08 77.6776.71
jpeg_copression-573.8574.5173.0474.6574.1375.2674.16 74.8574.19
pixelate-192.5786.9788.1987.3988.7387.4788.17 89.4287.26
pixelate-288.2381.9180.9582.3782.8281.9381.9983.80 80.95
pixelate-38478.475.2578.6378.0377.1577.2878.89 75.30
pixelate-468.5164.1158.1162.4960.8961.4360.0661.71 59.95
pixelate-550.5750.6844.7748.1845.4446.7443.2747.29 45.62
saturate-192.4184.9888.3885.3887.2685.3988.2389.34 86.82
saturate-290.1280.7485.2681.5782.6880.7484.2286.06 83.09
saturate-393.8387.8990.4588.1289.5388.0390.15 90.9788.73
saturate-491.6182.586.8882.684.6682.4486.04 87.5683.70
saturate-587.4876.0382.7675.5378.375.8580.62 82.6477.00
spatter-191.1787.589.7587.8389.3487.989.42 91.0088.98
spatter-285.283.8585.9883.5285.6484.7285.88 87.5985.00
spatter-380.6377.9480.3377.9580.1979.4180.07 82.6579.50
spatter-4
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AlgorithmEpochOptimizerLearning RateSchedulerAcc.Acc.1Acc.2Acc.3Acc.4meanstd
FP32200SGD0.1cosine94.5894.695.0594.6494.8494.740.20
200SGD0.1step92.6392.4292.1592.6292.3892.440.20
200SGD0.01cosine92.2391.7691.7691.9992.1791.980.22
200SGD0.01step83.9483.5082.8084.1383.8983.650.53
200Adam0.001cosine93.5192.9493.1293.3592.8693.160.27
200Adam0.001step93.3793.1593.3293.4193.3593.320.10
200 200Adam0.0001cosine89.9789.9289.9689.989.9289.930.03
Adam0.0001step90.5789.9190.4390.2590.3190.290.25
BNN200SGD0.1cosine87.6287.5387.9988.8687.8487.970.53
200SGD0.1step70.8773.8671.8373.172.8772.511.17
200SGD0.01cosine73.5272.6272.8271.1472.5972.540.87
200SGD0.01step52.8551.7752.0052.3453.1452.420.57
200Adam0.001cosine88.7688.9988.6788.8488.8188.810.12
200Adam0.001step88.8589.3488.7789.0289.0089.000.22
200Adam0.0001cosine83.4683.0983.2083.7083.2083.330.25
200Adam0.0001step84.0884.1184.2084.3183.5684.050.29
XNOR200SGD0.1cosine91.8391.9991.8792.0191.5691.850.18
200SGD0.1step90.0290.0190.1289.8289.789.930.17
200SGD0.01cosine90.0989.6890.0189.9790.0089.950.16
200SGD0.01step86.8686.6687.2186.9886.6186.860.24
200Adam0.001cosine89.3989.8189.7389.9189.7589.720.20
200Adam0.001step89.9289.7989.7390.0189.6189.810.16
200Adam0.0001cosine86.1886.2987.0386.3686.6286.500.34
200Adam0.0001step86.3287.0486.6886.9987.1886.840.34
DoReFa200SGD0.1cosine85.6485.6785.8986.0085.7985.800.15
200SGD0.1step86.9586.9886.6986.6286.6586.780.17
200SGD0.01cosine86.5686.5986.5286.6986.8886.650.14
200SGD0.01step78.7679.9780.7379.9480.4779.970.76
200Adam0.001cosine88.8589.0688.9288.8788.7588.890.11
200Adam0.001step89.0889.1688.9389.2388.8489.050.16
200Adam0.0001cosine83.5683.1783.6583.6083.6683.530.20
200Adam0.0001step83.7083.7484.2784.1984.0183.980.26
Bi-Real200SGD0.1cosine87.5587.8188.0687.3087.8887.720.30
200SGD0.1step87.9588.3588.1387.7388.2588.080.25
200SGD0.01cosine87.7687.9387.7387.7287.6487.760.11 0.40
200 200SGD Adam0.01 0.001step cosine83.75 88.7882.91 89.1582.82 89.0682.91 89.0083.39 89.283.16 89.04
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AlgorithmEpochOptimizerLearning RateSchedulerAcc. Acc.1Acc.2 Acc.3Acc.4meanstd
200SGD0.1cosine87.8288.4288.1288.5588.1988.220.28
200SGD0.1step73.5573.1175.0674.0573.7873.910.73
200SGD0.01cosine74.0375.0673.6474.5374.7174.390.56
200SGD0.01step53.5554.1654.0152.9154.3653.800.58
200Adam0.001cosine88.7788.6589.1088.6188.8188.790.19
200Adam0.001step89.1889.0589.2788.9389.0089.090.14
200Adam0.0001cosine83.8683.4983.5683.1683.6283.540.25
200 Adam0.0001step 83.4683.7784.4084.06 83.8283.900.35
ReActNet200SGD0.1cosine88.6088.5388.3888.4888.8988.580.19
200SGD0.1step88.4288.0188.1088.0288.4388.200.21
200SGD0.01cosine87.7587.8688.0087.8088.0287.890.12
200SGD0.01step83.2982.8983.6583.7683.2783.370.35
200Adam0.001cosine89.4789.2989.0189.0589.1489.190.19
200Adam0.001step89.2789.7489.4889.4089.3989.460.18
200Adam0.0001cosine84.6584.9384.4884.6584.6784.680.16
200Adam0.0001step84.6984.5584.9384.9485.3884.900.32
ReCU200SGD0.1cosine91.7291.9491.6891.6991.8191.770.11
200SGD0.1step87.7388.1487.8188.0287.9187.920.16
200SGD0.01cosine87.3287.2887.5387.4887.3287.390.11
200SGD0.01step71.8671.7271.7872.2671.5971.840.25
200Adam0.001cosine88.2489.9888.2688.4888.1388.620.77
200Adam0.001step88.3688.4888.5588.4288.6388.490.11
200Adam0.0001cosine80.0781.1080.6281.0979.9580.570.55
200 Adam0.0001step 81.2681.4281.0881.58 81.6981.410.24
FDA200SGD0.1cosine89.6989.5989.5689.5389.6589.600.07
200SGD0.1step80.3880.3480.8380.5280.5280.520.19
200SGD0.01cosine80.7280.9380.8980.7080.7980.810.10
200SGD0.01step63.4162.8563.0463.0463.1463.100.20
200Adam0.001cosine89.7089.5789.5789.8089.7689.680.11
200Adam0.001step89.8489.8590.1089.7990.0189.920.13
200Adam0.0001cosine89.5989.1089.3489.3189.5189.370.19
200Adam0.0001step89.5289.5989.5289.6489.5889.570.05
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HardwareThreadsArch.LarqReAct FP32
FP32BNNBNN
Kirin 9701ResNet18716.427 1449.67123.263 159.615126.457427.585 72.585
ResNet34171.227 16.401836.321124.091
VGG-Small242.44314.833
ResNet18372.642 72.69778.605
2ResNet34732.35596.711108.41
VGG-Small121.9110.304 11.935=
ResNet18191.517 42.98647.182
4ResNet34367.89161.413 73.101 8.387
VGG-Small57.7218.72
ResNet1896.937 37.45756.017
8ResNet34 VGG-Small212.982 33.64753.80967.667
18.64919.818二1
ResNet18 307.624 ResNet3449.00950.018158.36331.803
Kirin 980507.734 83.16371.90974.920 8.215308.53753.031
VGG-Small7.772
ResNet18 187.49452.05754.285
2ResNet34367.853 57.336 6.11660.483 5.604
VGG-Small49.264
ResNet18104.076 29.55635.539
4ResNet34 VGG-Small202.173 31.32435.911 3.291
22.6903.147
ResNet18 ResNet3460.307 45.68356.416二 =
VGG-Small120.738 18.14760.758 21.68886.887 23.3501
ResNet18173.23834.528
438.97127.429 58.16530.626 60.885164.55657.808
Kirin 98570.7976.1476.796323.439
VGG-Small
ResNet18 103.621 327.41625.67235.477
2ResNet3453.94962.865
VGG-Small55.328 5.9556.243
ResNet1892.387 26.72834.778
4ResNet34184.050 39.88152.153
VGG-Small28.076 8.91914.795
ResNet1882.77289.766=
8ResNet34 VGG-Small227.504119.586143.9581
44.33934.03443.8161
ResNet18 ResNet34114.23821.23522.066144.20529.239
Kirin 9901227.043 38.11831.545 3.33832.821275.50249.476 1
VGG-Small3.482
ResNet18 59.32913.91114.179一
ResNet34116.82223.45222.770
VGG-Small 20.0552.0802.194
ResNet1838.40310.28012.208
4ResNet3481.27315.57017.727
VGG-Small13.5081.5421.760
ResNet1837.70325.360
31.365 39.363
8ResNet34 VGG-Small78.753 12.70734.884 14.41421.749
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HardwareThreadsArch.LarqdaBNN
FP32BNNReAct FP32BNN
Kirin 9000E1ResNet18118.05919.865 20.547129.27024.781
ResNet34236.04731.82232.575250.68042.134
VGG-Small39.1143.595 3.832一=
2ResNet1868.35116.82116.115
ResNet34133.67125.06124.660
VGG-Small23.0182.6842.598一
4ResNet1845.59217.452 18.847
ResNet3491.64823.39528.022
VGG-Small14.3602.7622.782
8ResNet1843.36361.26342.328
ResNet34 VGG-Small89.405 19.07070.232 17.35193.558
ResNet18158.83532.63623.825
Dimensity 8201ResNet34328.020 57.13334.912 60.807323.035 629.49363.471 102.443
2VGG-Small82.4175.958 6.420一
122.1671
ResNet18 ResNet34250.08829.306 43.306 50.14334.384
4VGG-Small51.3204.6705.053
ResNet1894.63621.85030.027
ResNet34177.75733.80940.816
VGG-Small45.0564.2234.546
8ResNet18 ResNet3490.210 45.35761.981
VGG-Small 32.971166.989 68.444 21.34474.286 23.706
ResNet18106.38821.02324.770148.69029.030
1ResNet34 VGG-Small210.665 42.05732.84134.590284.43849.854
2ResNet1881.6064.410 22.6615.530 27.050
ResNet3427.666
VGG-Small143.349 26.5122.27337.910 2.410
ResNet1851.42113.07915.200
8ResNet34100.24923.31425.920
VGG-Small17.7353.0153.770
ResNet18 ResNet3443.35524.93930.740
1VGG-Small84.18230.212 14.25839.9901
14.85717.540131.174
90.430 19.76920.530163.293
Snapdragon 855+ResNet18 ResNet34186.69429.12630.512298.88249.948
VGG-Small29.7353.1533.25911
ResNet1858.510 25.78026.331
2ResNet34124.58031.02332.646
VGG-Small19.4082.2582.471
ResNet1839.26919.86523.297
ResNet3482.18030.24831.387
VGG-Small13.5662.0322.359
8
ResNet1836.630 49.06085.861
ResNet3473.51341.13188.101
VGG-Small12.86017.82823.489
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HardwareThreadsArch.LarqdaBNN
FP32BNNReActFP32BNN
Snapdragon 8701ResNet1888.14516.52717.020126.76225.240
ResNet34185.46825.48826.195237.36141.440
VGG-Small30.3182.8512.964
2ResNet1863.82918.35119.575
ResNet34159.17425.35226.340
VGG-Small27.6692.0942.308
442.79617.578
ResNet18 ResNet3489.96025.81621.083 27.201
VGG-Small14.8292.6142.215
846.79819.19228.579
ResNet18 ResNet3497.83425.06032.863
VGG-Small16.7999.71717.2931
Snapdragon 8881ResNet1877.20515.54716.111123.61825.240
ResNet34152.88722.90623.893234.97241.648
VGG-Small25.1332.4102.543
2ResNet1846.297 19.30919.321
4ResNet3493.61520.47322.489
VGG-Small16.0011.9202.213
ResNet1833.52413.69914.332
ResNet3467.49519.02021.157
8VGG-Small11.7432.8822.768
ResNet1833.76126.10858.989
ResNet3467.87637.01861.315
1VGG-Small11.75227.61516.774
ResNet18 ResNet34740.509 1536.915158.732 240.606175.2561460.723241.713 435.170
VGG-Small257.07924.479254.810 25.7902774.888
Raspberrypi 3B+2ResNet18 ResNet34667.012 143.716106.894
933.149144.287158.868
VGG-Small145.42714.50315.628
ResNet18562.567108.585116.640
ResNet34976.223159.258183.698
8VGG-Small191.47010.83910.196
ResNet18877.026279.660356.239
ResNet341638.035389.924485.260=
Raspberrypi 4BVGG-Small 399.338110.448142.978二-
ResNet18448.74480.82282.380688.838120.348209.276
ResNet34897.735112.837119.5361362.893
VGG-Small150.81411.17712.024
2ResNet18 ResNet34261.861 525.73549.079 67.48055.279 79.468
VGG-Small89.2846.6477.882
ResNet18270.19136.33145.903
ResNet34572.42353.86670.841
VGG-Small90.6505.0566.167
ResNet18466.585168.844226.771
ResNet34879.375 216.439264.638 114.064319.789 162.118
8
VGG-Small
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LarqdaBNN
HardwareThreadsArch.FP32BNNReActFP32BNN
Apple M1ResNet1844.3348.2198.355二1
ResNet3488.33412.50512.771二一
VGG-Small14.0931.4461.465
ResNet1824.7755.0375.1941
2ResNet3447.1797.4257.690
VGG-Small7.3980.8290.8541
ResNet1818.6123.4483.671
ResNet3427.5154.9655.254
VGG-Small4.2940.5260.551
ResNet1816.6535.0356.003
8ResNet3427.6806.4456.953
VGG-Small3.9960.7350.712
ResNet1890.32370.69773.729
16ResNet34162.057130.907125.362二
VGG-Small25.36623.05023.194
ResNet1846.0538.6538.486二
1ResNet3491.86112.59313.039
VGG-Small14.2851.4541.488
ResNet1825.0395.4505.361
2ResNet3451.8607.5798.925
VGG-Small7.6570.8550.896
ResNet1814.7083.6253.888一
4ResNet3427.9335.2666.021
VGG-Small4.2920.5760.620
ResNet1810.6603.7184.510
ResNet3418.9884.7455.457
VGG-Small3.4320.5600.629
ResNet1860.50047.72753.900
16ResNet34120.44991.46496.356
VGG-Small21.35413.86815.311
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However, despite being a general technique, recent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 469, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 469, + 258 + ], + "score": 1.0, + "content": "works reveal that applying binarization in various practical scenarios, including", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 256, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 256, + 469, + 268 + ], + "score": 1.0, + "content": "multiple tasks, architectures, and hardware, is not trivial. Moreover, common", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "challenges, such as severe degradation in accuracy and limited efficiency gains,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 469, + 289 + ], + "score": 1.0, + "content": "suggest that specific attributes of binarization are not thoroughly studied and ad-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "equately understood. To comprehensively understand binarization methods, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "score": 1.0, + "content": "present BiBench, a carefully engineered benchmark with in-depth analysis for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 322 + ], + "score": 1.0, + "content": "network binarization. We first inspect the requirements of binarization in the ac-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 469, + 334 + ], + "score": 1.0, + "content": "tual production setting. Then for the sake of fairness and systematic, we define", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 332, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 469, + 344 + ], + "score": 1.0, + "content": "the evaluation tracks and metrics. We also perform a comprehensive evaluation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 142, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "with a rich collection of milestone binarization algorithms. Our benchmark results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 367 + ], + "score": 1.0, + "content": "show that binarization still faces severe accuracy challenges, and newer state-of-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 365, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 469, + 378 + ], + "score": 1.0, + "content": "the-art binarization algorithms bring diminishing improvements, even at the ex-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 469, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 469, + 389 + ], + "score": 1.0, + "content": "pense of efficiency. Moreover, the actual deployment of certain binarization op-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 388, + 469, + 399 + ], + "score": 1.0, + "content": "erations reveals a surprisingly large deviation from their theoretical consumption.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 398, + 470, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 470, + 411 + ], + "score": 1.0, + "content": "Finally, based on our benchmark results and analysis, we suggest establishing a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 409, + 469, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 469, + 422 + ], + "score": 1.0, + "content": "paradigm for accurate and efficient binarization among existing techniques. 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Efficiency analysis remaining at the theoretical level. Network binarization is widely", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "score": 1.0, + "content": "recognized for its significant storage and computation savings. 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In addition, the training efficiency of the binarization algorithm is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "usually neglected in current research, which causes several negative phenomena in training a binary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "network, such as the increasing demand for computation resources and time consumption, being", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 430, + 426, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 426, + 444 + ], + "score": 1.0, + "content": "sensitive to hyperparameters, and requiring detailed tuning in optimization, etc.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 504, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 504, + 460 + ], + "score": 1.0, + "content": "In this paper, we present BiBench, a network Binarization Benchmark to evaluate binarization algo-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "rithms comprehensively from accuracy and efficiency perspectives (Table 1). Based on BiBench, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "benchmark 8 representative binarization algorithms on 9 deep learning datasets, 13 different neu-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "ral architectures, 2 deployment libraries, 14 hardware chips, and various hyperparameter settings.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 490, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 505, + 505 + ], + "score": 1.0, + "content": "It costs us about 4 GPU years of computation time to build our BiBench, devoted to promoting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "comprehensive evaluation for network binarization from the perspectives of accuracy and efficiency.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "Furthermore, we analyze the benchmark results in depth and reveal insights along evaluation tracks,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 524, + 382, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 382, + 537 + ], + "score": 1.0, + "content": "and give suggestions for designing practical binarization algorithms.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 551, + 200, + 564 + ], + "lines": [ + { + "bbox": [ + 104, + 550, + 202, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 202, + 567 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 576, + 243, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 244, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 244, + 588 + ], + "score": 1.0, + "content": "2.1 NETWORK BINARIZATION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 104, + 593, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 104, + 593, + 245, + 610 + ], + "score": 1.0, + "content": "Binarization compresses weights", + "type": "text" + }, + { + "bbox": [ + 245, + 595, + 326, + 606 + ], + "score": 0.91, + "content": "{ \\pmb w } \\in \\mathbb { R } ^ { c _ { \\mathrm { i n } } \\times c _ { \\mathrm { o u t } } \\times k \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 593, + 393, + 610 + ], + "score": 1.0, + "content": "and activations", + "type": "text" + }, + { + "bbox": [ + 394, + 595, + 457, + 606 + ], + "score": 0.94, + "content": "\\pmb { a } ~ \\in ~ \\mathbb { R } ^ { c _ { \\mathrm { i n } } \\times w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 593, + 506, + 610 + ], + "score": 1.0, + "content": "to 1-bit in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 281, + 620 + ], + "score": 1.0, + "content": "computationally dense convolution, where", + "type": "text" + }, + { + "bbox": [ + 282, + 609, + 293, + 618 + ], + "score": 0.72, + "content": "c _ { \\mathrm { i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 606, + 298, + 620 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 298, + 608, + 305, + 617 + ], + "score": 0.62, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 606, + 310, + 620 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 310, + 609, + 325, + 618 + ], + "score": 0.74, + "content": "c _ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 606, + 330, + 620 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 330, + 609, + 339, + 617 + ], + "score": 0.66, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 606, + 361, + 620 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 361, + 608, + 369, + 617 + ], + "score": 0.77, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "denote the input channel, kernel", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 618, + 463, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 463, + 631 + ], + "score": 1.0, + "content": "size, output channel, input width, and input height. 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In addition, the training efficiency of the binarization algorithm is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "usually neglected in current research, which causes several negative phenomena in training a binary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "network, such as the increasing demand for computation resources and time consumption, being", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 430, + 426, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 426, + 444 + ], + "score": 1.0, + "content": "sensitive to hyperparameters, and requiring detailed tuning in optimization, etc.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 342, + 506, + 444 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 504, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 504, + 460 + ], + "score": 1.0, + "content": "In this paper, we present BiBench, a network Binarization Benchmark to evaluate binarization algo-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "rithms comprehensively from accuracy and efficiency perspectives (Table 1). 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AlgorithmTechniqueAccurate BinarizationEfficient Binarization
STg#Task#Arch.Corruption RobustnessTraining ConsumptionTheoretical ComplexityHardware Inference
BNN (Courbariaux et al., 2016b)××√33*√√
XNOR (Rastegari et al., 2016)√××23*√√
DoReFa (Zhou et al., 2016b)√××22*×>>>>×
Bi-Real (Liu et al., 2018b)××√12×××
XNOR++ (Bulat et al., 2019)√××12××××
ReActNet (Liu et al., 2020)×√×12××√×
ReCU (Xu et al., 2021b)×√√24××××
FDA (Xu et al., 2021a)××√16××××
Our Benchmark (BiBench)√√√913√√√√
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We thereby consider 8 generic binarization algorithms in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "our BiBench (Courbariaux et al., 2016b; Rastegari et al., 2016; Zhou et al., 2016b; Liu et al., 2018b;", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 325, + 504, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 504, + 338 + ], + "score": 1.0, + "content": "Bulat et al., 2019; Liu et al., 2020; Xu et al., 2021b;a). And techniques of these binarization algo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "rithms focus on binarized operator improvement and can be categorized into several types broadly,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "i.e., scaling factors, parameter redistribution, and gradient approximation. Note that the techniques", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "requiring specified local structures or training pipelines are excluded for a fair comparison, such as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "the bi-real shortcut (Liu et al., 2018a) and duplicate activation (Liu et al., 2020). For more details of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 381, + 370, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 370, + 392 + ], + "score": 1.0, + "content": "binarization algorithms for our BiBench, please see Appendix. 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AlgorithmTechniqueAccurate BinarizationEfficient Binarization
STg#Task#Arch.Corruption RobustnessTraining ConsumptionTheoretical ComplexityHardware Inference
BNN (Courbariaux et al., 2016b)××√33*√√
XNOR (Rastegari et al., 2016)√××23*√√
DoReFa (Zhou et al., 2016b)√××22*×>>>>×
Bi-Real (Liu et al., 2018b)××√12×××
XNOR++ (Bulat et al., 2019)√××12××××
ReActNet (Liu et al., 2020)×√×12××√×
ReCU (Xu et al., 2021b)×√√24××××
FDA (Xu et al., 2021a)××√16××××
Our Benchmark (BiBench)√√√913√√√√
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Recent works", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "have revealed the capability of binarization algorithms evaluated on image classification is not al-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 504, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 504, + 492 + ], + "score": 1.0, + "content": "ways appliable to new learning tasks and neural architectures (Qin et al., 2020a; Wang et al., 2020b;", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Qin et al., 2021; Liu et al., 2022). And to achieve higher accuracy, some binarization algorithms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 502, + 504, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 504, + 515 + ], + "score": 1.0, + "content": "require several times of training resources compared with the training of full-precision networks.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Ideal binarized networks should be hardware-friendly and robust when deployed to edge devices.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "Unlike well-supported multi-bit (2-8 bit) quantization, most mainstream inference libraries have not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "yet supported the deployment of binarization on hardware (NVIDIA, 2022; HUAWEI, 2022; Qual-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 558 + ], + "score": 1.0, + "content": "comm, 2022), which also causes the theoretical efficiency of existing binarization algorithms cannot", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "align their performance on actual hardware. Moreover, the data collected by low-cost devices in nat-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "ural edge scenarios is not always clean and high-quality, which affects the robustness of binarized", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "models severely (Lin et al., 2018; Ye et al., 2019; Cygert & Czyzewski ˙ , 2021). However, recent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 590, + 416, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 416, + 602 + ], + "score": 1.0, + "content": "binarization algorithms rarely consider corruption robustness when designed.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 425, + 506, + 602 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 617, + 404, + 630 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 406, + 632 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 406, + 632 + ], + "score": 1.0, + "content": "3 BIBENCH: TRACKS AND METRICS FOR BINARIZATION", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 642, + 504, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 504, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 504, + 654 + ], + "score": 1.0, + "content": "In this section, we present BiBench towards accurate and efficient network binarization. As Figure 1", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "shows, we build 6 evaluation tracks and corresponding metrics upon the practical challenges in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 664, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 504, + 677 + ], + "score": 1.0, + "content": "production and deployment of binarization. For all tracks, higher metrics mean better performance.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 642, + 505, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 289, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 291, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 291, + 702 + ], + "score": 1.0, + "content": "3.1 TOWARDS ACCURATE BINARIZATION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "The evaluation tracks for accurate network binarization in our BiBench are “Learning Task”, “Neural", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 720, + 424, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 424, + 733 + ], + "score": 1.0, + "content": "Architecture” (for production), and “Corruption Robustness” (for deployment).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 116, + 92 + ], + "score": 0.74, + "content": "\\textcircled{1}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Learning Task. We selected 9 learning tasks for 4 different data modalities to comprehensively", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "evaluate network binarization algorithms. For the widely-evaluated 2D visual modality, in addition", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 504, + 116 + ], + "score": 1.0, + "content": "to image classification tasks on CIFAR-10 (Krizhevsky et al., 2014) and ImageNet (Krizhevsky et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "2012), we also include object detection tasks on PASCAL VOC (Hoiem et al., 2009) and COCO (Lin", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "et al., 2014) across all algorithms. For 3D visual modality, we evaluate the algorithm on ModelNet40", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "classification (Wu et al., 2015) and ShapeNet segmentation (Chang et al., 2015) tasks of 3D point", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "clouds. For textual modality, the natural language understanding tasks in GLUE benchmark (Wang", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 504, + 171 + ], + "score": 1.0, + "content": "et al., 2018) are applied for evaluation. For speech modality, we evaluate algorithms on Speech", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 169, + 477, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 477, + 183 + ], + "score": 1.0, + "content": "Commands KWS task (Warden, 2018). The details of tasks and datasets are in Appendix. B.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Then we build the evaluation metric for this track. For a particular binarization algorithm, we take", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "the accuracy of full-precision models as baselines and calculate the mean relative accuracy for all", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "score": 1.0, + "content": "architectures on each task. Then we calculate the Overall Metric (OM) of the task track as the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 473, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 473, + 232 + ], + "score": 1.0, + "content": "quadratic mean of all tasks (Curtis & Marshall, 2000). The equation of evaluation metric is", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 237, + 375, + 277 + ], + "lines": [ + { + "bbox": [ + 235, + 237, + 375, + 277 + ], + "spans": [ + { + "bbox": [ + 235, + 237, + 375, + 277 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { t a s k } } = \\sqrt { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { t a s k } _ { i } } ^ { b i } } { A _ { \\mathrm { t a s k } _ { i } } } \\right) } ,", + "type": "interline_equation", + "image_path": "89e5216f01e36140a7d8c2efe2d5593b59430895e96720180b2f7cff11a02451.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 237, + 375, + 257.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 235, + 257.0, + 375, + 277.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 104, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 282, + 133, + 297 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 282, + 157, + 296 + ], + "score": 0.92, + "content": "A _ { \\mathrm { t a s k } _ { i } } ^ { b i }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 282, + 175, + 297 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 284, + 199, + 295 + ], + "score": 0.92, + "content": "A _ { \\mathrm { t a s k } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 282, + 488, + 297 + ], + "score": 1.0, + "content": "denote the accuracy results of the binarized and full-precision models on", + "type": "text" + }, + { + "bbox": [ + 488, + 285, + 493, + 293 + ], + "score": 0.49, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "-th", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 128, + 308 + ], + "score": 1.0, + "content": "task,", + "type": "text" + }, + { + "bbox": [ + 128, + 296, + 138, + 306 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 295, + 253, + 308 + ], + "score": 1.0, + "content": "is the number of tasks, and", + "type": "text" + }, + { + "bbox": [ + 253, + 295, + 272, + 308 + ], + "score": 0.8, + "content": "\\mathbb { E } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "denote taking the mean value. The quadratic mean form", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "is uniformly applied in BiBench to unify all metrics, which prevents metrics from being unduly", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 477, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 477, + 330 + ], + "score": 1.0, + "content": "influenced by certain bad items and thus can measure the overall performance on each track.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 116, + 344 + ], + "score": 0.75, + "content": "\\textcircled{2}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 334, + 504, + 345 + ], + "score": 1.0, + "content": "Neural Architecture. We choose diverse neural architectures with mainstream CNN-based,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 345, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 504, + 357 + ], + "score": 1.0, + "content": "transformer-based, and MLP-based architectures to evaluate the generalizability of binarization al-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "score": 1.0, + "content": "gorithms from the neural architecture perspective. We adopt standard ResNet-18/20/34 (He et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "2016) and VGG (Simonyan & Zisserman, 2015) to evaluate CNN architectures, and the Faster-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "RCNN (Ren et al., 2015) and SSD300 (Liu et al., 2016) frameworks are applied as detectors. We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "binarize BERT-Tiny4/Tiny6/Base (Kenton & Toutanova, 2019) with the bi-attention mechanism", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "for convergence (Qin et al., 2021) to evaluate transformer-based architectures. And we evaluate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 409, + 507, + 425 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 507, + 425 + ], + "score": 1.0, + "content": "PointNetvanilla and PointNet (Qi et al., 2017) with EMA aggregator (Qin et al., 2020a), FSMN (Zhang", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 422, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 504, + 434 + ], + "score": 1.0, + "content": "et al., 2015), and Deep-FSMN (Zhang et al., 2018a) as typical MLP-based architectures for their lin-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 433, + 448, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 448, + 446 + ], + "score": 1.0, + "content": "ear unit composition. The details of these architectures are presented in Appendix. C.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 487, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 486, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 486, + 462 + ], + "score": 1.0, + "content": "Similar to the overall metric for learning task track, we build the metric for neural architecture:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 467, + 449, + 501 + ], + "lines": [ + { + "bbox": [ + 162, + 467, + 449, + 501 + ], + "spans": [ + { + "bbox": [ + 162, + 467, + 449, + 501 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { a r c h } } = \\sqrt { \\frac { 1 } { 3 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { C N N } } ^ { b i } } { A _ { \\mathrm { C N N } } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { T r a n s f o r m e r } } ^ { b i } } { A _ { \\mathrm { T r a n s f o r m e r } } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { M L P } } ^ { b i } } { A _ { \\mathrm { M L P } } } \\right) \\right) } .", + "type": "interline_equation", + "image_path": "c97b82b78d7f43bdf4ffe09febdf4ea251812857f1da88031980f22553303410.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 162, + 467, + 449, + 478.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 162, + 478.3333333333333, + 449, + 489.66666666666663 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 162, + 489.66666666666663, + 449, + 500.99999999999994 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 116, + 522 + ], + "score": 0.78, + "content": "\\textcircled{3}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Corruption Robustness. The corruption robustness of binarization on deployment is critical to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "deal with bad cases like perceptual device damage, a common problem with low-cost equipment in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "real-world implementations. We consider the robustness of binarized models to corruption of 2D", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "score": 1.0, + "content": "visual data and evaluate algorithms on the CIFAR10-C (Hendrycks & Dietterich, 2018) benchmark.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 503, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Therefore, we evaluate binarization algorithms’ performance on the corrupted data compared with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 572, + 417, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 417, + 585 + ], + "score": 1.0, + "content": "the normal data using the corruption generalization gap (Zhang et al., 2022a):", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 590, + 356, + 605 + ], + "lines": [ + { + "bbox": [ + 254, + 590, + 356, + 605 + ], + "spans": [ + { + "bbox": [ + 254, + 590, + 356, + 605 + ], + "score": 0.92, + "content": "\\begin{array} { r } { G _ { \\mathrm { t a s k } _ { i } } = A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { n o r m } } - A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { c o r r } } , } \\end{array}", + "type": "interline_equation", + "image_path": "4ddaabe321d3718c23d6fb945e70615308b2a72b3e41dbbaff1e41fb59ad6862.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 254, + 590, + 356, + 605 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 102, + 604, + 509, + 628 + ], + "spans": [ + { + "bbox": [ + 102, + 604, + 134, + 628 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 609, + 158, + 622 + ], + "score": 0.91, + "content": "A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { c o r r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 604, + 178, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 609, + 203, + 622 + ], + "score": 0.91, + "content": "A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { n o r m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 604, + 424, + 628 + ], + "score": 1.0, + "content": "denote the accuracy results under all architectures on", + "type": "text" + }, + { + "bbox": [ + 424, + 610, + 429, + 619 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 604, + 509, + 628 + ], + "score": 1.0, + "content": "-th corruption task", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 620, + 495, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 495, + 633 + ], + "score": 1.0, + "content": "and corresponding normal task, respectively. And the overall metric on this track is calculated by", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 637, + 379, + 677 + ], + "lines": [ + { + "bbox": [ + 232, + 637, + 379, + 677 + ], + "spans": [ + { + "bbox": [ + 232, + 637, + 379, + 677 + ], + "score": 0.94, + "content": "\\mathrm { O M } _ { \\mathrm { r o b u s t } } = \\sqrt { \\frac { 1 } { C } \\sum _ { i = 1 } ^ { C } \\mathbb { E } ^ { 2 } \\left( \\frac { G _ { \\mathrm { t a s k } _ { i } } } { G _ { \\mathrm { t a s k } _ { i } } ^ { b i } } \\right) } .", + "type": "interline_equation", + "image_path": "8547a0e7eb2fa8cb76c456b327a283eece2ff3f000cd22422758e62be3832e2a.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 637, + 379, + 657.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 232, + 657.0, + 379, + 677.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 288, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 689, + 289, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 289, + 702 + ], + "score": 1.0, + "content": "3.2 TOWARDS EFFICIENT BINARIZATION", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "As for the efficiency of network binarization, we evaluate “Training Consumption” for production,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 386, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 386, + 733 + ], + "score": 1.0, + "content": "“Theoretical Complexity” and “Hardware Inference” for deployment.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "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 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 116, + 92 + ], + "score": 0.74, + "content": "\\textcircled{1}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Learning Task. We selected 9 learning tasks for 4 different data modalities to comprehensively", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "evaluate network binarization algorithms. For the widely-evaluated 2D visual modality, in addition", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 504, + 116 + ], + "score": 1.0, + "content": "to image classification tasks on CIFAR-10 (Krizhevsky et al., 2014) and ImageNet (Krizhevsky et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "2012), we also include object detection tasks on PASCAL VOC (Hoiem et al., 2009) and COCO (Lin", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "et al., 2014) across all algorithms. For 3D visual modality, we evaluate the algorithm on ModelNet40", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "classification (Wu et al., 2015) and ShapeNet segmentation (Chang et al., 2015) tasks of 3D point", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "clouds. For textual modality, the natural language understanding tasks in GLUE benchmark (Wang", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 504, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 504, + 171 + ], + "score": 1.0, + "content": "et al., 2018) are applied for evaluation. For speech modality, we evaluate algorithms on Speech", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 169, + 477, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 477, + 183 + ], + "score": 1.0, + "content": "Commands KWS task (Warden, 2018). The details of tasks and datasets are in Appendix. B.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 81, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Then we build the evaluation metric for this track. For a particular binarization algorithm, we take", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "the accuracy of full-precision models as baselines and calculate the mean relative accuracy for all", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 504, + 221 + ], + "score": 1.0, + "content": "architectures on each task. Then we calculate the Overall Metric (OM) of the task track as the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 473, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 473, + 232 + ], + "score": 1.0, + "content": "quadratic mean of all tasks (Curtis & Marshall, 2000). The equation of evaluation metric is", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 187, + 505, + 232 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 237, + 375, + 277 + ], + "lines": [ + { + "bbox": [ + 235, + 237, + 375, + 277 + ], + "spans": [ + { + "bbox": [ + 235, + 237, + 375, + 277 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { t a s k } } = \\sqrt { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { t a s k } _ { i } } ^ { b i } } { A _ { \\mathrm { t a s k } _ { i } } } \\right) } ,", + "type": "interline_equation", + "image_path": "89e5216f01e36140a7d8c2efe2d5593b59430895e96720180b2f7cff11a02451.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 237, + 375, + 257.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 235, + 257.0, + 375, + 277.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 104, + 282, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 104, + 282, + 133, + 297 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 282, + 157, + 296 + ], + "score": 0.92, + "content": "A _ { \\mathrm { t a s k } _ { i } } ^ { b i }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 282, + 175, + 297 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 284, + 199, + 295 + ], + "score": 0.92, + "content": "A _ { \\mathrm { t a s k } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 282, + 488, + 297 + ], + "score": 1.0, + "content": "denote the accuracy results of the binarized and full-precision models on", + "type": "text" + }, + { + "bbox": [ + 488, + 285, + 493, + 293 + ], + "score": 0.49, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 282, + 505, + 297 + ], + "score": 1.0, + "content": "-th", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 128, + 308 + ], + "score": 1.0, + "content": "task,", + "type": "text" + }, + { + "bbox": [ + 128, + 296, + 138, + 306 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 295, + 253, + 308 + ], + "score": 1.0, + "content": "is the number of tasks, and", + "type": "text" + }, + { + "bbox": [ + 253, + 295, + 272, + 308 + ], + "score": 0.8, + "content": "\\mathbb { E } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "denote taking the mean value. The quadratic mean form", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "is uniformly applied in BiBench to unify all metrics, which prevents metrics from being unduly", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 477, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 477, + 330 + ], + "score": 1.0, + "content": "influenced by certain bad items and thus can measure the overall performance on each track.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 282, + 505, + 330 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 116, + 344 + ], + "score": 0.75, + "content": "\\textcircled{2}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 334, + 504, + 345 + ], + "score": 1.0, + "content": "Neural Architecture. We choose diverse neural architectures with mainstream CNN-based,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 345, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 504, + 357 + ], + "score": 1.0, + "content": "transformer-based, and MLP-based architectures to evaluate the generalizability of binarization al-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "score": 1.0, + "content": "gorithms from the neural architecture perspective. We adopt standard ResNet-18/20/34 (He et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "2016) and VGG (Simonyan & Zisserman, 2015) to evaluate CNN architectures, and the Faster-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "RCNN (Ren et al., 2015) and SSD300 (Liu et al., 2016) frameworks are applied as detectors. We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "binarize BERT-Tiny4/Tiny6/Base (Kenton & Toutanova, 2019) with the bi-attention mechanism", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 411 + ], + "score": 1.0, + "content": "for convergence (Qin et al., 2021) to evaluate transformer-based architectures. And we evaluate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 409, + 507, + 425 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 507, + 425 + ], + "score": 1.0, + "content": "PointNetvanilla and PointNet (Qi et al., 2017) with EMA aggregator (Qin et al., 2020a), FSMN (Zhang", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 422, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 504, + 434 + ], + "score": 1.0, + "content": "et al., 2015), and Deep-FSMN (Zhang et al., 2018a) as typical MLP-based architectures for their lin-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 433, + 448, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 448, + 446 + ], + "score": 1.0, + "content": "ear unit composition. The details of these architectures are presented in Appendix. C.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 334, + 507, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 487, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 486, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 486, + 462 + ], + "score": 1.0, + "content": "Similar to the overall metric for learning task track, we build the metric for neural architecture:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 449, + 486, + 462 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 467, + 449, + 501 + ], + "lines": [ + { + "bbox": [ + 162, + 467, + 449, + 501 + ], + "spans": [ + { + "bbox": [ + 162, + 467, + 449, + 501 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { a r c h } } = \\sqrt { \\frac { 1 } { 3 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { C N N } } ^ { b i } } { A _ { \\mathrm { C N N } } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { T r a n s f o r m e r } } ^ { b i } } { A _ { \\mathrm { T r a n s f o r m e r } } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { A _ { \\mathrm { M L P } } ^ { b i } } { A _ { \\mathrm { M L P } } } \\right) \\right) } .", + "type": "interline_equation", + "image_path": "c97b82b78d7f43bdf4ffe09febdf4ea251812857f1da88031980f22553303410.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 162, + 467, + 449, + 478.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 162, + 478.3333333333333, + 449, + 489.66666666666663 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 162, + 489.66666666666663, + 449, + 500.99999999999994 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 116, + 522 + ], + "score": 0.78, + "content": "\\textcircled{3}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Corruption Robustness. The corruption robustness of binarization on deployment is critical to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "deal with bad cases like perceptual device damage, a common problem with low-cost equipment in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "real-world implementations. We consider the robustness of binarized models to corruption of 2D", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "score": 1.0, + "content": "visual data and evaluate algorithms on the CIFAR10-C (Hendrycks & Dietterich, 2018) benchmark.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 511, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 503, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Therefore, we evaluate binarization algorithms’ performance on the corrupted data compared with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 572, + 417, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 417, + 585 + ], + "score": 1.0, + "content": "the normal data using the corruption generalization gap (Zhang et al., 2022a):", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 106, + 561, + 505, + 585 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 590, + 356, + 605 + ], + "lines": [ + { + "bbox": [ + 254, + 590, + 356, + 605 + ], + "spans": [ + { + "bbox": [ + 254, + 590, + 356, + 605 + ], + "score": 0.92, + "content": "\\begin{array} { r } { G _ { \\mathrm { t a s k } _ { i } } = A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { n o r m } } - A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { c o r r } } , } \\end{array}", + "type": "interline_equation", + "image_path": "4ddaabe321d3718c23d6fb945e70615308b2a72b3e41dbbaff1e41fb59ad6862.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 254, + 590, + 356, + 605 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 102, + 604, + 509, + 628 + ], + "spans": [ + { + "bbox": [ + 102, + 604, + 134, + 628 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 609, + 158, + 622 + ], + "score": 0.91, + "content": "A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { c o r r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 604, + 178, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 609, + 203, + 622 + ], + "score": 0.91, + "content": "A _ { \\mathrm { t a s k } _ { i } } ^ { \\mathrm { n o r m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 604, + 424, + 628 + ], + "score": 1.0, + "content": "denote the accuracy results under all architectures on", + "type": "text" + }, + { + "bbox": [ + 424, + 610, + 429, + 619 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 604, + 509, + 628 + ], + "score": 1.0, + "content": "-th corruption task", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 620, + 495, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 495, + 633 + ], + "score": 1.0, + "content": "and corresponding normal task, respectively. And the overall metric on this track is calculated by", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 102, + 604, + 509, + 633 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 637, + 379, + 677 + ], + "lines": [ + { + "bbox": [ + 232, + 637, + 379, + 677 + ], + "spans": [ + { + "bbox": [ + 232, + 637, + 379, + 677 + ], + "score": 0.94, + "content": "\\mathrm { O M } _ { \\mathrm { r o b u s t } } = \\sqrt { \\frac { 1 } { C } \\sum _ { i = 1 } ^ { C } \\mathbb { E } ^ { 2 } \\left( \\frac { G _ { \\mathrm { t a s k } _ { i } } } { G _ { \\mathrm { t a s k } _ { i } } ^ { b i } } \\right) } .", + "type": "interline_equation", + "image_path": "8547a0e7eb2fa8cb76c456b327a283eece2ff3f000cd22422758e62be3832e2a.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 637, + 379, + 657.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 232, + 657.0, + 379, + 677.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 288, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 689, + 289, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 289, + 702 + ], + "score": 1.0, + "content": "3.2 TOWARDS EFFICIENT BINARIZATION", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "As for the efficiency of network binarization, we evaluate “Training Consumption” for production,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 386, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 386, + 733 + ], + "score": 1.0, + "content": "“Theoretical Complexity” and “Hardware Inference” for deployment.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 116, + 93 + ], + "score": 0.74, + "content": "\\textcircled{4}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Training Consumption. We consider the occupied training resource and the hyperparameter", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "sensitivity of binarization algorithms, which affect the consumption of one training and the whole", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "tuning process, respectively. For each algorithm, we train its binarized networks with various hyper-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "parameter settings, including different learning rates, learning rate schedulers, optimizers, and even", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "random seeds, to evaluate whether the binarization algorithm is easy to tune to an optimal state. We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "align the epochs for binarized and full-precision networks and compare their consumption and time.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 105, + 154, + 503, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "score": 1.0, + "content": "The evaluation metric for the training consumption track is related to the training time and hyperpa-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 371, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 371, + 178 + ], + "score": 1.0, + "content": "rameter sensitivity. For a specific binarization algorithm, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 181, + 416, + 221 + ], + "lines": [ + { + "bbox": [ + 194, + 181, + 416, + 221 + ], + "spans": [ + { + "bbox": [ + 194, + 181, + 416, + 221 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { t r a i n } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { T _ { \\mathrm { t r a i n } } } { T _ { \\mathrm { t r a i n } } ^ { b i } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { \\mathrm { s t d } ( A _ { \\mathrm { h y p e r } } ) } { \\mathrm { s t d } ( A _ { \\mathrm { h y p e r } } ^ { b i } ) } \\right) \\right) } ,", + "type": "interline_equation", + "image_path": "a59e99a0fc5c5320a409a3bed69687db0c631fa5722fc945630d567b15d02224.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 194, + 181, + 416, + 194.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 194, + 194.33333333333334, + 416, + 207.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 194, + 207.66666666666669, + 416, + 221.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 502, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 133, + 236 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 224, + 154, + 235 + ], + "score": 0.91, + "content": "\\pmb { T } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 222, + 315, + 236 + ], + "score": 1.0, + "content": "denotes the set of time in once training,", + "type": "text" + }, + { + "bbox": [ + 316, + 223, + 342, + 236 + ], + "score": 0.91, + "content": "A _ { \\mathrm { h y p e r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 222, + 504, + 236 + ], + "score": 1.0, + "content": "is the set of results with different hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 370, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 201, + 247 + ], + "score": 1.0, + "content": "parameter settings, and", + "type": "text" + }, + { + "bbox": [ + 201, + 235, + 226, + 246 + ], + "score": 0.51, + "content": "\\operatorname { s t d } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 234, + 370, + 247 + ], + "score": 1.0, + "content": "is taking standard deviation values.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 504, + 274 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 116, + 261 + ], + "score": 0.79, + "content": "\\textcircled{5}", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 250, + 505, + 264 + ], + "score": 1.0, + "content": "Theoretical Complexity. When evaluating theoretical complexity, we calculate the compression", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 437, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 437, + 274 + ], + "score": 1.0, + "content": "and speedup ratio before and after binarization on architectures such as ResNet18.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "The evaluation metric relates to model size saving (MB) and computational floating-point operations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "(FLOPs) at inference. For binarized parameters, the storage occupation of binarized parameters is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "1/32 as their 32-bit floating-point counterparts (Rastegari et al., 2016). 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The compression ratio", + "type": "text" + }, + { + "bbox": [ + 263, + 336, + 273, + 345 + ], + "score": 0.87, + "content": "\\mathbf { \\nabla } _ { \\mathbf { r } _ { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 333, + 347, + 347 + ], + "score": 1.0, + "content": "and speedup ratio", + "type": "text" + }, + { + "bbox": [ + 347, + 336, + 358, + 345 + ], + "score": 0.87, + "content": "\\mathbf { \\Delta } _ { r _ { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 333, + 374, + 347 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 349, + 481, + 384 + ], + "lines": [ + { + "bbox": [ + 118, + 349, + 481, + 384 + ], + "spans": [ + { + "bbox": [ + 118, + 349, + 481, + 384 + ], + "score": 0.93, + "content": "r _ { c } = \\frac { | M | _ { \\ell 0 } } { \\frac { 1 } { 3 2 } \\left( | M | _ { \\ell 0 } - | \\hat { M } | _ { \\ell 0 } \\right) + | \\hat { M } | _ { \\ell 0 } } , \\quad r _ { s } = \\frac { \\mathrm { F L O P s } _ { M } } { \\frac { 1 } { 6 4 } \\left( \\mathrm { F L O P s } _ { M } - \\mathrm { F L O P s } _ { \\hat { M } } \\right) + \\mathrm { F L O P s } _ { \\hat { M } } } ,", + "type": "interline_equation", + "image_path": "786f54912324a3ca4ffee190faaad61ca39aa062b5d9ae7974b3c2c93f31060c.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 118, + 349, + 481, + 360.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 118, + 360.6666666666667, + 481, + 372.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 118, + 372.33333333333337, + 481, + 384.00000000000006 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 134, + 402 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 389, + 148, + 400 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 388, + 167, + 402 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 388, + 181, + 400 + ], + "score": 0.87, + "content": "\\hat { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "are the amount of full-precision parameters in the original and binarized mod-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 192, + 413 + ], + "score": 1.0, + "content": "els, respectively, and", + "type": "text" + }, + { + "bbox": [ + 192, + 401, + 234, + 412 + ], + "score": 0.89, + "content": "\\mathrm { F L O P s } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 401, + 253, + 413 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 401, + 295, + 413 + ], + "score": 0.9, + "content": "\\mathrm { F L O P s } _ { \\hat { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "denote the computation related to these parameters,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 367, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 367, + 424 + ], + "score": 1.0, + "content": "respectively. And the overall metric for theoretical complexity is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 427, + 379, + 455 + ], + "lines": [ + { + "bbox": [ + 232, + 427, + 379, + 455 + ], + "spans": [ + { + "bbox": [ + 232, + 427, + 379, + 455 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { c o m p } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } ( \\pmb { r } _ { c } ) + \\mathbb { E } ^ { 2 } ( \\pmb { r } _ { s } ) \\right) } .", + "type": "interline_equation", + "image_path": "a46db1217a1899ecf08ad4d0f099d7f3085ee76e09d55211d9f21042bce8c953.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 427, + 379, + 441.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 232, + 441.0, + 379, + 455.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 116, + 474 + ], + "score": 0.8, + "content": "\\textcircled{6}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "Hardware Inference. Since the limited support of binarization in hardware deployment, just two", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "score": 1.0, + "content": "inference libraries, Larq’s Compute Engine (Geiger & Team, 2020) and JD’s daBNN (Zhang et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "score": 1.0, + "content": "2019) can deploy and evaluate the binarized models on ARM hardware in practice. Regarding target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "hardware devices, we mainly focus on ARM CPU inference as this is the mainstream hardware", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "type for edge scenarios, including HUAWEI Kirin, Qualcomm Snapdragon, Apple M1, MediaTek", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 519, + 406, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 406, + 531 + ], + "score": 1.0, + "content": "Dimensity, and Raspberry Pi. We put the hardware details in Appendix. D.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 108, + 535, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "And for a certain binarization algorithm, we take the saving of storage and inference time under", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 546, + 363, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 363, + 559 + ], + "score": 1.0, + "content": "different inference libraries and hardware as evaluation metrics:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 560, + 402, + 594 + ], + "lines": [ + { + "bbox": [ + 207, + 560, + 402, + 594 + ], + "spans": [ + { + "bbox": [ + 207, + 560, + 402, + 594 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { i n f e r } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { T _ { \\mathrm { i n f e r } } } { T _ { \\mathrm { i n f e r } } ^ { b i } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { S _ { \\mathrm { i n f e r } } } { S _ { \\mathrm { i n f e r } } ^ { b i } } \\right) \\right) } ,", + "type": "interline_equation", + "image_path": "12e4456d6a4543908bcee133ea7d612dd9c7617a209192aad00ceaba442fa6ed.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 560, + 402, + 577.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 207, + 577.0, + 402, + 594.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 596, + 480, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 483, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 133, + 611 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 597, + 155, + 608 + ], + "score": 0.9, + "content": "\\mathbf { \\mathit { T } } _ { \\mathrm { i n f e r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 595, + 173, + 611 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 597, + 195, + 608 + ], + "score": 0.91, + "content": "S _ { \\mathrm { i n f e r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 595, + 483, + 611 + ], + "score": 1.0, + "content": "denote the inference time and storage on different devices, respectively.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 624, + 272, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 273, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 273, + 638 + ], + "score": 1.0, + "content": "4 BIBENCH IMPLEMENTATION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 105, + 648, + 503, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 504, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 504, + 662 + ], + "score": 1.0, + "content": "This section shows the implementation details and training and inference pipelines of our BiBench.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Implementation details. We implement BiBench with PyTorch (Paszke et al., 2019) packages. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "definitions of the binarized operators are entirely independent of corresponding single files. And", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "score": 1.0, + "content": "the corresponding operator in the original model can be flexibly replaced by the binarized one while", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "evaluating different tasks and architectures. When deployed, we export well-trained binarized mod-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "els of a binarization algorithm to the Open Neural Network Exchange (ONNX) format (developers,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 482, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 482, + 732 + ], + "score": 1.0, + "content": "2021) and then feed them to the inference libraries in BiBench (if it applies to this algorithm).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "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, + 81, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 116, + 93 + ], + "score": 0.74, + "content": "\\textcircled{4}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Training Consumption. We consider the occupied training resource and the hyperparameter", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "sensitivity of binarization algorithms, which affect the consumption of one training and the whole", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "tuning process, respectively. For each algorithm, we train its binarized networks with various hyper-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "parameter settings, including different learning rates, learning rate schedulers, optimizers, and even", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "random seeds, to evaluate whether the binarization algorithm is easy to tune to an optimal state. We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "align the epochs for binarized and full-precision networks and compare their consumption and time.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 81, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 154, + 503, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "score": 1.0, + "content": "The evaluation metric for the training consumption track is related to the training time and hyperpa-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 371, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 371, + 178 + ], + "score": 1.0, + "content": "rameter sensitivity. For a specific binarization algorithm, we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 152, + 505, + 178 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 181, + 416, + 221 + ], + "lines": [ + { + "bbox": [ + 194, + 181, + 416, + 221 + ], + "spans": [ + { + "bbox": [ + 194, + 181, + 416, + 221 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { t r a i n } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { T _ { \\mathrm { t r a i n } } } { T _ { \\mathrm { t r a i n } } ^ { b i } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { \\mathrm { s t d } ( A _ { \\mathrm { h y p e r } } ) } { \\mathrm { s t d } ( A _ { \\mathrm { h y p e r } } ^ { b i } ) } \\right) \\right) } ,", + "type": "interline_equation", + "image_path": "a59e99a0fc5c5320a409a3bed69687db0c631fa5722fc945630d567b15d02224.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 194, + 181, + 416, + 194.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 194, + 194.33333333333334, + 416, + 207.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 194, + 207.66666666666669, + 416, + 221.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 502, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 133, + 236 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 224, + 154, + 235 + ], + "score": 0.91, + "content": "\\pmb { T } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 222, + 315, + 236 + ], + "score": 1.0, + "content": "denotes the set of time in once training,", + "type": "text" + }, + { + "bbox": [ + 316, + 223, + 342, + 236 + ], + "score": 0.91, + "content": "A _ { \\mathrm { h y p e r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 222, + 504, + 236 + ], + "score": 1.0, + "content": "is the set of results with different hyper-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 370, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 201, + 247 + ], + "score": 1.0, + "content": "parameter settings, and", + "type": "text" + }, + { + "bbox": [ + 201, + 235, + 226, + 246 + ], + "score": 0.51, + "content": "\\operatorname { s t d } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 234, + 370, + 247 + ], + "score": 1.0, + "content": "is taking standard deviation values.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 222, + 504, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 504, + 274 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 116, + 261 + ], + "score": 0.79, + "content": "\\textcircled{5}", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 250, + 505, + 264 + ], + "score": 1.0, + "content": "Theoretical Complexity. When evaluating theoretical complexity, we calculate the compression", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 437, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 437, + 274 + ], + "score": 1.0, + "content": "and speedup ratio before and after binarization on architectures such as ResNet18.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 250, + 505, + 274 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "The evaluation metric relates to model size saving (MB) and computational floating-point operations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "(FLOPs) at inference. 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For binarized operations, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "multiplication between a 1-bit number (weight) and a 1-bit number (activation) approximately takes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 136, + 334 + ], + "score": 0.81, + "content": "1 ^ { * } 1 / 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "FLOPs for a CPU with the instruction size of 64 bits (Zhou et al., 2016b; Liu et al., 2018b;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 333, + 374, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 262, + 347 + ], + "score": 1.0, + "content": "Li et al., 2019). 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And the overall metric for theoretical complexity is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 388, + 506, + 424 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 427, + 379, + 455 + ], + "lines": [ + { + "bbox": [ + 232, + 427, + 379, + 455 + ], + "spans": [ + { + "bbox": [ + 232, + 427, + 379, + 455 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { c o m p } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } ( \\pmb { r } _ { c } ) + \\mathbb { E } ^ { 2 } ( \\pmb { r } _ { s } ) \\right) } .", + "type": "interline_equation", + "image_path": "a46db1217a1899ecf08ad4d0f099d7f3085ee76e09d55211d9f21042bce8c953.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 427, + 379, + 441.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 232, + 441.0, + 379, + 455.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 116, + 474 + ], + "score": 0.8, + "content": "\\textcircled{6}", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "Hardware Inference. Since the limited support of binarization in hardware deployment, just two", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 487 + ], + "score": 1.0, + "content": "inference libraries, Larq’s Compute Engine (Geiger & Team, 2020) and JD’s daBNN (Zhang et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "score": 1.0, + "content": "2019) can deploy and evaluate the binarized models on ARM hardware in practice. Regarding target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "hardware devices, we mainly focus on ARM CPU inference as this is the mainstream hardware", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "type for edge scenarios, including HUAWEI Kirin, Qualcomm Snapdragon, Apple M1, MediaTek", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 519, + 406, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 406, + 531 + ], + "score": 1.0, + "content": "Dimensity, and Raspberry Pi. We put the hardware details in Appendix. D.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 462, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 535, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "And for a certain binarization algorithm, we take the saving of storage and inference time under", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 546, + 363, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 363, + 559 + ], + "score": 1.0, + "content": "different inference libraries and hardware as evaluation metrics:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 536, + 505, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 560, + 402, + 594 + ], + "lines": [ + { + "bbox": [ + 207, + 560, + 402, + 594 + ], + "spans": [ + { + "bbox": [ + 207, + 560, + 402, + 594 + ], + "score": 0.95, + "content": "\\mathrm { O M } _ { \\mathrm { i n f e r } } = \\sqrt { \\frac { 1 } { 2 } \\left( \\mathbb { E } ^ { 2 } \\left( \\frac { T _ { \\mathrm { i n f e r } } } { T _ { \\mathrm { i n f e r } } ^ { b i } } \\right) + \\mathbb { E } ^ { 2 } \\left( \\frac { S _ { \\mathrm { i n f e r } } } { S _ { \\mathrm { i n f e r } } ^ { b i } } \\right) \\right) } ,", + "type": "interline_equation", + "image_path": "12e4456d6a4543908bcee133ea7d612dd9c7617a209192aad00ceaba442fa6ed.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 560, + 402, + 577.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 207, + 577.0, + 402, + 594.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 596, + 480, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 483, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 133, + 611 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 597, + 155, + 608 + ], + "score": 0.9, + "content": "\\mathbf { \\mathit { T } } _ { \\mathrm { i n f e r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 595, + 173, + 611 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 597, + 195, + 608 + ], + "score": 0.91, + "content": "S _ { \\mathrm { i n f e r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 595, + 483, + 611 + ], + "score": 1.0, + "content": "denote the inference time and storage on different devices, respectively.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 595, + 483, + 611 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 624, + 272, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 273, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 273, + 638 + ], + "score": 1.0, + "content": "4 BIBENCH IMPLEMENTATION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 105, + 648, + 503, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 504, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 504, + 662 + ], + "score": 1.0, + "content": "This section shows the implementation details and training and inference pipelines of our BiBench.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 648, + 504, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Implementation details. We implement BiBench with PyTorch (Paszke et al., 2019) packages. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "definitions of the binarized operators are entirely independent of corresponding single files. And", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "score": 1.0, + "content": "the corresponding operator in the original model can be flexibly replaced by the binarized one while", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "evaluating different tasks and architectures. When deployed, we export well-trained binarized mod-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "els of a binarization algorithm to the Open Neural Network Exchange (ONNX) format (developers,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 482, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 482, + 732 + ], + "score": 1.0, + "content": "2021) and then feed them to the inference libraries in BiBench (if it applies to this algorithm).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 665, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 78, + 504, + 248 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 110, + 66, + 500, + 77 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 65, + 500, + 79 + ], + "spans": [ + { + "bbox": [ + 110, + 65, + 500, + 79 + ], + "score": 1.0, + "content": "Table 2: Accuracy benchmark for network binarization. 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TrackMetricBinarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCUFDA
Learning Task (%)CIFAR1094.5494.7395.0395.6194.5295.9296.7294.66
ImageNet75.8177.2476.6178.3875.0178.6477.9878.15
VOC0776.9774.6176.3580.0774.4181.3881.6579.02
COC01777.9475.3778.3181.6279.4183.8285.6682.35
ModelNet4054.1993.8693.7493.2385.2092.4195.0794.38
ShapeNet48.9673.6270.7968.1341.1668.5140.6571.16
GLUE49.7559.6366.6069.4249.3367.6450.6670.61
SpeechCom.75.0376.9376.6482.4268.6581.8676.9877.90
OMtask70.8278.9779.8281.6372.8981.8177.9681.49
Neural Architecture (%)CNNs72.9083.7483.8685.0278.9586.2083.5086.34
Transformers49.7559.6366.6069.4249.3367.6450.6670.61
MLPs64.6185.4085.1987.8376.9287.1386.0286.14
OMarch63.1577.1679.0181.1669.7280.8275.1481.36
RobustnessCIFAR10-C95.26100.9781.4396.5692.6994.01103.2998.35
Corruption (%)OMcorr95.26100.9781.4396.5692.6994.01103.2998.35
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Hype-parameters: We train the binarized networks with the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "same number of epochs as their full-precision counterparts. Inspired by results in Section 5.2.1,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "score": 1.0, + "content": "we use the Adam optimizer for all binarized models for better convergence, the initial learning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 144, + 312 + ], + "score": 1.0, + "content": "rates are", + "type": "text" + }, + { + "bbox": [ + 144, + 300, + 174, + 311 + ], + "score": 0.86, + "content": "1 e - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 300, + 233, + 312 + ], + "score": 1.0, + "content": "as default (or", + "type": "text" + }, + { + "bbox": [ + 234, + 300, + 256, + 311 + ], + "score": 0.87, + "content": "0 . 1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "of the default learning rate), and the learning rate scheduler", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "is CosineAnnealingLR (Loshchilov & Hutter, 2017). Architecture: In BiBench, we thoroughly", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "follow the original architectures of full-precision models and binarize their convolution, linear, and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "multiplication units with the binarization algorithms. Hardtanh is uniformly used as the activation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "function to avoid the all-one feature. 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TrackMetricBinarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCUFDA
Learning Task (%)CIFAR1094.5494.7395.0395.6194.5295.9296.7294.66
ImageNet75.8177.2476.6178.3875.0178.6477.9878.15
VOC0776.9774.6176.3580.0774.4181.3881.6579.02
COC01777.9475.3778.3181.6279.4183.8285.6682.35
ModelNet4054.1993.8693.7493.2385.2092.4195.0794.38
ShapeNet48.9673.6270.7968.1341.1668.5140.6571.16
GLUE49.7559.6366.6069.4249.3367.6450.6670.61
SpeechCom.75.0376.9376.6482.4268.6581.8676.9877.90
OMtask70.8278.9779.8281.6372.8981.8177.9681.49
Neural Architecture (%)CNNs72.9083.7483.8685.0278.9586.2083.5086.34
Transformers49.7559.6366.6069.4249.3367.6450.6670.61
MLPs64.6185.4085.1987.8376.9287.1386.0286.14
OMarch63.1577.1679.0181.1669.7280.8275.1481.36
RobustnessCIFAR10-C95.26100.9781.4396.5692.6994.01103.2998.35
Corruption (%)OMcorr95.26100.9781.4396.5692.6994.01103.2998.35
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When comparing various tasks, an interesting phenomenon is that the binarized networks", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "suffer a huge accuracy drop in language understanding GLUE benchmark, but it can almost approach", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "full-precision performance on the ModelNet40 point cloud classification task. Similar phenomena", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 671, + 499, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 499, + 683 + ], + "score": 1.0, + "content": "suggest that the direct transfer of binarization studies’ insights across different tasks is non-trivial.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 626, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "As for the overall performance, both ReCU and ReActeNet show high accuracy across different", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "learning tasks. Surprisingly, although ReCU wins the championship on most four individual tasks,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "ReActNet stands out in the overall metric comparison finally. They both apply reparameterization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 721, + 447, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 447, + 734 + ], + "score": 1.0, + "content": "in the forward propagation and gradient approximation in the backward propagation.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 688, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 70, + 503, + 188 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 59, + 501, + 70 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 57, + 502, + 72 + ], + "spans": [ + { + "bbox": [ + 109, + 57, + 502, + 72 + ], + "score": 1.0, + "content": "Table 3: Efficiency benchmark for network binarization. Blue: best in a row. Red: worst in a row.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 70, + 503, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 70, + 503, + 188 + ], + "spans": [ + { + "bbox": [ + 108, + 70, + 503, + 188 + ], + "score": 0.983, + "html": "
TrackMetricBinarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCUFDA
Training Consumption (%)Sensitivity27.28175.53113.40144.5928.66146.0653.3336.62
Time Cost82.1971.4376.9245.8068.1845.4558.2520.62
OMtrain61.23134.0096.89107.2552.30108.1655.8429.72
Theoretical Complexity (x）Speedup12.6012.2612.3712.3712.2612.2612.3712.37
Compression13.2713.2013.2013.2013.1613.2013.2013.20
OMcomp12.9412.7412.7912.7912.7112.7412.7912.79
Hardware Deployment (x）Speedup5.45False5.455.45False4.895.45 5.45
Compression15.62False15.6215.62False15.5215.6215.62
OMinfer11.70False11.7011.70False11.5111.7011.70
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Since being widely studied, the advanced binarization algorithms can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 165, + 264 + ], + "score": 1.0, + "content": "achieve about", + "type": "text" + }, + { + "bbox": [ + 165, + 252, + 206, + 263 + ], + "score": 0.68, + "content": "78 \\% - 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "of the full-precision accuracy in CNNs, and the binarized networks with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 435, + 275 + ], + "score": 1.0, + "content": "MLP architectures even approach the full-precision performance (e.g., Bi-Real Net", + "type": "text" + }, + { + "bbox": [ + 435, + 263, + 468, + 274 + ], + "score": 0.76, + "content": "8 7 . 8 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "). In con-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "trast, the transformer-based one suffers from extremely significant performance degradation when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 454, + 297 + ], + "score": 1.0, + "content": "binarized, and none of the algorithms achieves an overall accuracy metric higher than", + "type": "text" + }, + { + "bbox": [ + 455, + 285, + 474, + 295 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 285, + 505, + 297 + ], + "score": 1.0, + "content": ". Com-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "pared to CNNs and MLPs, the results show that transformer-based architectures constructed by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 307, + 483, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 483, + 320 + ], + "score": 1.0, + "content": "unique attention mechanisms require specific binarization designs instead of direct binarizing.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 324, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "The overall winner on the architecture track is the FDA algorithm. In this track, FDA has achieved", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "the best results in both CNN and Transformer. The evaluation of these two tracks proves that these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "binarization algorithms, which apply statistical channel-wise scaling factor and custom gradient", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 470, + 370 + ], + "score": 1.0, + "content": "approximation like FDA and ReActNet, have the advantage of stability to a certain degree.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 109, + 381, + 289, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 290, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 290, + 394 + ], + "score": 1.0, + "content": "5.1.3 CORRUPTION ROBUSTNESS TRACK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 504, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "The binarized network can approach full-precision level robustness for corruption. Surpris-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "ingly, binarized networks show robustness close to full-precision counterparts in corruption evalua-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "tion. Evaluation results on the CIFAR10-C dataset show that the binarized network performs close", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "to the full-precision network in the typical 2D image corruption. ReCU and XNOR-Net even out-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "perform their full-precision counterparts. If corruption robustness requirements are the same, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "binarized version network requires little additional designs or supervision for robustness. Thus, bi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "narized networks usually show comparable robustness for corruption against full-precision counter-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 504, + 491 + ], + "score": 1.0, + "content": "parts, which can be seen as a general property of binarized networks rather than specific algorithms.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 314, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 316, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 316, + 518 + ], + "score": 1.0, + "content": "5.2 EFFICIENCY ANALYSIS FOR BINARIZATION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 109, + 527, + 503, + 548 + ], + "lines": [ + { + "bbox": [ + 107, + 525, + 504, + 539 + ], + "spans": [ + { + "bbox": [ + 107, + 525, + 504, + 539 + ], + "score": 1.0, + "content": "As for efficiency, we discuss and analysis the metrics of training consumption, theoretical complex-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 537, + 272, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 272, + 550 + ], + "score": 1.0, + "content": "ity, and hardware inference tracks below.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 563, + 283, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 285, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 285, + 576 + ], + "score": 1.0, + "content": "5.2.1 TRAINING CONSUMPTION TRACK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 108, + 583, + 504, + 616 + ], + "lines": [ + { + "bbox": [ + 107, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 504, + 595 + ], + "score": 1.0, + "content": "We comprehensively investigate the training cost of binarization algorithms on ResNet18 of CI-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "FAR10 and present the sensitivity and training time results for different binarization algorithms in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 604, + 245, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 245, + 618 + ], + "score": 1.0, + "content": "Table 3 and Figure 3, respectively.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 164, + 635 + ], + "score": 1.0, + "content": "“Binarization", + "type": "text" + }, + { + "bbox": [ + 165, + 622, + 173, + 633 + ], + "score": 0.53, + "content": "\\nVDash", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "Sensitivity”: existing techniques can stabilize binarization-aware training. An", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "existing common intuition is that the training of binarized networks is usually more sensitive to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "the training settings than full-precision networks, caused by the representation limits and gradi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "ent approximation errors brought by the high degree of discretization. However, we find that the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "hyperparameter sensitivities of existing binarization algorithms are polarized. Some of them are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "even more hyperparameter-stable than the training of full-precision networks, while others fluctuate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "hugely. The reason for this problem is the difference in the techniques applied by the binarized", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "operators of these algorithms. The training-stable binarization algorithms often have the follow-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ing commonalities: (1) Channel-wise floating-point scaling factors based on learning or statistics;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "(2) Soft gradient approximation to reduce gradient error. These hyperparameter-stable binarization", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 70, + 503, + 188 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 59, + 501, + 70 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 57, + 502, + 72 + ], + "spans": [ + { + "bbox": [ + 109, + 57, + 502, + 72 + ], + "score": 1.0, + "content": "Table 3: Efficiency benchmark for network binarization. Blue: best in a row. Red: worst in a row.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 70, + 503, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 70, + 503, + 188 + ], + "spans": [ + { + "bbox": [ + 108, + 70, + 503, + 188 + ], + "score": 0.983, + "html": "
TrackMetricBinarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCUFDA
Training Consumption (%)Sensitivity27.28175.53113.40144.5928.66146.0653.3336.62
Time Cost82.1971.4376.9245.8068.1845.4558.2520.62
OMtrain61.23134.0096.89107.2552.30108.1655.8429.72
Theoretical Complexity (x）Speedup12.6012.2612.3712.3712.2612.2612.3712.37
Compression13.2713.2013.2013.2013.1613.2013.2013.20
OMcomp12.9412.7412.7912.7912.7112.7412.7912.79
Hardware Deployment (x）Speedup5.45False5.455.45False4.895.45 5.45
Compression15.62False15.6215.62False15.5215.6215.62
OMinfer11.70False11.7011.70False11.5111.7011.70
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Since being widely studied, the advanced binarization algorithms can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 165, + 264 + ], + "score": 1.0, + "content": "achieve about", + "type": "text" + }, + { + "bbox": [ + 165, + 252, + 206, + 263 + ], + "score": 0.68, + "content": "78 \\% - 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "of the full-precision accuracy in CNNs, and the binarized networks with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 435, + 275 + ], + "score": 1.0, + "content": "MLP architectures even approach the full-precision performance (e.g., Bi-Real Net", + "type": "text" + }, + { + "bbox": [ + 435, + 263, + 468, + 274 + ], + "score": 0.76, + "content": "8 7 . 8 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "). In con-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "trast, the transformer-based one suffers from extremely significant performance degradation when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 454, + 297 + ], + "score": 1.0, + "content": "binarized, and none of the algorithms achieves an overall accuracy metric higher than", + "type": "text" + }, + { + "bbox": [ + 455, + 285, + 474, + 295 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 285, + 505, + 297 + ], + "score": 1.0, + "content": ". Com-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "pared to CNNs and MLPs, the results show that transformer-based architectures constructed by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 307, + 483, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 483, + 320 + ], + "score": 1.0, + "content": "unique attention mechanisms require specific binarization designs instead of direct binarizing.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 230, + 505, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 324, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "The overall winner on the architecture track is the FDA algorithm. In this track, FDA has achieved", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "the best results in both CNN and Transformer. The evaluation of these two tracks proves that these", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "binarization algorithms, which apply statistical channel-wise scaling factor and custom gradient", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 470, + 370 + ], + "score": 1.0, + "content": "approximation like FDA and ReActNet, have the advantage of stability to a certain degree.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 324, + 505, + 370 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 381, + 289, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 290, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 290, + 394 + ], + "score": 1.0, + "content": "5.1.3 CORRUPTION ROBUSTNESS TRACK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 504, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "The binarized network can approach full-precision level robustness for corruption. Surpris-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "ingly, binarized networks show robustness close to full-precision counterparts in corruption evalua-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "tion. Evaluation results on the CIFAR10-C dataset show that the binarized network performs close", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "to the full-precision network in the typical 2D image corruption. ReCU and XNOR-Net even out-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "perform their full-precision counterparts. If corruption robustness requirements are the same, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "binarized version network requires little additional designs or supervision for robustness. Thus, bi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "narized networks usually show comparable robustness for corruption against full-precision counter-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 504, + 491 + ], + "score": 1.0, + "content": "parts, which can be seen as a general property of binarized networks rather than specific algorithms.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 401, + 506, + 491 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 314, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 316, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 316, + 518 + ], + "score": 1.0, + "content": "5.2 EFFICIENCY ANALYSIS FOR BINARIZATION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 109, + 527, + 503, + 548 + ], + "lines": [ + { + "bbox": [ + 107, + 525, + 504, + 539 + ], + "spans": [ + { + "bbox": [ + 107, + 525, + 504, + 539 + ], + "score": 1.0, + "content": "As for efficiency, we discuss and analysis the metrics of training consumption, theoretical complex-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 537, + 272, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 272, + 550 + ], + "score": 1.0, + "content": "ity, and hardware inference tracks below.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 525, + 504, + 550 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 563, + 283, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 285, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 285, + 576 + ], + "score": 1.0, + "content": "5.2.1 TRAINING CONSUMPTION TRACK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 108, + 583, + 504, + 616 + ], + "lines": [ + { + "bbox": [ + 107, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 504, + 595 + ], + "score": 1.0, + "content": "We comprehensively investigate the training cost of binarization algorithms on ResNet18 of CI-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "FAR10 and present the sensitivity and training time results for different binarization algorithms in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 604, + 245, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 245, + 618 + ], + "score": 1.0, + "content": "Table 3 and Figure 3, respectively.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 583, + 505, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 164, + 635 + ], + "score": 1.0, + "content": "“Binarization", + "type": "text" + }, + { + "bbox": [ + 165, + 622, + 173, + 633 + ], + "score": 0.53, + "content": "\\nVDash", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "Sensitivity”: existing techniques can stabilize binarization-aware training. An", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "existing common intuition is that the training of binarized networks is usually more sensitive to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "the training settings than full-precision networks, caused by the representation limits and gradi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "ent approximation errors brought by the high degree of discretization. However, we find that the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "hyperparameter sensitivities of existing binarization algorithms are polarized. Some of them are", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "even more hyperparameter-stable than the training of full-precision networks, while others fluctuate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "hugely. The reason for this problem is the difference in the techniques applied by the binarized", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "operators of these algorithms. The training-stable binarization algorithms often have the follow-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "ing commonalities: (1) Channel-wise floating-point scaling factors based on learning or statistics;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "(2) Soft gradient approximation to reduce gradient error. These hyperparameter-stable binarization", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "algorithms may not certainly outperform other algorithms but can simplify the tuning process in", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 218, + 330, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 330, + 234 + ], + "score": 1.0, + "content": "production and obtain reliable accuracy in one training.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 620, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 66, + 502, + 171 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 66, + 502, + 171 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 66, + 502, + 171 + ], + "spans": [ + { + "bbox": [ + 107, + 66, + 502, + 171 + ], + "score": 0.969, + "type": "image", + "image_path": "ce936e7dd0211ceaee8c117b81d59f14136795fbe8fdbfb74f2225ad2e3a2987.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 66, + 502, + 101.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 101.0, + 502, + 136.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 136.0, + 502, + 171.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 168, + 175, + 441, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 172, + 442, + 189 + ], + "spans": [ + { + "bbox": [ + 168, + 172, + 442, + 189 + ], + "score": 1.0, + "content": "Figure 2: Comparisons of accuracy under different training settings.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 209, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "algorithms may not certainly outperform other algorithms but can simplify the tuning process in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 218, + 330, + 234 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 330, + 234 + ], + "score": 1.0, + "content": "production and obtain reliable accuracy in one training.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "The preference for hyperparameter settings is evident for the training of binarization. From the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 248, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 345, + 259 + ], + "score": 1.0, + "content": "statistical results in Figure 2, training with Adam optimizer,", + "type": "text" + }, + { + "bbox": [ + 346, + 248, + 360, + 258 + ], + "score": 0.85, + "content": "1 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 248, + 504, + 259 + ], + "score": 1.0, + "content": "(identical to full-precision network)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "learning rate and CosineAnnealingLR scheduler is more stable than that with other settings. Inspired", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 270, + 491, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 491, + 282 + ], + "score": 1.0, + "content": "by this, we adopt this setting in evaluating binarization as part of the standard training pipelines.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "The soft gradient approximation in binarization brings a significant training time increase.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "Comparing the time consumed by each binarization algorithm, we found that the training time of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "algorithms using the custom gradient approximation techniques such as Bi-Real and ReActNet in-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 377, + 332 + ], + "score": 1.0, + "content": "creased significantly, and the metric of FDA is even as worse as", + "type": "text" + }, + { + "bbox": [ + 377, + 320, + 409, + 330 + ], + "score": 0.88, + "content": "2 0 . 6 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 319, + 505, + 332 + ], + "score": 1.0, + "content": ", which means that the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 395, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 237, + 345 + ], + "score": 1.0, + "content": "training time he spent is close to", + "type": "text" + }, + { + "bbox": [ + 238, + 331, + 252, + 341 + ], + "score": 0.86, + "content": "5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 329, + 395, + 345 + ], + "score": 1.0, + "content": "the full-precision network training.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 354, + 294, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 296, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 296, + 367 + ], + "score": 1.0, + "content": "5.2.2 THEORETICAL COMPLEXITY TRACK", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "There is no obvious difference in the theoretical complexity among binarization algorithms.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "The leading cause of the difference in compression rate is the definition of the static scaling factor", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "of each model, e.g., BNN does not apply any factors and enjoys the most compression. 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These results show binarization algorithms should have similar inference efficiency.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 476, + 274, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 276, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 276, + 488 + ], + "score": 1.0, + "content": "5.2.3 HARDWARE INFERENCE TRACK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 295, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 297, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 297, + 507 + ], + "score": 1.0, + "content": "Compared to other tracks, the hardware infer-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 506, + 297, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 297, + 518 + ], + "score": 1.0, + "content": "ence track brings us some insights for binariza-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 517, + 237, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 237, + 529 + ], + "score": 1.0, + "content": "tion in a real-world deployment.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 296, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 297, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 297, + 546 + ], + "score": 1.0, + "content": "Limited inference libraries lead to an al-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 544, + 297, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 297, + 558 + ], + "score": 1.0, + "content": "most fixed paradigm of binarization deploy-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 555, + 297, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 297, + 568 + ], + "score": 1.0, + "content": "ment. 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Both inference libraries support a", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "channel-wise scaling factor in floating-point form and force it to fuse into the BN layer (BN must", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "follow every convolution of the binarized model). Neither supports dynamic activation of activations", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "statistics nor re-scaling in inference. 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The scale factor shape of", + "type": "text" + }, + { + "bbox": [ + 331, + 699, + 373, + 709 + ], + "score": 0.63, + "content": "\\mathrm { X N O R + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "led to its failed deployment, and", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "XNOR also failed because of the activation re-scaling technique. The vast majority of binarization", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "methods have almost identical inference performance, and the mean-shifting operation of ReAct-", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 54.5, + "bbox_fs": [ + 105, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 70, + 502, + 197 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 143, + 59, + 465, + 69 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 58, + 467, + 72 + ], + "spans": [ + { + "bbox": [ + 144, + 58, + 467, + 72 + ], + "score": 1.0, + "content": "Table 4: Deployment capability of different inference libraries on real hardware.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 70, + 502, + 197 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 70, + 502, + 197 + ], + "spans": [ + { + "bbox": [ + 108, + 70, + 502, + 197 + ], + "score": 0.984, + "html": "
Infer. Lib.Providers Granularitys FormFlod BNAct. Re-scalingAct.Mean-shifting
LarqLarqChannel-wiseFP32√×√
daBNNJDChannel-wiseFP32√××
AlgorithmDeployables GranularitysFormFlod BNAct. Re-scalingAct. Mean-shifting
BNN√N/AN/AN/A××
XNOR×Channel-wiseFP32>>>×√>>√×
DoReFa√Channel-wiseFP32××
Bi-Real√Channel-wiseFP32××
XNOR++×Spatial-wiseFP32××
ReActNetChannel-wiseFP32×√
ReCU√V√Channel-wiseFP32××
FDAChannel-wiseFP32××
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A counterintuitive finding is that the higher", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "the chip capability, the worse the speedup of binarization on the chip (Figure 3). Further observation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "showed that the contradiction is mainly because higher-performance chips have more acceleration", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "brought by multi-threading when running floating-point models. 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The scenarios where network binarization technology comes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "into play better are edge chips with low performance and cost, and the extreme compression and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 334, + 502, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 502, + 345 + ], + "score": 1.0, + "content": "acceleration of binarization are making deployment of advanced neural networks on edge possible.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 358, + 334, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 335, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 335, + 370 + ], + "score": 1.0, + "content": "5.3 SUGGESTIONS FOR BINARIZATION ALGORITHM", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "Based on the above evaluation and analysis, we attempt to summarize a paradigm for accurate and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "efficient network binarization among existing techniques: (1) Soft quantization approximation is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "an undisputed must-have technique. This binarization technique does not affect hardware inference", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "efficiency and is adopted by all winning binarization algorithms on accuracy tracks, including Bi-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "Real, ReActNet, and ReCU. (2) Channel-wise scaling factors are the only option available for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "practical binarization. The results of the learning task and neural architecture tracks demonstrate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "the advantage of floating-point scaling factors, and analysis of efficiency tracks further limits it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "to the channel-wise form. (3) Mean-shifting the input with a fixed bias is an optional helpful", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "operation. Our results show that this technique in ReActNet effectively improves accuracy and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 427, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 427, + 490 + ], + "score": 1.0, + "content": "consumes almost no extra computation, but not all inference libraries support it.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "score": 1.0, + "content": "We have to stress that although the benchmarking on evaluation tracks leads us to several ground", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "rules towards accurate and efficient binarization, none of the binarization techniques or algo-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "rithms work well across all scenarios so far. In the future, binarization research should focus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "on breaking the mutual restrictions between production and deployment, and the binarization al-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "gorithms should consider deployability and efficiency. The inference libraries are also expected to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 283, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 283, + 561 + ], + "score": 1.0, + "content": "support more advanced binarized operators.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 576, + 190, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 192, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 192, + 591 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "In this paper, we present BiBench, a versatile and comprehensive benchmark that delves into the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 504, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 504, + 624 + ], + "score": 1.0, + "content": "most fundamental questions of model binarization. BiBench covers 8 network binarization algo-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "rithms, 9 deep learning datasets (including one corruption benchmark), 13 different neural architec-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "tures, 2 deployment libraries, 14 real-world hardware, and various hyperparameter settings. More-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "over, BiBench proposes evaluation tracks specifically designed to measure critical aspects such as", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "accuracy under multiple conditions and efficiency when deployed on actual hardware. More im-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "portantly, by collating experiment results and analysis, BiBench hopes to establish an empirically", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "optimized paradigm with several critical considerations for designing accurate and efficient binariza-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "tion methods. 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Infer. Lib.Providers Granularitys FormFlod BNAct. Re-scalingAct.Mean-shifting
LarqLarqChannel-wiseFP32√×√
daBNNJDChannel-wiseFP32√××
AlgorithmDeployables GranularitysFormFlod BNAct. Re-scalingAct. Mean-shifting
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XNOR×Channel-wiseFP32>>>×√>>√×
DoReFa√Channel-wiseFP32××
Bi-Real√Channel-wiseFP32××
XNOR++×Spatial-wiseFP32××
ReActNetChannel-wiseFP32×√
ReCU√V√Channel-wiseFP32××
FDAChannel-wiseFP32××
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A counterintuitive finding is that the higher", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "the chip capability, the worse the speedup of binarization on the chip (Figure 3). Further observation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "showed that the contradiction is mainly because higher-performance chips have more acceleration", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "brought by multi-threading when running floating-point models. 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The scenarios where network binarization technology comes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "into play better are edge chips with low performance and cost, and the extreme compression and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 334, + 502, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 502, + 345 + ], + "score": 1.0, + "content": "acceleration of binarization are making deployment of advanced neural networks on edge possible.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 244, + 506, + 345 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 358, + 334, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 335, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 335, + 370 + ], + "score": 1.0, + "content": "5.3 SUGGESTIONS FOR BINARIZATION ALGORITHM", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "Based on the above evaluation and analysis, we attempt to summarize a paradigm for accurate and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "efficient network binarization among existing techniques: (1) Soft quantization approximation is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "an undisputed must-have technique. This binarization technique does not affect hardware inference", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "efficiency and is adopted by all winning binarization algorithms on accuracy tracks, including Bi-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "Real, ReActNet, and ReCU. (2) Channel-wise scaling factors are the only option available for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "practical binarization. The results of the learning task and neural architecture tracks demonstrate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "the advantage of floating-point scaling factors, and analysis of efficiency tracks further limits it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "to the channel-wise form. (3) Mean-shifting the input with a fixed bias is an optional helpful", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "operation. Our results show that this technique in ReActNet effectively improves accuracy and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 477, + 427, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 427, + 490 + ], + "score": 1.0, + "content": "consumes almost no extra computation, but not all inference libraries support it.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 378, + 506, + 490 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "score": 1.0, + "content": "We have to stress that although the benchmarking on evaluation tracks leads us to several ground", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "rules towards accurate and efficient binarization, none of the binarization techniques or algo-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "rithms work well across all scenarios so far. In the future, binarization research should focus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "on breaking the mutual restrictions between production and deployment, and the binarization al-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "gorithms should consider deployability and efficiency. The inference libraries are also expected to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 283, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 283, + 561 + ], + "score": 1.0, + "content": "support more advanced binarized operators.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 494, + 505, + 561 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 576, + 190, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 192, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 192, + 591 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "In this paper, we present BiBench, a versatile and comprehensive benchmark that delves into the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 612, + 504, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 504, + 624 + ], + "score": 1.0, + "content": "most fundamental questions of model binarization. BiBench covers 8 network binarization algo-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "rithms, 9 deep learning datasets (including one corruption benchmark), 13 different neural architec-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "tures, 2 deployment libraries, 14 real-world hardware, and various hyperparameter settings. 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More im-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "portantly, by collating experiment results and analysis, BiBench hopes to establish an empirically", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "optimized paradigm with several critical considerations for designing accurate and efficient binariza-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "tion methods. We hope BiBench can facilitate a fair comparison of algorithms through a systematic", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "score": 1.0, + "content": "investigation with metrics that reflect the fundamental requirements and serve as a foundation for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 384, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 384, + 724 + ], + "score": 1.0, + "content": "applying model binarization in broader and more practical scenarios.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 601, + 505, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 74, + 507, + 736 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 114 + ], + "score": 1.0, + "content": "Milad Alizadeh, Javier Fernandez-Marqu ´ es, Nicholas D. 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AlgorithmYear ConferenceCitation (to 2022/11/08) Techniques SourceOperatorOpen Specified Structure / Training-pipeline
BNN (Courbariaux et al., 2016a)2016NeurIPS1846YesYesNo
XNOR-Net (Rastegari et al., 2016) 2016ECCV4313YesYesNo
DoReFa (Zhou et al., 2016a)2016ArXiv174YesYesNo
Bi-Real Net (Liu et al., 2018a)2018ECCV371YesYesOptional
CI-BCNN (Wang et al., 2019)2019CVPR77YesYesYes
XNOR-Net++ (Bulat et al., 2019)2019BMVC118YesYesNo
RBNN (Lin et al., 2020)2020NeurIPS62YesYesNo
ReActNet (Liu et al., 2020)2020ECCV151YesYesOptional
Si-BNN (Wang et al., 2020a)2020AAAI24YesNoNo
ProxyBNN (He et al., 2020)2020ECCV24YesNoYes
FDA (Xu et al., 2021a)2021NeurIPS11YesYesNo
ReCU (Xu et al., 2021b)2021ICCV15YesYesNo
LCR-BNN (Shang et al., 2022a)2022ECCV0YesYesYes
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Therefore, binarization is considered to take advantage of the hardware and can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 313, + 307, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 307, + 325 + ], + "score": 1.0, + "content": "achieve more speedup than multi-bit quantization.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 107, + 329, + 174, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 177, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 177, + 342 + ], + "score": 1.0, + "content": "Selection Rules:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "First of all, we state that we obey some general rules for selecting binarization algorithms, i.e., the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 357, + 504, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 504, + 369 + ], + "score": 1.0, + "content": "selected binarization algorithms should improve the binarized operator since it is the fundamental", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "difference between binarized and full-precision models (as discussed in Section 2.1). And we thus", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "also exclude the algorithms and techniques requiring specified local structures or training pipelines", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 390, + 194, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 194, + 402 + ], + "score": 1.0, + "content": "for a fair comparison.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "Then, we explain in detail the choice of binarization algorithms and why they are representative.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 431 + ], + "score": 1.0, + "content": "When we built the BiBench, we considered various binarization algorithms with improved operator", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 429, + 504, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 504, + 441 + ], + "score": 1.0, + "content": "techniques in binarization research, and now we list them in detail in Table 5. We consider from the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 440, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 504, + 453 + ], + "score": 1.0, + "content": "following perspectives, the purposes are making the selected binarization algorithms representative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "and should complete all evaluations in BiBench fairly: Operator Techniques (Yes/No), Year, Con-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "ference, Citation (to 2022/11/08), Open source (Yes/No), and Specified Structure / Training-pipeline", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 474, + 185, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 185, + 485 + ], + "score": 1.0, + "content": "(Yes/No/Optional).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "(1) We analyze the techniques proposed in these works. 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Bold means", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 312, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 312, + 103 + ], + "score": 1.0, + "content": "that the algorithm has an advantage in that column.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 102, + 510, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 102, + 510, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 510, + 264 + ], + "score": 0.984, + "html": "
AlgorithmYear ConferenceCitation (to 2022/11/08) Techniques SourceOperatorOpen Specified Structure / Training-pipeline
BNN (Courbariaux et al., 2016a)2016NeurIPS1846YesYesNo
XNOR-Net (Rastegari et al., 2016) 2016ECCV4313YesYesNo
DoReFa (Zhou et al., 2016a)2016ArXiv174YesYesNo
Bi-Real Net (Liu et al., 2018a)2018ECCV371YesYesOptional
CI-BCNN (Wang et al., 2019)2019CVPR77YesYesYes
XNOR-Net++ (Bulat et al., 2019)2019BMVC118YesYesNo
RBNN (Lin et al., 2020)2020NeurIPS62YesYesNo
ReActNet (Liu et al., 2020)2020ECCV151YesYesOptional
Si-BNN (Wang et al., 2020a)2020AAAI24YesNoNo
ProxyBNN (He et al., 2020)2020ECCV24YesNoYes
FDA (Xu et al., 2021a)2021NeurIPS11YesYesNo
ReCU (Xu et al., 2021b)2021ICCV15YesYesNo
LCR-BNN (Shang et al., 2022a)2022ECCV0YesYesYes
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First, representative perception modalities are selected", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "in our deep learning, including (2D/3D) vision, text, and speech. Research on these modalities has", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "progressed rapidly and has a broad impact, so we choose specific tasks and datasets in these modal-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "ities. Specifically, (1) on the 2D vision modality, we choose the basic image classification task and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "object detection task (one of the most popular downstream tasks), the former including CIFAR10", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "and ImageNet datasets, the latter including Pascal VOC and COCO datasets. These datasets Ima-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "geNet and COCO are both more challenging large datasets, while CIFAR10 and Pascal VOC are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "more basic. For other modalities, binarization is still challenging even with the underlying tasks and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "datasets in the field, since there are few related binarization studies: (2) In the 3D vision modal-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "ity, the basic point cloud classification ModelNet40 dataset is selected to evaluate the binarization", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "performance, which is regarded as one of the most fundamental tasks in 3D point cloud research", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "and is widely studied. 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This dataset is widely used for image classification tasks. There are 60,000 color im-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 231, + 393 + ], + "score": 1.0, + "content": "ages, each of which measures", + "type": "text" + }, + { + "bbox": [ + 231, + 380, + 258, + 390 + ], + "score": 0.69, + "content": "3 2 \\mathrm { x } 3 2 ", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "pixels. All images are categorized into 10 different classes:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks. Each class has 6000 images,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "where 5000 are for training and 1000 are for testing. 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The", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "dataset consists of 328K images. According to community feedback, in the 2017 release, the train-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "ing/validation split was changed from 83K/41K to 118K/5K. And the images and annotations are the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "same. The 2017 test set is a subset of 41K images from the 2015 test set. 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It benchmarks", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "the frailty of classifiers under corruption, including noise, blur, weather, and digital influence. And", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "each type of corruption has five levels of severity, resulting in 75 distinct corruptions. We report the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "accuracy of the classifiers under each level of severity and each corruption. 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The classifier’s aggregate performance across the five severities of the corruption types.ruption Errors of a certain type of Corruption is computed with the formula:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 154, + 510, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 249, + 403, + 283 + ], + "lines": [ + { + "bbox": [ + 208, + 249, + 403, + 283 + ], + "spans": [ + { + "bbox": [ + 208, + 249, + 403, + 283 + ], + "score": 0.94, + "content": "C E _ { C o r r u p t i o n } ^ { N e t w o r k } = \\sum _ { s = 1 } ^ { 5 } E _ { s , C o r r u p t i o n } ^ { N e t w o r k } / \\sum _ { s = 1 } ^ { 5 } E _ { s , C o r r u p t i o n } ^ { A l e x N e t } .", + "type": "interline_equation", + "image_path": "3a3acb2bee88d6a8517ed0ce149f7383132e5eec470929935f1437e92593170e.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 249, + 403, + 266.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 208, + 266.0, + 403, + 283.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 301 + ], + "score": 1.0, + "content": "To make Corruption Errors comparable across corruption types, the difficulty is usually adjusted by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 298, + 225, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 225, + 311 + ], + "score": 1.0, + "content": "dividing by AlexNet’s errors.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 286, + 505, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 326, + 327, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 329, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 329, + 340 + ], + "score": 1.0, + "content": "C DETAILS OF NEURAL ARCHITECTURES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "ResNet (He et al., 2016): Residual Networks, or ResNets, learn residual functions concerning the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "layer inputs instead of learning unreferenced functions. 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There is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "empirical evidence that these networks are easier to optimize and can achieve higher accuracy with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 227, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 227, + 408 + ], + "score": 1.0, + "content": "considerably increased depth.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 350, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 504, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "VGG (Simonyan & Zisserman, 2015): VGG is a classical convolutional neural network architecture.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "It is proposed by an analysis of how to increase the depth of such networks. It is characterized by its", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 257, + 446 + ], + "score": 1.0, + "content": "simplicity: the network utilizes small", + "type": "text" + }, + { + "bbox": [ + 257, + 434, + 277, + 444 + ], + "score": 0.87, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "filters, and the only other components are pooling layers", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 444, + 218, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 218, + 457 + ], + "score": 1.0, + "content": "and a fully connected layer.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 411, + 505, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 504, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "MobileNetV2 (Sandler et al., 2018): MobileNetV2 is a convolutional neural network architecture", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "that performs well on mobile devices. 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By masking some tokens from the input, the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "masked language model attempts to estimate the original vocabulary id of the masked word based", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "solely on its context. An MLM objective differs from a left-to-right language model in that it en-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "ables the representation to integrate the left and right contexts, which facilitates pre-training a deep", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "bidirectional Transformer. Additionally, BERT uses a next-sentence prediction task that pre-trains", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "text-pair representations along with the masked language model. Note that we replace the direct bi-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "narized attention with a bi-attention mechanism to prevent the model from completely crashing (Qin", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 159, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 159, + 150 + ], + "score": 1.0, + "content": "et al., 2021).", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "solely on its context. An MLM objective differs from a left-to-right language model in that it en-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "ables the representation to integrate the left and right contexts, which facilitates pre-training a deep", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "bidirectional Transformer. Additionally, BERT uses a next-sentence prediction task that pre-trains", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "text-pair representations along with the masked language model. Note that we replace the direct bi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "narized attention with a bi-attention mechanism to prevent the model from completely crashing (Qin", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 159, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 159, + 150 + ], + "score": 1.0, + "content": "et al., 2021).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "PointNet (Qi et al., 2017): PointNet is a unified architecture for applications ranging from object", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "classification and part segmentation to scene semantic parsing. The architecture directly receives", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "point clouds as input and outputs either class labels for the entire input or point segment/part labels.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "PointNet-Vanilla is a variant of PointNet, which drops off the T-Net module. And for all PointNet", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 196, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 506, + 212 + ], + "score": 1.0, + "content": "models, we apply the EMA-Max (Qin et al., 2020a) as the aggregator, because directly following", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 434, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 434, + 223 + ], + "score": 1.0, + "content": "the max pooling aggregator will cause the binarized PointNets to fail to converge.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "FSMN (Zhang et al., 2015): Feedforward sequential memory networks or FSMN is a novel neural", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "network structure to model long-term dependency in time series without using recurrent feedback. It", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "is a standard fully connected feedforward neural network containing some learnable memory blocks.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "As a short-term memory mechanism, the memory blocks encode long context information using a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 218, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 218, + 283 + ], + "score": 1.0, + "content": "tapped-delay line structure.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "Deep-FSMN (Zhang et al., 2018a): The Deep-FSMN architecture is an improved feedforward se-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "quential memory network (FSMN) with skip connections between memory blocks in adjacent lay-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "ers. 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Their products include Kirin 970, Kirin 980, Kirin 985, etc.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 398, + 503, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 412 + ], + "score": 1.0, + "content": "MediaTek Dimensity (MediaTek, 2022): Dimensity is a series of ARM-based systems-on-a-chip", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 410, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 504, + 423 + ], + "score": 1.0, + "content": "(SoCs) produced by MediaTek. 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As a result of its low cost, modularity, and open design, it is used", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "in many applications, including weather monitoring, and is sold outside the intended market. It is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 565, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 576 + ], + "score": 1.0, + "content": "typically used by computer hobbyists and electronic enthusiasts due to the adoption of HDMI and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 171, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 171, + 587 + ], + "score": 1.0, + "content": "USB standards.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 592, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "Apple M1 (Wikipedia, 2022a): Apple M1 is a series of ARM-based systems-on-a-chip (SoCs)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 616 + ], + "score": 1.0, + "content": "designed by Apple Inc. 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It", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "is a standard fully connected feedforward neural network containing some learnable memory blocks.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "As a short-term memory mechanism, the memory blocks encode long context information using a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 218, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 218, + 283 + ], + "score": 1.0, + "content": "tapped-delay line structure.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 225, + 506, + 283 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "Deep-FSMN (Zhang et al., 2018a): The Deep-FSMN architecture is an improved feedforward se-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "quential memory network (FSMN) with skip connections between memory blocks in adjacent lay-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "ers. 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TaskArch.FP32Binarization Algorithm
BNNXNOR DoReFa Bi-Real XNOR++ ReActNet ReCU FDA
CIFAR10ResNet2091.9985.3185.5385.1885.5685.4186.1886.4286.38
ResNet1894.8289.6991.491.5591.2090.0491.5592.7990.42
ResNet3495.3490.8289.5890.9592.5090.5992.6993.6489.59
VGG-Small93.8089.6689.6589.6690.2589.3490.2790.8489.48
ImageNetResNet1869.9052.9953.9953.5554.7952.4354.9754.5154.63
VOC07Faster-RCNN76.0658.5456.7558.0760.9056.6061.9062.1060.10
SSD30077.349.0933.7230.7031.909.4138.419.8043.68
COC014Faster-RCNN27.2021.2020.5021.3022.2021.6022.8023.3022.40
ModelNet40PointNetvanilla86.8085.1383.4785.2185.3785.6685.1385.2185.49
PointNet88.209.0880.7578.7777.7163.2576.5081.1279.62
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TaskArch.FP32Binarization Algorithm
BNNXNOR DoReFa Bi-Real XNOR++ ReActNet ReCU FDA
CIFAR10ResNet2091.9985.3185.5385.1885.5685.4186.1886.4286.38
ResNet1894.8289.6991.491.5591.2090.0491.5592.7990.42
ResNet3495.3490.8289.5890.9592.5090.5992.6993.6489.59
VGG-Small93.8089.6689.6589.6690.2589.3490.2790.8489.48
ImageNetResNet1869.9052.9953.9953.5554.7952.4354.9754.5154.63
VOC07Faster-RCNN76.0658.5456.7558.0760.9056.6061.9062.1060.10
SSD30077.349.0933.7230.7031.909.4138.419.8043.68
COC014Faster-RCNN27.2021.2020.5021.3022.2021.6022.8023.3022.40
ModelNet40PointNetvanilla86.8085.1383.4785.2185.3785.6685.1385.2185.49
PointNet88.209.0880.7578.7777.7163.2576.5081.1279.62
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TaskArch.CategoryFP32Binarization Algorithm
BNN XNOR DoReFa Bi-Real XNOR++ ReActNet ReCU FDA
Airplane83.737.5 74.1467.267.6130.3666.1231.6165.34
Bag79.644.2 4955.3447.1137.4450.2838.5848.62
Cap92.344.3 73.3251.2161.4140.3756.7340.1356
Car76.824.3 55.2752.2449.3924.0749.1123.9258.5
Chair90.961.6 85.6283.9683.641.8983.8341.583.27
Earphone 70.238.530.9734.9435.2426.336.7223.01 34.46
Guitar91.132.969.1767.965.9923.4564.1828.3878.69
Knife85.74378.1676.1675.5337.6275.0138.8177.07
ShapeNet PointNet Lamp8251.26968.756049.4566.1348.41 67.45
Laptop95.549.493.2992.9392.7941.8992.9342.28 93.66
Motorbike 64.416.319.0418.8818.6913.1818.5911.2620.38
Mug93.649.164.3253.5652.0147.5852.5146.83 53.48
Pistol80.825.562.2959.1551.4326.9653.7927.81 62.61
Rocket54.426.930.9527.9226.6122.3826.0119.32 23.08
Skateboard 70.741.245.750.1543.7828.6343.7426.71 45.81
Table81.451.373.6872.6969.7245.7469.5645.2173.45
Overall80.81875 39.8260.8758.3156.3133.5856.5833.3658.68
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TaskArch.CategoryFP32Binarization Algorithm
BNN XNOR DoReFa Bi-Real XNOR++ ReActNet ReCU FDA
Airplane83.737.5 74.1467.267.6130.3666.1231.6165.34
Bag79.644.2 4955.3447.1137.4450.2838.5848.62
Cap92.344.3 73.3251.2161.4140.3756.7340.1356
Car76.824.3 55.2752.2449.3924.0749.1123.9258.5
Chair90.961.6 85.6283.9683.641.8983.8341.583.27
Earphone 70.238.530.9734.9435.2426.336.7223.01 34.46
Guitar91.132.969.1767.965.9923.4564.1828.3878.69
Knife85.74378.1676.1675.5337.6275.0138.8177.07
ShapeNet PointNet Lamp8251.26968.756049.4566.1348.41 67.45
Laptop95.549.493.2992.9392.7941.8992.9342.28 93.66
Motorbike 64.416.319.0418.8818.6913.1818.5911.2620.38
Mug93.649.164.3253.5652.0147.5852.5146.83 53.48
Pistol80.825.562.2959.1551.4326.9653.7927.81 62.61
Rocket54.426.930.9527.9226.6122.3826.0119.32 23.08
Skateboard 70.741.245.750.1543.7828.6343.7426.71 45.81
Table81.451.373.6872.6969.7245.7469.5645.2173.45
Overall80.81875 39.8260.8758.3156.3133.5856.5833.3658.68
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TaskArch.FP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNet ReCUFDA
BERT-Tiny4L82.8136.9041.2052.3155.0937.2755.5238.5559.41
MNLI-mBERT-Tiny6L84.7637.0151.1763.0966.8137.9866.4737.9568.46
BERT-Base84.8835.4541.4060.6762.4735.4560.2235.4563.49
BERT-Tiny4L83.0836.5441.5553.0155.5736.0755.8937.6259.76
MNLI-mmBERT-Tiny6L84.4236.4750.9263.8766.8238.1167.6436.9168.98
BERT-Base85.4535.2241.1860.9663.1735.2261.1935.2263.72
BERT-Tiny4L90.4766.1973.6975.7977.3864.9776.9267.3278.92
QQPBERT-Tiny6L85.9863.1878.9080.9382.4263.1982.9563.383.19
BERT-Base91.5163.1871.9377.0780.0163.1881.1663.1883.26
BERT-Tiny4L87.4651.7160.5961.1561.9252.7962.6753.9962.29
QNLIBERT-Tiny6L90.7952.2262.7566.8869.7251.8470.2751.3272.72
BERT-Base92.1451.860.2970.7870.1454.0769.4451.8772.43
GLUEBERT-Tiny4L92.4352.9879.9382.4584.0654.0184.1754.2486.12
SST-2BERT-Tiny6L90.2558.1484.7486.2387.7369.3887.9552.4087.72
BERT-Base93.2352.2978.7886.0186.3553.3284.452.4087.93
BERT-Tiny4L49.616.557.2212.6916.86014.716.2517.80
CoLABERT-Tiny6L54.172.5712.5715.9717.94015.242.2422.21
BERT-Base59.714.6304.7415.9504.630.404.63
BERT-Tiny4L86.354.3118.0518.7422.657.4522.738.2027.56
STS-BBERT-Tiny6L89.791.0414.7222.3124.595.7023.408.2237.15
BERT-Base90.066.9412.1918.2620.764.998.736.5910.14
BERT-Tiny4L85.5068.3071.7471.9971.7468.3071.7471.2571.49
MRPCBERT-Tiny6L87.7168.3070.7671.7471.4968.3071.7469.0471.74
BERT-Base86.2468.3068.370.0270.2768.3071.2568.3069.04
BERT-Tiny4L65.3456.3153.4356.3155.5954.1557.7661.0159.20
RTEBERT-Tiny6L68.9556.3154.5154.5158.1249.0953.4358.8454.87
BERT-Base72.2053.4357.0455.2354.5154.8754.5155.2355.23
FSMN94.8956.4556.4568.6573.6075.0473.8056.45
Speech CommandsD-FSMN97.5188.3292.0378.9285.1156.7783.8092.1156.45 93.91
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TaskArch.FP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNet ReCUFDA
BERT-Tiny4L82.8136.9041.2052.3155.0937.2755.5238.5559.41
MNLI-mBERT-Tiny6L84.7637.0151.1763.0966.8137.9866.4737.9568.46
BERT-Base84.8835.4541.4060.6762.4735.4560.2235.4563.49
BERT-Tiny4L83.0836.5441.5553.0155.5736.0755.8937.6259.76
MNLI-mmBERT-Tiny6L84.4236.4750.9263.8766.8238.1167.6436.9168.98
BERT-Base85.4535.2241.1860.9663.1735.2261.1935.2263.72
BERT-Tiny4L90.4766.1973.6975.7977.3864.9776.9267.3278.92
QQPBERT-Tiny6L85.9863.1878.9080.9382.4263.1982.9563.383.19
BERT-Base91.5163.1871.9377.0780.0163.1881.1663.1883.26
BERT-Tiny4L87.4651.7160.5961.1561.9252.7962.6753.9962.29
QNLIBERT-Tiny6L90.7952.2262.7566.8869.7251.8470.2751.3272.72
BERT-Base92.1451.860.2970.7870.1454.0769.4451.8772.43
GLUEBERT-Tiny4L92.4352.9879.9382.4584.0654.0184.1754.2486.12
SST-2BERT-Tiny6L90.2558.1484.7486.2387.7369.3887.9552.4087.72
BERT-Base93.2352.2978.7886.0186.3553.3284.452.4087.93
BERT-Tiny4L49.616.557.2212.6916.86014.716.2517.80
CoLABERT-Tiny6L54.172.5712.5715.9717.94015.242.2422.21
BERT-Base59.714.6304.7415.9504.630.404.63
BERT-Tiny4L86.354.3118.0518.7422.657.4522.738.2027.56
STS-BBERT-Tiny6L89.791.0414.7222.3124.595.7023.408.2237.15
BERT-Base90.066.9412.1918.2620.764.998.736.5910.14
BERT-Tiny4L85.5068.3071.7471.9971.7468.3071.7471.2571.49
MRPCBERT-Tiny6L87.7168.3070.7671.7471.4968.3071.7469.0471.74
BERT-Base86.2468.3068.370.0270.2768.3071.2568.3069.04
BERT-Tiny4L65.3456.3153.4356.3155.5954.1557.7661.0159.20
RTEBERT-Tiny6L68.9556.3154.5154.5158.1249.0953.4358.8454.87
BERT-Base72.2053.4357.0455.2354.5154.8754.5155.2355.23
FSMN94.8956.4556.4568.6573.6075.0473.8056.45
Speech CommandsD-FSMN97.5188.3292.0378.9285.1156.7783.8092.1156.45 93.91
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NoiseFP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCU
Origin94.8289.6991.4091.55 91.2090.0491.55 92.7990.42
gaussian_noise-178.2374.2276.0074.9774.9575.0775.15 78.25 61.3277.36
gaussian_noise-256.7256.7362.4455.9458.3357.5255.9760.48
gaussian_noise-336.9342.6947.5839.56 43.4740.7937.99 43.3244.26
gaussian_noise-431.0338.3541.4333.24 36.6534.6831.47 35.9137.30
gaussian_noise-525.5434.0536.2228.6631.7830.1325.49 30.1932.09
ipulse_nosie-182.5484.5786.9484.6887.3085.7286.89 88.7385.8
ipulse_nosie-270.1277.1380.7477.3581.1479.6280.25 82.8080.3
ipulse_nosie-359.8870.5875.0169.2074.5971.8272.16 76.0572.6
ipulse_nosie-440.5954.4259.3949.4856.6652.6149.79 58.4456.45
ipulse_nosie-526.0339.8641.5432.7237.2835.1228.42 38.2639.98
shot_noise-185.7581.5181.3181.8481.5880.6681.88 83.9882.42
shot_noise-276.6172.0474.0272.2172.8173.0372.32 76.7075.1
shot_noise-352.2153.9057.0850.6654.5953.7651.31 57.2256.56
shot_noise-444.1347.5851.2943.5948.3646.6444.21 48.7848.91
shot_noise-532.7339.9340.7933.8038.5036.4731.79 36.4637.8
speckle_noise-186.3081.2981.9480.9380.7781.1482.17 84.1782.62
speckle_noise-271.9468.0770.1467.569.2269.3568.26 72.9471.70
speckle_noise-364.4762.1264.1360.2463.4462.5061.14 66.8964.27
speckle_noise-449.8151.9353.7747.9352.7550.5948.39 54.1352.40
speckle_noise-538.7044.2543.6038.6543.1642.0937.78 42.1342.57
gaussian_blur-194.1789.0390.589.3390.5689.0091.05 92.1689.33
gaussian_blur-287.0478.381.9878.8180.4277.7581.20 84.8078.93
gaussian_blur-375.1567.7468.2767.6768.1664.5467.42 73.6266.29
gaussian_blur-459.555.1753.6355.7454.0852.4452.72 60.3253.37
gaussian_blur-536.0337.3133.9637.5037.5436.7734.08 39.2234.93
defocus_blur-194.288.7391.0689.190.3288.9190.92 91.9889.58
defocus_blur-292.7585.9788.9986.5988.3185.5887.91 90.4787.01
defocus_blur-387.3879.0282.4378.8880.7177.5880.88 84.8579.52
defocus_blur-476.9969.1371.0268.2968.3365.9668.42 74.4068.22
defocus_blur-552.0948.8551.9948.8249.1748.4546.92 55.7048.27
glass_blur-154.9356.5751.7257.9456.7857.2956.27 58.8258.92
glass-blur-256.3757.9353.4660.4259.2159.3258.03 60.2560.56
glass_blur-359.2161.4356.9864.1161.7262.4160.3962.84 63.32
glass_blur-445.6546.5042.7248.4847.1947.8346.8849.09 49.23
glass_blur-549.1949.5246.4052.0649.8350.0249.0851.14 51.82
otion_blur-189.4081.5783.2782.0083.1181.4884.1982.83
otion_blur-281.9571.5274.7173.3872.4870.9974.3577.75 74.09
otion_blur-372.4861.8766.2163.8663.3961.5763.8568.31 64.36
otion_blur-472.7962.4066.1863.9462.8462.0364.54 67.8864.13
otion_blur-563.9154.1457.9856.0755.9054.3555.60 59.7156.65
zoo_blur-187.3678.2581.3178.5679.4577.2080.22 83.6978.55
zoo_blur-283.89 77.7374.84 69.0077.73 70.9875.3975.8872.8375.74 80.46 68.34 74.3374.72 68.21
zoo_blur-3 zoo_blur-4
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NoiseFP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCU
Origin94.8289.6991.4091.55 91.2090.0491.55 92.7990.42
gaussian_noise-178.2374.2276.0074.9774.9575.0775.15 78.25 61.3277.36
gaussian_noise-256.7256.7362.4455.9458.3357.5255.9760.48
gaussian_noise-336.9342.6947.5839.56 43.4740.7937.99 43.3244.26
gaussian_noise-431.0338.3541.4333.24 36.6534.6831.47 35.9137.30
gaussian_noise-525.5434.0536.2228.6631.7830.1325.49 30.1932.09
ipulse_nosie-182.5484.5786.9484.6887.3085.7286.89 88.7385.8
ipulse_nosie-270.1277.1380.7477.3581.1479.6280.25 82.8080.3
ipulse_nosie-359.8870.5875.0169.2074.5971.8272.16 76.0572.6
ipulse_nosie-440.5954.4259.3949.4856.6652.6149.79 58.4456.45
ipulse_nosie-526.0339.8641.5432.7237.2835.1228.42 38.2639.98
shot_noise-185.7581.5181.3181.8481.5880.6681.88 83.9882.42
shot_noise-276.6172.0474.0272.2172.8173.0372.32 76.7075.1
shot_noise-352.2153.9057.0850.6654.5953.7651.31 57.2256.56
shot_noise-444.1347.5851.2943.5948.3646.6444.21 48.7848.91
shot_noise-532.7339.9340.7933.8038.5036.4731.79 36.4637.8
speckle_noise-186.3081.2981.9480.9380.7781.1482.17 84.1782.62
speckle_noise-271.9468.0770.1467.569.2269.3568.26 72.9471.70
speckle_noise-364.4762.1264.1360.2463.4462.5061.14 66.8964.27
speckle_noise-449.8151.9353.7747.9352.7550.5948.39 54.1352.40
speckle_noise-538.7044.2543.6038.6543.1642.0937.78 42.1342.57
gaussian_blur-194.1789.0390.589.3390.5689.0091.05 92.1689.33
gaussian_blur-287.0478.381.9878.8180.4277.7581.20 84.8078.93
gaussian_blur-375.1567.7468.2767.6768.1664.5467.42 73.6266.29
gaussian_blur-459.555.1753.6355.7454.0852.4452.72 60.3253.37
gaussian_blur-536.0337.3133.9637.5037.5436.7734.08 39.2234.93
defocus_blur-194.288.7391.0689.190.3288.9190.92 91.9889.58
defocus_blur-292.7585.9788.9986.5988.3185.5887.91 90.4787.01
defocus_blur-387.3879.0282.4378.8880.7177.5880.88 84.8579.52
defocus_blur-476.9969.1371.0268.2968.3365.9668.42 74.4068.22
defocus_blur-552.0948.8551.9948.8249.1748.4546.92 55.7048.27
glass_blur-154.9356.5751.7257.9456.7857.2956.27 58.8258.92
glass-blur-256.3757.9353.4660.4259.2159.3258.03 60.2560.56
glass_blur-359.2161.4356.9864.1161.7262.4160.3962.84 63.32
glass_blur-445.6546.5042.7248.4847.1947.8346.8849.09 49.23
glass_blur-549.1949.5246.4052.0649.8350.0249.0851.14 51.82
otion_blur-189.4081.5783.2782.0083.1181.4884.1982.83
otion_blur-281.9571.5274.7173.3872.4870.9974.3577.75 74.09
otion_blur-372.4861.8766.2163.8663.3961.5763.8568.31 64.36
otion_blur-472.7962.4066.1863.9462.8462.0364.54 67.8864.13
otion_blur-563.9154.1457.9856.0755.9054.3555.60 59.7156.65
zoo_blur-187.3678.2581.3178.5679.4577.2080.22 83.6978.55
zoo_blur-283.89 77.7374.84 69.0077.73 70.9875.3975.8872.8375.74 80.46 68.34 74.3374.72 68.21
zoo_blur-3 zoo_blur-4
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NoiseFP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCU
fog-194.0488.1790.8988.84 89.9188.5190.84 92.0889.43
fog-293.0384.5888.8585.48 87.2684.7788 89.8786.76
fog-390.6978.0785.280.0783.3278.9483.77 86.8282.78
fog-486.7269.5678.9272.27 75.8971.0177.96 81.5675.62
fog-568.649.0453.952.33 52.8849.6857.67 62.2955.18
frost-189.9783.6684.784.0785.7683.6485.51 87.7584.85
frost-284.4277.8878.9777.4778.9677.2679.14 81.479.16
frost-374.8567.6769.367.14 68.7666.0369.58 72.5470.14
frost-473.3265.9367.1465.97 68.5265.3767.93 71.4469.41
frost-562.1355.0256.6755.62 56.7754.2657.69 61.1159.88
snow-189.2684.5885.4484.59 86.4385.0785.95 87.8285.67
snow-278.9672.0173.3773.14 73.4272.4373.65 78.0573.84
snow-382.8575.676.8475.74 76.9575.7477.76 79.9875.95
snow-480.2970.8472.5671.93 72.3971.2572.97 76.1272.16
snow-574.9463.8566.9465.9 65.7164.2466.29 70.3366.53
contrast-193.8287.2389.9687.93 89.6887.8890.16 91.5388.84
contrast-290.5376.028477.2782.176.382.8 86.0281.37
contrast-385.8463.9777.165.7772.7764.1874.62 79.3071.89
contrast-475.0844.0762.3747.35 55.5744.9459.87 65.7255.14
contrast-529.362025.6720.18 22.1821.0425.28 25.6624.04
elastic_transfor-189.9782.9784.583.3884.7482.4884.42 86.5483.25
elastic_transfor-289.4382.1284.7982.4484.0781.9784.61 86.2083.15
elastic_transfor-385.5277.5680.7177.9279.117779.53 82.2778.35
elastic_transfor-479.4873.7574.8373.9273.4172.7773.98 77.5374.17
elastic_transfor-575.0270.9770.2271.3671.0371.6371.65 75.3171.49
jpeg_copression-187.3683.2883.9384.0783.8383.7784.3 85.6583.63
jpeg_copression-281.6880.0979.6679.7780.0380.2980.59 81.6679.8
jpeg_copression-379.9878.5578.2178.3278.2778.9978.51 79.9478.44
jpeg_copression-477.1777.1275.7877.4477.0477.4677.08 77.6776.71
jpeg_copression-573.8574.5173.0474.6574.1375.2674.16 74.8574.19
pixelate-192.5786.9788.1987.3988.7387.4788.17 89.4287.26
pixelate-288.2381.9180.9582.3782.8281.9381.9983.80 80.95
pixelate-38478.475.2578.6378.0377.1577.2878.89 75.30
pixelate-468.5164.1158.1162.4960.8961.4360.0661.71 59.95
pixelate-550.5750.6844.7748.1845.4446.7443.2747.29 45.62
saturate-192.4184.9888.3885.3887.2685.3988.2389.34 86.82
saturate-290.1280.7485.2681.5782.6880.7484.2286.06 83.09
saturate-393.8387.8990.4588.1289.5388.0390.15 90.9788.73
saturate-491.6182.586.8882.684.6682.4486.04 87.5683.70
saturate-587.4876.0382.7675.5378.375.8580.62 82.6477.00
spatter-191.1787.589.7587.8389.3487.989.42 91.0088.98
spatter-285.283.8585.9883.5285.6484.7285.88 87.5985.00
spatter-380.6377.9480.3377.9580.1979.4180.07 82.6579.50
spatter-4
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NoiseFP32Binarization Algorithm
BNNXNORDoReFaBi-RealXNOR++ReActNetReCU
fog-194.0488.1790.8988.84 89.9188.5190.84 92.0889.43
fog-293.0384.5888.8585.48 87.2684.7788 89.8786.76
fog-390.6978.0785.280.0783.3278.9483.77 86.8282.78
fog-486.7269.5678.9272.27 75.8971.0177.96 81.5675.62
fog-568.649.0453.952.33 52.8849.6857.67 62.2955.18
frost-189.9783.6684.784.0785.7683.6485.51 87.7584.85
frost-284.4277.8878.9777.4778.9677.2679.14 81.479.16
frost-374.8567.6769.367.14 68.7666.0369.58 72.5470.14
frost-473.3265.9367.1465.97 68.5265.3767.93 71.4469.41
frost-562.1355.0256.6755.62 56.7754.2657.69 61.1159.88
snow-189.2684.5885.4484.59 86.4385.0785.95 87.8285.67
snow-278.9672.0173.3773.14 73.4272.4373.65 78.0573.84
snow-382.8575.676.8475.74 76.9575.7477.76 79.9875.95
snow-480.2970.8472.5671.93 72.3971.2572.97 76.1272.16
snow-574.9463.8566.9465.9 65.7164.2466.29 70.3366.53
contrast-193.8287.2389.9687.93 89.6887.8890.16 91.5388.84
contrast-290.5376.028477.2782.176.382.8 86.0281.37
contrast-385.8463.9777.165.7772.7764.1874.62 79.3071.89
contrast-475.0844.0762.3747.35 55.5744.9459.87 65.7255.14
contrast-529.362025.6720.18 22.1821.0425.28 25.6624.04
elastic_transfor-189.9782.9784.583.3884.7482.4884.42 86.5483.25
elastic_transfor-289.4382.1284.7982.4484.0781.9784.61 86.2083.15
elastic_transfor-385.5277.5680.7177.9279.117779.53 82.2778.35
elastic_transfor-479.4873.7574.8373.9273.4172.7773.98 77.5374.17
elastic_transfor-575.0270.9770.2271.3671.0371.6371.65 75.3171.49
jpeg_copression-187.3683.2883.9384.0783.8383.7784.3 85.6583.63
jpeg_copression-281.6880.0979.6679.7780.0380.2980.59 81.6679.8
jpeg_copression-379.9878.5578.2178.3278.2778.9978.51 79.9478.44
jpeg_copression-477.1777.1275.7877.4477.0477.4677.08 77.6776.71
jpeg_copression-573.8574.5173.0474.6574.1375.2674.16 74.8574.19
pixelate-192.5786.9788.1987.3988.7387.4788.17 89.4287.26
pixelate-288.2381.9180.9582.3782.8281.9381.9983.80 80.95
pixelate-38478.475.2578.6378.0377.1577.2878.89 75.30
pixelate-468.5164.1158.1162.4960.8961.4360.0661.71 59.95
pixelate-550.5750.6844.7748.1845.4446.7443.2747.29 45.62
saturate-192.4184.9888.3885.3887.2685.3988.2389.34 86.82
saturate-290.1280.7485.2681.5782.6880.7484.2286.06 83.09
saturate-393.8387.8990.4588.1289.5388.0390.15 90.9788.73
saturate-491.6182.586.8882.684.6682.4486.04 87.5683.70
saturate-587.4876.0382.7675.5378.375.8580.62 82.6477.00
spatter-191.1787.589.7587.8389.3487.989.42 91.0088.98
spatter-285.283.8585.9883.5285.6484.7285.88 87.5985.00
spatter-380.6377.9480.3377.9580.1979.4180.07 82.6579.50
spatter-4
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AlgorithmEpochOptimizerLearning RateSchedulerAcc.Acc.1Acc.2Acc.3Acc.4meanstd
FP32200SGD0.1cosine94.5894.695.0594.6494.8494.740.20
200SGD0.1step92.6392.4292.1592.6292.3892.440.20
200SGD0.01cosine92.2391.7691.7691.9992.1791.980.22
200SGD0.01step83.9483.5082.8084.1383.8983.650.53
200Adam0.001cosine93.5192.9493.1293.3592.8693.160.27
200Adam0.001step93.3793.1593.3293.4193.3593.320.10
200 200Adam0.0001cosine89.9789.9289.9689.989.9289.930.03
Adam0.0001step90.5789.9190.4390.2590.3190.290.25
BNN200SGD0.1cosine87.6287.5387.9988.8687.8487.970.53
200SGD0.1step70.8773.8671.8373.172.8772.511.17
200SGD0.01cosine73.5272.6272.8271.1472.5972.540.87
200SGD0.01step52.8551.7752.0052.3453.1452.420.57
200Adam0.001cosine88.7688.9988.6788.8488.8188.810.12
200Adam0.001step88.8589.3488.7789.0289.0089.000.22
200Adam0.0001cosine83.4683.0983.2083.7083.2083.330.25
200Adam0.0001step84.0884.1184.2084.3183.5684.050.29
XNOR200SGD0.1cosine91.8391.9991.8792.0191.5691.850.18
200SGD0.1step90.0290.0190.1289.8289.789.930.17
200SGD0.01cosine90.0989.6890.0189.9790.0089.950.16
200SGD0.01step86.8686.6687.2186.9886.6186.860.24
200Adam0.001cosine89.3989.8189.7389.9189.7589.720.20
200Adam0.001step89.9289.7989.7390.0189.6189.810.16
200Adam0.0001cosine86.1886.2987.0386.3686.6286.500.34
200Adam0.0001step86.3287.0486.6886.9987.1886.840.34
DoReFa200SGD0.1cosine85.6485.6785.8986.0085.7985.800.15
200SGD0.1step86.9586.9886.6986.6286.6586.780.17
200SGD0.01cosine86.5686.5986.5286.6986.8886.650.14
200SGD0.01step78.7679.9780.7379.9480.4779.970.76
200Adam0.001cosine88.8589.0688.9288.8788.7588.890.11
200Adam0.001step89.0889.1688.9389.2388.8489.050.16
200Adam0.0001cosine83.5683.1783.6583.6083.6683.530.20
200Adam0.0001step83.7083.7484.2784.1984.0183.980.26
Bi-Real200SGD0.1cosine87.5587.8188.0687.3087.8887.720.30
200SGD0.1step87.9588.3588.1387.7388.2588.080.25
200SGD0.01cosine87.7687.9387.7387.7287.6487.760.11 0.40
200 200SGD Adam0.01 0.001step cosine83.75 88.7882.91 89.1582.82 89.0682.91 89.0083.39 89.283.16 89.04
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AlgorithmEpochOptimizerLearning RateSchedulerAcc.Acc.1Acc.2Acc.3Acc.4meanstd
FP32200SGD0.1cosine94.5894.695.0594.6494.8494.740.20
200SGD0.1step92.6392.4292.1592.6292.3892.440.20
200SGD0.01cosine92.2391.7691.7691.9992.1791.980.22
200SGD0.01step83.9483.5082.8084.1383.8983.650.53
200Adam0.001cosine93.5192.9493.1293.3592.8693.160.27
200Adam0.001step93.3793.1593.3293.4193.3593.320.10
200 200Adam0.0001cosine89.9789.9289.9689.989.9289.930.03
Adam0.0001step90.5789.9190.4390.2590.3190.290.25
BNN200SGD0.1cosine87.6287.5387.9988.8687.8487.970.53
200SGD0.1step70.8773.8671.8373.172.8772.511.17
200SGD0.01cosine73.5272.6272.8271.1472.5972.540.87
200SGD0.01step52.8551.7752.0052.3453.1452.420.57
200Adam0.001cosine88.7688.9988.6788.8488.8188.810.12
200Adam0.001step88.8589.3488.7789.0289.0089.000.22
200Adam0.0001cosine83.4683.0983.2083.7083.2083.330.25
200Adam0.0001step84.0884.1184.2084.3183.5684.050.29
XNOR200SGD0.1cosine91.8391.9991.8792.0191.5691.850.18
200SGD0.1step90.0290.0190.1289.8289.789.930.17
200SGD0.01cosine90.0989.6890.0189.9790.0089.950.16
200SGD0.01step86.8686.6687.2186.9886.6186.860.24
200Adam0.001cosine89.3989.8189.7389.9189.7589.720.20
200Adam0.001step89.9289.7989.7390.0189.6189.810.16
200Adam0.0001cosine86.1886.2987.0386.3686.6286.500.34
200Adam0.0001step86.3287.0486.6886.9987.1886.840.34
DoReFa200SGD0.1cosine85.6485.6785.8986.0085.7985.800.15
200SGD0.1step86.9586.9886.6986.6286.6586.780.17
200SGD0.01cosine86.5686.5986.5286.6986.8886.650.14
200SGD0.01step78.7679.9780.7379.9480.4779.970.76
200Adam0.001cosine88.8589.0688.9288.8788.7588.890.11
200Adam0.001step89.0889.1688.9389.2388.8489.050.16
200Adam0.0001cosine83.5683.1783.6583.6083.6683.530.20
200Adam0.0001step83.7083.7484.2784.1984.0183.980.26
Bi-Real200SGD0.1cosine87.5587.8188.0687.3087.8887.720.30
200SGD0.1step87.9588.3588.1387.7388.2588.080.25
200SGD0.01cosine87.7687.9387.7387.7287.6487.760.11 0.40
200 200SGD Adam0.01 0.001step cosine83.75 88.7882.91 89.1582.82 89.0682.91 89.0083.39 89.283.16 89.04
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AlgorithmEpochOptimizerLearning RateSchedulerAcc. Acc.1Acc.2 Acc.3Acc.4meanstd
200SGD0.1cosine87.8288.4288.1288.5588.1988.220.28
200SGD0.1step73.5573.1175.0674.0573.7873.910.73
200SGD0.01cosine74.0375.0673.6474.5374.7174.390.56
200SGD0.01step53.5554.1654.0152.9154.3653.800.58
200Adam0.001cosine88.7788.6589.1088.6188.8188.790.19
200Adam0.001step89.1889.0589.2788.9389.0089.090.14
200Adam0.0001cosine83.8683.4983.5683.1683.6283.540.25
200 Adam0.0001step 83.4683.7784.4084.06 83.8283.900.35
ReActNet200SGD0.1cosine88.6088.5388.3888.4888.8988.580.19
200SGD0.1step88.4288.0188.1088.0288.4388.200.21
200SGD0.01cosine87.7587.8688.0087.8088.0287.890.12
200SGD0.01step83.2982.8983.6583.7683.2783.370.35
200Adam0.001cosine89.4789.2989.0189.0589.1489.190.19
200Adam0.001step89.2789.7489.4889.4089.3989.460.18
200Adam0.0001cosine84.6584.9384.4884.6584.6784.680.16
200Adam0.0001step84.6984.5584.9384.9485.3884.900.32
ReCU200SGD0.1cosine91.7291.9491.6891.6991.8191.770.11
200SGD0.1step87.7388.1487.8188.0287.9187.920.16
200SGD0.01cosine87.3287.2887.5387.4887.3287.390.11
200SGD0.01step71.8671.7271.7872.2671.5971.840.25
200Adam0.001cosine88.2489.9888.2688.4888.1388.620.77
200Adam0.001step88.3688.4888.5588.4288.6388.490.11
200Adam0.0001cosine80.0781.1080.6281.0979.9580.570.55
200 Adam0.0001step 81.2681.4281.0881.58 81.6981.410.24
FDA200SGD0.1cosine89.6989.5989.5689.5389.6589.600.07
200SGD0.1step80.3880.3480.8380.5280.5280.520.19
200SGD0.01cosine80.7280.9380.8980.7080.7980.810.10
200SGD0.01step63.4162.8563.0463.0463.1463.100.20
200Adam0.001cosine89.7089.5789.5789.8089.7689.680.11
200Adam0.001step89.8489.8590.1089.7990.0189.920.13
200Adam0.0001cosine89.5989.1089.3489.3189.5189.370.19
200Adam0.0001step89.5289.5989.5289.6489.5889.570.05
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AlgorithmEpochOptimizerLearning RateSchedulerAcc. Acc.1Acc.2 Acc.3Acc.4meanstd
200SGD0.1cosine87.8288.4288.1288.5588.1988.220.28
200SGD0.1step73.5573.1175.0674.0573.7873.910.73
200SGD0.01cosine74.0375.0673.6474.5374.7174.390.56
200SGD0.01step53.5554.1654.0152.9154.3653.800.58
200Adam0.001cosine88.7788.6589.1088.6188.8188.790.19
200Adam0.001step89.1889.0589.2788.9389.0089.090.14
200Adam0.0001cosine83.8683.4983.5683.1683.6283.540.25
200 Adam0.0001step 83.4683.7784.4084.06 83.8283.900.35
ReActNet200SGD0.1cosine88.6088.5388.3888.4888.8988.580.19
200SGD0.1step88.4288.0188.1088.0288.4388.200.21
200SGD0.01cosine87.7587.8688.0087.8088.0287.890.12
200SGD0.01step83.2982.8983.6583.7683.2783.370.35
200Adam0.001cosine89.4789.2989.0189.0589.1489.190.19
200Adam0.001step89.2789.7489.4889.4089.3989.460.18
200Adam0.0001cosine84.6584.9384.4884.6584.6784.680.16
200Adam0.0001step84.6984.5584.9384.9485.3884.900.32
ReCU200SGD0.1cosine91.7291.9491.6891.6991.8191.770.11
200SGD0.1step87.7388.1487.8188.0287.9187.920.16
200SGD0.01cosine87.3287.2887.5387.4887.3287.390.11
200SGD0.01step71.8671.7271.7872.2671.5971.840.25
200Adam0.001cosine88.2489.9888.2688.4888.1388.620.77
200Adam0.001step88.3688.4888.5588.4288.6388.490.11
200Adam0.0001cosine80.0781.1080.6281.0979.9580.570.55
200 Adam0.0001step 81.2681.4281.0881.58 81.6981.410.24
FDA200SGD0.1cosine89.6989.5989.5689.5389.6589.600.07
200SGD0.1step80.3880.3480.8380.5280.5280.520.19
200SGD0.01cosine80.7280.9380.8980.7080.7980.810.10
200SGD0.01step63.4162.8563.0463.0463.1463.100.20
200Adam0.001cosine89.7089.5789.5789.8089.7689.680.11
200Adam0.001step89.8489.8590.1089.7990.0189.920.13
200Adam0.0001cosine89.5989.1089.3489.3189.5189.370.19
200Adam0.0001step89.5289.5989.5289.6489.5889.570.05
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HardwareThreadsArch.LarqReAct FP32
FP32BNNBNN
Kirin 9701ResNet18716.427 1449.67123.263 159.615126.457427.585 72.585
ResNet34171.227 16.401836.321124.091
VGG-Small242.44314.833
ResNet18372.642 72.69778.605
2ResNet34732.35596.711108.41
VGG-Small121.9110.304 11.935=
ResNet18191.517 42.98647.182
4ResNet34367.89161.413 73.101 8.387
VGG-Small57.7218.72
ResNet1896.937 37.45756.017
8ResNet34 VGG-Small212.982 33.64753.80967.667
18.64919.818二1
ResNet18 307.624 ResNet3449.00950.018158.36331.803
Kirin 980507.734 83.16371.90974.920 8.215308.53753.031
VGG-Small7.772
ResNet18 187.49452.05754.285
2ResNet34367.853 57.336 6.11660.483 5.604
VGG-Small49.264
ResNet18104.076 29.55635.539
4ResNet34 VGG-Small202.173 31.32435.911 3.291
22.6903.147
ResNet18 ResNet3460.307 45.68356.416二 =
VGG-Small120.738 18.14760.758 21.68886.887 23.3501
ResNet18173.23834.528
438.97127.429 58.16530.626 60.885164.55657.808
Kirin 98570.7976.1476.796323.439
VGG-Small
ResNet18 103.621 327.41625.67235.477
2ResNet3453.94962.865
VGG-Small55.328 5.9556.243
ResNet1892.387 26.72834.778
4ResNet34184.050 39.88152.153
VGG-Small28.076 8.91914.795
ResNet1882.77289.766=
8ResNet34 VGG-Small227.504119.586143.9581
44.33934.03443.8161
ResNet18 ResNet34114.23821.23522.066144.20529.239
Kirin 9901227.043 38.11831.545 3.33832.821275.50249.476 1
VGG-Small3.482
ResNet18 59.32913.91114.179一
ResNet34116.82223.45222.770
VGG-Small 20.0552.0802.194
ResNet1838.40310.28012.208
4ResNet3481.27315.57017.727
VGG-Small13.5081.5421.760
ResNet1837.70325.360
31.365 39.363
8ResNet34 VGG-Small78.753 12.70734.884 14.41421.749
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HardwareThreadsArch.LarqReAct FP32
FP32BNNBNN
Kirin 9701ResNet18716.427 1449.67123.263 159.615126.457427.585 72.585
ResNet34171.227 16.401836.321124.091
VGG-Small242.44314.833
ResNet18372.642 72.69778.605
2ResNet34732.35596.711108.41
VGG-Small121.9110.304 11.935=
ResNet18191.517 42.98647.182
4ResNet34367.89161.413 73.101 8.387
VGG-Small57.7218.72
ResNet1896.937 37.45756.017
8ResNet34 VGG-Small212.982 33.64753.80967.667
18.64919.818二1
ResNet18 307.624 ResNet3449.00950.018158.36331.803
Kirin 980507.734 83.16371.90974.920 8.215308.53753.031
VGG-Small7.772
ResNet18 187.49452.05754.285
2ResNet34367.853 57.336 6.11660.483 5.604
VGG-Small49.264
ResNet18104.076 29.55635.539
4ResNet34 VGG-Small202.173 31.32435.911 3.291
22.6903.147
ResNet18 ResNet3460.307 45.68356.416二 =
VGG-Small120.738 18.14760.758 21.68886.887 23.3501
ResNet18173.23834.528
438.97127.429 58.16530.626 60.885164.55657.808
Kirin 98570.7976.1476.796323.439
VGG-Small
ResNet18 103.621 327.41625.67235.477
2ResNet3453.94962.865
VGG-Small55.328 5.9556.243
ResNet1892.387 26.72834.778
4ResNet34184.050 39.88152.153
VGG-Small28.076 8.91914.795
ResNet1882.77289.766=
8ResNet34 VGG-Small227.504119.586143.9581
44.33934.03443.8161
ResNet18 ResNet34114.23821.23522.066144.20529.239
Kirin 9901227.043 38.11831.545 3.33832.821275.50249.476 1
VGG-Small3.482
ResNet18 59.32913.91114.179一
ResNet34116.82223.45222.770
VGG-Small 20.0552.0802.194
ResNet1838.40310.28012.208
4ResNet3481.27315.57017.727
VGG-Small13.5081.5421.760
ResNet1837.70325.360
31.365 39.363
8ResNet34 VGG-Small78.753 12.70734.884 14.41421.749
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HardwareThreadsArch.LarqdaBNN
FP32BNNReAct FP32BNN
Kirin 9000E1ResNet18118.05919.865 20.547129.27024.781
ResNet34236.04731.82232.575250.68042.134
VGG-Small39.1143.595 3.832一=
2ResNet1868.35116.82116.115
ResNet34133.67125.06124.660
VGG-Small23.0182.6842.598一
4ResNet1845.59217.452 18.847
ResNet3491.64823.39528.022
VGG-Small14.3602.7622.782
8ResNet1843.36361.26342.328
ResNet34 VGG-Small89.405 19.07070.232 17.35193.558
ResNet18158.83532.63623.825
Dimensity 8201ResNet34328.020 57.13334.912 60.807323.035 629.49363.471 102.443
2VGG-Small82.4175.958 6.420一
122.1671
ResNet18 ResNet34250.08829.306 43.306 50.14334.384
4VGG-Small51.3204.6705.053
ResNet1894.63621.85030.027
ResNet34177.75733.80940.816
VGG-Small45.0564.2234.546
8ResNet18 ResNet3490.210 45.35761.981
VGG-Small 32.971166.989 68.444 21.34474.286 23.706
ResNet18106.38821.02324.770148.69029.030
1ResNet34 VGG-Small210.665 42.05732.84134.590284.43849.854
2ResNet1881.6064.410 22.6615.530 27.050
ResNet3427.666
VGG-Small143.349 26.5122.27337.910 2.410
ResNet1851.42113.07915.200
8ResNet34100.24923.31425.920
VGG-Small17.7353.0153.770
ResNet18 ResNet3443.35524.93930.740
1VGG-Small84.18230.212 14.25839.9901
14.85717.540131.174
90.430 19.76920.530163.293
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VGG-Small29.7353.1533.25911
ResNet1858.510 25.78026.331
2ResNet34124.58031.02332.646
VGG-Small19.4082.2582.471
ResNet1839.26919.86523.297
ResNet3482.18030.24831.387
VGG-Small13.5662.0322.359
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ResNet1836.630 49.06085.861
ResNet3473.51341.13188.101
VGG-Small12.86017.82823.489
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FP32BNNReAct FP32BNN
Kirin 9000E1ResNet18118.05919.865 20.547129.27024.781
ResNet34236.04731.82232.575250.68042.134
VGG-Small39.1143.595 3.832一=
2ResNet1868.35116.82116.115
ResNet34133.67125.06124.660
VGG-Small23.0182.6842.598一
4ResNet1845.59217.452 18.847
ResNet3491.64823.39528.022
VGG-Small14.3602.7622.782
8ResNet1843.36361.26342.328
ResNet34 VGG-Small89.405 19.07070.232 17.35193.558
ResNet18158.83532.63623.825
Dimensity 8201ResNet34328.020 57.13334.912 60.807323.035 629.49363.471 102.443
2VGG-Small82.4175.958 6.420一
122.1671
ResNet18 ResNet34250.08829.306 43.306 50.14334.384
4VGG-Small51.3204.6705.053
ResNet1894.63621.85030.027
ResNet34177.75733.80940.816
VGG-Small45.0564.2234.546
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VGG-Small 32.971166.989 68.444 21.34474.286 23.706
ResNet18106.38821.02324.770148.69029.030
1ResNet34 VGG-Small210.665 42.05732.84134.590284.43849.854
2ResNet1881.6064.410 22.6615.530 27.050
ResNet3427.666
VGG-Small143.349 26.5122.27337.910 2.410
ResNet1851.42113.07915.200
8ResNet34100.24923.31425.920
VGG-Small17.7353.0153.770
ResNet18 ResNet3443.35524.93930.740
1VGG-Small84.18230.212 14.25839.9901
14.85717.540131.174
90.430 19.76920.530163.293
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VGG-Small29.7353.1533.25911
ResNet1858.510 25.78026.331
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VGG-Small19.4082.2582.471
ResNet1839.26919.86523.297
ResNet3482.18030.24831.387
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HardwareThreadsArch.LarqdaBNN
FP32BNNReActFP32BNN
Snapdragon 8701ResNet1888.14516.52717.020126.76225.240
ResNet34185.46825.48826.195237.36141.440
VGG-Small30.3182.8512.964
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ResNet34159.17425.35226.340
VGG-Small27.6692.0942.308
442.79617.578
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VGG-Small14.8292.6142.215
846.79819.19228.579
ResNet18 ResNet3497.83425.06032.863
VGG-Small16.7999.71717.2931
Snapdragon 8881ResNet1877.20515.54716.111123.61825.240
ResNet34152.88722.90623.893234.97241.648
VGG-Small25.1332.4102.543
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4ResNet3493.61520.47322.489
VGG-Small16.0011.9202.213
ResNet1833.52413.69914.332
ResNet3467.49519.02021.157
8VGG-Small11.7432.8822.768
ResNet1833.76126.10858.989
ResNet3467.87637.01861.315
1VGG-Small11.75227.61516.774
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VGG-Small257.07924.479254.810 25.7902774.888
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933.149144.287158.868
VGG-Small145.42714.50315.628
ResNet18562.567108.585116.640
ResNet34976.223159.258183.698
8VGG-Small191.47010.83910.196
ResNet18877.026279.660356.239
ResNet341638.035389.924485.260=
Raspberrypi 4BVGG-Small 399.338110.448142.978二-
ResNet18448.74480.82282.380688.838120.348209.276
ResNet34897.735112.837119.5361362.893
VGG-Small150.81411.17712.024
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VGG-Small89.2846.6477.882
ResNet18270.19136.33145.903
ResNet34572.42353.86670.841
VGG-Small90.6505.0566.167
ResNet18466.585168.844226.771
ResNet34879.375 216.439264.638 114.064319.789 162.118
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FP32BNNReActFP32BNN
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ResNet34185.46825.48826.195237.36141.440
VGG-Small30.3182.8512.964
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ResNet34159.17425.35226.340
VGG-Small27.6692.0942.308
442.79617.578
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VGG-Small14.8292.6142.215
846.79819.19228.579
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VGG-Small16.7999.71717.2931
Snapdragon 8881ResNet1877.20515.54716.111123.61825.240
ResNet34152.88722.90623.893234.97241.648
VGG-Small25.1332.4102.543
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4ResNet3493.61520.47322.489
VGG-Small16.0011.9202.213
ResNet1833.52413.69914.332
ResNet3467.49519.02021.157
8VGG-Small11.7432.8822.768
ResNet1833.76126.10858.989
ResNet3467.87637.01861.315
1VGG-Small11.75227.61516.774
ResNet18 ResNet34740.509 1536.915158.732 240.606175.2561460.723241.713 435.170
VGG-Small257.07924.479254.810 25.7902774.888
Raspberrypi 3B+2ResNet18 ResNet34667.012 143.716106.894
933.149144.287158.868
VGG-Small145.42714.50315.628
ResNet18562.567108.585116.640
ResNet34976.223159.258183.698
8VGG-Small191.47010.83910.196
ResNet18877.026279.660356.239
ResNet341638.035389.924485.260=
Raspberrypi 4BVGG-Small 399.338110.448142.978二-
ResNet18448.74480.82282.380688.838120.348209.276
ResNet34897.735112.837119.5361362.893
VGG-Small150.81411.17712.024
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VGG-Small89.2846.6477.882
ResNet18270.19136.33145.903
ResNet34572.42353.86670.841
VGG-Small90.6505.0566.167
ResNet18466.585168.844226.771
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LarqdaBNN
HardwareThreadsArch.FP32BNNReActFP32BNN
Apple M1ResNet1844.3348.2198.355二1
ResNet3488.33412.50512.771二一
VGG-Small14.0931.4461.465
ResNet1824.7755.0375.1941
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VGG-Small7.3980.8290.8541
ResNet1818.6123.4483.671
ResNet3427.5154.9655.254
VGG-Small4.2940.5260.551
ResNet1816.6535.0356.003
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VGG-Small3.9960.7350.712
ResNet1890.32370.69773.729
16ResNet34162.057130.907125.362二
VGG-Small25.36623.05023.194
ResNet1846.0538.6538.486二
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VGG-Small14.2851.4541.488
ResNet1825.0395.4505.361
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VGG-Small7.6570.8550.896
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ResNet1810.6603.7184.510
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LarqdaBNN
HardwareThreadsArch.FP32BNNReActFP32BNN
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ResNet1818.6123.4483.671
ResNet3427.5154.9655.254
VGG-Small4.2940.5260.551
ResNet1816.6535.0356.003
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VGG-Small3.9960.7350.712
ResNet1890.32370.69773.729
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VGG-Small25.36623.05023.194
ResNet1846.0538.6538.486二
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VGG-Small14.2851.4541.488
ResNet1825.0395.4505.361
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VGG-Small7.6570.8550.896
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VGG-Small4.2920.5760.620
ResNet1810.6603.7184.510
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ResNet1860.50047.72753.900
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VGG-Small21.35413.86815.311
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b/parse/dev/jmopGajkFY/images/e8f2183f465de6cfcd424821086443bc2909eaf40e29cb342a0af67e20074a66.jpg new file mode 100644 index 0000000000000000000000000000000000000000..20103ed386509d8d179c16e97b1bf1c1519f26b2 --- /dev/null +++ b/parse/dev/jmopGajkFY/images/e8f2183f465de6cfcd424821086443bc2909eaf40e29cb342a0af67e20074a66.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9be7a6efea0c0d36594e3b4119e77a077424a03a6a0a451456af406e77b2d34a +size 128834 diff --git a/parse/dev/k9bx1EfHI_-/k9bx1EfHI_-.md b/parse/dev/k9bx1EfHI_-/k9bx1EfHI_-.md new file mode 100644 index 0000000000000000000000000000000000000000..4426aae4ac95803c4cfd558a528fb0d901a97f6a --- /dev/null +++ b/parse/dev/k9bx1EfHI_-/k9bx1EfHI_-.md @@ -0,0 +1,389 @@ +# SELF-SUPERVISED GRAPH NEURAL NETWORKS FOR IMPROVED ELECTROENCEPHALOGRAPHIC SEIZURE ANALYSIS + +Siyi Tang, Jared Dunnmon, Khaled Saab, Xuan Zhang†, Qianying Huang†, Florian Dubost, Daniel Rubin‡, Christopher Lee-Messer‡ +Stanford University, CA, USA +{siyitang,jdunnmon,ksaab,kayleez,qyhuang,fdubost,rubin,cleemess} @stanford.edu +† Equal contributions. ‡ Equal contributions. + +# ABSTRACT + +Automated seizure detection and classification from electroencephalography (EEG) can greatly improve seizure diagnosis and treatment. However, several modeling challenges remain unaddressed in prior automated seizure detection and classification studies: (1) representing non-Euclidean data structure in EEGs, (2) accurately classifying rare seizure types, and (3) lacking a quantitative interpretability approach to measure model ability to localize seizures. In this study, we address these challenges by (1) representing the spatiotemporal dependencies in EEGs using a graph neural network (GNN) and proposing two EEG graph structures that capture the electrode geometry or dynamic brain connectivity, (2) proposing a self-supervised pre-training method that predicts preprocessed signals for the next time period to further improve model performance, particularly on rare seizure types, and (3) proposing a quantitative model interpretability approach to assess a model’s ability to localize seizures within EEGs. When evaluating our approach on seizure detection and classification on a large public dataset (5,499 EEGs), we find that our GNN with self-supervised pre-training achieves 0.875 Area Under the Receiver Operating Characteristic Curve on seizure detection and 0.749 weighted F1-score on seizure classification, outperforming previous methods for both seizure detection and classification. Moreover, our self-supervised pre-training strategy significantly improves classification of rare seizure types (e.g. 47 points increase in combined tonic seizure accuracy over baselines). Furthermore, quantitative interpretability analysis shows that our GNN with self-supervised pre-training precisely localizes $2 5 . 4 \%$ focal seizures, a 21.9 point improvement over existing CNNs. Finally, by superimposing the identified seizure locations on both raw EEG signals and EEG graphs, our approach could provide clinicians with an intuitive visualization of localized seizure regions. + +# 1 INTRODUCTION + +Seizures are among the most common neurological emergencies in the world (Strein et al., 2019). Seizures can be chronic as in the case of epilepsy, a neurological disease affecting 50 million people worldwide (WHO, 2019). Clinically, definitive detection of a seizure is only the first step in seizure diagnosis. An important subsequent step is to classify seizures into finer-grained types––such as focal versus generalized seizures––for identifying epilepsy syndromes, targeted therapies, and eligibility for epilepsy surgery (Fisher et al., 2017). + +Scalp electroencephalography (or “EEG”) plays a critical role in seizure detection and classification. Clinically, EEG-based seizure detection and classification are performed by a trained EEG reader who visually examines a patient’s EEG signals over time periods ranging from hours to days. However, this manual analysis is extremely resource- and time-intensive, and thus automated algorithms could greatly accelerate seizure diagnosis and improve outcomes for severely ill patients. + +![](images/48275fbc969bc964f0d407ba19c38d1c7a2054aea09af82c5827a6f3345e271f.jpg) +Figure 1: (a) EEG electrode placement in the standard 10-20 system. (b) Distance-based EEG graph. (c) An example correlation-based EEG graph. (d) Overview of our methods. The inputs to the models are the EEG graphs, where each node feature corresponds to the preprocessed EEG signals in the respective channel. Self-edges are not shown for better visualization. + +Although a large number of studies have attempted automated seizure detection (Rasheed et al., 2020; Siddiqui et al., 2020; Shoeibi et al., 2021; O’Shea et al., 2020; Saab et al., 2020) or seizure classification (Raghu et al., 2020; Asif et al., 2020; Iesmantas & Alzbutas, 2020; Ahmedt-Aristizabal et al., 2020; Roy et al., 2019), several challenges remain largely unaddressed. First, most recent studies use convolutional neural networks (CNNs) that assume Euclidean structures in EEG signals or spectrograms (Rasheed et al., 2020; Shoeibi et al., 2021; Raghu et al., 2020; Asif et al., 2020; Iesmantas & Alzbutas, 2020; Ahmedt-Aristizabal et al., 2020; Roy et al., 2019; O’Shea et al., 2020; Saab et al., 2020). However, assumption of Euclidean structure ignores the natural geometry of EEG electrodes and the connectivity in brain networks. EEGs are measured by electrodes placed on a manifold (i.e. patient’s scalp) (Figure 1a), and thus have inherent non-Euclidean structures. Graphs are a data structure that can represent complex, non-Euclidean data (Chami et al., 2020; Bronstein et al., 2017), and graph theory has been extensively used in modeling brain networks (Bullmore & Sporns, 2009). We therefore hypothesize that graph-based modeling approaches can better represent the inherent non-Euclidean structures in EEGs in a manner that improves both the performance and the clinical utility of seizure detection and classification models. Although traditional graph theory has been used (Supriya et al., 2021), only a few deep learning studies have modeled EEGs as graphs for seizure detection. However, these graph-based studies were limited to nonpublic (Covert et al., 2019) or small datasets (Craley et al., 2019; Li et al., 2021), did not leverage modern self-supervised approaches or examine seizure classification (Cisotto et al., 2020; Zhao et al., 2021; Li et al., 2021). + +Second, certain seizure types (e.g. clonic seizures) are rare by nature. Training machine learning models that perform well on these rarer seizure classes using traditional supervised learning approaches is challenging, which could explain the performance difference between major and minority seizure types in prior studies (Raghu et al., 2020; Iesmantas & Alzbutas, 2020; AhmedtAristizabal et al., 2020). Several studies have investigated an alternative, self-supervised training strategy (Banville et al., 2020; Mohsenvand et al., 2020; Kostas et al., 2021; Martini et al., 2021; Xu et al., 2020), but they did not model EEGs as graphs or address automated seizure classification. Prior works have shown that self-supervised pre-training significantly improves model performance on data with imbalanced labels in the field of computer vision (Yang & Xu, 2020; Liu et al., 2021). Hence, we hypothesize that self-supervised pre-training can help improve our graph model performance on rare seizure types. Moreover, a large portion of EEG signals generally do not have seizures. Self-supervised pre-training strategy would allow the model to leverage the abundant nonseizure EEGs that are readily available in the dataset. + +Lastly, for seizure detection and classification models, the ability to not only provide a single prediction across all EEG channels, but also to provide the interpretability and the ability to localize seizures would be clinically useful for informing treatment strategy. While prior studies (Saab et al., 2020; Covert et al., 2019) have shown qualitative visualization for model interpretability, none have quantitatively assessed the model’s ability to localize seizures. + +In this work, we aim to address these limitations in prior automated seizure detection and classification studies. First, we propose a graph-based modeling approach for EEG-based seizure detection and classification. Specifically, we propose two EEG graph structures that capture EEG sensor geometry (Figure 1b) or dynamic brain connectivity (Figure 1c), and we extend Diffusion Convolutional Recurrent Neural Network (DCRNN) (Li et al., 2018), an RNN with graph diffusion convolutions, to model the spatiotemporal dependencies in EEGs (Figure 1d). Second, we improve DCRNN performance using a self-supervised pre-training strategy of predicting the preprocessed EEG signals for the next time period without requiring additional data or labels. Finally, we propose quantitative metrics to assess our model’s ability to localize seizures. In summary: + +• We propose two EEG graph structures that capture (1) the natural geometry of EEG sensors or (2) dynamic connectivity in the brain, and show that building a recurrent graph neural network (GNN) based on these representations yields models for seizure detection and classification that outperform previous approaches on a large public dataset (5,499 EEGs). • We propose a self-supervised pre-training strategy to further improve our recurrent GNN model performance, particularly on rare seizure types. To our knowledge, our study is the first to date that combines graph-based modeling and self-supervised pre-training for EEGs. By leveraging graph structure and self-supervision, our method achieves 0.875 Area Under the Receiver Operating Characteristic Curve (AUROC) on seizure detection and 0.749 weighted F1-score on seizure classification, outperforming previous approaches on both seizure detection and classification on this large public dataset. Moreover, our selfsupervised pre-training method substantially improves classification of rare seizure types (e.g. 47 points increase in combined tonic seizure accuracy over baselines). • We propose a quantitative model interpretability analysis that can be used to assess a model’s ability to localize seizures, which is critical to determining the course of treatment for seizures. We show that by leveraging graph structure and self-supervision, our method precisely localizes $2 5 . 4 \%$ focal seizures, providing 21.9 points improvement over a prior state-of-the-art CNN. Finally, by displaying the identified seizure regions on raw EEG signals and EEG graphs, our approach could provide valuable insights that support more effective seizure diagnosis in real-world clinical settings. + +# 2 METHODS + +# 2.1 SEIZURE DETECTION AND CLASSIFICATION PROBLEM FORMULATION + +The goal of seizure detection is to predict whether a seizure exists within an EEG clip, and the goal of seizure classification is to predict the seizure type given a seizure EEG clip. Following a prior study (Saab et al., 2020), we examine our model’s capability for fast and slow detection and classification over 12-s and 60-s EEG clips, respectively. + +# 2.2 GRAPH-BASED MODELING FOR EEGS + +# 2.2.1 REPRESENTING EEGS AS GRAPHS + +We represent an EEG clip as a graph $\mathcal { G } = \{ \mathcal { V } , \mathcal { E } , W \}$ , where $\nu$ denotes the set of nodes (i.e. EEG electrodes/channels), $\mathcal { E }$ denotes the set of edges, and $W$ is the adjacency matrix. We propose the following two methods of constructing the EEG graph. + +Distance graph. To represent the natural geometry of EEG electrodes, we compute edge weight $W _ { i j }$ by applying a thresholded Gaussian kernel (Shuman et al., 2013) to the pairwise Euclidean distance between $v _ { i }$ and $v _ { j }$ , i.e., $W _ { i j } = \exp \biggl ( - \frac { \mathrm { d i s t } ( v _ { i } , v _ { j } ) ^ { 2 } } { \sigma ^ { 2 } } \biggr )$ if $\mathrm { d i s t } ( v _ { i } , v _ { j } ) \le \kappa$ , else 0. Here, $\mathrm { d i s t } ( v _ { i } , v _ { j } )$ is the Euclidean distance between electrodes $v _ { i }$ and $v _ { j }$ according to the standard 10-20 EEG electrode placement (Jasper, 1958), $\sigma$ is the standard deviation of the distances, and $\kappa$ is the threshold for sparsity. This results in a universal undirected, weighted graph for all EEG clips. Based on preliminary experiments and EEG domain knowledge, we chose $\kappa = 0 . 9$ because it results in a reasonable graph that also resembles the EEG montage (longitudinal bipolar and transverse bipolar) widely used clinically (Acharya et al., 2016). Figure 1b shows the distance graph with $\kappa = 0 . 9$ . In Appendix K, we explore different values of $\kappa$ as well as constructing the distance graph using a Gaussian kernel with a pre-specified bandwidth. + +Correlation graph. To capture dynamic brain connectivity, we define the edge weight $W _ { i j }$ as the absolute value of the normalized cross-correlation between the preprocessed signals in $v _ { i }$ and $v _ { j }$ . To introduce sparsity to the graph, only the edges whose weights are among the top- $\tau$ neighbors of each node are kept (plus self-edges), i.e., $\bar { W _ { i j } } = | X _ { : , i , : } * \mathbf { \bar { X } } _ { : , j , : } |$ if $v _ { j } \in \mathcal { N } ( v _ { i } )$ , else 0. Here, $X _ { : , i , : }$ : and $X _ { : , j , }$ : are preprocessed signals in $v _ { i }$ and $v _ { j }$ , $^ *$ represents the normalized cross-correlation, and $\mathcal { N } ( v _ { i } )$ represents the top- $\tau$ neighbors of $v _ { i }$ . This method results in a unique directed, weighted graph for each input EEG clip. Figure 1c shows an example correlation graph with $\tau = 3$ . + +# 2.2.2 GRAPH NEURAL NETWORK + +We adapt DCRNN (Li et al., 2018), a recurrent neural network with graph diffusion convolutions, to model the spatiotemporal dependencies in EEG signals. DCRNN was initially developed for traffic forecasting, where the dynamics of traffic flow are modeled as a diffusion process. Similarly, we can also model the spatial dependency in EEG signals as a diffusion process, because an electrode can be influenced more by electrodes in its anatomical proximity (measured by distance) (Acharya et al., 2016) or functional proximity (measured by correlation) (Sakkalis, 2011). Specifically, the diffusion process is characterized by a bidirectional random walk on a directed graph $\mathcal { G }$ , which results in the following diffusion convolution (Li et al., 2018): + +$$ +X _ { : , m \star \mathcal { G } } f _ { \theta } = \sum _ { k = 0 } ^ { K - 1 } \big ( \theta _ { k , 1 } ( D _ { \mathcal { O } } ^ { - 1 } W ) ^ { k } + \theta _ { k , 2 } ( D _ { I } ^ { - 1 } W ^ { \top } ) ^ { k } \big ) X _ { : , m } \mathrm { f o r } m \in \{ 1 , . . . , M \} +$$ + +where $\pmb { X } \in \mathbb { R } ^ { N \times M }$ is the preprocessed EEG clip at time step $t \in \{ 1 , . . . , T \}$ with $N$ nodes and $M$ features, $f _ { \theta }$ is the convolution filter with parameters $\boldsymbol { \theta } \in \mathbb { R } ^ { K \times 2 }$ , $D _ { O }$ and $D _ { I }$ are the out-degree and in-degree diagonal matrices of the graph, respectively, $D _ { O } ^ { - 1 } W$ and $D _ { I } ^ { - 1 } W ^ { \top }$ are the state transition matrices of the outward and inward diffusion processes, respectively, and $K$ is the number of maximum diffusion steps. + +For undirected graphs, the diffusion convolution is similar to ChebNet spectral graph convolution (Defferrard et al., 2016) up to a constant scaling factor, and thus can be computed using stable Chebyshev polynomial bases as follows (Li et al., 2018): + +$$ +{ \cal X } _ { : , m \star \mathcal { G } } f _ { \theta } = \Phi \bigg ( \sum _ { k = 0 } ^ { K - 1 } \theta _ { k } { \bf A } ^ { k } \bigg ) \Phi ^ { \top } { \cal X } _ { : , m } = \sum _ { k = 0 } ^ { K - 1 } \theta _ { k } L ^ { k } { \cal X } _ { : , m } = \sum _ { k = 0 } ^ { K - 1 } \tilde { \theta } _ { k } T _ { k } ( \tilde { L } ) { \cal X } _ { : , m } \mathrm { f o r } m \in \{ 1 , . . . , M \} +$$ + +where $T _ { 0 } ( x ) = 1 , T _ { 1 } ( x ) = x$ , and $T _ { k } ( x ) = 2 x T _ { k - 1 } ( x ) - T _ { k - 2 } ( x )$ for $k \geq 2$ are the bases of the Chebyshev polynomial, $L = D ^ { - { \frac { 1 } { 2 } } } ( D - W ) D ^ { - { \frac { 1 } { 2 } } } = \Phi \Lambda \Phi ^ { \top }$ is the normalized graph Laplacian, and $\begin{array} { r } { \tilde { \pmb { L } } = \frac { 2 } { \lambda _ { m a x } } \pmb { L } - \pmb { I } } \end{array}$ is the scaled graph Laplacian mapping eigenvalues from $[ 0 , \lambda _ { m a x } ]$ to $[ - 1 , 1 ]$ . We use Equation 1 for directed correlation graphs, and Equation 2 for undirected distance graph. + +Next, to model the temporal dependency in EEGs, we employ Gated Recurrent Units (GRUs) (Cho et al., 2014), a variant of RNN with a gating mechanism. Specifically, the matrix multiplications in GRUs are replaced with diffusion convolutions (or ChebNet spectral graph convolutions for undirected distance-based graph) (Li et al., 2018), allowing spatiotemporal modeling of EEG signals (referred to as “DCGRU”): + +$$ +\pmb { r } ^ { ( t ) } = \sigma \big ( \Theta _ { r \star \mathscr { G } } [ \pmb { X } ^ { ( t ) } , \pmb { H } ^ { ( t - 1 ) } ] + \mathbf { b } _ { r } \big ) \qquad \pmb { u } ^ { ( t ) } = \sigma \big ( \Theta _ { u \star \mathscr { G } } [ \pmb { X } ^ { ( t ) } , \pmb { H } ^ { ( t - 1 ) } ] + \mathbf { b } _ { u } \big ) +$$ + +$$ +C ^ { ( t ) } = \mathrm { t a n h } \big ( \Theta _ { C \times \mathcal { G } } [ \mathbf { X } ^ { ( t ) } , ( { \boldsymbol { r } } ^ { ( t ) } \odot \pmb { H } ^ { ( t - 1 ) } ) ] + \mathbf { b } _ { C } \big ) \quad \pmb { H } ^ { ( t ) } = \pmb { u } ^ { ( t ) } \odot \pmb { H } ^ { ( t - 1 ) } + ( 1 - \pmb { u } ^ { ( t ) } ) \odot \pmb { C } ^ { ( t ) } +$$ + +Here, $\mathbf { \boldsymbol { X } } ^ { ( t ) } , \mathbf { \boldsymbol { H } } ^ { ( t ) }$ denote the input and output of DCGRU at time step $t$ respectively, $\sigma$ denotes Sigmoid function, $\odot$ represents the Hadamard product, $\boldsymbol { r } ^ { ( t ) } , \boldsymbol { u } ^ { ( t ) }$ , $C ^ { ( t ) }$ denote reset gate, update gate and candidate at time step $t$ respectively, $\star \mathcal { G }$ denotes the diffusion convolution (or ChebNet spectral graph convolution), $\Theta _ { r }$ , ${ \bf b } _ { r }$ , $\Theta _ { u }$ $\mathbf { \Delta } ) _ { u } , \mathbf { b } _ { u } , \mathbf { e }$ $\Theta _ { C }$ and ${ \bf b } _ { C }$ are the weights and biases for the corresponding convolutional filters. Finally, for seizure detection and classification, the models consist of several stacked DCGRUs followed by a fully-connected layer. + +# 2.3 SELF-SUPERVISED PRE-TRAINING + +To further improve DCRNN performance, we propose a self-supervised pre-training strategy for EEGs. Specifically, we pre-train the model for predicting the next $T ^ { \prime }$ second preprocessed EEG clips given a preprocessed 12-s (60-s) EEG clip. We hypothesize that by learning to predict the EEG signals for the next time period, the model would learn task-agnostic representations and improve downstream seizure detection and classification tasks. The model for self-supervised pre-training has a sequence-to-sequence architecture with an encoder and a decoder (Sutskever et al., 2014), each of which has several stacked DCGRUs (Figure 1d). We use mean absolute error between the true preprocessed EEG clips and the predicted clips as the loss function. Preliminary experiments suggest that predicting future $T ^ { \prime } = 1 2$ second preprocessed EEG clips results in low regression loss on the validation set, and thus we use $T ^ { \prime } = 1 2$ in all self-supervised pre-training experiments. + +# 2.4 MODEL INTERPRETABILITY AND ABILITY TO LOCALIZE SEIZURES + +We perform model interpretability analyses using an occlusion-based approach (Zeiler & Fergus, 2013). First, for seizure detection, we zero-fill one second EEG signals in one channel at a time and compute the relative change in the model’s output logit with respect to the original non-occluded output (original − occluded). Furthermore, we scale the values in each clip to $[ 0 , 1 ]$ using the minimum and maximum values in that clip. This results in an occlusion map $M \in \mathbb { R } ^ { \bar { N } \times T }$ , where $N$ is the number of EEG channels, $T$ is the clip length, and $M _ { i j }$ indicates the relative change in the model output when the $j$ -th second EEG clip in the $i$ -th channel is occluded. A larger $M _ { i j }$ indicates that the occluded region is more important for predicting seizure, and vice versa. We visualize $M$ by superimposing it over raw EEG signals and graph structures. + +In addition to visually analyzing the aforementioned occlusion maps, we propose quantitative metrics to evaluate the model’s capability of localizing seizures based on the occlusion maps. Specifically, we define a coverage metric that quantifies how many true seizure regions are detected, as well as a localization metric that quantifies how many detected seizure regions are true seizure regions. Note that coverage and localization scores are analogous to recall and precision, respectively, in binary classification problems. Mathematically, let $M ^ { \bar { \mathrm { a n n o t } } } \in \mathbb { R } ^ { N \times T }$ be detailed annotations of seizure duration and location, where $M _ { i j } ^ { \mathrm { a n n o t } } = \mathrm { i }$ if there is a seizure at the $j$ -th second in the $i$ -th channel, otherwise $M _ { i j } ^ { \mathrm { a n n o t } } = 0$ . Let $\boldsymbol { M } \in \mathbb { R } ^ { N \times T }$ be the occlusion map described above. Then $\mathrm { c o v e r a g e } = \frac { \sum _ { i , j } \mathbf { 1 } _ { M _ { i j } } > 0 . 5 } { \sum _ { i , j } M _ { i j } ^ { \mathrm { a n n o t } } } \mathrm { a n d l o c a l i z a t i o n } = \frac { \sum _ { i , j } \mathbf { 1 } _ { M _ { i j } > 0 . 5 } M _ { i j } ^ { \mathrm { a n n o t } } } { \sum _ { i , j } \mathbf { 1 } _ { M _ { i j } > 0 . 5 } } .$ + +Finally, we also perform occlusion-based interpretability analysis for seizure classification. Since the goal of seizure classification is to predict the seizure type given a seizure EEG clip, we hypothesize that the difference in signals between EEG channels are more important for predicting seizure types. Therefore, we completely drop one EEG channel at a time, and compute the relative change in the model output with respect to the original output. This results in an occlusion map $M ^ { \prime } \in \mathbb { R } ^ { \tilde { N } }$ , where $M _ { i } ^ { \prime }$ indicates the relative change in the model output when the $i$ -th channel is dropped. + +# 3 EXPERIMENTS + +# 3.1 EXPERIMENTAL SETUP + +Dataset. We use the public Temple University Hospital EEG Seizure Corpus (TUSZ) v1.5.2 (Shah et al., 2018; Obeid & Picone, 2016), the largest public EEG seizure database to date with 5,612 EEGs, 3,050 annotated seizures from clinical recordings, and eight seizure types1. We include 19 EEG channels in the standard 10-20 system (Figure 1b–1c). To evaluate model generalizability to unseen patients, we exclude five patients from the official TUSZ test set who exist in both the official TUSZ train and test sets. Moreover, we use the detailed annotations of seizure duration and location available in TUSZ for our interpretability analyses. Table 1 summarizes the TUSZ data. + +Data preprocessing. Because seizures are associated with brain electrical activity in certain frequency bands (Tzallas et al., 2009), we perform data preprocessing to transform raw EEG signals to the frequency domain. Similar to prior studies (Asif et al., 2020; Ahmedt-Aristizabal et al., 2020; Covert et al., 2019), we obtain the log-amplitudes of the fast Fourier transform of raw EEG signals. For seizure detection and self-supervised pre-training, we use both seizure and non-seizure + +EEGs, and obtain the 12-s (60-s) EEG clips using non-overlapping 12-s (60-s) sliding windows. For seizure classification, we use only seizure EEGs, and obtain one 12-s (60-s) EEG clip from each seizure event, such that each EEG clip has exactly one seizure type. Appendix A presents details of data preprocessing, and Appendix B compares results of frequency-domain vs time-domain inputs. + +Refined seizure classification scheme. Because simple partial (SP) seizures and complex partial (CP) seizures are focal seizures characterized by a clinical behavior (consciousness during a seizure) and are not distinguishable by EEG signals alone (Fisher et al., 2017), we combine focal non-specific (FN), SP, and CP seizures to form a combined focal (CF) seizure class. We provide extensive experiments in Appendix C showing that these focal seizure types cannot be distinguished by a variety of models. We also exclude myoclonic seizures because only three myoclonic seizures are available in TUSZ. In addition, since tonic and tonic-clonic seizures are rare in the dataset (only 18 tonic seizures and 30 tonic-clonic seizures in TUSZ train set), and tonic-clonic seizures always start with a tonic phase (Fisher et al., 2017), we combine tonic-clonic seizures with tonic seizures to form a combined tonic (CT) seizure class. In summary, there are four seizure classes in total: CF, generalized non-specific (GN), absence (AB), and CT seizures (Table 1). In Appendix L, we show seizure classification results on the original eight seizure types in TUSZ. + +Data splits. We randomly split the official TUSZ train set by patients into train and validation sets by 90/10 for model training and hyperparameter tuning, respectively, and we hold-out the official TUSZ test set for model evaluation (excluding five patients who exist in both the official TUSZ train and test sets). The train, validation, and test sets consist of distinct patients. See Appendix D for the number of preprocessed EEG clips and patients in each split. + +Table 1: Summary of data in train and test sets of TUSZ v1.5.2. used in our study. + +
EEG Files (% Seizure)Patients (% Seizure)Total Duration (% Seizure)CF Seizures (Patients)GN Seizures (Patients)AB Seizures (Patients)CT Seizures (Patients)
Train Set4,599 (18.9%)592 (34.1%)45,174.72 min (6.3%)1,868 (148)409 (68)50(7)48 (11)
Test Set900 (25.6%)45 (77.8%)9,031.58 min (9.8%)297 (24)114 (11)49 (5)61 (4)
+ +Baselines. To compare our DCRNN to traditional CNNs/RNNs, we include three primary baselines: (a) Dense-CNN (Saab et al., 2020), a previous state-of-the-art CNN for seizure detection, (b) LSTM (Hochreiter & Schmidhuber, 1997), and (c) CNN-LSTM (implemented following AhmedtAristizabal et al. (2020)). The baselines are trained and evaluated on the same preprocessed data. Additionally, we compare our method to the reported results of two CNNs for seizure classification that use TUSZ and test on unseen patients (Asif et al., 2020; Iesmantas & Alzbutas, 2020). + +Model training. Training for all models was accomplished using the Adam optimizer (Kingma & Ba, 2014) in PyTorch on a single NVIDIA Titan RTX GPU. Model parameters were randomly initialized for models without self-supervised pre-training, and were initialized with the pre-trained weights of the encoder for models with self-supervised pre-training. All models were run for five runs with different random seeds. Detailed hyperparameter settings are shown in Appendix E. During training, we perform data augmentation as described in Appendix F. + +# 3.2 EXPERIMENTAL RESULTS + +Graph neural network performance. Following recent studies (Asif et al., 2020; AhmedtAristizabal et al., 2020; O’Shea et al., 2020; Saab et al., 2020; Covert et al., 2019), we use AUROC and weighted F1-score as the main evaluation metrics for seizure detection and classification, respectively. Table 2 (3rd–7th rows) shows the performance of our DCRNN (without self-supervised pre-training) and the baselines. Distance graph-based DCRNN (or “Dist-DCRNN”) and correlation graph-based DCRNN (or “Corr-DCRNN”) without self-supervised pre-training perform on par with or better than the baselines. See Appendix G for additional evaluation scores. + +Moreover, DCRNN outperforms the reported results of two existing CNNs (Asif et al., 2020; Ahmedt-Aristizabal et al., 2020) on seizure classification (Table 3). For fair comparison to Asif et al. (2020), we conduct 7-class seizure classification2 on the same 3-fold patient-wise split. + +In addition, we conduct an ablation experiment to examine the effectiveness of Fourier transform in our preprocessing step for DCRNNs. We find that frequency-domain inputs substantially outperform time-domain inputs on both seizure detection and classification (Appendix B). + +Table 2: Seizure detection and seizure classification results. Mean and standard deviations are from five random runs. Best non-pretrained and pre-trained mean results are highlighted in bold. + +
ModelSeizure Detection AUROCSeizure Classification Weighted F1-Score
12-s60-s12-s60-s
Dense-CNN0.812 ± 0.0140.796 ± 0.0140.576 ± 0.1010.626 ± 0.073
LSTM0.786 ± 0.0140.715 ± 0.0160.652 ± 0.0190.686 ± 0.020
CNN-LSTM0.749 ± 0.0060.682 ± 0.0030.633 ± 0.0250.641 ± 0.019
Corr-DCRNN w/o Pre-training0.812 ± 0.0120.804 ± 0.0150.710 ± 0.0230.701 ± 0.030
Dist-DCRNN w/o Pre-training0.824 ± 0.0200.793 ± 0.0220.703 ± 0.0250.690 ± 0.035
Corr-DCRNN w/ Pre-training0.861 ± 0.0050.850 ± 0.0140.723 ± 0.0170.749 ± 0.017
Dist-DCRNN w/Pre-training0.866 ± 0.0160.875 ± 0.0160.746 ± 0.0240.749 ± 0.028
+ +Table 3: Comparison between DCRNNs (w/o pre-training) and existing CNNs on seizure classification. Mean and standard deviations are from five random runs. Best mean results are in bold. + +
Model7-Class Classification Weighted F1-ScoreCF Seizure AUROCGN Seizure AUROCAB Seizure AUROCCT Seizure AUROC
Asif et al. (2020)0.62=1--
Iesmantas & Alzbutas (2020)-0.780.72-
Corr-DCRNN,12-s0.619 ± 0.0060.907 ± 0.0080.815 ± 0.0270.972 ± 0.0130.908 ± 0.005
Dist-DCRNN,12-s0.585 ± 0.0060.896 ± 0.0110.814 ± 0.0270.983 ± 0.0080.890 ± 0.014
Corr-DCRNN, 60-s0.650 ± 0.0080.914 ± 0.0070.795 ± 0.0310.971 ± 0.0200.939 ± 0.008
Dist-DCRNN, 60-s0.606 ± 0.0090.920 ± 0.0040.811 ± 0.0320.973 ± 0.0090.926 ± 0.020
+ +Self-supervised pre-training improves graph neural network performance. To assess the effectiveness of self-supervised pre-training in improving DCRNN’s performance, we compare the results of DCRNN without and with self-supervised pre-training. As shown in Table 2 (last two rows), DCRNN with self-supervised pre-training outperforms its non-pretrained counterpart on both seizure detection and classification. Notably, Dist-DCRNN with self-supervised pre-training achieves an AUROC of 0.875 for 60-s seizure detection, matching the performance of a CNN (AU$\mathrm { R O C } { = } 0 . 8 8 $ that was pre-trained using supervised learning on a labeled dataset that was five times larger than TUSZ (Saab et al., 2020). Importantly, the pre-trained model weights are used for both seizure detection and classification, which indicates that our self-supervised pre-training method provides good model initialization that generalizes across tasks. + +Figure 2a shows the ROC curves of median DCRNNs and baselines for seizure detection. At a low false positive rate (FPR), such as $2 5 \%$ FPR, pre-trained Dist-DCRNN achieves $8 4 . 3 \%$ true positive rate (TPR) and pre-trained Corr-DCRNN achieves $8 1 . 7 \%$ TPR on 12-s clips. Conversely, DenseCNN, LSTM, and CNN-LSTM only achieve $7 2 . 8 \%$ , $6 7 . 0 \%$ , and $6 2 . 5 \%$ TPRs, respectively. Figure 2b shows the confusion matrices for Dist-DCRNNs and baselines for 12-s seizure classification. Dist-DCRNN without self-supervised pre-training achieves $93 \%$ accuracy on the rare AB seizures, providing 2 points increase over the best baseline. Furthermore, Dist-DCRNN with self-supervised pre-training achieves $74 \%$ accuracy on the rare CT seizures, providing 47 points increase in accuracy over the best baseline (Dense-CNN) and 48 points increase over non-pretrained Dist-DCRNN. + +Surprisingly, despite being a majority class, many GN seizures are misclassified as CF seizures (Figure 2b). A board-certified neurologist manually analyzed the EEGs of 32 misclassified test GN seizures. We find that 27 seizures are in fact focal seizures but are mislabeled as GN seizures. In contrast, only 5 seizures are indeed generalized seizures but are misclassified by our models. + +Comparison between self-supervised pre-training and transfer learning. To compare our selfsupervised pre-training strategy to traditional transfer learning approaches, we pre-train DCRNNs for seizure detection and self-supervised prediction, respectively, on a large in-house dataset (40,316 EEGs, Table 7) and finetune the models for seizure detection and classification on TUSZ. We find that self-supervised pre-training consistently outperforms transfer learning (Appendix J). + +Comparison between self-supervised pre-training and auxiliary learning. We also investigate whether using the self-supervised task as an auxiliary task can outperform self-supervised pretraining. We find that auxiliary learning performs comparable to self-supervised pre-training on 12-s seizure detection, whereas self-supervised pre-training significantly outperforms auxiliary learning on 60-s seizure detection. See Appendix M and Table 10 for details. + +![](images/f68ffb6e6422a2e2d70bc9f339a5c9f170d1acf2a7b09a3ff8b4281232e7eb16.jpg) +Figure 2: (a) ROC curves of median models for seizure detection. (b) Confusion matrices (averaged across five random runs) for the baselines and Dist-DCRNN without and with self-supervised pre-training for 12-s seizure classification. Each row of the confusion matrices is normalized by dividing by the number of examples in the corresponding class, such that each row sums up to one. +Figure 3: Occlusion maps for seizure detection obtained from Corr-DCRNN model for (a)–(b) focal, and (c)–(d) generalized seizures. In each subfigure, left panel shows the occlusion map for 12-s when a seizure occurs and right panel shows the occlusion map values averaged along time and overlaid on the corresponding correlation graph. + +Improved model interpretability and ability to localize seizures. We leverage occlusion-based techniques to localize seizures using our model predictions. Figure 3 shows example occlusion maps of correctly predicted 60-s test EEG clips from Corr-DCRNN with self-supervised pre-training. We observe that high saliency for focal seizures (Figures 3a–3b) is localized in the more abnormal EEG channels, whereas high saliency for generalized seizures (Figures 3c–3d) is more diffuse across channels. These patterns reflect the underlying brain activity of focal seizures (i.e. localized in one brain area) and generalized seizures (i.e. occur in all areas of the brain) (Fisher et al., 2017). Moreover, one can also interpret the seizure locations from the occlusion map overlaid on the graphs. In Figure 3a, Fp2, F4 and F8 are highlighted on the graph occlusion map, which correspond to the right frontal, central frontal, and anterior frontal brain regions that show the most abnormality in the EEG. In Figure 3b, while the seizure starts in P3, Fp1, F3, and F7, the graph occlusion map mainly highlights the central parietal region (P3), which is likely due to the ongoing artifact in channels Fp1-F7, Fp1-F3, and F3-C3. Figure 3c–3d show two generalized seizures that occur in all regions of the brain, and the highlighted nodes on the graphs are indeed more spread out across channels. In contrast, high saliency from Dense-CNN does not localize in any seizure regions (Appendix H). + +Additionally, we leverage detailed annotations of seizure duration and location available in TUSZ dataset, and use coverage and localization scores to quantify the models’ ability to accurately localize seizures. Figure 4 shows coverage and localization distributions for Dense-CNN and DCRNNs (w/o and with pre-training) on correctly predicted 60-s test EEG clips. Both Dist-DCRNN and Corr-DCRNN have many more occlusion maps with high coverage and high localization scores than Dense-CNN. This suggests that DCRNN more accurately localizes seizures than Dense-CNN, which is intuitive given that DCRNN captures the connectivity of EEG electrodes. + +Clinically, localizing focal seizure onset regions is key to epilepsy surgery for patients with focal seizures. Thus, the model’s ability to identify focal seizure regions as precisely as possible would be highly desirable. Notably, pre-trained Corr-DCRNN precisely localizes $2 5 . 4 \%$ focal seizures and pre-trained Dist-DCRNN precisely localizes $2 1 . 8 \%$ focal seizures (localization score $> 0 . 8$ , Figure 4c). Conversely, both non-pretrained Dist-DCRNN and Corr-DCRNN precisely localize $6 . 3 \%$ focal seizures, and Dense-CNN only localizes $3 . 5 \%$ focal seizures precisely (localization score $> 0 . 8$ ). Figures 4e–4f show example occlusion maps of test focal seizures whose localization scores $> 0 . 9$ from pre-trained Corr-DCRNN, where high saliency overlaps well with annotated seizure regions. + +Lastly, interpretability experiments on seizure classification show that high saliency EEG channels correspond to where the seizures are localized in CF seizures (Appendix I). This again suggests that our method could inform seizure locations for treatment strategy if used clinically. + +![](images/4a4a126e8f76d0849f3cae8db0656244a5c998ac8d4ae2d07790dd894e907c15.jpg) +Figure 4: Distributions of (a)–(b) coverage and (c)–(d) localization scores for Dense-CNN and DCRNNs on correctly predicted 60-s EEG clips for focal and generalized seizures. (e)–(f) Example occlusion maps with focal seizure whose localization ${ > } 0 . 9$ from pre-trained Corr-DCRNN. Only regions with normalized occlusion value ${ > } 0 . 5$ are colored. Red boxes are annotated seizure regions. + +Comparison between graph structures. While Dist-DCRNN and Corr-DCRNN perform comparable to each other on seizure detection and classification (Table 2), we observe that Corr-DCRNN better localizes focal seizures than Dist-DCRNN, particularly when combined with self-supervised pre-training (e.g., coverage and localization scores $> 0 . 9$ in Figure 4a, 4c). This suggests that the correlation-based graph structure could provide better interpretability and representation of focal seizure EEGs. Moreover, the correlation graph structure has two particular advantages: (a) it can be used even when the physical locations of electrodes are unknown and (b) it captures dynamic brain connectivity rather than replying purely on spatial sensor information, which is particularly desirable for automated seizure detection and classification models. + +# 4 CONCLUSION + +In conclusion, we present a novel method combining graph-based modeling and self-supervised pretraining for EEG-based seizure detection and classification, as well as an interpretability method to quantify model ability to localize seizures. Our method sets new state-of-the-art on both seizure detection and classification on a large public dataset, significantly improves classification of rare seizure classes, and more accurately localizes seizures. We also find that the correlation-based graph more accurately localizes focal seizures than distance-based graph. The improved ability to localize seizures and the novel graph visualizations could provide clinicians with valuable insights about localized seizure regions in real-world clinical settings. Looking to the future, because our methods are not confined to EEG alone, our study opens exciting opportunities to build graph-based representations for a wide variety of medical time series. + +# ACKNOWLEDGMENTS + +This work was supported by a Wu Tsai Neurosciences Institute Neuroscience Translate Grant. The authors would like to thank Bibek Paudel, Liangqiong Qu, Jean Benoit Delbrouck, and Nandita Bhaskhar for their feedback on the paper. + +# ETHICS STATEMENT + +The Temple University Hospital EEG Seizure Corpus used in our study is anonymized and publicly available3 with full IRB approval (Shah et al., 2018; Obeid & Picone, 2016). No conflict of interest is reported from the authors. No harmful insights are provided by the seizure detection and classification models described in this study. Although we show that our methods could provide improved performance and clinical utility for seizure detection and classification, additional model validations are needed before they can be used in real-world clinical settings, including (a) validation on datasets from multiple institutions, (b) validation on different populations and age groups, and (c) validation on a EEG waveform-based seizure classification scheme that is approved by a consensus of board-certified neurologists. + +# REPRODUCIBILITY STATEMENT + +The Temple University Hospital EEG Seizure Corpus used in our study is publicly available. Detailed data preprocessing steps are provided in Appendix A. Source code is publicly available at https://github.com/tsy935/eeg-gnn-ssl. + +# REFERENCES + +Jayant N Acharya, Abeer J Hani, Parthasarathy Thirumala, and Tammy N Tsuchida. American clinical neurophysiology society guideline 3: a proposal for standard montages to be used in clinical eeg. 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We perform the following preprocessing steps to obtain EEG clips in the frequency domain and their corresponding labels. + +First, for seizure detection, we use both seizure and non-seizure EEGs. We obtain the EEG clips by sliding a 12-s (or 60-s) window over the EEG signals without overlaps, and ignore the last window if it is shorter than the clip length. The label for each clip is $y = 1$ if at least one seizure event occurs within this clip, otherwise $y = 0$ . + +Second, for seizure classification, we use only seizure EEGs. We obtain one 12-s (or 60-s) EEG clip for each seizure event starting at 2-s before the annotated seizure onset time, where a 2-s offset accounts for tolerance in the annotations. If a seizure event is shorter than 12-s (or 60-s), the EEG clip is truncated at the end of the seizure to prevent a clip from having multiple seizure types. The label for each clip is the index of the corresponding seizure class, i.e. $y ~ \in ~ \{ 0 , 1 , 2 , 3 \}$ , which corresponds to combined focal (CF) seizures, generalized non-specific (GN) seizures, absence (AB) seizures, and combined tonic (CT) seizures. + +Third, for self-supervised pre-training, we use the same EEG clips as seizure detection. + +Fourth, for each EEG clip in each of the seizure detection/seizure classification/self-supervised pretraining tasks, we perform the following preprocessing steps to transform the signals in time domain to frequency domain: (a) slide a $t$ second window over the EEG clip without overlap, where $t$ is the time step size for networks involving recurrent layers; (b) apply fast Fourier transform (FFT) to each $t$ second window using the “fft” function in Scipy python package (Virtanen et al., 2020b), and retain the log amplitudes of the non-negative frequency components similar to prior studies (Asif et al., 2020; Ahmedt-Aristizabal et al., 2020; Covert et al., 2019); (c) $\mathbf { Z }$ -normalize the EEG clip with respect to the mean and standard deviation of the training data. Because EEG clips for seizure classification may have variable lengths due to short seizures, we pad the clips with 0’s to facilitate model training in batches. We use $t = 1$ second as a natural choice of the time step size. After preprocessing, each EEG clip can be denoted as $\pmb { X } \in \mathbb { R } ^ { T \times N \times M }$ , where $T = 1 2$ (or $T = 6 0$ ) represents the clip length, $N = 1 9$ represents the number of EEG channels/electrodes, and $M = 1 0 0$ represents the feature dimension after the aforementioned Fourier transform. + +# B EFFECTIVENESS OF FOURIER TRANSFORM + +To evaluate the effectiveness of Fourier transform (Appendix A), we compare the performance of our DCRNN (without self-supervised pre-training) on inputs without and with Fourier transform on both seizure detection and seizure classification. As shown in Table 4, frequency-domain inputs (with Fourier transform) result in significantly better performance than time-domain inputs (without Fourier transform). This is likely because seizures are associated with electrical activity in certain frequency bands (Tzallas et al., 2009), and thus short-time-interval frequency-domain inputs could be more informative than time-domain inputs. + +# C DISTINGUISHABILITY OF FOCAL NON-SPECIFIC, SIMPLE PARTIAL, AND COMPLEX PARTIAL SEIZURES + +Because simple partial and complex partial seizures are focal seizures characterized by the clinical behavior, consciousness during seizure (Fisher et al., 2017), they are not distinguishable from other focal seizures from EEG signals alone. In this study, we merge focal non-specific (FN), simple partial (SP), and complex partial (CP) seizures into a combined focal seizure class. To justify our decision, we perform classification of these focal seizure types using our DCRNNs and baselines. As shown in Figure 5, SP and CP seizures are largely misclassified as FN seizures, suggesting that machine learning models may not be able to distinguish these focal seizure types from EEG signals alone. This supports our decision of combining FN, SP, and CP seizures into one class. + +Table 4: Seizure detection and classification results from DCRNNs (without self-supervised pretraining) on time-domain inputs and frequency-domain inputs. Mean and standard deviations are obtained from five random runs. + +
ModelInput DomainSeizure DetectionAUROC12-s 60-sSeizure ClassificationWeighted F1-Score12-s 60-s
Corr-DCRNNWithout Pre-TrainingTimeFrequency0.717 ± 0.003 0.704 ± 0.0230.812 ± 0.012 0.804 ± 0.0150.597 ± 0.023 0.618 ± 0.0180.710 ± 0.023 0.701 ± 0.030
Dist-DCRNNWithout Pre-TrainingTimeFrequency0.733 ± 0.0120.825 ± 0.0190.698 ± 0.0030.793 ± 0.0220.592 ± 0.0220.703 ± 0.0250.608 ± 0.0150.690 ± 0.035
+ +# D TRAIN, VALIDATION, AND TEST SETS + +Table 5 shows the number of EEG clips and number of patients in our train, validation, and test sets for self-supervised pre-training, seizure detection, and seizure classification tasks. Train, validation, and test sets consist of distinct patients. + +Table 5: Number of EEG clips and patients in the train, validation, and test sets in our study. Train, validation, and test sets consist of distinct patients. + +
EEG ClipLength(Secs)Train SetValidation SetTest Set
EEG Clips(% Seizure)Patients(% Seizure)EEG Clips(% Seizure)Patients(% Seizure)EEG Clips(% Seizure)Patients(% Seizure)
Pre-training& SeizureDetection60-s12-s38.613(9.3%)196.646(6.9%)530(34.0%)531(33.9%)5,503(11.4%)28.057(8.7%)61(36.1%)61(36.1%)8.848(14.7%)44,959(10.9%)45(77.8%)45(77.8%)
SeizureClassification60-s&12-s1,925(100.0%)179(100.0%)450(100.0%)22(100.0%)521(100.0%)34(100.0%)
+ +# E DETAILS OF MODEL TRAINING PROCEDURES AND HYPERPARAMETERS + +We performed the following hyperparameter search on the validation set: (a) initial learning rate within range [5e-5, 1e-3]; (b) $\bar { \tau } \in \bar { \{ 2 , 3 , 4 \} }$ , the number of neighbors to keep for each node in the correlation graphs; (c) the number of Diffusion Convolutional Gated Recurrent Units (DCGRU) layers within range $\{ 2 , 3 , 4 , 5 \}$ and hidden units within range $\{ 3 2 , 6 4 , 1 2 8 \}$ ; (d) the maximum diffusion step $K \in \{ 2 , 3 , 4 \}$ ; (e) dropout probability in the last fully connected layer. We used a batch size of 40 EEG clips, the maximum possible across all models and baselines on a single Titan RTX GPU. The hyperparameters were selected based on the best performance on the validation set, and are detailed in the next sections. We used the cosine annealing learning rate scheduler (Loshchilov & Hutter, 2017) in PyTorch for all model training. We ran five runs with different random seeds for all models. In all experiments, model training was early stopped when the validation loss did not decrease for five consecutive epochs. + +Model training for seizure detection. For seizure detection, the plentiful negative examples in the train set were undersampled such that the train set had $50 \%$ positive examples, which resulted in 27,292 training examples for 12-s clips and 7,188 training examples for 60-s clips. We used binary cross-entropy as the loss function to train the seizure detection models. The models were trained for 100 epochs with an initial learning rate of 1e-4. For the correlation graphs, the top-3 neighbors’ edges were kept for each node. The maximum number of diffusion step was 2, and the dropout probability was 0 (i.e. no dropout). The model consists of two stacked DCGRU layers with 64 hidden units, resulting in 168,641 trainable parameters for the distance graph and 280,769 trainable parameters for the correlation graphs. Model training for seizure detection took about $2 0 \mathrm { - m i n }$ for 12-s EEG clips, and about 30-min for 60-s EEG clips. + +![](images/6e69085ea0475e338a3a51b915c47e73bc6469ba1595b05e67a4c8141cab5237.jpg) +Figure 5: Confusion matrices (averaged across five random runs) of focal seizure classification on (a) 60-s EEG clips, and (b) 12-s EEG clips. All models fail to distinguish well among these focal seizure types, supporting our decision of merging these seizure types into one combined focal seizure class. Note that each row of the confusion matrices is normalized by dividing by the number of examples in the corresponding class, such that each row sums up to one. + +To obtain the final seizure/non-seizure prediction, we performed decision threshold search on the validation set. More specifically, to balance between precision and recall scores, we selected the decision threshold that results in the highest F1-score on the validation set. When evaluating the models on the test set, EEG clips with probabilities above this decision threshold are predicted as seizures, while clips with probabilities below this decision threshold are predicted as non-seizures. + +Model training for seizure classification. For seizure classification, we used multi-class crossentropy as the loss function during training. The models were trained for 60 epochs with an initial learning rate of 3e-4. For the correlation graphs, the top-3 neighbors’ edges were kept for each node. The maximum number of diffusion step was 2, and the dropout probability was 0.5. The model consists of two stacked DCGRU layers with 64 hidden units, resulting in 168,836 trainable parameters for the distance graph and 280,964 trainable parameters for the correlation graphs. Model training for seizure classification took about 3-min for 12-s EEG clips, and about 7-min for 60-s EEG clips. + +Model training for self-supervised task. Preliminary experiments suggested that predicting future $T ^ { \prime } = 1 2$ second preprocessed EEG clips results in low regression loss on the validation set given previous 12-s (60-s) preprocessed clips, and thus we used $\bar { T ^ { \prime } } = 1 2$ in all self-supervised pre-training experiments. For self-supervised pre-training, we used mean absolute error (MAE) as the loss function. The models were trained for 350 epochs with an initial learning rate of 5e-4. For the correlation graphs, the top-3 neighbors’ edges were kept for each node. The maximum number of diffusion step was 2. The model consists of three stacked DCGRU layers with 64 hidden units in both the encoder and decoder, resulting in 417,572 trainable parameters for the distance graph and 690,980 trainable parameters for the correlation graphs. Model training for self-supervised prediction took about 10-h for 12-s EEG clips, and about $^ { 2 4 - \mathrm { h } }$ for 60-s EEG clips. + +Model training for baselines. For baseline Dense-CNN, we employ the same model architecture as that described in Saab et al. (2020). For baseline LSTM (Hochreiter & Schmidhuber, 1997), we have the number of LSTM layers and hidden units the same as the number of DCGRU layers and hidden units in our DCRNN model. For baseline CNN-LSTM, we use the same model architecture described in Ahmedt-Aristizabal et al. (2020), i.e., two stacked convolutional layers $3 2 3 \times 3$ kernels), one max-pooling layer $( 2 \times 2 )$ , one fully-connected layer (output neuron $= 5 1 2$ ), two stacked LSTM layers (hidden size $= 1 2 8$ ), and one fully connected layer. + +# F DATA AUGMENTATION + +During training, we performed the following data augmentations based on EEG domain knowledge: (a) randomly scaling the amplitude of the raw EEG signals by a scale within [0.8, 1.2] and (b) randomly reflecting the signals along the scalp midline. + +# G ADDITIONAL EVALUATION RESULTS ON SEIZURE DETECTION + +Table 6 shows F1-score, Area Under the Precision-Recall Curve (AUPR), sensitivity, and specificity of seizure detection models. + +# H SEIZURE DETECTION OCCLUSION MAPS FROM BASELINE DENSE-CNN + +Figure 6 shows example seizure detection occlusion maps obtained from baseline Dense-CNN on correctly predicted 60-s EEG clips in the test set. Unlike our model (Figure 3), high saliency from Dense-CNN does not localize in any seizure regions. + +# I SEIZURE CLASSIFICATION OCCLUSION MAPS FROM DCRNN + +Figure 7 shows example seizure classification occlusion maps from Corr-DCRNN with selfsupervised pre-training on correctly predicted 60-s EEG clips in the test set. The occlusion maps are obtained by completely dropping one EEG channel at a time and calculating the relative change in the model output. For CF seizures (a–b), high saliency regions correspond to brain areas where the focal seizures are localized. For the other generalized seizure types (c–e), less salient regions correspond to less abnormal areas. + +Table 6: Additional evaluation scores for seizure detection on (a) 12-s EEG clips and (b) $6 0 \mathrm { - } \mathrm { s }$ EEG clips. Mean and standard deviations are obtained from five random runs. Best non-pretrained and pre-trained mean results are highlighted in bold. + +(a) 12-s EEG clips + +
ModelF1-Score(mean ± std)AUPR(mean ± std)Sensitivity(mean ± std)Specificity(mean ± std)
Dense-CNNLSTMCNN-LSTM0.326 ± 0.0190.376 ± 0.0210.337 ± 0.0090.328 ± 0.0430.354 ± 0.0230.309 ± 0.0150.293 ± 0.0210.357 ± 0.0450.333 ± 0.0280.938 ± 0.0140.934 ± 0.0150.920 ± 0.021
Corr-DCRNNWithout Pre-trainingDist-DCRNNWithout Pre-training0.392 ± 0.0270.437 ± 0.0290.370 ± 0.0270.411 ± 0.0410.373 ± 0.0350.411 ± 0.0380.935 ± 0.0120.943 ± 0.006
Corr-DCRNNWith Pre-trainingDist-DCRNNWith Pre-training0.484 ± 0.0110.487 ± 0.0420.454 ± 0.0200.463 ± 0.0480.524 ± 0.0120.592 ± 0.0520.922 ± 0.0040.897 ± 0.012
+ +(b) 60-s EEG clips + +
ModelF1-Score(mean ± std)AUPR(mean ± std)Sensitivity(mean ± std)Specificity(mean ± std)
Dense-CNNLSTMCNN-LSTM0.404 ± 0.0220.365 ± 0.0090.330 ± 0.0160.399 ± 0.0170.287 ± 0.0260.276 ± 0.0090.451 ± 0.1340.463 ± 0.0600.363 ± 0.0440.869 ± 0.0710.814 ± 0.0530.857 ± 0.023
Corr-DCRNNWithout Pre-trainingDist-DCRNNWithout Pre-training0.448 ± 0.0290.341 ± 0.1700.440 ± 0.0210.418 ± 0.0460.457 ± 0.0580.326 ± 0.1830.900 ± 0.0280.932 ± 0.058
Corr-DCRNN With Pre-trainingDist-DCRNNWith Pre-training0.514 ± 0.0280.571 ± 0.0290.539 ± 0.0240.593 ± 0.0310.502 ± 0.0470.570 ± 0.0470.923 ± 0.0080.927 ± 0.012
+ +# J COMPARISON BETWEEN SELF-SUPERVISED PRE-TRAINING AND TRANSFER LEARNING + +To compare our self-supervised pre-training strategy to traditional transfer learning, we perform transfer learning by pre-training DCRNNs for seizure detection on an in-house dataset (40,316 EEGs, Table 7) and finetuning on TUSZ data for seizure detection and classification. Due to the lack of fine-grained seizure type labels, we do not pre-train DCRNNs for seizure classification on the inhouse dataset. Moreover, we pre-train DCRNNs on the in-house dataset using our self-supervised pre-training strategy and finetuned them for seizure detection and classification on TUSZ. + +Table 8 shows DCRNN results with self-supervised pre-training and transfer learning from the inhouse dataset. Self-supervised pre-training from the in-house dataset (3rd-4th rows) consistently outperforms trasnfer learning from the in-house dataset (5th-6th rows). We speculate that transfer learning does not perform comparably to self-supervised pre-training because it suffers from distribution shift in the data (i.e., the in-house dataset comes from a different population and uses a slightly different EEG acquisition protocol). In contrast, by learning to predict the EEGs for the next time period, self-supervised pre-training encourages the model to learn task-agnostic representations and thus mitigates the problem of distribution shift. + +![](images/6e0d377f9625210f0abab6d5575190ebe819810f3fecb396c941993df0e710a1.jpg) +Figure 6: Example occlusion maps for seizure detection obtained from baseline Dense-CNN on correctly predicted 60-s clips for (a)–(b) focal seizures, and (c)–(d) generalized seizures. Red boxes indicate the duration of the seizures. Note that the values within an occlusion map are normalized, and thus should not be compared across different occlusion maps. + +Table 7: Summary of in-house dataset. Only annotations for the start of seizure are available. For EEG files with seizures, EEG clips are obtained by taking 12-s (or 60-s) signals starting from the annotated seizure start time. For EEG files without seizures, EEG clips are obtained by taking 12- s (or 60-s) signals randomly from the entire signal. We apply the same data preprocessing steps described in Appendix A. + +
EEG Files (% Seizure) Total DurationPatients (% Seizure)EEG Clips (% Seizure)
In-House Train Set40,316 (24.1%)853,141.87 min5,355 (25.7%)46,613 (34.4%)
+ +# K CHOICE OF DISTANCE-BASED GRAPH STRUCTURE + +For the distance graph, we used a threholded Gaussian kernel (Shuman et al., 2013) to compute the edge weight between two electrodes (Section 2.2.1). We experimented with $\kappa$ , the threshold for graph sparsity, within range [0.1, 2]. Based on preliminary experiments and EEG domain knowledge, we chose $\kappa = 0 . 9$ because it results in a reasonable graph (e.g. no long-range connection) that resembles the EEG montage (longitudinal bipolar and transverse bipolar) widely used clinically (Acharya et al., 2016). Figure 8 shows distance graphs resulting from different thresholds $\kappa$ . In Figure 8, we can see that a smaller threshold (e.g., 0.7 or 0.8) results in missing edges between nearby electrodes. For example, there is no edge between FP1 and FZ for threshold 0.8 and no edge between T3 and C3 for threshold 0.7. In contrast, a larger threshold (e.g., 1.0, 1.1, or 1.2) results in edges connecting electrodes that are spatially far apart. For example, there is an edge connecting C3 and $\mathrm { F Z }$ , as well as an edge between F7 and T5 for threshold 1.2. Using EEG domain knowledge provided by a board certified neurologist, we believe that a threshold of 0.9 results in a more reasonable distance-based EEG graph compared to other thresholding values. + +Table 8: Comparison of results between self-supervised pre-training and transfer learning pretrained on an in-house dataset. 3rd-4th rows: DCRNN results with self-supervised (SS) pre-training on the in-house dataset. 5th-6th rows: DCRNN results with transfer learning pre-trained for seizure detection on the in-house dataset. Mean and standard deviations are from five random runs. Best mean results for each column are highlighted in bold. + +
ModelSeizure Detection AUROCSeizure Classification Weighted F1-Score
12-s60-s12-s60-s
Corr-DCRNN w/ SS Pre-Training Dist-DCRNN w/ SS0.863 ± 0.0050.856 ± 0.0130.736 ± 0.0070.723 ± 0.010
Pre-Training Corr-DCRNN w/ Transfer0.879 ± 0.0060.886 ± 0.0060.767 ± 0.0380.718 ± 0.018
Learning Dist-DCRNN w/ Transfer Learning0.848 ± 0.018 0.866 ± 0.0010.850 ± 0.002 0.847 ± 0.0040.733 ± 0.027 0.720 ± 0.0460.711 ± 0.025 0.691 ± 0.038
+ +In addition, we explore using a Gaussian kernel with a specified bandwidth for building the distance graph, i.e., $W _ { i j } = \frac { 1 } { \sqrt { 2 \pi h ^ { 2 } } } \mathrm { e x p } \bigg ( - \frac { \mathrm { d i s t } ( v _ { i } , v _ { j } ) ^ { 2 } } { 2 h ^ { 2 } } \bigg )$ , where $W _ { i j }$ is the edge weight between electrodes $v _ { i }$ and $v _ { j }$ , $\mathrm { d i s t } ( v _ { i } , v _ { j } )$ is the Euclidean distance between $v _ { i }$ and $v _ { j }$ , and $h$ is the Gaussian kernel bandwidth. Figure 9 shows the distance graph structures resulting from different values of bandwidth. With bandwidth $= 0 . 0 6$ , the distance graph structure resembles the original graph structure using thresholded Gaussian kernel with a threshold of 0.9 (Figure 1b). + +# L 8-CLASS SEIZURE CLASSIFICATION + +We also perform seizure classification on the original eight seizure types4 available in TUSZ (see Table 9). Note that only one patient’s two myoclonic seizures are available in the official TUSZ train set and only one patient’s one myoclonic seizure is available in the official TUSZ test set. Hence, patient-wise train/validation splits for myoclonic seizure is not possible, and we randomly assign one myoclonic seizure in the official TUSZ train set to our train split and the other to our validation split. As shown in Table 9, DCRNNs consistently outperform the baselines on 8-class seizure classification. + +# M USING SELF-SUPERVISED PREDICTION AS AN AUXILIARY TASK + +Here, instead of pre-training DCRNNs for the self-supervised prediction task (i.e., predicting preprocessed EEG clips for the next time period), we conduct experiments using the self-supervised prediction task as an auxiliary task. Because EEG clips for the seizure classification task have variable lengths, we only perform this experiment for seizure detection to facilitate training in batches. Specifically, the loss function is $\mathcal { L } = \mathcal { L } _ { \mathrm { s e i z u r e ~ d e t e c t i o n } } + \lambda \mathcal { L } _ { \mathrm { S S } }$ prediction where $\lambda$ is a hyperparameter balancing the seizure detection loss and the self-supervised loss and is tuned on the validation set. For 12-s (or 60-s) seizure detection, the auxiliary task is to predict the next 6-s preprocessed EEG clips given the first 6-s (or 30-s) EEG clips. + +Table 9: Results (weighted F1-scores) of seizure classification on original 8 seizure types in TUSZ. Mean and standard deviations are from five random runs. Best mean results for both non-pretrained and pretrained models are highlighted in bold. + +
Model12-s60-s
Dense-CNNLSTMCNN-LSTM0.431 ± 0.0370.515 ± 0.0250.489 ± 0.0360.427 ± 0.0470.525 ± 0.0170.509 ± 0.021
Corr-DCRNNWithout Pre-trainingDist-DCRNNWithout Pre-training0.553 ± 0.0250.570 ± 0.0270.577 ± 0.0280.600 ± 0.022
Corr-DCRNNWith Pre-trainingDist-DCRNNWith Pre-training0.582 ± 0.0140.583 ± 0.0090.591 ± 0.0080.607 ± 0.017
+ +Table 10 shows the seizure detection results for DCRNNs when the self-supervised (SS) prediction task is used as an auxiliary task during training. Compared to self-supervised pre-training (Table 2 last two rows), auxiliary learning only marginally improves Dist-DCRNN performance on 12- s seizure detection (no statistical significance), whereas self-supervised pre-training significantly outperforms auxiliary learning for 60-s seizure detection. + +Table 10: Seizure detection results for DCRNNs with self-supervised pre-training (same as Table 2) and auxiliary learning. Mean and standard deviations are from five random runs. Best mean results for Dist-DCRNN/Corr-DCRNN are in bold. + +
ModelSeizure Detection AUROC
12-s60-s
Corr-DCRNN w/ SS Pre-Training (Table 2)0.861 ± 0.0050.850 ± 0.013
Corr-DCRNN w/ SS Auxiliary Task0.851 ± 0.0080.811 ± 0.005
Dist-DCRNN w/ SS Pre-Training (Table 2) Dist-DCRNN w/ SS Auxiliary Task0.866 ± 0.016 0.875 ± 0.0050.875 ± 0.016 0.840 ± 0.013
+ +![](images/ea3f2f8b86ff27ddf606d1573755de6ac09c4d00c7782dea554e1f1e21c8f90a.jpg) +Figure 7: Example occlusion maps for seizure classification obtained from pre-trained Corr-DCRNN model for (a)–(b) combined focal (CF) seizures, (c) generalized non-specific (GN) seizure, (d) absence (AB) seizure, and (e) combined tonic (CT) seizure. In each subfigure, the bottom panel shows the occlusion values for each channel replicated along the time dimension and overlaid on 60-s EEG signals, the red boxes indicate the duration of seizures in the EEG clips, and the top panel shows the occlusion values overlaid on the correlation graph structure. To visualize occlusion map values in this “double banana” montage, we subtract the occlusion values between the corresponding channels in the montage, which results in different values between occlusion maps shown on the EEG signals (bottom panel in each subfigure) and that shown on the graph structures (top panel in each subfigure). Note that the values within an occlusion map are normalized, and thus should not be compared across different occlusion maps. + +![](images/b6c52e09286733a3355b5ae80fc11331e96176ddbf306c8eeb9bf6421a1f0537.jpg) +Figure 8: Distance graphs with different threshold $\kappa$ in the thresholded Gaussian kernel. Small thresholds (e.g., 0.7 and 0.8) result in missing edges between nearby electrodes, whereas large thresholds (e.g., 1.0, 1.1, 1.2) result in edges connecting electrodes that are spatially far away. + +![](images/83d4c3506315b7feb93843473c7173f9438f4005f183f02d52fa22501f96de9b.jpg) +Figure 9: Distance-based graphs constructed using Gaussian kernels with different bandwidths. When bandwidth $= 0 . 0 6$ , the distance graph structure resembles the original graph structure using thresholded Gaussian kernel with a threshold of 0.9 (Figure 1b). \ No newline at end of file diff --git a/parse/dev/k9bx1EfHI_-/k9bx1EfHI_-_content_list.json b/parse/dev/k9bx1EfHI_-/k9bx1EfHI_-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..efe843a4ab2322a6e3244852fb6874b658775042 --- /dev/null +++ b/parse/dev/k9bx1EfHI_-/k9bx1EfHI_-_content_list.json @@ -0,0 +1,2132 @@ +[ + { + "type": "text", + "text": "SELF-SUPERVISED GRAPH NEURAL NETWORKS FOR IMPROVED ELECTROENCEPHALOGRAPHIC SEIZURE ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Siyi Tang, Jared Dunnmon, Khaled Saab, Xuan Zhang†, Qianying Huang†, Florian Dubost, Daniel Rubin‡, Christopher Lee-Messer‡ \nStanford University, CA, USA \n{siyitang,jdunnmon,ksaab,kayleez,qyhuang,fdubost,rubin,cleemess} @stanford.edu \n† Equal contributions. ‡ Equal contributions. ", + "bbox": [ + 181, + 194, + 820, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 318, + 544, + 333 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Automated seizure detection and classification from electroencephalography (EEG) can greatly improve seizure diagnosis and treatment. However, several modeling challenges remain unaddressed in prior automated seizure detection and classification studies: (1) representing non-Euclidean data structure in EEGs, (2) accurately classifying rare seizure types, and (3) lacking a quantitative interpretability approach to measure model ability to localize seizures. In this study, we address these challenges by (1) representing the spatiotemporal dependencies in EEGs using a graph neural network (GNN) and proposing two EEG graph structures that capture the electrode geometry or dynamic brain connectivity, (2) proposing a self-supervised pre-training method that predicts preprocessed signals for the next time period to further improve model performance, particularly on rare seizure types, and (3) proposing a quantitative model interpretability approach to assess a model’s ability to localize seizures within EEGs. When evaluating our approach on seizure detection and classification on a large public dataset (5,499 EEGs), we find that our GNN with self-supervised pre-training achieves 0.875 Area Under the Receiver Operating Characteristic Curve on seizure detection and 0.749 weighted F1-score on seizure classification, outperforming previous methods for both seizure detection and classification. Moreover, our self-supervised pre-training strategy significantly improves classification of rare seizure types (e.g. 47 points increase in combined tonic seizure accuracy over baselines). Furthermore, quantitative interpretability analysis shows that our GNN with self-supervised pre-training precisely localizes $2 5 . 4 \\%$ focal seizures, a 21.9 point improvement over existing CNNs. Finally, by superimposing the identified seizure locations on both raw EEG signals and EEG graphs, our approach could provide clinicians with an intuitive visualization of localized seizure regions. ", + "bbox": [ + 232, + 352, + 764, + 698 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 729, + 336, + 746 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Seizures are among the most common neurological emergencies in the world (Strein et al., 2019). Seizures can be chronic as in the case of epilepsy, a neurological disease affecting 50 million people worldwide (WHO, 2019). Clinically, definitive detection of a seizure is only the first step in seizure diagnosis. An important subsequent step is to classify seizures into finer-grained types––such as focal versus generalized seizures––for identifying epilepsy syndromes, targeted therapies, and eligibility for epilepsy surgery (Fisher et al., 2017). ", + "bbox": [ + 174, + 763, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Scalp electroencephalography (or “EEG”) plays a critical role in seizure detection and classification. Clinically, EEG-based seizure detection and classification are performed by a trained EEG reader who visually examines a patient’s EEG signals over time periods ranging from hours to days. However, this manual analysis is extremely resource- and time-intensive, and thus automated algorithms could greatly accelerate seizure diagnosis and improve outcomes for severely ill patients. ", + "bbox": [ + 176, + 854, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/48275fbc969bc964f0d407ba19c38d1c7a2054aea09af82c5827a6f3345e271f.jpg", + "image_caption": [ + "Figure 1: (a) EEG electrode placement in the standard 10-20 system. (b) Distance-based EEG graph. (c) An example correlation-based EEG graph. (d) Overview of our methods. The inputs to the models are the EEG graphs, where each node feature corresponds to the preprocessed EEG signals in the respective channel. Self-edges are not shown for better visualization. " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 624, + 309 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Although a large number of studies have attempted automated seizure detection (Rasheed et al., 2020; Siddiqui et al., 2020; Shoeibi et al., 2021; O’Shea et al., 2020; Saab et al., 2020) or seizure classification (Raghu et al., 2020; Asif et al., 2020; Iesmantas & Alzbutas, 2020; Ahmedt-Aristizabal et al., 2020; Roy et al., 2019), several challenges remain largely unaddressed. First, most recent studies use convolutional neural networks (CNNs) that assume Euclidean structures in EEG signals or spectrograms (Rasheed et al., 2020; Shoeibi et al., 2021; Raghu et al., 2020; Asif et al., 2020; Iesmantas & Alzbutas, 2020; Ahmedt-Aristizabal et al., 2020; Roy et al., 2019; O’Shea et al., 2020; Saab et al., 2020). However, assumption of Euclidean structure ignores the natural geometry of EEG electrodes and the connectivity in brain networks. EEGs are measured by electrodes placed on a manifold (i.e. patient’s scalp) (Figure 1a), and thus have inherent non-Euclidean structures. Graphs are a data structure that can represent complex, non-Euclidean data (Chami et al., 2020; Bronstein et al., 2017), and graph theory has been extensively used in modeling brain networks (Bullmore & Sporns, 2009). We therefore hypothesize that graph-based modeling approaches can better represent the inherent non-Euclidean structures in EEGs in a manner that improves both the performance and the clinical utility of seizure detection and classification models. Although traditional graph theory has been used (Supriya et al., 2021), only a few deep learning studies have modeled EEGs as graphs for seizure detection. However, these graph-based studies were limited to nonpublic (Covert et al., 2019) or small datasets (Craley et al., 2019; Li et al., 2021), did not leverage modern self-supervised approaches or examine seizure classification (Cisotto et al., 2020; Zhao et al., 2021; Li et al., 2021). ", + "bbox": [ + 174, + 347, + 825, + 609 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Second, certain seizure types (e.g. clonic seizures) are rare by nature. Training machine learning models that perform well on these rarer seizure classes using traditional supervised learning approaches is challenging, which could explain the performance difference between major and minority seizure types in prior studies (Raghu et al., 2020; Iesmantas & Alzbutas, 2020; AhmedtAristizabal et al., 2020). Several studies have investigated an alternative, self-supervised training strategy (Banville et al., 2020; Mohsenvand et al., 2020; Kostas et al., 2021; Martini et al., 2021; Xu et al., 2020), but they did not model EEGs as graphs or address automated seizure classification. Prior works have shown that self-supervised pre-training significantly improves model performance on data with imbalanced labels in the field of computer vision (Yang & Xu, 2020; Liu et al., 2021). Hence, we hypothesize that self-supervised pre-training can help improve our graph model performance on rare seizure types. Moreover, a large portion of EEG signals generally do not have seizures. Self-supervised pre-training strategy would allow the model to leverage the abundant nonseizure EEGs that are readily available in the dataset. ", + "bbox": [ + 174, + 617, + 825, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lastly, for seizure detection and classification models, the ability to not only provide a single prediction across all EEG channels, but also to provide the interpretability and the ability to localize seizures would be clinically useful for informing treatment strategy. While prior studies (Saab et al., 2020; Covert et al., 2019) have shown qualitative visualization for model interpretability, none have quantitatively assessed the model’s ability to localize seizures. ", + "bbox": [ + 174, + 805, + 823, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we aim to address these limitations in prior automated seizure detection and classification studies. First, we propose a graph-based modeling approach for EEG-based seizure detection and classification. Specifically, we propose two EEG graph structures that capture EEG sensor geometry (Figure 1b) or dynamic brain connectivity (Figure 1c), and we extend Diffusion Convolutional Recurrent Neural Network (DCRNN) (Li et al., 2018), an RNN with graph diffusion convolutions, to model the spatiotemporal dependencies in EEGs (Figure 1d). Second, we improve DCRNN performance using a self-supervised pre-training strategy of predicting the preprocessed EEG signals for the next time period without requiring additional data or labels. Finally, we propose quantitative metrics to assess our model’s ability to localize seizures. In summary: ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 825, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• We propose two EEG graph structures that capture (1) the natural geometry of EEG sensors or (2) dynamic connectivity in the brain, and show that building a recurrent graph neural network (GNN) based on these representations yields models for seizure detection and classification that outperform previous approaches on a large public dataset (5,499 EEGs). • We propose a self-supervised pre-training strategy to further improve our recurrent GNN model performance, particularly on rare seizure types. To our knowledge, our study is the first to date that combines graph-based modeling and self-supervised pre-training for EEGs. By leveraging graph structure and self-supervision, our method achieves 0.875 Area Under the Receiver Operating Characteristic Curve (AUROC) on seizure detection and 0.749 weighted F1-score on seizure classification, outperforming previous approaches on both seizure detection and classification on this large public dataset. Moreover, our selfsupervised pre-training method substantially improves classification of rare seizure types (e.g. 47 points increase in combined tonic seizure accuracy over baselines). • We propose a quantitative model interpretability analysis that can be used to assess a model’s ability to localize seizures, which is critical to determining the course of treatment for seizures. We show that by leveraging graph structure and self-supervision, our method precisely localizes $2 5 . 4 \\%$ focal seizures, providing 21.9 points improvement over a prior state-of-the-art CNN. Finally, by displaying the identified seizure regions on raw EEG signals and EEG graphs, our approach could provide valuable insights that support more effective seizure diagnosis in real-world clinical settings. ", + "bbox": [ + 215, + 200, + 825, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 METHODS ", + "text_level": 1, + "bbox": [ + 174, + 515, + 290, + 531 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 SEIZURE DETECTION AND CLASSIFICATION PROBLEM FORMULATION ", + "text_level": 1, + "bbox": [ + 174, + 547, + 697, + 563 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The goal of seizure detection is to predict whether a seizure exists within an EEG clip, and the goal of seizure classification is to predict the seizure type given a seizure EEG clip. Following a prior study (Saab et al., 2020), we examine our model’s capability for fast and slow detection and classification over 12-s and 60-s EEG clips, respectively. ", + "bbox": [ + 174, + 575, + 825, + 631 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 GRAPH-BASED MODELING FOR EEGS ", + "text_level": 1, + "bbox": [ + 178, + 650, + 480, + 665 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2.1 REPRESENTING EEGS AS GRAPHS ", + "text_level": 1, + "bbox": [ + 176, + 678, + 470, + 693 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We represent an EEG clip as a graph $\\mathcal { G } = \\{ \\mathcal { V } , \\mathcal { E } , W \\}$ , where $\\nu$ denotes the set of nodes (i.e. EEG electrodes/channels), $\\mathcal { E }$ denotes the set of edges, and $W$ is the adjacency matrix. We propose the following two methods of constructing the EEG graph. ", + "bbox": [ + 176, + 702, + 823, + 746 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Distance graph. To represent the natural geometry of EEG electrodes, we compute edge weight $W _ { i j }$ by applying a thresholded Gaussian kernel (Shuman et al., 2013) to the pairwise Euclidean distance between $v _ { i }$ and $v _ { j }$ , i.e., $W _ { i j } = \\exp \\biggl ( - \\frac { \\mathrm { d i s t } ( v _ { i } , v _ { j } ) ^ { 2 } } { \\sigma ^ { 2 } } \\biggr )$ if $\\mathrm { d i s t } ( v _ { i } , v _ { j } ) \\le \\kappa$ , else 0. Here, $\\mathrm { d i s t } ( v _ { i } , v _ { j } )$ is the Euclidean distance between electrodes $v _ { i }$ and $v _ { j }$ according to the standard 10-20 EEG electrode placement (Jasper, 1958), $\\sigma$ is the standard deviation of the distances, and $\\kappa$ is the threshold for sparsity. This results in a universal undirected, weighted graph for all EEG clips. Based on preliminary experiments and EEG domain knowledge, we chose $\\kappa = 0 . 9$ because it results in a reasonable graph that also resembles the EEG montage (longitudinal bipolar and transverse bipolar) widely used clinically (Acharya et al., 2016). Figure 1b shows the distance graph with $\\kappa = 0 . 9$ . In Appendix K, we explore different values of $\\kappa$ as well as constructing the distance graph using a Gaussian kernel with a pre-specified bandwidth. ", + "bbox": [ + 173, + 752, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Correlation graph. To capture dynamic brain connectivity, we define the edge weight $W _ { i j }$ as the absolute value of the normalized cross-correlation between the preprocessed signals in $v _ { i }$ and $v _ { j }$ . To introduce sparsity to the graph, only the edges whose weights are among the top- $\\tau$ neighbors of each node are kept (plus self-edges), i.e., $\\bar { W _ { i j } } = | X _ { : , i , : } * \\mathbf { \\bar { X } } _ { : , j , : } |$ if $v _ { j } \\in \\mathcal { N } ( v _ { i } )$ , else 0. Here, $X _ { : , i , : }$ : and $X _ { : , j , }$ : are preprocessed signals in $v _ { i }$ and $v _ { j }$ , $^ *$ represents the normalized cross-correlation, and $\\mathcal { N } ( v _ { i } )$ represents the top- $\\tau$ neighbors of $v _ { i }$ . This method results in a unique directed, weighted graph for each input EEG clip. Figure 1c shows an example correlation graph with $\\tau = 3$ . ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2.2 GRAPH NEURAL NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 215, + 419, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We adapt DCRNN (Li et al., 2018), a recurrent neural network with graph diffusion convolutions, to model the spatiotemporal dependencies in EEG signals. DCRNN was initially developed for traffic forecasting, where the dynamics of traffic flow are modeled as a diffusion process. Similarly, we can also model the spatial dependency in EEG signals as a diffusion process, because an electrode can be influenced more by electrodes in its anatomical proximity (measured by distance) (Acharya et al., 2016) or functional proximity (measured by correlation) (Sakkalis, 2011). Specifically, the diffusion process is characterized by a bidirectional random walk on a directed graph $\\mathcal { G }$ , which results in the following diffusion convolution (Li et al., 2018): ", + "bbox": [ + 173, + 239, + 825, + 352 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d205a3fa2c76fb673ba5201219cf8233959eebb858876509763a574881b0c592.jpg", + "text": "$$\nX _ { : , m \\star \\mathcal { G } } f _ { \\theta } = \\sum _ { k = 0 } ^ { K - 1 } \\big ( \\theta _ { k , 1 } ( D _ { \\mathcal { O } } ^ { - 1 } W ) ^ { k } + \\theta _ { k , 2 } ( D _ { I } ^ { - 1 } W ^ { \\top } ) ^ { k } \\big ) X _ { : , m } \\mathrm { f o r } m \\in \\{ 1 , . . . , M \\}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 354, + 766, + 398 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\pmb { X } \\in \\mathbb { R } ^ { N \\times M }$ is the preprocessed EEG clip at time step $t \\in \\{ 1 , . . . , T \\}$ with $N$ nodes and $M$ features, $f _ { \\theta }$ is the convolution filter with parameters $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { K \\times 2 }$ , $D _ { O }$ and $D _ { I }$ are the out-degree and in-degree diagonal matrices of the graph, respectively, $D _ { O } ^ { - 1 } W$ and $D _ { I } ^ { - 1 } W ^ { \\top }$ are the state transition matrices of the outward and inward diffusion processes, respectively, and $K$ is the number of maximum diffusion steps. ", + "bbox": [ + 173, + 401, + 825, + 473 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For undirected graphs, the diffusion convolution is similar to ChebNet spectral graph convolution (Defferrard et al., 2016) up to a constant scaling factor, and thus can be computed using stable Chebyshev polynomial bases as follows (Li et al., 2018): ", + "bbox": [ + 176, + 478, + 823, + 522 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/39f729c57d06fc7c58db91bfb9ddf5f9e6a6d169ecf2e7bc3edc076b96f92177.jpg", + "text": "$$\n{ \\cal X } _ { : , m \\star \\mathcal { G } } f _ { \\theta } = \\Phi \\bigg ( \\sum _ { k = 0 } ^ { K - 1 } \\theta _ { k } { \\bf A } ^ { k } \\bigg ) \\Phi ^ { \\top } { \\cal X } _ { : , m } = \\sum _ { k = 0 } ^ { K - 1 } \\theta _ { k } L ^ { k } { \\cal X } _ { : , m } = \\sum _ { k = 0 } ^ { K - 1 } \\tilde { \\theta } _ { k } T _ { k } ( \\tilde { L } ) { \\cal X } _ { : , m } \\mathrm { f o r } m \\in \\{ 1 , . . . , M \\}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 523, + 834, + 569 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $T _ { 0 } ( x ) = 1 , T _ { 1 } ( x ) = x$ , and $T _ { k } ( x ) = 2 x T _ { k - 1 } ( x ) - T _ { k - 2 } ( x )$ for $k \\geq 2$ are the bases of the Chebyshev polynomial, $L = D ^ { - { \\frac { 1 } { 2 } } } ( D - W ) D ^ { - { \\frac { 1 } { 2 } } } = \\Phi \\Lambda \\Phi ^ { \\top }$ is the normalized graph Laplacian, and $\\begin{array} { r } { \\tilde { \\pmb { L } } = \\frac { 2 } { \\lambda _ { m a x } } \\pmb { L } - \\pmb { I } } \\end{array}$ is the scaled graph Laplacian mapping eigenvalues from $[ 0 , \\lambda _ { m a x } ]$ to $[ - 1 , 1 ]$ . We use Equation 1 for directed correlation graphs, and Equation 2 for undirected distance graph. ", + "bbox": [ + 174, + 578, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Next, to model the temporal dependency in EEGs, we employ Gated Recurrent Units (GRUs) (Cho et al., 2014), a variant of RNN with a gating mechanism. Specifically, the matrix multiplications in GRUs are replaced with diffusion convolutions (or ChebNet spectral graph convolutions for undirected distance-based graph) (Li et al., 2018), allowing spatiotemporal modeling of EEG signals (referred to as “DCGRU”): ", + "bbox": [ + 173, + 648, + 825, + 719 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2b47a8d1ed73d5b9ffef8c7d9001d3578ffa617023d93402c9d1f0173e5564e7.jpg", + "text": "$$\n\\pmb { r } ^ { ( t ) } = \\sigma \\big ( \\Theta _ { r \\star \\mathscr { G } } [ \\pmb { X } ^ { ( t ) } , \\pmb { H } ^ { ( t - 1 ) } ] + \\mathbf { b } _ { r } \\big ) \\qquad \\pmb { u } ^ { ( t ) } = \\sigma \\big ( \\Theta _ { u \\star \\mathscr { G } } [ \\pmb { X } ^ { ( t ) } , \\pmb { H } ^ { ( t - 1 ) } ] + \\mathbf { b } _ { u } \\big )\n$$", + "text_format": "latex", + "bbox": [ + 230, + 722, + 767, + 742 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f7d502682b37f54d33462da447fe540e524a72c9cace0c2936c61d4665c1c327.jpg", + "text": "$$\nC ^ { ( t ) } = \\mathrm { t a n h } \\big ( \\Theta _ { C \\times \\mathcal { G } } [ \\mathbf { X } ^ { ( t ) } , ( { \\boldsymbol { r } } ^ { ( t ) } \\odot \\pmb { H } ^ { ( t - 1 ) } ) ] + \\mathbf { b } _ { C } \\big ) \\quad \\pmb { H } ^ { ( t ) } = \\pmb { u } ^ { ( t ) } \\odot \\pmb { H } ^ { ( t - 1 ) } + ( 1 - \\pmb { u } ^ { ( t ) } ) \\odot \\pmb { C } ^ { ( t ) }\n$$", + "text_format": "latex", + "bbox": [ + 178, + 746, + 799, + 765 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, $\\mathbf { \\boldsymbol { X } } ^ { ( t ) } , \\mathbf { \\boldsymbol { H } } ^ { ( t ) }$ denote the input and output of DCGRU at time step $t$ respectively, $\\sigma$ denotes Sigmoid function, $\\odot$ represents the Hadamard product, $\\boldsymbol { r } ^ { ( t ) } , \\boldsymbol { u } ^ { ( t ) }$ , $C ^ { ( t ) }$ denote reset gate, update gate and candidate at time step $t$ respectively, $\\star \\mathcal { G }$ denotes the diffusion convolution (or ChebNet spectral graph convolution), $\\Theta _ { r }$ , ${ \\bf b } _ { r }$ , $\\Theta _ { u }$ $\\mathbf { \\Delta } ) _ { u } , \\mathbf { b } _ { u } , \\mathbf { e }$ $\\Theta _ { C }$ and ${ \\bf b } _ { C }$ are the weights and biases for the corresponding convolutional filters. Finally, for seizure detection and classification, the models consist of several stacked DCGRUs followed by a fully-connected layer. ", + "bbox": [ + 173, + 766, + 825, + 853 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 SELF-SUPERVISED PRE-TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 869, + 454, + 883 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To further improve DCRNN performance, we propose a self-supervised pre-training strategy for EEGs. Specifically, we pre-train the model for predicting the next $T ^ { \\prime }$ second preprocessed EEG clips given a preprocessed 12-s (60-s) EEG clip. We hypothesize that by learning to predict the EEG signals for the next time period, the model would learn task-agnostic representations and improve downstream seizure detection and classification tasks. The model for self-supervised pre-training has a sequence-to-sequence architecture with an encoder and a decoder (Sutskever et al., 2014), each of which has several stacked DCGRUs (Figure 1d). We use mean absolute error between the true preprocessed EEG clips and the predicted clips as the loss function. Preliminary experiments suggest that predicting future $T ^ { \\prime } = 1 2$ second preprocessed EEG clips results in low regression loss on the validation set, and thus we use $T ^ { \\prime } = 1 2$ in all self-supervised pre-training experiments. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 214 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 MODEL INTERPRETABILITY AND ABILITY TO LOCALIZE SEIZURES ", + "text_level": 1, + "bbox": [ + 176, + 232, + 678, + 246 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We perform model interpretability analyses using an occlusion-based approach (Zeiler & Fergus, 2013). First, for seizure detection, we zero-fill one second EEG signals in one channel at a time and compute the relative change in the model’s output logit with respect to the original non-occluded output (original − occluded). Furthermore, we scale the values in each clip to $[ 0 , 1 ]$ using the minimum and maximum values in that clip. This results in an occlusion map $M \\in \\mathbb { R } ^ { \\bar { N } \\times T }$ , where $N$ is the number of EEG channels, $T$ is the clip length, and $M _ { i j }$ indicates the relative change in the model output when the $j$ -th second EEG clip in the $i$ -th channel is occluded. A larger $M _ { i j }$ indicates that the occluded region is more important for predicting seizure, and vice versa. We visualize $M$ by superimposing it over raw EEG signals and graph structures. ", + "bbox": [ + 174, + 257, + 825, + 385 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In addition to visually analyzing the aforementioned occlusion maps, we propose quantitative metrics to evaluate the model’s capability of localizing seizures based on the occlusion maps. Specifically, we define a coverage metric that quantifies how many true seizure regions are detected, as well as a localization metric that quantifies how many detected seizure regions are true seizure regions. Note that coverage and localization scores are analogous to recall and precision, respectively, in binary classification problems. Mathematically, let $M ^ { \\bar { \\mathrm { a n n o t } } } \\in \\mathbb { R } ^ { N \\times T }$ be detailed annotations of seizure duration and location, where $M _ { i j } ^ { \\mathrm { a n n o t } } = \\mathrm { i }$ if there is a seizure at the $j$ -th second in the $i$ -th channel, otherwise $M _ { i j } ^ { \\mathrm { a n n o t } } = 0$ . Let $\\boldsymbol { M } \\in \\mathbb { R } ^ { N \\times T }$ be the occlusion map described above. Then $\\mathrm { c o v e r a g e } = \\frac { \\sum _ { i , j } \\mathbf { 1 } _ { M _ { i j } } > 0 . 5 } { \\sum _ { i , j } M _ { i j } ^ { \\mathrm { a n n o t } } } \\mathrm { a n d l o c a l i z a t i o n } = \\frac { \\sum _ { i , j } \\mathbf { 1 } _ { M _ { i j } > 0 . 5 } M _ { i j } ^ { \\mathrm { a n n o t } } } { \\sum _ { i , j } \\mathbf { 1 } _ { M _ { i j } > 0 . 5 } } .$ ", + "bbox": [ + 161, + 391, + 825, + 547 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, we also perform occlusion-based interpretability analysis for seizure classification. Since the goal of seizure classification is to predict the seizure type given a seizure EEG clip, we hypothesize that the difference in signals between EEG channels are more important for predicting seizure types. Therefore, we completely drop one EEG channel at a time, and compute the relative change in the model output with respect to the original output. This results in an occlusion map $M ^ { \\prime } \\in \\mathbb { R } ^ { \\tilde { N } }$ , where $M _ { i } ^ { \\prime }$ indicates the relative change in the model output when the $i$ -th channel is dropped. ", + "bbox": [ + 174, + 553, + 825, + 637 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 656, + 326, + 672 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 688, + 375, + 702 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dataset. We use the public Temple University Hospital EEG Seizure Corpus (TUSZ) v1.5.2 (Shah et al., 2018; Obeid & Picone, 2016), the largest public EEG seizure database to date with 5,612 EEGs, 3,050 annotated seizures from clinical recordings, and eight seizure types1. We include 19 EEG channels in the standard 10-20 system (Figure 1b–1c). To evaluate model generalizability to unseen patients, we exclude five patients from the official TUSZ test set who exist in both the official TUSZ train and test sets. Moreover, we use the detailed annotations of seizure duration and location available in TUSZ for our interpretability analyses. Table 1 summarizes the TUSZ data. ", + "bbox": [ + 174, + 713, + 825, + 811 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Data preprocessing. Because seizures are associated with brain electrical activity in certain frequency bands (Tzallas et al., 2009), we perform data preprocessing to transform raw EEG signals to the frequency domain. Similar to prior studies (Asif et al., 2020; Ahmedt-Aristizabal et al., 2020; Covert et al., 2019), we obtain the log-amplitudes of the fast Fourier transform of raw EEG signals. For seizure detection and self-supervised pre-training, we use both seizure and non-seizure ", + "bbox": [ + 174, + 819, + 823, + 888 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "EEGs, and obtain the 12-s (60-s) EEG clips using non-overlapping 12-s (60-s) sliding windows. For seizure classification, we use only seizure EEGs, and obtain one 12-s (60-s) EEG clip from each seizure event, such that each EEG clip has exactly one seizure type. Appendix A presents details of data preprocessing, and Appendix B compares results of frequency-domain vs time-domain inputs. ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Refined seizure classification scheme. Because simple partial (SP) seizures and complex partial (CP) seizures are focal seizures characterized by a clinical behavior (consciousness during a seizure) and are not distinguishable by EEG signals alone (Fisher et al., 2017), we combine focal non-specific (FN), SP, and CP seizures to form a combined focal (CF) seizure class. We provide extensive experiments in Appendix C showing that these focal seizure types cannot be distinguished by a variety of models. We also exclude myoclonic seizures because only three myoclonic seizures are available in TUSZ. In addition, since tonic and tonic-clonic seizures are rare in the dataset (only 18 tonic seizures and 30 tonic-clonic seizures in TUSZ train set), and tonic-clonic seizures always start with a tonic phase (Fisher et al., 2017), we combine tonic-clonic seizures with tonic seizures to form a combined tonic (CT) seizure class. In summary, there are four seizure classes in total: CF, generalized non-specific (GN), absence (AB), and CT seizures (Table 1). In Appendix L, we show seizure classification results on the original eight seizure types in TUSZ. ", + "bbox": [ + 174, + 166, + 825, + 333 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data splits. We randomly split the official TUSZ train set by patients into train and validation sets by 90/10 for model training and hyperparameter tuning, respectively, and we hold-out the official TUSZ test set for model evaluation (excluding five patients who exist in both the official TUSZ train and test sets). The train, validation, and test sets consist of distinct patients. See Appendix D for the number of preprocessed EEG clips and patients in each split. ", + "bbox": [ + 174, + 340, + 823, + 410 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/781e99ed9ce9e7eb5a2d511dbaec18d20af07a76a57183e1560a3c482a740442.jpg", + "table_caption": [ + "Table 1: Summary of data in train and test sets of TUSZ v1.5.2. used in our study. " + ], + "table_footnote": [], + "table_body": "
EEG Files (% Seizure)Patients (% Seizure)Total Duration (% Seizure)CF Seizures (Patients)GN Seizures (Patients)AB Seizures (Patients)CT Seizures (Patients)
Train Set4,599 (18.9%)592 (34.1%)45,174.72 min (6.3%)1,868 (148)409 (68)50(7)48 (11)
Test Set900 (25.6%)45 (77.8%)9,031.58 min (9.8%)297 (24)114 (11)49 (5)61 (4)
", + "bbox": [ + 173, + 452, + 825, + 507 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. To compare our DCRNN to traditional CNNs/RNNs, we include three primary baselines: (a) Dense-CNN (Saab et al., 2020), a previous state-of-the-art CNN for seizure detection, (b) LSTM (Hochreiter & Schmidhuber, 1997), and (c) CNN-LSTM (implemented following AhmedtAristizabal et al. (2020)). The baselines are trained and evaluated on the same preprocessed data. Additionally, we compare our method to the reported results of two CNNs for seizure classification that use TUSZ and test on unseen patients (Asif et al., 2020; Iesmantas & Alzbutas, 2020). ", + "bbox": [ + 173, + 523, + 823, + 608 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Model training. Training for all models was accomplished using the Adam optimizer (Kingma & Ba, 2014) in PyTorch on a single NVIDIA Titan RTX GPU. Model parameters were randomly initialized for models without self-supervised pre-training, and were initialized with the pre-trained weights of the encoder for models with self-supervised pre-training. All models were run for five runs with different random seeds. Detailed hyperparameter settings are shown in Appendix E. During training, we perform data augmentation as described in Appendix F. ", + "bbox": [ + 174, + 614, + 825, + 699 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 719, + 392, + 733 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Graph neural network performance. Following recent studies (Asif et al., 2020; AhmedtAristizabal et al., 2020; O’Shea et al., 2020; Saab et al., 2020; Covert et al., 2019), we use AUROC and weighted F1-score as the main evaluation metrics for seizure detection and classification, respectively. Table 2 (3rd–7th rows) shows the performance of our DCRNN (without self-supervised pre-training) and the baselines. Distance graph-based DCRNN (or “Dist-DCRNN”) and correlation graph-based DCRNN (or “Corr-DCRNN”) without self-supervised pre-training perform on par with or better than the baselines. See Appendix G for additional evaluation scores. ", + "bbox": [ + 174, + 746, + 825, + 844 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Moreover, DCRNN outperforms the reported results of two existing CNNs (Asif et al., 2020; Ahmedt-Aristizabal et al., 2020) on seizure classification (Table 3). For fair comparison to Asif et al. (2020), we conduct 7-class seizure classification2 on the same 3-fold patient-wise split. ", + "bbox": [ + 176, + 852, + 823, + 893 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In addition, we conduct an ablation experiment to examine the effectiveness of Fourier transform in our preprocessing step for DCRNNs. We find that frequency-domain inputs substantially outperform time-domain inputs on both seizure detection and classification (Appendix B). ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/7f066763f7716b716c5d43c332b5f59e9d55805802d81f9f77d50b5701dfd9be.jpg", + "table_caption": [ + "Table 2: Seizure detection and seizure classification results. Mean and standard deviations are from five random runs. Best non-pretrained and pre-trained mean results are highlighted in bold. " + ], + "table_footnote": [], + "table_body": "
ModelSeizure Detection AUROCSeizure Classification Weighted F1-Score
12-s60-s12-s60-s
Dense-CNN0.812 ± 0.0140.796 ± 0.0140.576 ± 0.1010.626 ± 0.073
LSTM0.786 ± 0.0140.715 ± 0.0160.652 ± 0.0190.686 ± 0.020
CNN-LSTM0.749 ± 0.0060.682 ± 0.0030.633 ± 0.0250.641 ± 0.019
Corr-DCRNN w/o Pre-training0.812 ± 0.0120.804 ± 0.0150.710 ± 0.0230.701 ± 0.030
Dist-DCRNN w/o Pre-training0.824 ± 0.0200.793 ± 0.0220.703 ± 0.0250.690 ± 0.035
Corr-DCRNN w/ Pre-training0.861 ± 0.0050.850 ± 0.0140.723 ± 0.0170.749 ± 0.017
Dist-DCRNN w/Pre-training0.866 ± 0.0160.875 ± 0.0160.746 ± 0.0240.749 ± 0.028
", + "bbox": [ + 204, + 196, + 794, + 316 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d1517b9178541b50c1b63f8401f27d5b280c70f1c0ba67e9eb13edf4eee74757.jpg", + "table_caption": [ + "Table 3: Comparison between DCRNNs (w/o pre-training) and existing CNNs on seizure classification. Mean and standard deviations are from five random runs. Best mean results are in bold. " + ], + "table_footnote": [], + "table_body": "
Model7-Class Classification Weighted F1-ScoreCF Seizure AUROCGN Seizure AUROCAB Seizure AUROCCT Seizure AUROC
Asif et al. (2020)0.62=1--
Iesmantas & Alzbutas (2020)-0.780.72-
Corr-DCRNN,12-s0.619 ± 0.0060.907 ± 0.0080.815 ± 0.0270.972 ± 0.0130.908 ± 0.005
Dist-DCRNN,12-s0.585 ± 0.0060.896 ± 0.0110.814 ± 0.0270.983 ± 0.0080.890 ± 0.014
Corr-DCRNN, 60-s0.650 ± 0.0080.914 ± 0.0070.795 ± 0.0310.971 ± 0.0200.939 ± 0.008
Dist-DCRNN, 60-s0.606 ± 0.0090.920 ± 0.0040.811 ± 0.0320.973 ± 0.0090.926 ± 0.020
", + "bbox": [ + 205, + 372, + 792, + 465 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Self-supervised pre-training improves graph neural network performance. To assess the effectiveness of self-supervised pre-training in improving DCRNN’s performance, we compare the results of DCRNN without and with self-supervised pre-training. As shown in Table 2 (last two rows), DCRNN with self-supervised pre-training outperforms its non-pretrained counterpart on both seizure detection and classification. Notably, Dist-DCRNN with self-supervised pre-training achieves an AUROC of 0.875 for 60-s seizure detection, matching the performance of a CNN (AU$\\mathrm { R O C } { = } 0 . 8 8 $ that was pre-trained using supervised learning on a labeled dataset that was five times larger than TUSZ (Saab et al., 2020). Importantly, the pre-trained model weights are used for both seizure detection and classification, which indicates that our self-supervised pre-training method provides good model initialization that generalizes across tasks. ", + "bbox": [ + 173, + 478, + 825, + 617 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 2a shows the ROC curves of median DCRNNs and baselines for seizure detection. At a low false positive rate (FPR), such as $2 5 \\%$ FPR, pre-trained Dist-DCRNN achieves $8 4 . 3 \\%$ true positive rate (TPR) and pre-trained Corr-DCRNN achieves $8 1 . 7 \\%$ TPR on 12-s clips. Conversely, DenseCNN, LSTM, and CNN-LSTM only achieve $7 2 . 8 \\%$ , $6 7 . 0 \\%$ , and $6 2 . 5 \\%$ TPRs, respectively. Figure 2b shows the confusion matrices for Dist-DCRNNs and baselines for 12-s seizure classification. Dist-DCRNN without self-supervised pre-training achieves $93 \\%$ accuracy on the rare AB seizures, providing 2 points increase over the best baseline. Furthermore, Dist-DCRNN with self-supervised pre-training achieves $74 \\%$ accuracy on the rare CT seizures, providing 47 points increase in accuracy over the best baseline (Dense-CNN) and 48 points increase over non-pretrained Dist-DCRNN. ", + "bbox": [ + 173, + 623, + 825, + 750 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Surprisingly, despite being a majority class, many GN seizures are misclassified as CF seizures (Figure 2b). A board-certified neurologist manually analyzed the EEGs of 32 misclassified test GN seizures. We find that 27 seizures are in fact focal seizures but are mislabeled as GN seizures. In contrast, only 5 seizures are indeed generalized seizures but are misclassified by our models. ", + "bbox": [ + 174, + 756, + 825, + 811 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison between self-supervised pre-training and transfer learning. To compare our selfsupervised pre-training strategy to traditional transfer learning approaches, we pre-train DCRNNs for seizure detection and self-supervised prediction, respectively, on a large in-house dataset (40,316 EEGs, Table 7) and finetune the models for seizure detection and classification on TUSZ. We find that self-supervised pre-training consistently outperforms transfer learning (Appendix J). ", + "bbox": [ + 174, + 819, + 823, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison between self-supervised pre-training and auxiliary learning. We also investigate whether using the self-supervised task as an auxiliary task can outperform self-supervised pretraining. We find that auxiliary learning performs comparable to self-supervised pre-training on 12-s seizure detection, whereas self-supervised pre-training significantly outperforms auxiliary learning on 60-s seizure detection. See Appendix M and Table 10 for details. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f68ffb6e6422a2e2d70bc9f339a5c9f170d1acf2a7b09a3ff8b4281232e7eb16.jpg", + "image_caption": [ + "Figure 2: (a) ROC curves of median models for seizure detection. (b) Confusion matrices (averaged across five random runs) for the baselines and Dist-DCRNN without and with self-supervised pre-training for 12-s seizure classification. Each row of the confusion matrices is normalized by dividing by the number of examples in the corresponding class, such that each row sums up to one. ", + "Figure 3: Occlusion maps for seizure detection obtained from Corr-DCRNN model for (a)–(b) focal, and (c)–(d) generalized seizures. In each subfigure, left panel shows the occlusion map for 12-s when a seizure occurs and right panel shows the occlusion map values averaged along time and overlaid on the corresponding correlation graph. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 596, + 531 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 563, + 825, + 603 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Improved model interpretability and ability to localize seizures. We leverage occlusion-based techniques to localize seizures using our model predictions. Figure 3 shows example occlusion maps of correctly predicted 60-s test EEG clips from Corr-DCRNN with self-supervised pre-training. We observe that high saliency for focal seizures (Figures 3a–3b) is localized in the more abnormal EEG channels, whereas high saliency for generalized seizures (Figures 3c–3d) is more diffuse across channels. These patterns reflect the underlying brain activity of focal seizures (i.e. localized in one brain area) and generalized seizures (i.e. occur in all areas of the brain) (Fisher et al., 2017). Moreover, one can also interpret the seizure locations from the occlusion map overlaid on the graphs. In Figure 3a, Fp2, F4 and F8 are highlighted on the graph occlusion map, which correspond to the right frontal, central frontal, and anterior frontal brain regions that show the most abnormality in the EEG. In Figure 3b, while the seizure starts in P3, Fp1, F3, and F7, the graph occlusion map mainly highlights the central parietal region (P3), which is likely due to the ongoing artifact in channels Fp1-F7, Fp1-F3, and F3-C3. Figure 3c–3d show two generalized seizures that occur in all regions of the brain, and the highlighted nodes on the graphs are indeed more spread out across channels. In contrast, high saliency from Dense-CNN does not localize in any seizure regions (Appendix H). ", + "bbox": [ + 173, + 611, + 825, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Additionally, we leverage detailed annotations of seizure duration and location available in TUSZ dataset, and use coverage and localization scores to quantify the models’ ability to accurately localize seizures. Figure 4 shows coverage and localization distributions for Dense-CNN and DCRNNs (w/o and with pre-training) on correctly predicted 60-s test EEG clips. Both Dist-DCRNN and Corr-DCRNN have many more occlusion maps with high coverage and high localization scores than Dense-CNN. This suggests that DCRNN more accurately localizes seizures than Dense-CNN, which is intuitive given that DCRNN captures the connectivity of EEG electrodes. ", + "bbox": [ + 174, + 827, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Clinically, localizing focal seizure onset regions is key to epilepsy surgery for patients with focal seizures. Thus, the model’s ability to identify focal seizure regions as precisely as possible would be highly desirable. Notably, pre-trained Corr-DCRNN precisely localizes $2 5 . 4 \\%$ focal seizures and pre-trained Dist-DCRNN precisely localizes $2 1 . 8 \\%$ focal seizures (localization score $> 0 . 8$ , Figure 4c). Conversely, both non-pretrained Dist-DCRNN and Corr-DCRNN precisely localize $6 . 3 \\%$ focal seizures, and Dense-CNN only localizes $3 . 5 \\%$ focal seizures precisely (localization score $> 0 . 8$ ). Figures 4e–4f show example occlusion maps of test focal seizures whose localization scores $> 0 . 9$ from pre-trained Corr-DCRNN, where high saliency overlaps well with annotated seizure regions. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lastly, interpretability experiments on seizure classification show that high saliency EEG channels correspond to where the seizures are localized in CF seizures (Appendix I). This again suggests that our method could inform seizure locations for treatment strategy if used clinically. ", + "bbox": [ + 174, + 222, + 825, + 263 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/4a4a126e8f76d0849f3cae8db0656244a5c998ac8d4ae2d07790dd894e907c15.jpg", + "image_caption": [ + "Figure 4: Distributions of (a)–(b) coverage and (c)–(d) localization scores for Dense-CNN and DCRNNs on correctly predicted 60-s EEG clips for focal and generalized seizures. (e)–(f) Example occlusion maps with focal seizure whose localization ${ > } 0 . 9$ from pre-trained Corr-DCRNN. Only regions with normalized occlusion value ${ > } 0 . 5$ are colored. Red boxes are annotated seizure regions. " + ], + "image_footnote": [], + "bbox": [ + 240, + 290, + 758, + 487 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Comparison between graph structures. While Dist-DCRNN and Corr-DCRNN perform comparable to each other on seizure detection and classification (Table 2), we observe that Corr-DCRNN better localizes focal seizures than Dist-DCRNN, particularly when combined with self-supervised pre-training (e.g., coverage and localization scores $> 0 . 9$ in Figure 4a, 4c). This suggests that the correlation-based graph structure could provide better interpretability and representation of focal seizure EEGs. Moreover, the correlation graph structure has two particular advantages: (a) it can be used even when the physical locations of electrodes are unknown and (b) it captures dynamic brain connectivity rather than replying purely on spatial sensor information, which is particularly desirable for automated seizure detection and classification models. ", + "bbox": [ + 174, + 584, + 825, + 709 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 744, + 318, + 761 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In conclusion, we present a novel method combining graph-based modeling and self-supervised pretraining for EEG-based seizure detection and classification, as well as an interpretability method to quantify model ability to localize seizures. Our method sets new state-of-the-art on both seizure detection and classification on a large public dataset, significantly improves classification of rare seizure classes, and more accurately localizes seizures. We also find that the correlation-based graph more accurately localizes focal seizures than distance-based graph. The improved ability to localize seizures and the novel graph visualizations could provide clinicians with valuable insights about localized seizure regions in real-world clinical settings. Looking to the future, because our methods are not confined to EEG alone, our study opens exciting opportunities to build graph-based representations for a wide variety of medical time series. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 356, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported by a Wu Tsai Neurosciences Institute Neuroscience Translate Grant. The authors would like to thank Bibek Paudel, Liangqiong Qu, Jean Benoit Delbrouck, and Nandita Bhaskhar for their feedback on the paper. ", + "bbox": [ + 176, + 133, + 825, + 176 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ETHICS STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 196, + 343, + 213 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The Temple University Hospital EEG Seizure Corpus used in our study is anonymized and publicly available3 with full IRB approval (Shah et al., 2018; Obeid & Picone, 2016). No conflict of interest is reported from the authors. No harmful insights are provided by the seizure detection and classification models described in this study. Although we show that our methods could provide improved performance and clinical utility for seizure detection and classification, additional model validations are needed before they can be used in real-world clinical settings, including (a) validation on datasets from multiple institutions, (b) validation on different populations and age groups, and (c) validation on a EEG waveform-based seizure classification scheme that is approved by a consensus of board-certified neurologists. ", + "bbox": [ + 174, + 229, + 825, + 354 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REPRODUCIBILITY STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 377, + 436, + 392 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The Temple University Hospital EEG Seizure Corpus used in our study is publicly available. Detailed data preprocessing steps are provided in Appendix A. Source code is publicly available at https://github.com/tsy935/eeg-gnn-ssl. 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Anomaly detection on electroencephalography with self-supervised learning. In 2020 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), pp. 363–368. IEEE, 2020. ", + "bbox": [ + 173, + 313, + 825, + 357 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Yuzhe Yang and Zhi Xu. Rethinking the value of labels for improving class-imbalanced learning. In Conference on Neural Information Processing Systems (NeurIPS), 2020. ", + "bbox": [ + 171, + 364, + 820, + 393 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks, 2013. ", + "bbox": [ + 171, + 402, + 810, + 419 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Yanna Zhao, Gaobo Zhang, Changxu Dong, Qi Yuan, Fangzhou Xu, and Yuanjie Zheng. Graph attention network with focal loss for seizure detection on electroencephalography signals. International Journal of Neural Systems, pp. 2150027, 2021. ", + "bbox": [ + 176, + 426, + 821, + 469 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A DATA PREPROCESSING ", + "text_level": 1, + "bbox": [ + 176, + 137, + 400, + 155 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Because the EEG signals are sampled at different frequencies in the Temple University EEG Seizure Corpus (TUSZ), we resample them to the same frequency of $2 0 0 \\mathrm { H z }$ using the “resample” function in SciPy python package (Virtanen et al., 2020a). We perform the following preprocessing steps to obtain EEG clips in the frequency domain and their corresponding labels. ", + "bbox": [ + 174, + 172, + 825, + 228 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "First, for seizure detection, we use both seizure and non-seizure EEGs. We obtain the EEG clips by sliding a 12-s (or 60-s) window over the EEG signals without overlaps, and ignore the last window if it is shorter than the clip length. The label for each clip is $y = 1$ if at least one seizure event occurs within this clip, otherwise $y = 0$ . ", + "bbox": [ + 174, + 236, + 823, + 291 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Second, for seizure classification, we use only seizure EEGs. We obtain one 12-s (or 60-s) EEG clip for each seizure event starting at 2-s before the annotated seizure onset time, where a 2-s offset accounts for tolerance in the annotations. If a seizure event is shorter than 12-s (or 60-s), the EEG clip is truncated at the end of the seizure to prevent a clip from having multiple seizure types. The label for each clip is the index of the corresponding seizure class, i.e. $y ~ \\in ~ \\{ 0 , 1 , 2 , 3 \\}$ , which corresponds to combined focal (CF) seizures, generalized non-specific (GN) seizures, absence (AB) seizures, and combined tonic (CT) seizures. ", + "bbox": [ + 174, + 297, + 825, + 396 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Third, for self-supervised pre-training, we use the same EEG clips as seizure detection. ", + "bbox": [ + 174, + 402, + 743, + 417 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Fourth, for each EEG clip in each of the seizure detection/seizure classification/self-supervised pretraining tasks, we perform the following preprocessing steps to transform the signals in time domain to frequency domain: (a) slide a $t$ second window over the EEG clip without overlap, where $t$ is the time step size for networks involving recurrent layers; (b) apply fast Fourier transform (FFT) to each $t$ second window using the “fft” function in Scipy python package (Virtanen et al., 2020b), and retain the log amplitudes of the non-negative frequency components similar to prior studies (Asif et al., 2020; Ahmedt-Aristizabal et al., 2020; Covert et al., 2019); (c) $\\mathbf { Z }$ -normalize the EEG clip with respect to the mean and standard deviation of the training data. Because EEG clips for seizure classification may have variable lengths due to short seizures, we pad the clips with 0’s to facilitate model training in batches. We use $t = 1$ second as a natural choice of the time step size. After preprocessing, each EEG clip can be denoted as $\\pmb { X } \\in \\mathbb { R } ^ { T \\times N \\times M }$ , where $T = 1 2$ (or $T = 6 0$ ) represents the clip length, $N = 1 9$ represents the number of EEG channels/electrodes, and $M = 1 0 0$ represents the feature dimension after the aforementioned Fourier transform. ", + "bbox": [ + 174, + 424, + 825, + 606 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B EFFECTIVENESS OF FOURIER TRANSFORM ", + "text_level": 1, + "bbox": [ + 174, + 631, + 563, + 647 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To evaluate the effectiveness of Fourier transform (Appendix A), we compare the performance of our DCRNN (without self-supervised pre-training) on inputs without and with Fourier transform on both seizure detection and seizure classification. As shown in Table 4, frequency-domain inputs (with Fourier transform) result in significantly better performance than time-domain inputs (without Fourier transform). This is likely because seizures are associated with electrical activity in certain frequency bands (Tzallas et al., 2009), and thus short-time-interval frequency-domain inputs could be more informative than time-domain inputs. ", + "bbox": [ + 174, + 665, + 825, + 763 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C DISTINGUISHABILITY OF FOCAL NON-SPECIFIC, SIMPLE PARTIAL, AND COMPLEX PARTIAL SEIZURES ", + "text_level": 1, + "bbox": [ + 178, + 787, + 807, + 821 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Because simple partial and complex partial seizures are focal seizures characterized by the clinical behavior, consciousness during seizure (Fisher et al., 2017), they are not distinguishable from other focal seizures from EEG signals alone. In this study, we merge focal non-specific (FN), simple partial (SP), and complex partial (CP) seizures into a combined focal seizure class. To justify our decision, we perform classification of these focal seizure types using our DCRNNs and baselines. As shown in Figure 5, SP and CP seizures are largely misclassified as FN seizures, suggesting that machine learning models may not be able to distinguish these focal seizure types from EEG signals alone. This supports our decision of combining FN, SP, and CP seizures into one class. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/077fdc82df1796e64747623085b3f0443a74e4d5057728509e27750df881b39f.jpg", + "table_caption": [ + "Table 4: Seizure detection and classification results from DCRNNs (without self-supervised pretraining) on time-domain inputs and frequency-domain inputs. Mean and standard deviations are obtained from five random runs. " + ], + "table_footnote": [], + "table_body": "
ModelInput DomainSeizure DetectionAUROC12-s 60-sSeizure ClassificationWeighted F1-Score12-s 60-s
Corr-DCRNNWithout Pre-TrainingTimeFrequency0.717 ± 0.003 0.704 ± 0.0230.812 ± 0.012 0.804 ± 0.0150.597 ± 0.023 0.618 ± 0.0180.710 ± 0.023 0.701 ± 0.030
Dist-DCRNNWithout Pre-TrainingTimeFrequency0.733 ± 0.0120.825 ± 0.0190.698 ± 0.0030.793 ± 0.0220.592 ± 0.0220.703 ± 0.0250.608 ± 0.0150.690 ± 0.035
", + "bbox": [ + 173, + 155, + 825, + 261 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 287, + 825, + 315 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D TRAIN, VALIDATION, AND TEST SETS ", + "text_level": 1, + "bbox": [ + 174, + 337, + 524, + 354 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 5 shows the number of EEG clips and number of patients in our train, validation, and test sets for self-supervised pre-training, seizure detection, and seizure classification tasks. Train, validation, and test sets consist of distinct patients. ", + "bbox": [ + 174, + 369, + 825, + 412 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/b81995571e247396e037af4a88fe0feb76e5b9e3311aed4dbfb96b7d6e3ef17d.jpg", + "table_caption": [ + "Table 5: Number of EEG clips and patients in the train, validation, and test sets in our study. Train, validation, and test sets consist of distinct patients. " + ], + "table_footnote": [], + "table_body": "
EEG ClipLength(Secs)Train SetValidation SetTest Set
EEG Clips(% Seizure)Patients(% Seizure)EEG Clips(% Seizure)Patients(% Seizure)EEG Clips(% Seizure)Patients(% Seizure)
Pre-training& SeizureDetection60-s12-s38.613(9.3%)196.646(6.9%)530(34.0%)531(33.9%)5,503(11.4%)28.057(8.7%)61(36.1%)61(36.1%)8.848(14.7%)44,959(10.9%)45(77.8%)45(77.8%)
SeizureClassification60-s&12-s1,925(100.0%)179(100.0%)450(100.0%)22(100.0%)521(100.0%)34(100.0%)
", + "bbox": [ + 173, + 467, + 825, + 590 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E DETAILS OF MODEL TRAINING PROCEDURES AND HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 619, + 802, + 637 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We performed the following hyperparameter search on the validation set: (a) initial learning rate within range [5e-5, 1e-3]; (b) $\\bar { \\tau } \\in \\bar { \\{ 2 , 3 , 4 \\} }$ , the number of neighbors to keep for each node in the correlation graphs; (c) the number of Diffusion Convolutional Gated Recurrent Units (DCGRU) layers within range $\\{ 2 , 3 , 4 , 5 \\}$ and hidden units within range $\\{ 3 2 , 6 4 , 1 2 8 \\}$ ; (d) the maximum diffusion step $K \\in \\{ 2 , 3 , 4 \\}$ ; (e) dropout probability in the last fully connected layer. We used a batch size of 40 EEG clips, the maximum possible across all models and baselines on a single Titan RTX GPU. The hyperparameters were selected based on the best performance on the validation set, and are detailed in the next sections. We used the cosine annealing learning rate scheduler (Loshchilov & Hutter, 2017) in PyTorch for all model training. We ran five runs with different random seeds for all models. In all experiments, model training was early stopped when the validation loss did not decrease for five consecutive epochs. ", + "bbox": [ + 173, + 652, + 825, + 805 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Model training for seizure detection. For seizure detection, the plentiful negative examples in the train set were undersampled such that the train set had $50 \\%$ positive examples, which resulted in 27,292 training examples for 12-s clips and 7,188 training examples for 60-s clips. We used binary cross-entropy as the loss function to train the seizure detection models. The models were trained for 100 epochs with an initial learning rate of 1e-4. For the correlation graphs, the top-3 neighbors’ edges were kept for each node. The maximum number of diffusion step was 2, and the dropout probability was 0 (i.e. no dropout). The model consists of two stacked DCGRU layers with 64 hidden units, resulting in 168,641 trainable parameters for the distance graph and 280,769 trainable parameters for the correlation graphs. Model training for seizure detection took about $2 0 \\mathrm { - m i n }$ for 12-s EEG clips, and about 30-min for 60-s EEG clips. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/6e69085ea0475e338a3a51b915c47e73bc6469ba1595b05e67a4c8141cab5237.jpg", + "image_caption": [ + "Figure 5: Confusion matrices (averaged across five random runs) of focal seizure classification on (a) 60-s EEG clips, and (b) 12-s EEG clips. All models fail to distinguish well among these focal seizure types, supporting our decision of merging these seizure types into one combined focal seizure class. Note that each row of the confusion matrices is normalized by dividing by the number of examples in the corresponding class, such that each row sums up to one. " + ], + "image_footnote": [], + "bbox": [ + 207, + 106, + 792, + 746 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 859, + 823, + 888 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "To obtain the final seizure/non-seizure prediction, we performed decision threshold search on the validation set. More specifically, to balance between precision and recall scores, we selected the decision threshold that results in the highest F1-score on the validation set. When evaluating the models on the test set, EEG clips with probabilities above this decision threshold are predicted as seizures, while clips with probabilities below this decision threshold are predicted as non-seizures. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 145 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Model training for seizure classification. For seizure classification, we used multi-class crossentropy as the loss function during training. The models were trained for 60 epochs with an initial learning rate of 3e-4. For the correlation graphs, the top-3 neighbors’ edges were kept for each node. The maximum number of diffusion step was 2, and the dropout probability was 0.5. The model consists of two stacked DCGRU layers with 64 hidden units, resulting in 168,836 trainable parameters for the distance graph and 280,964 trainable parameters for the correlation graphs. Model training for seizure classification took about 3-min for 12-s EEG clips, and about 7-min for 60-s EEG clips. ", + "bbox": [ + 174, + 152, + 825, + 263 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Model training for self-supervised task. Preliminary experiments suggested that predicting future $T ^ { \\prime } = 1 2$ second preprocessed EEG clips results in low regression loss on the validation set given previous 12-s (60-s) preprocessed clips, and thus we used $\\bar { T ^ { \\prime } } = 1 2$ in all self-supervised pre-training experiments. For self-supervised pre-training, we used mean absolute error (MAE) as the loss function. The models were trained for 350 epochs with an initial learning rate of 5e-4. For the correlation graphs, the top-3 neighbors’ edges were kept for each node. The maximum number of diffusion step was 2. The model consists of three stacked DCGRU layers with 64 hidden units in both the encoder and decoder, resulting in 417,572 trainable parameters for the distance graph and 690,980 trainable parameters for the correlation graphs. Model training for self-supervised prediction took about 10-h for 12-s EEG clips, and about $^ { 2 4 - \\mathrm { h } }$ for 60-s EEG clips. ", + "bbox": [ + 173, + 270, + 825, + 410 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Model training for baselines. For baseline Dense-CNN, we employ the same model architecture as that described in Saab et al. (2020). For baseline LSTM (Hochreiter & Schmidhuber, 1997), we have the number of LSTM layers and hidden units the same as the number of DCGRU layers and hidden units in our DCRNN model. For baseline CNN-LSTM, we use the same model architecture described in Ahmedt-Aristizabal et al. (2020), i.e., two stacked convolutional layers $3 2 3 \\times 3$ kernels), one max-pooling layer $( 2 \\times 2 )$ , one fully-connected layer (output neuron $= 5 1 2$ ), two stacked LSTM layers (hidden size $= 1 2 8$ ), and one fully connected layer. ", + "bbox": [ + 174, + 416, + 825, + 515 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F DATA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 535, + 395, + 551 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "During training, we performed the following data augmentations based on EEG domain knowledge: (a) randomly scaling the amplitude of the raw EEG signals by a scale within [0.8, 1.2] and (b) randomly reflecting the signals along the scalp midline. ", + "bbox": [ + 174, + 568, + 825, + 609 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "G ADDITIONAL EVALUATION RESULTS ON SEIZURE DETECTION ", + "text_level": 1, + "bbox": [ + 176, + 631, + 725, + 647 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table 6 shows F1-score, Area Under the Precision-Recall Curve (AUPR), sensitivity, and specificity of seizure detection models. ", + "bbox": [ + 173, + 662, + 821, + 690 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "H SEIZURE DETECTION OCCLUSION MAPS FROM BASELINE DENSE-CNN ", + "text_level": 1, + "bbox": [ + 169, + 712, + 808, + 728 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Figure 6 shows example seizure detection occlusion maps obtained from baseline Dense-CNN on correctly predicted 60-s EEG clips in the test set. Unlike our model (Figure 3), high saliency from Dense-CNN does not localize in any seizure regions. ", + "bbox": [ + 174, + 744, + 825, + 786 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "I SEIZURE CLASSIFICATION OCCLUSION MAPS FROM DCRNN ", + "text_level": 1, + "bbox": [ + 174, + 808, + 714, + 824 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Figure 7 shows example seizure classification occlusion maps from Corr-DCRNN with selfsupervised pre-training on correctly predicted 60-s EEG clips in the test set. The occlusion maps are obtained by completely dropping one EEG channel at a time and calculating the relative change in the model output. For CF seizures (a–b), high saliency regions correspond to brain areas where the focal seizures are localized. For the other generalized seizure types (c–e), less salient regions correspond to less abnormal areas. ", + "bbox": [ + 173, + 840, + 825, + 922 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table 6: Additional evaluation scores for seizure detection on (a) 12-s EEG clips and (b) $6 0 \\mathrm { - } \\mathrm { s }$ EEG clips. Mean and standard deviations are obtained from five random runs. Best non-pretrained and pre-trained mean results are highlighted in bold. ", + "bbox": [ + 173, + 101, + 826, + 143 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/ac6855daf5f4ca0c49c286de4adf2a73eecc775a15c5f23950ee05975d1e874b.jpg", + "table_caption": [ + "(a) 12-s EEG clips " + ], + "table_footnote": [], + "table_body": "
ModelF1-Score(mean ± std)AUPR(mean ± std)Sensitivity(mean ± std)Specificity(mean ± std)
Dense-CNNLSTMCNN-LSTM0.326 ± 0.0190.376 ± 0.0210.337 ± 0.0090.328 ± 0.0430.354 ± 0.0230.309 ± 0.0150.293 ± 0.0210.357 ± 0.0450.333 ± 0.0280.938 ± 0.0140.934 ± 0.0150.920 ± 0.021
Corr-DCRNNWithout Pre-trainingDist-DCRNNWithout Pre-training0.392 ± 0.0270.437 ± 0.0290.370 ± 0.0270.411 ± 0.0410.373 ± 0.0350.411 ± 0.0380.935 ± 0.0120.943 ± 0.006
Corr-DCRNNWith Pre-trainingDist-DCRNNWith Pre-training0.484 ± 0.0110.487 ± 0.0420.454 ± 0.0200.463 ± 0.0480.524 ± 0.0120.592 ± 0.0520.922 ± 0.0040.897 ± 0.012
", + "bbox": [ + 173, + 175, + 825, + 393 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/81484830b45f9fed4cb2d4f973f9bb08e8fddfc2e7b663c4fc8f472078972b19.jpg", + "table_caption": [ + "(b) 60-s EEG clips " + ], + "table_footnote": [], + "table_body": "
ModelF1-Score(mean ± std)AUPR(mean ± std)Sensitivity(mean ± std)Specificity(mean ± std)
Dense-CNNLSTMCNN-LSTM0.404 ± 0.0220.365 ± 0.0090.330 ± 0.0160.399 ± 0.0170.287 ± 0.0260.276 ± 0.0090.451 ± 0.1340.463 ± 0.0600.363 ± 0.0440.869 ± 0.0710.814 ± 0.0530.857 ± 0.023
Corr-DCRNNWithout Pre-trainingDist-DCRNNWithout Pre-training0.448 ± 0.0290.341 ± 0.1700.440 ± 0.0210.418 ± 0.0460.457 ± 0.0580.326 ± 0.1830.900 ± 0.0280.932 ± 0.058
Corr-DCRNN With Pre-trainingDist-DCRNNWith Pre-training0.514 ± 0.0280.571 ± 0.0290.539 ± 0.0240.593 ± 0.0310.502 ± 0.0470.570 ± 0.0470.923 ± 0.0080.927 ± 0.012
", + "bbox": [ + 173, + 433, + 825, + 651 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "J COMPARISON BETWEEN SELF-SUPERVISED PRE-TRAINING AND TRANSFER LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 690, + 740, + 724 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To compare our self-supervised pre-training strategy to traditional transfer learning, we perform transfer learning by pre-training DCRNNs for seizure detection on an in-house dataset (40,316 EEGs, Table 7) and finetuning on TUSZ data for seizure detection and classification. Due to the lack of fine-grained seizure type labels, we do not pre-train DCRNNs for seizure classification on the inhouse dataset. Moreover, we pre-train DCRNNs on the in-house dataset using our self-supervised pre-training strategy and finetuned them for seizure detection and classification on TUSZ. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Table 8 shows DCRNN results with self-supervised pre-training and transfer learning from the inhouse dataset. Self-supervised pre-training from the in-house dataset (3rd-4th rows) consistently outperforms trasnfer learning from the in-house dataset (5th-6th rows). We speculate that transfer learning does not perform comparably to self-supervised pre-training because it suffers from distribution shift in the data (i.e., the in-house dataset comes from a different population and uses a slightly different EEG acquisition protocol). In contrast, by learning to predict the EEGs for the next time period, self-supervised pre-training encourages the model to learn task-agnostic representations and thus mitigates the problem of distribution shift. ", + "bbox": [ + 174, + 840, + 823, + 924 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/6e0d377f9625210f0abab6d5575190ebe819810f3fecb396c941993df0e710a1.jpg", + "image_caption": [ + "Figure 6: Example occlusion maps for seizure detection obtained from baseline Dense-CNN on correctly predicted 60-s clips for (a)–(b) focal seizures, and (c)–(d) generalized seizures. Red boxes indicate the duration of the seizures. Note that the values within an occlusion map are normalized, and thus should not be compared across different occlusion maps. " + ], + "image_footnote": [], + "bbox": [ + 178, + 113, + 820, + 482 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 588, + 823, + 617 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/d97b26932a0026f7c65908204c1f1535327e76614170a6baf7eaa2c4970a3e3f.jpg", + "table_caption": [ + "Table 7: Summary of in-house dataset. Only annotations for the start of seizure are available. For EEG files with seizures, EEG clips are obtained by taking 12-s (or 60-s) signals starting from the annotated seizure start time. For EEG files without seizures, EEG clips are obtained by taking 12- s (or 60-s) signals randomly from the entire signal. We apply the same data preprocessing steps described in Appendix A. " + ], + "table_footnote": [], + "table_body": "
EEG Files (% Seizure) Total DurationPatients (% Seizure)EEG Clips (% Seizure)
In-House Train Set40,316 (24.1%)853,141.87 min5,355 (25.7%)46,613 (34.4%)
", + "bbox": [ + 184, + 712, + 812, + 768 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "K CHOICE OF DISTANCE-BASED GRAPH STRUCTURE ", + "text_level": 1, + "bbox": [ + 174, + 795, + 633, + 813 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For the distance graph, we used a threholded Gaussian kernel (Shuman et al., 2013) to compute the edge weight between two electrodes (Section 2.2.1). We experimented with $\\kappa$ , the threshold for graph sparsity, within range [0.1, 2]. Based on preliminary experiments and EEG domain knowledge, we chose $\\kappa = 0 . 9$ because it results in a reasonable graph (e.g. no long-range connection) that resembles the EEG montage (longitudinal bipolar and transverse bipolar) widely used clinically (Acharya et al., 2016). Figure 8 shows distance graphs resulting from different thresholds $\\kappa$ . In Figure 8, we can see that a smaller threshold (e.g., 0.7 or 0.8) results in missing edges between nearby electrodes. For example, there is no edge between FP1 and FZ for threshold 0.8 and no edge between T3 and C3 for threshold 0.7. In contrast, a larger threshold (e.g., 1.0, 1.1, or 1.2) results in edges connecting electrodes that are spatially far apart. For example, there is an edge connecting C3 and $\\mathrm { F Z }$ , as well as an edge between F7 and T5 for threshold 1.2. Using EEG domain knowledge provided by a board certified neurologist, we believe that a threshold of 0.9 results in a more reasonable distance-based EEG graph compared to other thresholding values. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/1ed952354041b4f418a708fd0fb19d993725ec5389aad0c54c3eb6387044a8e7.jpg", + "table_caption": [ + "Table 8: Comparison of results between self-supervised pre-training and transfer learning pretrained on an in-house dataset. 3rd-4th rows: DCRNN results with self-supervised (SS) pre-training on the in-house dataset. 5th-6th rows: DCRNN results with transfer learning pre-trained for seizure detection on the in-house dataset. Mean and standard deviations are from five random runs. Best mean results for each column are highlighted in bold. " + ], + "table_footnote": [], + "table_body": "
ModelSeizure Detection AUROCSeizure Classification Weighted F1-Score
12-s60-s12-s60-s
Corr-DCRNN w/ SS Pre-Training Dist-DCRNN w/ SS0.863 ± 0.0050.856 ± 0.0130.736 ± 0.0070.723 ± 0.010
Pre-Training Corr-DCRNN w/ Transfer0.879 ± 0.0060.886 ± 0.0060.767 ± 0.0380.718 ± 0.018
Learning Dist-DCRNN w/ Transfer Learning0.848 ± 0.018 0.866 ± 0.0010.850 ± 0.002 0.847 ± 0.0040.733 ± 0.027 0.720 ± 0.0460.711 ± 0.025 0.691 ± 0.038
", + "bbox": [ + 173, + 183, + 825, + 356 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 382, + 825, + 467 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In addition, we explore using a Gaussian kernel with a specified bandwidth for building the distance graph, i.e., $W _ { i j } = \\frac { 1 } { \\sqrt { 2 \\pi h ^ { 2 } } } \\mathrm { e x p } \\bigg ( - \\frac { \\mathrm { d i s t } ( v _ { i } , v _ { j } ) ^ { 2 } } { 2 h ^ { 2 } } \\bigg )$ , where $W _ { i j }$ is the edge weight between electrodes $v _ { i }$ and $v _ { j }$ , $\\mathrm { d i s t } ( v _ { i } , v _ { j } )$ is the Euclidean distance between $v _ { i }$ and $v _ { j }$ , and $h$ is the Gaussian kernel bandwidth. Figure 9 shows the distance graph structures resulting from different values of bandwidth. With bandwidth $= 0 . 0 6$ , the distance graph structure resembles the original graph structure using thresholded Gaussian kernel with a threshold of 0.9 (Figure 1b). ", + "bbox": [ + 173, + 472, + 825, + 574 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "L 8-CLASS SEIZURE CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 594, + 506, + 611 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We also perform seizure classification on the original eight seizure types4 available in TUSZ (see Table 9). Note that only one patient’s two myoclonic seizures are available in the official TUSZ train set and only one patient’s one myoclonic seizure is available in the official TUSZ test set. Hence, patient-wise train/validation splits for myoclonic seizure is not possible, and we randomly assign one myoclonic seizure in the official TUSZ train set to our train split and the other to our validation split. As shown in Table 9, DCRNNs consistently outperform the baselines on 8-class seizure classification. ", + "bbox": [ + 174, + 626, + 825, + 723 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "M USING SELF-SUPERVISED PREDICTION AS AN AUXILIARY TASK ", + "text_level": 1, + "bbox": [ + 173, + 743, + 746, + 760 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Here, instead of pre-training DCRNNs for the self-supervised prediction task (i.e., predicting preprocessed EEG clips for the next time period), we conduct experiments using the self-supervised prediction task as an auxiliary task. Because EEG clips for the seizure classification task have variable lengths, we only perform this experiment for seizure detection to facilitate training in batches. Specifically, the loss function is $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { s e i z u r e ~ d e t e c t i o n } } + \\lambda \\mathcal { L } _ { \\mathrm { S S } }$ prediction where $\\lambda$ is a hyperparameter balancing the seizure detection loss and the self-supervised loss and is tuned on the validation set. For 12-s (or 60-s) seizure detection, the auxiliary task is to predict the next 6-s preprocessed EEG clips given the first 6-s (or 30-s) EEG clips. ", + "bbox": [ + 173, + 775, + 825, + 887 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/e3b4b6562d2007d47f4fbaba13304ce6f69e7776bdd870d1b04e94b34fa5295d.jpg", + "table_caption": [ + "Table 9: Results (weighted F1-scores) of seizure classification on original 8 seizure types in TUSZ. Mean and standard deviations are from five random runs. Best mean results for both non-pretrained and pretrained models are highlighted in bold. " + ], + "table_footnote": [], + "table_body": "
Model12-s60-s
Dense-CNNLSTMCNN-LSTM0.431 ± 0.0370.515 ± 0.0250.489 ± 0.0360.427 ± 0.0470.525 ± 0.0170.509 ± 0.021
Corr-DCRNNWithout Pre-trainingDist-DCRNNWithout Pre-training0.553 ± 0.0250.570 ± 0.0270.577 ± 0.0280.600 ± 0.022
Corr-DCRNNWith Pre-trainingDist-DCRNNWith Pre-training0.582 ± 0.0140.583 ± 0.0090.591 ± 0.0080.607 ± 0.017
", + "bbox": [ + 305, + 155, + 692, + 348 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 10 shows the seizure detection results for DCRNNs when the self-supervised (SS) prediction task is used as an auxiliary task during training. Compared to self-supervised pre-training (Table 2 last two rows), auxiliary learning only marginally improves Dist-DCRNN performance on 12- s seizure detection (no statistical significance), whereas self-supervised pre-training significantly outperforms auxiliary learning for 60-s seizure detection. ", + "bbox": [ + 173, + 373, + 825, + 444 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/bfa1e02275a653a5381fd9251603856b234d47305bd8b59502103b0cc88372e0.jpg", + "table_caption": [ + "Table 10: Seizure detection results for DCRNNs with self-supervised pre-training (same as Table 2) and auxiliary learning. Mean and standard deviations are from five random runs. Best mean results for Dist-DCRNN/Corr-DCRNN are in bold. " + ], + "table_footnote": [], + "table_body": "
ModelSeizure Detection AUROC
12-s60-s
Corr-DCRNN w/ SS Pre-Training (Table 2)0.861 ± 0.0050.850 ± 0.013
Corr-DCRNN w/ SS Auxiliary Task0.851 ± 0.0080.811 ± 0.005
Dist-DCRNN w/ SS Pre-Training (Table 2) Dist-DCRNN w/ SS Auxiliary Task0.866 ± 0.016 0.875 ± 0.0050.875 ± 0.016 0.840 ± 0.013
", + "bbox": [ + 230, + 511, + 766, + 614 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/ea3f2f8b86ff27ddf606d1573755de6ac09c4d00c7782dea554e1f1e21c8f90a.jpg", + "image_caption": [ + "Figure 7: Example occlusion maps for seizure classification obtained from pre-trained Corr-DCRNN model for (a)–(b) combined focal (CF) seizures, (c) generalized non-specific (GN) seizure, (d) absence (AB) seizure, and (e) combined tonic (CT) seizure. In each subfigure, the bottom panel shows the occlusion values for each channel replicated along the time dimension and overlaid on 60-s EEG signals, the red boxes indicate the duration of seizures in the EEG clips, and the top panel shows the occlusion values overlaid on the correlation graph structure. To visualize occlusion map values in this “double banana” montage, we subtract the occlusion values between the corresponding channels in the montage, which results in different values between occlusion maps shown on the EEG signals (bottom panel in each subfigure) and that shown on the graph structures (top panel in each subfigure). Note that the values within an occlusion map are normalized, and thus should not be compared across different occlusion maps. " + ], + "image_footnote": [], + "bbox": [ + 209, + 94, + 782, + 750 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/b6c52e09286733a3355b5ae80fc11331e96176ddbf306c8eeb9bf6421a1f0537.jpg", + "image_caption": [ + "Figure 8: Distance graphs with different threshold $\\kappa$ in the thresholded Gaussian kernel. Small thresholds (e.g., 0.7 and 0.8) result in missing edges between nearby electrodes, whereas large thresholds (e.g., 1.0, 1.1, 1.2) result in edges connecting electrodes that are spatially far away. 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Specifically, we examine the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 373, + 470, + 384 + ], + "spans": [ + { + "bbox": [ + 141, + 373, + 470, + 384 + ], + "score": 1.0, + "content": "expressivity of LLMs with CoT in solving fundamental mathematical and decision-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 383, + 469, + 395 + ], + "spans": [ + { + "bbox": [ + 142, + 383, + 469, + 395 + ], + "score": 1.0, + "content": "making problems. By using circuit complexity theory, we first give impossibility", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 394, + 469, + 406 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 469, + 406 + ], + "score": 1.0, + "content": "results showing that bounded-depth Transformers are unable to directly produce", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 405, + 469, + 417 + ], + "spans": [ + { + "bbox": [ + 141, + 405, + 469, + 417 + ], + "score": 1.0, + "content": "correct answers for basic arithmetic/equation tasks unless the model size grows", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 416, + 469, + 429 + ], + "spans": [ + { + "bbox": [ + 141, + 416, + 469, + 429 + ], + "score": 1.0, + "content": "super-polynomially with respect to the input length. In contrast, we then prove by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 426, + 469, + 438 + ], + "spans": [ + { + "bbox": [ + 141, + 426, + 469, + 438 + ], + "score": 1.0, + "content": "construction that autoregressive Transformers of constant size suffice to solve both", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 438, + 470, + 450 + ], + "spans": [ + { + "bbox": [ + 141, + 438, + 470, + 450 + ], + "score": 1.0, + "content": "tasks by generating CoT derivations using a commonly used math language format.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 448, + 469, + 461 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 469, + 461 + ], + "score": 1.0, + "content": "Moreover, we show LLMs with CoT can handle a general class of decision-making", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 459, + 470, + 473 + ], + "spans": [ + { + "bbox": [ + 141, + 459, + 470, + 473 + ], + "score": 1.0, + "content": "problems known as Dynamic Programming, thus justifying their power in tackling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 470, + 469, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 470, + 469, + 482 + ], + "score": 1.0, + "content": "complex real-world tasks. Finally, an extensive set of experiments show that, while", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 481, + 469, + 493 + ], + "spans": [ + { + "bbox": [ + 142, + 481, + 469, + 493 + ], + "score": 1.0, + "content": "Transformers always fail to directly predict the answers, they can consistently learn", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 492, + 458, + 504 + ], + "spans": [ + { + "bbox": [ + 141, + 492, + 458, + 504 + ], + "score": 1.0, + "content": "to generate correct solutions step-by-step given sufficient CoT demonstrations.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 18.5, + "bbox_fs": [ + 141, + 306, + 470, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 190, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 192, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 192, + 538 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "Transformer-based Large Language Models (LLMs) have emerged as a foundation model in natural", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "language processing. Among them, the autoregressive paradigm has gained arguably the most", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 568, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 506, + 583 + ], + "score": 1.0, + "content": "popularity [51, 9, 46, 69, 57, 16, 52, 54], based on the philosophy that all different tasks can be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "uniformly treated as sequence generation problems. Specifically, given any task, the input along with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "the task description can be together encoded as a sequence of tokens (called the prompt); the answer is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 601, + 507, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 507, + 617 + ], + "score": 1.0, + "content": "then generated by predicting subsequent tokens conditioned on the prompt in an autoregressive way.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 547, + 507, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "Previous studies highlighted that a carefully designed prompt greatly matters LLMs’ performance", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "[32, 38]. In particular, the so-called Chain-of-Thought prompting (CoT) [61] has been found crucial", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 506, + 653 + ], + "score": 1.0, + "content": "for tasks involving arithmetic or reasoning, where the correctness of generated answers can be dramat-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 652, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 506, + 664 + ], + "score": 1.0, + "content": "ically improved via a modified prompt that triggers LLMs to output intermediate derivations. Practi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 507, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 507, + 676 + ], + "score": 1.0, + "content": "cally, this can be achieved by either adding special phrases such as “let’s think step by step” or by giv-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 686 + ], + "score": 1.0, + "content": "ing few-shot CoT demonstrations [34, 61, 56, 44, 70, 63]. However, despite the striking performance,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "the underlying mechanism behind CoT remains largely unclear and mysterious. On one hand, are there", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "indeed inherent limitations of LLMs in directly answering math/reasoning questions? On the other", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 326, + 96 + ], + "score": 1.0, + "content": "hand, what is the essential reason behind the success of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 326, + 83, + 349, + 94 + ], + "score": 0.78, + "content": "\\mathrm { C o } \\mathbf { \\check { T } } ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 349, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "in boosting the performance of LLMs?", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 618, + 507, + 697 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "indeed inherent limitations of LLMs in directly answering math/reasoning questions? On the other", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 326, + 96 + ], + "score": 1.0, + "content": "hand, what is the essential reason behind the success of", + "type": "text" + }, + { + "bbox": [ + 326, + 83, + 349, + 94 + ], + "score": 0.78, + "content": "\\mathrm { C o } \\mathbf { \\check { T } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "in boosting the performance of LLMs?", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 114 + ], + "score": 1.0, + "content": "This paper takes a step towards theoretically answering the above questions. We begin with studying", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 109, + 506, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 125 + ], + "score": 1.0, + "content": "the capability of LLMs on two basic mathematical tasks: evaluating arithmetic expressions and solving", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "linear equations. Both tasks are extensively employed and serve as elementary building blocks in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "solving complex real-world math problems [10]. We first provide fundamental impossibility results,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "showing that none of these tasks can be solved using bounded-depth Transformer models without", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "CoT unless the model size grows super-polynomially with respect to the input length (Theorems 3.1", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "and 3.2). Remarkably, our proofs provide insights into why this happens: the reason is not due to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "(serialized) computational cost of these problems but rather to their parallel complexity [2]. We next", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "score": 1.0, + "content": "show that the community may largely undervalue the strength of autoregressive generation: we prove", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "by construction that autoregressive Transformers of constant size can already perfectly solve both", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "tasks by generating intermediate derivations in a step-by-step manner using a commonly-used math", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "language format (Theorems 3.3 and 3.4). Intuitively, this result hinges on the recursive nature of CoT,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 503, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 503, + 244 + ], + "score": 1.0, + "content": "which increases the “effective depth” of the Transformer to be proportional to the generation steps.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "Besides mathematics, CoT also exhibits remarkable performance across a wide range of reasoning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "tasks. To gain a systematic understanding of why CoT is beneficial, we next turn to a fundamental", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "class of problems known as Dynamic Programming (DP) [6]. DP represents a golden framework", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "for solving sequential decision-making tasks: it decomposes a complex problem into a sequence (or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "chain) of subproblems, and by following the reasoning chain step by step, each subproblem can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "solved based on the results of previous subproblems. Our main finding demonstrates that, for general", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "score": 1.0, + "content": "DP problems of the form (5), LLMs with CoT can generate the complete chain and output the correct", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "answer (Theorem 4.7). However, it is impossible to directly generate the answer in general: as a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "counterexample, we prove that bounded-depth Transformers of polynomial size cannot solve a classic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 346, + 440, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 440, + 357 + ], + "score": 1.0, + "content": "DP problem known as Context-Free Grammar Membership Testing (Theorem 4.8).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "Our theoretical findings are complemented by an extensive set of experiments. We consider the two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "aforementioned math tasks plus two celebrated DP problems listed in the “Introduction to Algorithms”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "book [17], known as longest increasing subsequence (LIS) and edit distance (ED). For all these tasks,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "our experimental results show that directly predicting the answers without CoT always fails (accuracy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 164, + 420 + ], + "score": 1.0, + "content": "mostly below", + "type": "text" + }, + { + "bbox": [ + 164, + 406, + 185, + 416 + ], + "score": 0.88, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "). In contrast, autoregressive Transformers equipped with CoT can learn entire", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "solutions given sufficient training demonstrations. Moreover, they even generalize well to longer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "input sequences, suggesting that the models have learned the underlying reasoning process rather", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "than statistically memorizing input-output distributions. These results verify our theory and reveal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 450, + 452, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 452, + 461 + ], + "score": 1.0, + "content": "the strength of autoregressive LLMs and the importance of CoT in practical scenarios.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 187, + 489 + ], + "lines": [ + { + "bbox": [ + 104, + 473, + 189, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 189, + 493 + ], + "score": 1.0, + "content": "2 Preliminary", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "score": 1.0, + "content": "An (autoregressive) Transformer [58, 50] is a neural network architecture designed to process a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 497, + 524 + ], + "score": 1.0, + "content": "sequence of input tokens and generate tokens for subsequent positions. Given an input sequence", + "type": "text" + }, + { + "bbox": [ + 498, + 513, + 504, + 520 + ], + "score": 0.63, + "content": "\\pmb { s }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 145, + 535 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 145, + 524, + 153, + 532 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 521, + 448, + 535 + ], + "score": 1.0, + "content": ", a Transformer operates the sequence as follows. 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In other words, this paper studies why an LLM equipped", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 455, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 455, + 723 + ], + "score": 1.0, + "content": "with CoT can succeed in math/reasoning tasks rather than which prompt can trigger this process.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 506, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 114 + ], + "score": 1.0, + "content": "This paper takes a step towards theoretically answering the above questions. We begin with studying", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 109, + 506, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 125 + ], + "score": 1.0, + "content": "the capability of LLMs on two basic mathematical tasks: evaluating arithmetic expressions and solving", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "linear equations. Both tasks are extensively employed and serve as elementary building blocks in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "solving complex real-world math problems [10]. We first provide fundamental impossibility results,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "showing that none of these tasks can be solved using bounded-depth Transformer models without", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "CoT unless the model size grows super-polynomially with respect to the input length (Theorems 3.1", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "and 3.2). Remarkably, our proofs provide insights into why this happens: the reason is not due to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "(serialized) computational cost of these problems but rather to their parallel complexity [2]. We next", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "score": 1.0, + "content": "show that the community may largely undervalue the strength of autoregressive generation: we prove", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "by construction that autoregressive Transformers of constant size can already perfectly solve both", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "tasks by generating intermediate derivations in a step-by-step manner using a commonly-used math", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "language format (Theorems 3.3 and 3.4). Intuitively, this result hinges on the recursive nature of CoT,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 230, + 503, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 503, + 244 + ], + "score": 1.0, + "content": "which increases the “effective depth” of the Transformer to be proportional to the generation steps.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 99, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "Besides mathematics, CoT also exhibits remarkable performance across a wide range of reasoning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "tasks. To gain a systematic understanding of why CoT is beneficial, we next turn to a fundamental", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "class of problems known as Dynamic Programming (DP) [6]. DP represents a golden framework", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "for solving sequential decision-making tasks: it decomposes a complex problem into a sequence (or", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "chain) of subproblems, and by following the reasoning chain step by step, each subproblem can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "solved based on the results of previous subproblems. Our main finding demonstrates that, for general", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "score": 1.0, + "content": "DP problems of the form (5), LLMs with CoT can generate the complete chain and output the correct", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "answer (Theorem 4.7). However, it is impossible to directly generate the answer in general: as a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "counterexample, we prove that bounded-depth Transformers of polynomial size cannot solve a classic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 346, + 440, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 440, + 357 + ], + "score": 1.0, + "content": "DP problem known as Context-Free Grammar Membership Testing (Theorem 4.8).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 246, + 506, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "Our theoretical findings are complemented by an extensive set of experiments. We consider the two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "aforementioned math tasks plus two celebrated DP problems listed in the “Introduction to Algorithms”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "book [17], known as longest increasing subsequence (LIS) and edit distance (ED). For all these tasks,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "our experimental results show that directly predicting the answers without CoT always fails (accuracy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 164, + 420 + ], + "score": 1.0, + "content": "mostly below", + "type": "text" + }, + { + "bbox": [ + 164, + 406, + 185, + 416 + ], + "score": 0.88, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "). In contrast, autoregressive Transformers equipped with CoT can learn entire", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "solutions given sufficient training demonstrations. Moreover, they even generalize well to longer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "input sequences, suggesting that the models have learned the underlying reasoning process rather", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "than statistically memorizing input-output distributions. These results verify our theory and reveal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 450, + 452, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 452, + 461 + ], + "score": 1.0, + "content": "the strength of autoregressive LLMs and the importance of CoT in practical scenarios.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 362, + 506, + 461 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 187, + 489 + ], + "lines": [ + { + "bbox": [ + 104, + 473, + 189, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 189, + 493 + ], + "score": 1.0, + "content": "2 Preliminary", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "score": 1.0, + "content": "An (autoregressive) Transformer [58, 50] is a neural network architecture designed to process a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 497, + 524 + ], + "score": 1.0, + "content": "sequence of input tokens and generate tokens for subsequent positions. 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The process continues iteratively", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 208, + 503, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 503, + 221 + ], + "score": 1.0, + "content": "until a designated End-of-Sentence token is generated, signifying the completion of the process.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "Chain-of-Thought prompting. Autoregressive Transformers possess the ability to tackle a wide", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "range of tasks by encoding the task description into a partial sentence, with the answer being derived", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "by complementing the subsequent sentence [9]. However, for some challenging tasks involving math", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "or general reasoning, a direct generation often struggles to yield a correct answer. To address this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "shortcoming, researchers proposed the CoT prompting that induces LLMs to generate intermediate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "reasoning steps before reaching the answer [61, 34, 56, 44, 70, 11]. In this paper, our primary focus", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "lies in understanding the mechanism behind CoT, while disregarding the aspect of how prompting", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "facilitates its triggering. Specifically, we examine CoT from an expressivity perspective: for both", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "mathematical problems and general decision-making tasks studied in Sections 3 and 4, we will", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "investigate whether autoregressive Transformers are expressive for (i) directly generating the answer,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 334, + 299, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 299, + 346 + ], + "score": 1.0, + "content": "and (ii) generating a CoT solution for the tasks.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 106, + 359, + 381, + 373 + ], + "lines": [ + { + "bbox": [ + 104, + 358, + 382, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 382, + 375 + ], + "score": 1.0, + "content": "3 CoT is the Key to Solving Mathematical Problems", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "Previous studies have observed that Transformer-based LLMs exhibit surprising math abilities in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "various aspects [46, 10]. In this section, we begin to explore this intriguing phenomenon via two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "well-chosen tasks: arithmetic and equation. 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As shown in Fig-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 481, + 290, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 290, + 494 + ], + "score": 1.0, + "content": "ure 1 (left), the input of this task is a sequence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 290, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 236, + 504 + ], + "score": 1.0, + "content": "consisting of numbers, addition", + "type": "text" + }, + { + "bbox": [ + 237, + 493, + 252, + 504 + ], + "score": 0.73, + "content": "( + )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 492, + 290, + 504 + ], + "score": 1.0, + "content": ", subtrac-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 503, + 290, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 124, + 515 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 124, + 504, + 140, + 515 + ], + "score": 0.73, + "content": "( - )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 503, + 200, + 515 + ], + "score": 1.0, + "content": ", multiplication", + "type": "text" + }, + { + "bbox": [ + 201, + 504, + 216, + 515 + ], + "score": 0.85, + "content": "( \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 503, + 253, + 515 + ], + "score": 1.0, + "content": ", division", + "type": "text" + }, + { + "bbox": [ + 254, + 504, + 268, + 514 + ], + "score": 0.7, + "content": "( \\div )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 503, + 290, + 515 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 514, + 290, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 290, + 526 + ], + "score": 1.0, + "content": "brackets, followed by an equal sign. The goal", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 525, + 289, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 289, + 537 + ], + "score": 1.0, + "content": "is to calculate the arithmetic expression and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 291, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 291, + 547 + ], + "score": 1.0, + "content": "generate the correct result. This task has a nat-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 547, + 289, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 289, + 559 + ], + "score": 1.0, + "content": "ural CoT solution, where each step performs", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 559, + 290, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 290, + 570 + ], + "score": 1.0, + "content": "an intermediate computation, gradually reduc-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 290, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 290, + 582 + ], + "score": 1.0, + "content": "ing one atomic operation at a time while copy-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 289, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 289, + 591 + ], + "score": 1.0, + "content": "ing down other unrelated items. 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As shown in Figure 1 (right), the", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 283, + 641 + ], + "score": 1.0, + "content": "input of this task is a sequence consisting of", + "type": "text" + }, + { + "bbox": [ + 283, + 630, + 293, + 639 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 629, + 455, + 641 + ], + "score": 1.0, + "content": "linear equations, each of which involves", + "type": "text" + }, + { + "bbox": [ + 455, + 631, + 466, + 639 + ], + "score": 0.64, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "variables.", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 230, + 653 + ], + "score": 1.0, + "content": "The input ends with a special", + "type": "text" + }, + { + "bbox": [ + 230, + 640, + 280, + 651 + ], + "score": 0.32, + "content": "\\mathbf { s y m b o l } \\implies", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 639, + 506, + 653 + ], + "score": 1.0, + "content": ". The goal is to output the value of these variables that", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "satisfies the set of equations (assuming the answer exists and is unique). 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The process continues iteratively", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 208, + 503, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 503, + 221 + ], + "score": 1.0, + "content": "until a designated End-of-Sentence token is generated, signifying the completion of the process.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 170, + 505, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "Chain-of-Thought prompting. Autoregressive Transformers possess the ability to tackle a wide", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "range of tasks by encoding the task description into a partial sentence, with the answer being derived", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "by complementing the subsequent sentence [9]. However, for some challenging tasks involving math", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "or general reasoning, a direct generation often struggles to yield a correct answer. To address this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "shortcoming, researchers proposed the CoT prompting that induces LLMs to generate intermediate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "reasoning steps before reaching the answer [61, 34, 56, 44, 70, 11]. In this paper, our primary focus", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "lies in understanding the mechanism behind CoT, while disregarding the aspect of how prompting", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "facilitates its triggering. Specifically, we examine CoT from an expressivity perspective: for both", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "mathematical problems and general decision-making tasks studied in Sections 3 and 4, we will", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "investigate whether autoregressive Transformers are expressive for (i) directly generating the answer,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 334, + 299, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 299, + 346 + ], + "score": 1.0, + "content": "and (ii) generating a CoT solution for the tasks.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 225, + 506, + 346 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 359, + 381, + 373 + ], + "lines": [ + { + "bbox": [ + 104, + 358, + 382, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 382, + 375 + ], + "score": 1.0, + "content": "3 CoT is the Key to Solving Mathematical Problems", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "Previous studies have observed that Transformer-based LLMs exhibit surprising math abilities in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "various aspects [46, 10]. In this section, we begin to explore this intriguing phenomenon via two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "well-chosen tasks: arithmetic and equation. We will give concrete evidence that LLMs are capable of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 416, + 486, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 486, + 429 + ], + "score": 1.0, + "content": "solving both tasks when equipped with CoT, while LLMs without CoT are provably incapable.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 383, + 505, + 429 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 439, + 219, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 221, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 221, + 452 + ], + "score": 1.0, + "content": "3.1 Problem formulation", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 289, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 290, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 290, + 471 + ], + "score": 1.0, + "content": "Arithmetic. The first task focuses on evaluat-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 471, + 290, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 290, + 483 + ], + "score": 1.0, + "content": "ing arithmetic expressions. As shown in Fig-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 481, + 290, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 290, + 494 + ], + "score": 1.0, + "content": "ure 1 (left), the input of this task is a sequence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 290, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 236, + 504 + ], + "score": 1.0, + "content": "consisting of numbers, addition", + "type": "text" + }, + { + "bbox": [ + 237, + 493, + 252, + 504 + ], + "score": 0.73, + "content": "( + )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 492, + 290, + 504 + ], + "score": 1.0, + "content": ", subtrac-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 503, + 290, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 124, + 515 + ], + "score": 1.0, + "content": "tion", + "type": "text" + }, + { + "bbox": [ + 124, + 504, + 140, + 515 + ], + "score": 0.73, + "content": "( - )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 503, + 200, + 515 + ], + "score": 1.0, + "content": ", multiplication", + "type": "text" + }, + { + "bbox": [ + 201, + 504, + 216, + 515 + ], + "score": 0.85, + "content": "( \\times )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 503, + 253, + 515 + ], + "score": 1.0, + "content": ", division", + "type": "text" + }, + { + "bbox": [ + 254, + 504, + 268, + 514 + ], + "score": 0.7, + "content": "( \\div )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 503, + 290, + 515 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 514, + 290, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 290, + 526 + ], + "score": 1.0, + "content": "brackets, followed by an equal sign. The goal", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 525, + 289, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 289, + 537 + ], + "score": 1.0, + "content": "is to calculate the arithmetic expression and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 291, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 291, + 547 + ], + "score": 1.0, + "content": "generate the correct result. This task has a nat-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 547, + 289, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 289, + 559 + ], + "score": 1.0, + "content": "ural CoT solution, where each step performs", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 559, + 290, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 290, + 570 + ], + "score": 1.0, + "content": "an intermediate computation, gradually reduc-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 290, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 290, + 582 + ], + "score": 1.0, + "content": "ing one atomic operation at a time while copy-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 289, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 289, + 591 + ], + "score": 1.0, + "content": "ing down other unrelated items. Figure 1 (left)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 591, + 289, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 289, + 602 + ], + "score": 1.0, + "content": "gives an illustration, and the formal definition", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 601, + 289, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 289, + 614 + ], + "score": 1.0, + "content": "of the CoT format is deferred to Appendix B.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 460, + 291, + 614 + ] + }, + { + "type": "image", + "bbox": [ + 299, + 437, + 502, + 599 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 299, + 437, + 502, + 599 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 299, + 437, + 502, + 599 + ], + "spans": [ + { + "bbox": [ + 299, + 437, + 502, + 599 + ], + "score": 0.931, + "type": "image", + "image_path": "014b174a0c8ec1f8653018150a3f018cba6056cf439977be4e772ff7c6d1c75a.jpg" + } + ] + } + ], + "index": 49, + "virtual_lines": [ + { + "bbox": [ + 299, + 437, + 502, + 449.46153846153845 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 299, + 449.46153846153845, + 502, + 461.9230769230769 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 299, + 461.9230769230769, + 502, + 474.38461538461536 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 299, + 474.38461538461536, + 502, + 486.8461538461538 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 299, + 486.8461538461538, + 502, + 499.30769230769226 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 299, + 499.30769230769226, + 502, + 511.7692307692307 + ], + "spans": [], + "index": 48 + }, + { + "bbox": [ + 299, + 511.7692307692307, + 502, + 524.2307692307692 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 299, + 524.2307692307692, + 502, + 536.6923076923076 + ], + "spans": [], + "index": 50 + }, + { + "bbox": [ + 299, + 536.6923076923076, + 502, + 549.1538461538461 + ], + "spans": [], + "index": 51 + }, + { + "bbox": [ + 299, + 549.1538461538461, + 502, + 561.6153846153845 + ], + "spans": [], + "index": 52 + }, + { + "bbox": [ + 299, + 561.6153846153845, + 502, + 574.076923076923 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 299, + 574.076923076923, + 502, + 586.5384615384614 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 299, + 586.5384615384614, + 502, + 598.9999999999999 + ], + "spans": [], + "index": 55 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 303, + 602, + 498, + 613 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 300, + 601, + 500, + 615 + ], + "spans": [ + { + "bbox": [ + 300, + 601, + 500, + 615 + ], + "score": 1.0, + "content": "Figure 1: Illustrations of CoT on two math tasks.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 56 + } + ], + "index": 52.5 + }, + { + "type": "text", + "bbox": [ + 106, + 618, + 506, + 695 + ], + "lines": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "Equation. The second task considers solving linear equations. As shown in Figure 1 (right), the", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 283, + 641 + ], + "score": 1.0, + "content": "input of this task is a sequence consisting of", + "type": "text" + }, + { + "bbox": [ + 283, + 630, + 293, + 639 + ], + "score": 0.72, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 629, + 455, + 641 + ], + "score": 1.0, + "content": "linear equations, each of which involves", + "type": "text" + }, + { + "bbox": [ + 455, + 631, + 466, + 639 + ], + "score": 0.64, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "variables.", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 230, + 653 + ], + "score": 1.0, + "content": "The input ends with a special", + "type": "text" + }, + { + "bbox": [ + 230, + 640, + 280, + 651 + ], + "score": 0.32, + "content": "\\mathbf { s y m b o l } \\implies", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 639, + 506, + 653 + ], + "score": 1.0, + "content": ". The goal is to output the value of these variables that", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "satisfies the set of equations (assuming the answer exists and is unique). A natural CoT solution is the", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 662, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 506, + 674 + ], + "score": 1.0, + "content": "Gaussian elimination algorithm: at each step, it eliminates a certain variable in all but one equations.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 130, + 685 + ], + "score": 1.0, + "content": "After", + "type": "text" + }, + { + "bbox": [ + 131, + 673, + 158, + 683 + ], + "score": 0.89, + "content": "m - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "steps, all equations will have only one variable and the problem is solved. Figure 1", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 684, + 494, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 494, + 695 + ], + "score": 1.0, + "content": "(right) gives an illustration, and we defer the formal definition of the CoT format to Appendix B.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 60, + "bbox_fs": [ + 105, + 618, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "Number field. Ideally, for both tasks, the input sequences involve not only symbol tokens but", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "also floating-point numbers. This complicates the definitions of the model’s input/output format", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "and further entails intricate precision considerations when dealing with floating-point divisions. To", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "simplify our subsequent analysis, here we turn to a more convenient setting by transitioning to", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 290, + 108 + ], + "score": 1.0, + "content": "the finite field generated by integers modulo", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 290, + 96, + 297, + 106 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 297, + 94, + 380, + 108 + ], + "score": 1.0, + "content": "for a prime number", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 380, + 96, + 387, + 106 + ], + "score": 0.74, + "content": "p", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 387, + 94, + 506, + 108 + ], + "score": 1.0, + "content": ". Importantly, the finite field", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 162, + 118 + ], + "score": 1.0, + "content": "contains only", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 163, + 107, + 169, + 117 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 170, + 106, + 283, + 118 + ], + "score": 1.0, + "content": "numbers (ranging from 0 to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 284, + 106, + 309, + 117 + ], + "score": 0.87, + "content": "p - 1 )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 310, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "and thus can be uniformly treated as tokens in a", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 507, + 129 + ], + "score": 1.0, + "content": "pre-defined dictionary (like other operators or brackets), making the problem setting much cleaner.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 235, + 140 + ], + "score": 1.0, + "content": "Moreover, arithmetic operations", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 235, + 128, + 287, + 138 + ], + "score": 0.92, + "content": "( + , - , \\times , \\div )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 288, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "are well-defined and parallel the real number field (see", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 410, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 410, + 150 + ], + "score": 1.0, + "content": "Appendix A.1 for details). Therefore, this setting does not lose generalities.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 64.5, + "bbox_fs": [ + 106, + 700, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "and further entails intricate precision considerations when dealing with floating-point divisions. To", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "simplify our subsequent analysis, here we turn to a more convenient setting by transitioning to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 290, + 108 + ], + "score": 1.0, + "content": "the finite field generated by integers modulo", + "type": "text" + }, + { + "bbox": [ + 290, + 96, + 297, + 106 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 94, + 380, + 108 + ], + "score": 1.0, + "content": "for a prime number", + "type": "text" + }, + { + "bbox": [ + 380, + 96, + 387, + 106 + ], + "score": 0.74, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 94, + 506, + 108 + ], + "score": 1.0, + "content": ". Importantly, the finite field", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 162, + 118 + ], + "score": 1.0, + "content": "contains only", + "type": "text" + }, + { + "bbox": [ + 163, + 107, + 169, + 117 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 106, + 283, + 118 + ], + "score": 1.0, + "content": "numbers (ranging from 0 to", + "type": "text" + }, + { + "bbox": [ + 284, + 106, + 309, + 117 + ], + "score": 0.87, + "content": "p - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "and thus can be uniformly treated as tokens in a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 507, + 129 + ], + "score": 1.0, + "content": "pre-defined dictionary (like other operators or brackets), making the problem setting much cleaner.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 235, + 140 + ], + "score": 1.0, + "content": "Moreover, arithmetic operations", + "type": "text" + }, + { + "bbox": [ + 235, + 128, + 287, + 138 + ], + "score": 0.92, + "content": "( + , - , \\times , \\div )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "are well-defined and parallel the real number field (see", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 410, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 410, + 150 + ], + "score": 1.0, + "content": "Appendix A.1 for details). Therefore, this setting does not lose generalities.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 506, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 507, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 297, + 167 + ], + "score": 1.0, + "content": "In subsequent sections, we denote by Arithmetic", + "type": "text" + }, + { + "bbox": [ + 297, + 154, + 321, + 166 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 153, + 507, + 167 + ], + "score": 1.0, + "content": "the arithmetic evaluation task defined on the fi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 174, + 179 + ], + "score": 1.0, + "content": "nite field modulo", + "type": "text" + }, + { + "bbox": [ + 174, + 167, + 181, + 177 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 164, + 335, + 179 + ], + "score": 1.0, + "content": ", where the input length does not exceed", + "type": "text" + }, + { + "bbox": [ + 336, + 167, + 343, + 175 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 164, + 478, + 179 + ], + "score": 1.0, + "content": ". Similarly, we denote by Equation", + "type": "text" + }, + { + "bbox": [ + 478, + 165, + 505, + 177 + ], + "score": 0.91, + "content": "( m , p )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 471, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 336, + 189 + ], + "score": 1.0, + "content": "the linear equation task defined on the finite field modulo", + "type": "text" + }, + { + "bbox": [ + 336, + 178, + 343, + 188 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 177, + 419, + 189 + ], + "score": 1.0, + "content": "with no more than", + "type": "text" + }, + { + "bbox": [ + 420, + 178, + 430, + 186 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 177, + 471, + 189 + ], + "score": 1.0, + "content": "variables.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 200, + 210, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 210, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 210, + 213 + ], + "score": 1.0, + "content": "3.2 Theoretical results", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 506, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "We begin by investigating whether Transformers can directly produce answers to the aforemen-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 424, + 243 + ], + "score": 1.0, + "content": "tioned problems. This corresponds to generating, for instance, the number", + "type": "text" + }, + { + "bbox": [ + 424, + 231, + 440, + 241 + ], + "score": 0.37, + "content": "\\mathbf { \\ddot { \\delta } } 3 ^ { \\mathbf { \\mathit { , } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "or the solution", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 108, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 108, + 242, + 194, + 253 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { } ^ { \\cdot } x = 1 ; y = 1 ; z = 0 ^ { , , } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "in Figure 1 immediately after the input sequence (without outputting intermedi-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "ate steps). This question can be examined via different theoretical perspectives. One natural approach", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "is to employ the classic representation theory, which states that multi-layer perceptrons with suffi-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "cient size (e.g., the depth or width approaches infinity) are already universal function approximators", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "[18, 35, 40]. Recently, such results have been well extended to Transformer models [67]: it is not hard", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 295, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 104, + 295, + 507, + 309 + ], + "score": 1.0, + "content": "to show that a constant-depth Transformer with sufficient size can solve the above tasks3. However,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "the above results become elusive when taking the representation efficiency into account, since it says", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "nothing about the required model size for any specific task. Below, we would like to give a more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "fine-grained analysis of how large the network needs to be by leveraging the tool of complexity theory.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 506, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 507, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 507, + 358 + ], + "score": 1.0, + "content": "We focus on a realistic setting called the log-precision Transformer [42, 37]: it refers to a Trans-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 452, + 369 + ], + "score": 1.0, + "content": "former whose internal neurons can only store floating-point numbers within a finite", + "type": "text" + }, + { + "bbox": [ + 453, + 357, + 490, + 369 + ], + "score": 0.92, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "bit", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 175, + 380 + ], + "score": 1.0, + "content": "precision where", + "type": "text" + }, + { + "bbox": [ + 175, + 369, + 183, + 377 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "is the maximal length of the input sequence (see Appendix A.3 for a formal", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "definition). Such an assumption well-resembles practical situations, in which the machine precision", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "(e.g., 16 or 32 bits) is typically much smaller than the input length (e.g., 2048 in GPT), avoiding the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "unrealistic (but crucial) assumption of infinite precision made in several prior works [49, 20]. Further-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "more, log-precision implies that the number of values each neuron can take is polynomial in the input", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 420, + 507, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 507, + 435 + ], + "score": 1.0, + "content": "length, which is a necessary condition for representing important quantities like positional embedding.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 433, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 507, + 446 + ], + "score": 1.0, + "content": "Equipped with the concept of log-precision, we are ready to present a central impossibility result,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 444, + 471, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 471, + 455 + ], + "score": 1.0, + "content": "showing that the required network size must be prohibitively large for both math problems:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 108, + 456, + 504, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 202, + 472 + ], + "score": 1.0, + "content": "Theorem 3.1. Assume", + "type": "text" + }, + { + "bbox": [ + 202, + 456, + 252, + 470 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 455, + 351, + 472 + ], + "score": 1.0, + "content": ". For any prime number", + "type": "text" + }, + { + "bbox": [ + 351, + 461, + 357, + 470 + ], + "score": 0.57, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 455, + 410, + 472 + ], + "score": 1.0, + "content": ", any integer", + "type": "text" + }, + { + "bbox": [ + 410, + 459, + 418, + 469 + ], + "score": 0.55, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 455, + 506, + 472 + ], + "score": 1.0, + "content": ", and any polynomial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 115, + 481 + ], + "score": 0.69, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 469, + 505, + 482 + ], + "score": 1.0, + "content": ", there exists a problem size n such that no log-precision autoregressive Transformer defined in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 494, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 190, + 493 + ], + "score": 1.0, + "content": "Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 190, + 480, + 198, + 490 + ], + "score": 0.43, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 479, + 289, + 493 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 290, + 480, + 331, + 492 + ], + "score": 0.93, + "content": "d \\leq Q ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 479, + 466, + 493 + ], + "score": 1.0, + "content": "can solve the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 467, + 480, + 490, + 492 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 479, + 494, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 493, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 202, + 509 + ], + "score": 1.0, + "content": "Theorem 3.2. Assume", + "type": "text" + }, + { + "bbox": [ + 202, + 493, + 252, + 506 + ], + "score": 0.91, + "content": "{ \\sf T C } ^ { 0 } \\neq { \\sf N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 492, + 351, + 509 + ], + "score": 1.0, + "content": ". For any prime number", + "type": "text" + }, + { + "bbox": [ + 351, + 497, + 357, + 506 + ], + "score": 0.61, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 492, + 410, + 509 + ], + "score": 1.0, + "content": ", any integer", + "type": "text" + }, + { + "bbox": [ + 410, + 495, + 417, + 505 + ], + "score": 0.58, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 492, + 506, + 509 + ], + "score": 1.0, + "content": ", and any polynomial", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 115, + 517 + ], + "score": 0.68, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 506, + 505, + 518 + ], + "score": 1.0, + "content": ", there exists a problem size m such that no log-precision autoregressive Transformer defined in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 515, + 492, + 530 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 190, + 530 + ], + "score": 1.0, + "content": "Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 190, + 517, + 198, + 527 + ], + "score": 0.4, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 515, + 289, + 530 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 290, + 517, + 333, + 529 + ], + "score": 0.93, + "content": "d \\leq Q ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 515, + 462, + 530 + ], + "score": 1.0, + "content": "can solve the problem Equation", + "type": "text" + }, + { + "bbox": [ + 462, + 517, + 488, + 529 + ], + "score": 0.9, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 515, + 492, + 530 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "Why does this happen? As presented in Appendices D.2 and E.2, the crux of our proof lies in apply-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "ing circuit complexity theory [2]. By framing the finite-precision Transformer as a computation model,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 554, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 554, + 507, + 569 + ], + "score": 1.0, + "content": "one can precisely delineate its expressivity limitation through an analysis of its circuit complexity.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "Here, bounded-depth log-precision Transformers of polynomial size represent a class of shallow cir-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 264, + 591 + ], + "score": 1.0, + "content": "cuits with complexity upper bounded by", + "type": "text" + }, + { + "bbox": [ + 264, + 577, + 283, + 589 + ], + "score": 0.83, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "[42]. On the other hand, we prove that the complexity of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 589, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 296, + 604 + ], + "score": 1.0, + "content": "both math problems above are lower bounded by", + "type": "text" + }, + { + "bbox": [ + 296, + 589, + 315, + 601 + ], + "score": 0.87, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 590, + 424, + 604 + ], + "score": 1.0, + "content": "by applying reduction from", + "type": "text" + }, + { + "bbox": [ + 424, + 589, + 443, + 601 + ], + "score": 0.82, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 590, + 507, + 604 + ], + "score": 1.0, + "content": "-complete prob-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "lems. Consequently, they are intrinsically hard to be solved by a well-parallelized Transformer unless", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 612, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 267, + 627 + ], + "score": 1.0, + "content": "the two complexity classes collapse (i.e.,", + "type": "text" + }, + { + "bbox": [ + 267, + 612, + 317, + 624 + ], + "score": 0.9, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } = { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 612, + 507, + 627 + ], + "score": 1.0, + "content": "), a scenario widely regarded as impossible [65].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "How about generating a CoT solution? We next turn to the setting of generating CoT solutions", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "for these problems. From an expressivity perspective, one might intuitively perceive this problem as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "more challenging as the model is required to express the entire problem solving process, potentially", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "necessitating a larger model size. However, we show this is not the case: a constant-size autoregressive", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 673, + 406, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 406, + 686 + ], + "score": 1.0, + "content": "Transformer already suffices to generate solutions for both math problems.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 691, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 118, + 690, + 298, + 704 + ], + "score": 1.0, + "content": "3For example, given an input sequence of length", + "type": "text" + }, + { + "bbox": [ + 299, + 694, + 305, + 701 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 690, + 319, + 704 + ], + "score": 1.0, + "content": ", an", + "type": "text" + }, + { + "bbox": [ + 320, + 694, + 326, + 701 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 690, + 506, + 704 + ], + "score": 1.0, + "content": "-head self-attention layer with hidden dimension", + "type": "text" + } + ] + }, + { + "bbox": [ + 107, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 107, + 701, + 127, + 712 + ], + "score": 0.89, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "can extract the information of the entire sequence into the representation of the last position. Then, an MLP", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 501, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 501, + 724 + ], + "score": 1.0, + "content": "operating on the last position can universally approximate any (continuous) function over the input sequence.", + "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, + 149 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 72, + 507, + 150 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 506, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 507, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 297, + 167 + ], + "score": 1.0, + "content": "In subsequent sections, we denote by Arithmetic", + "type": "text" + }, + { + "bbox": [ + 297, + 154, + 321, + 166 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 153, + 507, + 167 + ], + "score": 1.0, + "content": "the arithmetic evaluation task defined on the fi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 174, + 179 + ], + "score": 1.0, + "content": "nite field modulo", + "type": "text" + }, + { + "bbox": [ + 174, + 167, + 181, + 177 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 164, + 335, + 179 + ], + "score": 1.0, + "content": ", where the input length does not exceed", + "type": "text" + }, + { + "bbox": [ + 336, + 167, + 343, + 175 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 164, + 478, + 179 + ], + "score": 1.0, + "content": ". Similarly, we denote by Equation", + "type": "text" + }, + { + "bbox": [ + 478, + 165, + 505, + 177 + ], + "score": 0.91, + "content": "( m , p )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 471, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 336, + 189 + ], + "score": 1.0, + "content": "the linear equation task defined on the finite field modulo", + "type": "text" + }, + { + "bbox": [ + 336, + 178, + 343, + 188 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 177, + 419, + 189 + ], + "score": 1.0, + "content": "with no more than", + "type": "text" + }, + { + "bbox": [ + 420, + 178, + 430, + 186 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 177, + 471, + 189 + ], + "score": 1.0, + "content": "variables.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 153, + 507, + 189 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 200, + 210, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 210, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 210, + 213 + ], + "score": 1.0, + "content": "3.2 Theoretical results", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 506, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "We begin by investigating whether Transformers can directly produce answers to the aforemen-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 424, + 243 + ], + "score": 1.0, + "content": "tioned problems. This corresponds to generating, for instance, the number", + "type": "text" + }, + { + "bbox": [ + 424, + 231, + 440, + 241 + ], + "score": 0.37, + "content": "\\mathbf { \\ddot { \\delta } } 3 ^ { \\mathbf { \\mathit { , } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "or the solution", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 108, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 108, + 242, + 194, + 253 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { } ^ { \\cdot } x = 1 ; y = 1 ; z = 0 ^ { , , } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "in Figure 1 immediately after the input sequence (without outputting intermedi-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "ate steps). This question can be examined via different theoretical perspectives. One natural approach", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "is to employ the classic representation theory, which states that multi-layer perceptrons with suffi-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "cient size (e.g., the depth or width approaches infinity) are already universal function approximators", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "[18, 35, 40]. Recently, such results have been well extended to Transformer models [67]: it is not hard", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 295, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 104, + 295, + 507, + 309 + ], + "score": 1.0, + "content": "to show that a constant-depth Transformer with sufficient size can solve the above tasks3. However,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "the above results become elusive when taking the representation efficiency into account, since it says", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "nothing about the required model size for any specific task. Below, we would like to give a more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "fine-grained analysis of how large the network needs to be by leveraging the tool of complexity theory.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 219, + 507, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 506, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 507, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 507, + 358 + ], + "score": 1.0, + "content": "We focus on a realistic setting called the log-precision Transformer [42, 37]: it refers to a Trans-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 452, + 369 + ], + "score": 1.0, + "content": "former whose internal neurons can only store floating-point numbers within a finite", + "type": "text" + }, + { + "bbox": [ + 453, + 357, + 490, + 369 + ], + "score": 0.92, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "bit", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 175, + 380 + ], + "score": 1.0, + "content": "precision where", + "type": "text" + }, + { + "bbox": [ + 175, + 369, + 183, + 377 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "is the maximal length of the input sequence (see Appendix A.3 for a formal", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "definition). Such an assumption well-resembles practical situations, in which the machine precision", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "(e.g., 16 or 32 bits) is typically much smaller than the input length (e.g., 2048 in GPT), avoiding the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "unrealistic (but crucial) assumption of infinite precision made in several prior works [49, 20]. Further-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "more, log-precision implies that the number of values each neuron can take is polynomial in the input", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 420, + 507, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 507, + 435 + ], + "score": 1.0, + "content": "length, which is a necessary condition for representing important quantities like positional embedding.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 433, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 507, + 446 + ], + "score": 1.0, + "content": "Equipped with the concept of log-precision, we are ready to present a central impossibility result,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 444, + 471, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 471, + 455 + ], + "score": 1.0, + "content": "showing that the required network size must be prohibitively large for both math problems:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 345, + 507, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 456, + 504, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 202, + 472 + ], + "score": 1.0, + "content": "Theorem 3.1. Assume", + "type": "text" + }, + { + "bbox": [ + 202, + 456, + 252, + 470 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 455, + 351, + 472 + ], + "score": 1.0, + "content": ". For any prime number", + "type": "text" + }, + { + "bbox": [ + 351, + 461, + 357, + 470 + ], + "score": 0.57, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 455, + 410, + 472 + ], + "score": 1.0, + "content": ", any integer", + "type": "text" + }, + { + "bbox": [ + 410, + 459, + 418, + 469 + ], + "score": 0.55, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 455, + 506, + 472 + ], + "score": 1.0, + "content": ", and any polynomial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 115, + 481 + ], + "score": 0.69, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 469, + 505, + 482 + ], + "score": 1.0, + "content": ", there exists a problem size n such that no log-precision autoregressive Transformer defined in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 494, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 190, + 493 + ], + "score": 1.0, + "content": "Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 190, + 480, + 198, + 490 + ], + "score": 0.43, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 479, + 289, + 493 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 290, + 480, + 331, + 492 + ], + "score": 0.93, + "content": "d \\leq Q ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 479, + 466, + 493 + ], + "score": 1.0, + "content": "can solve the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 467, + 480, + 490, + 492 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 479, + 494, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 455, + 506, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 493, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 202, + 509 + ], + "score": 1.0, + "content": "Theorem 3.2. Assume", + "type": "text" + }, + { + "bbox": [ + 202, + 493, + 252, + 506 + ], + "score": 0.91, + "content": "{ \\sf T C } ^ { 0 } \\neq { \\sf N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 492, + 351, + 509 + ], + "score": 1.0, + "content": ". For any prime number", + "type": "text" + }, + { + "bbox": [ + 351, + 497, + 357, + 506 + ], + "score": 0.61, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 492, + 410, + 509 + ], + "score": 1.0, + "content": ", any integer", + "type": "text" + }, + { + "bbox": [ + 410, + 495, + 417, + 505 + ], + "score": 0.58, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 492, + 506, + 509 + ], + "score": 1.0, + "content": ", and any polynomial", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 115, + 517 + ], + "score": 0.68, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 506, + 505, + 518 + ], + "score": 1.0, + "content": ", there exists a problem size m such that no log-precision autoregressive Transformer defined in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 515, + 492, + 530 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 190, + 530 + ], + "score": 1.0, + "content": "Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 190, + 517, + 198, + 527 + ], + "score": 0.4, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 515, + 289, + 530 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 290, + 517, + 333, + 529 + ], + "score": 0.93, + "content": "d \\leq Q ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 515, + 462, + 530 + ], + "score": 1.0, + "content": "can solve the problem Equation", + "type": "text" + }, + { + "bbox": [ + 462, + 517, + 488, + 529 + ], + "score": 0.9, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 515, + 492, + 530 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 492, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "Why does this happen? As presented in Appendices D.2 and E.2, the crux of our proof lies in apply-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "ing circuit complexity theory [2]. By framing the finite-precision Transformer as a computation model,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 554, + 507, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 554, + 507, + 569 + ], + "score": 1.0, + "content": "one can precisely delineate its expressivity limitation through an analysis of its circuit complexity.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 579 + ], + "score": 1.0, + "content": "Here, bounded-depth log-precision Transformers of polynomial size represent a class of shallow cir-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 264, + 591 + ], + "score": 1.0, + "content": "cuits with complexity upper bounded by", + "type": "text" + }, + { + "bbox": [ + 264, + 577, + 283, + 589 + ], + "score": 0.83, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "[42]. On the other hand, we prove that the complexity of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 589, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 296, + 604 + ], + "score": 1.0, + "content": "both math problems above are lower bounded by", + "type": "text" + }, + { + "bbox": [ + 296, + 589, + 315, + 601 + ], + "score": 0.87, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 590, + 424, + 604 + ], + "score": 1.0, + "content": "by applying reduction from", + "type": "text" + }, + { + "bbox": [ + 424, + 589, + 443, + 601 + ], + "score": 0.82, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 590, + 507, + 604 + ], + "score": 1.0, + "content": "-complete prob-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "lems. Consequently, they are intrinsically hard to be solved by a well-parallelized Transformer unless", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 612, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 267, + 627 + ], + "score": 1.0, + "content": "the two complexity classes collapse (i.e.,", + "type": "text" + }, + { + "bbox": [ + 267, + 612, + 317, + 624 + ], + "score": 0.9, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } = { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 612, + 507, + 627 + ], + "score": 1.0, + "content": "), a scenario widely regarded as impossible [65].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 533, + 507, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "How about generating a CoT solution? We next turn to the setting of generating CoT solutions", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "for these problems. From an expressivity perspective, one might intuitively perceive this problem as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "more challenging as the model is required to express the entire problem solving process, potentially", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "necessitating a larger model size. However, we show this is not the case: a constant-size autoregressive", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 673, + 406, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 406, + 686 + ], + "score": 1.0, + "content": "Transformer already suffices to generate solutions for both math problems.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 629, + 505, + 686 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 301, + 86 + ], + "score": 1.0, + "content": "Theorem 3.3. Fix any prime p. For any integer", + "type": "text" + }, + { + "bbox": [ + 302, + 73, + 327, + 83 + ], + "score": 0.88, + "content": "n > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 72, + 506, + 86 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 286, + 96 + ], + "score": 1.0, + "content": "defined in Section 2 with constant hidden size", + "type": "text" + }, + { + "bbox": [ + 286, + 84, + 293, + 94 + ], + "score": 0.54, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 83, + 357, + 96 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 357, + 86, + 364, + 93 + ], + "score": 0.55, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 83, + 395, + 96 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 395, + 84, + 421, + 94 + ], + "score": 0.88, + "content": "L = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 83, + 442, + 96 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 442, + 84, + 448, + 93 + ], + "score": 0.36, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "heads in each", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 358, + 107 + ], + "score": 1.0, + "content": "layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 359, + 95, + 367, + 104 + ], + "score": 0.51, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 95, + 479, + 107 + ], + "score": 1.0, + "content": "for all inputs in Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 95, + 502, + 106 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 95, + 506, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 427, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 375, + 118 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 106, + 423, + 118 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 105, + 427, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 120, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 300, + 133 + ], + "score": 1.0, + "content": "Theorem 3.4. Fix any prime p. For any integer", + "type": "text" + }, + { + "bbox": [ + 300, + 121, + 329, + 131 + ], + "score": 0.89, + "content": "m > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 119, + 505, + 133 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 294, + 144 + ], + "score": 1.0, + "content": "defined in Section 2 with constant hidden size", + "type": "text" + }, + { + "bbox": [ + 295, + 132, + 302, + 141 + ], + "score": 0.29, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 132, + 369, + 144 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 369, + 133, + 379, + 142 + ], + "score": 0.63, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 132, + 411, + 144 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 412, + 132, + 439, + 142 + ], + "score": 0.88, + "content": "L = 4 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 132, + 505, + 144 + ], + "score": 1.0, + "content": ", and 5 heads in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 142, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 367, + 155 + ], + "score": 1.0, + "content": "each layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 367, + 144, + 375, + 152 + ], + "score": 0.34, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 142, + 477, + 155 + ], + "score": 1.0, + "content": "for all inputs in Equation", + "type": "text" + }, + { + "bbox": [ + 477, + 142, + 502, + 154 + ], + "score": 0.92, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 142, + 506, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 153, + 430, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 375, + 166 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 153, + 426, + 165 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( m ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 153, + 430, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "Remark 3.5. The polynomial upper bound for parameters in Theorems 3.3 and 3.4 readily implies", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "that these Transformers can be implemented using log-precision without loss of accuracy. See", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 381, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 381, + 201 + ], + "score": 1.0, + "content": "Appendix A.3 for a detailed discussion on how this can be achieved.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "Proof sketch. The proofs of Theorems 3.3 and 3.4 are based on construction. We begin by building", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 233 + ], + "score": 1.0, + "content": "a set of fundamental operations in Appendix C that can be implemented by Transformer layers.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "Specifically, the softmax attention head can perform two types of operations called the (conditional)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "COPY and MEAN (Lemmas C.7 and C.8). Here, conditional COPY extracts the content of the unique", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "previous position that satisfies certain conditions, while Conditional MEAN averages the values of a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "set of previous positions that satisfy certain conditions. These two operations can be seen as a form", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "of “gather/scatter” operator in parallel computing. On the other hand, the FFN in a Transformer", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "layer can perform basic computations within each position, such as multiplication (Lemma C.1),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "conditional selection (Lemma C.4), and lookup tables (Lemma C.5). With these basic operations as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 307, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 507, + 322 + ], + "score": 1.0, + "content": "“instructions” and by treating autoregressive generation as a loop, it is possible to write “programs”", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "that can solve fairly complex tasks. As detailed in Appendices D.1 and E.1, we construct parallel", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "algorithms that can generate CoT sequences for both math problems, thus concluding the proof.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 506, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "Several discussions are made as follows. Firstly, the constructions in our proof reveal the significance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "of several key components in the Transformer design, such as softmax attention, multi-head, feed-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "forward networks, and residual connection. Our proofs offer deep insights into the inner workings of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "Transformer models when dealing with complex tasks, significantly advancing prior understandings", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "such as the “induction head” mechanism [45]. Moreover, our results identify an inherent advantage", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "of Transformers compared to other sequence models like RNNs: indeed, as shown in Appendix F.2,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "constant-size RNNs cannot solve any of the above math tasks using the same CoT format. Secondly,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 428, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 439 + ], + "score": 1.0, + "content": "we highlight that in our setting, the CoT derivations of both math problems are purely written in a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 507, + 452 + ], + "score": 1.0, + "content": "readable math language format, largely resembling how humans write solutions. In a broad sense,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "our findings justify that LLMs have the potential to convey meaningful human thoughts through", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "grammatically precise sentences. Finally, one may ask how LLMs equipped with CoT can bypass", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "the impossibility results outlined in Theorems 3.1 and 3.2. Actually, this can be understood via the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 473, + 495 + ], + "score": 1.0, + "content": "effective depth of the Transformer circuit. By employing CoT, the effective depth is no longer", + "type": "text" + }, + { + "bbox": [ + 473, + 483, + 482, + 492 + ], + "score": 0.73, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "since", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "the generated outputs are repeatedly looped back to the input. The dependency between output tokens", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "leads to a significantly deeper circuit with depth proportional to the length of the CoT solution. Note", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "that even if the recursive procedure is repeated within a fixed Transformer (or circuit), the expressivity", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 524, + 508, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 199, + 542 + ], + "score": 1.0, + "content": "can still be far beyond", + "type": "text" + }, + { + "bbox": [ + 199, + 525, + 218, + 537 + ], + "score": 0.88, + "content": "{ \\bar { \\mathsf { T } } } { \\mathsf { C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 524, + 508, + 542 + ], + "score": 1.0, + "content": ": as will be shown in Section 4, with a sufficient number of CoT steps,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 538, + 373, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 282, + 551 + ], + "score": 1.0, + "content": "autoregressive Transformers can even solve", + "type": "text" + }, + { + "bbox": [ + 282, + 538, + 290, + 548 + ], + "score": 0.28, + "content": "\\mathsf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 538, + 373, + 551 + ], + "score": 1.0, + "content": "-complete problems.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 440, + 578 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 442, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 442, + 581 + ], + "score": 1.0, + "content": "4 CoT is the Key to Solving General Decision-Making Problems", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "The previous section has delineated the critical role of CoT in solving math problems. In this section,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "we will switch our attention to a more general setting beyond mathematics. Remarkably, we find that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "LLMs with CoT are theoretically capable of emulating a powerful decision-making framework called", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 622, + 491, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 491, + 635 + ], + "score": 1.0, + "content": "Dynamic Programming [6], thus strongly justifying the ability of CoT in solving complex tasks.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 231, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 233, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 233, + 662 + ], + "score": 1.0, + "content": "4.1 Dynamic Programming", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Dynamic programming (DP) is widely regarded as a core technique to solve decision-making", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "problems [55]. The basic idea of DP lies in breaking down a complex problem into a series of small", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "subproblems that can be tackled in a sequential manner. Here, the decomposition ensures that there is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "a significant interconnection (overlap) among various subproblems, so that each subproblem can be", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 507, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 507, + 724 + ], + "score": 1.0, + "content": "efficiently solved by utilizing the answers (or other relevant information) obtained from previous ones.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 301, + 86 + ], + "score": 1.0, + "content": "Theorem 3.3. Fix any prime p. For any integer", + "type": "text" + }, + { + "bbox": [ + 302, + 73, + 327, + 83 + ], + "score": 0.88, + "content": "n > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 72, + 506, + 86 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 286, + 96 + ], + "score": 1.0, + "content": "defined in Section 2 with constant hidden size", + "type": "text" + }, + { + "bbox": [ + 286, + 84, + 293, + 94 + ], + "score": 0.54, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 83, + 357, + 96 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 357, + 86, + 364, + 93 + ], + "score": 0.55, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 83, + 395, + 96 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 395, + 84, + 421, + 94 + ], + "score": 0.88, + "content": "L = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 83, + 442, + 96 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 442, + 84, + 448, + 93 + ], + "score": 0.36, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "heads in each", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 358, + 107 + ], + "score": 1.0, + "content": "layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 359, + 95, + 367, + 104 + ], + "score": 0.51, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 95, + 479, + 107 + ], + "score": 1.0, + "content": "for all inputs in Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 95, + 502, + 106 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 95, + 506, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 427, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 375, + 118 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 106, + 423, + 118 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 105, + 427, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 506, + 118 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 120, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 300, + 133 + ], + "score": 1.0, + "content": "Theorem 3.4. Fix any prime p. For any integer", + "type": "text" + }, + { + "bbox": [ + 300, + 121, + 329, + 131 + ], + "score": 0.89, + "content": "m > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 119, + 505, + 133 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 294, + 144 + ], + "score": 1.0, + "content": "defined in Section 2 with constant hidden size", + "type": "text" + }, + { + "bbox": [ + 295, + 132, + 302, + 141 + ], + "score": 0.29, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 132, + 369, + 144 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 369, + 133, + 379, + 142 + ], + "score": 0.63, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 132, + 411, + 144 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 412, + 132, + 439, + 142 + ], + "score": 0.88, + "content": "L = 4 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 132, + 505, + 144 + ], + "score": 1.0, + "content": ", and 5 heads in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 142, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 367, + 155 + ], + "score": 1.0, + "content": "each layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 367, + 144, + 375, + 152 + ], + "score": 0.34, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 142, + 477, + 155 + ], + "score": 1.0, + "content": "for all inputs in Equation", + "type": "text" + }, + { + "bbox": [ + 477, + 142, + 502, + 154 + ], + "score": 0.92, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 142, + 506, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 153, + 430, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 375, + 166 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 153, + 426, + 165 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( m ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 153, + 430, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 119, + 506, + 166 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "Remark 3.5. The polynomial upper bound for parameters in Theorems 3.3 and 3.4 readily implies", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "that these Transformers can be implemented using log-precision without loss of accuracy. See", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 381, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 381, + 201 + ], + "score": 1.0, + "content": "Appendix A.3 for a detailed discussion on how this can be achieved.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 168, + 505, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "Proof sketch. The proofs of Theorems 3.3 and 3.4 are based on construction. We begin by building", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 233 + ], + "score": 1.0, + "content": "a set of fundamental operations in Appendix C that can be implemented by Transformer layers.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "Specifically, the softmax attention head can perform two types of operations called the (conditional)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "COPY and MEAN (Lemmas C.7 and C.8). Here, conditional COPY extracts the content of the unique", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "previous position that satisfies certain conditions, while Conditional MEAN averages the values of a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "set of previous positions that satisfy certain conditions. These two operations can be seen as a form", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "of “gather/scatter” operator in parallel computing. On the other hand, the FFN in a Transformer", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "layer can perform basic computations within each position, such as multiplication (Lemma C.1),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "conditional selection (Lemma C.4), and lookup tables (Lemma C.5). With these basic operations as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 307, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 507, + 322 + ], + "score": 1.0, + "content": "“instructions” and by treating autoregressive generation as a loop, it is possible to write “programs”", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "that can solve fairly complex tasks. As detailed in Appendices D.1 and E.1, we construct parallel", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "algorithms that can generate CoT sequences for both math problems, thus concluding the proof.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 209, + 507, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 506, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "Several discussions are made as follows. Firstly, the constructions in our proof reveal the significance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "of several key components in the Transformer design, such as softmax attention, multi-head, feed-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "forward networks, and residual connection. Our proofs offer deep insights into the inner workings of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "Transformer models when dealing with complex tasks, significantly advancing prior understandings", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "such as the “induction head” mechanism [45]. Moreover, our results identify an inherent advantage", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "of Transformers compared to other sequence models like RNNs: indeed, as shown in Appendix F.2,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "constant-size RNNs cannot solve any of the above math tasks using the same CoT format. Secondly,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 428, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 439 + ], + "score": 1.0, + "content": "we highlight that in our setting, the CoT derivations of both math problems are purely written in a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 507, + 452 + ], + "score": 1.0, + "content": "readable math language format, largely resembling how humans write solutions. In a broad sense,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "our findings justify that LLMs have the potential to convey meaningful human thoughts through", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "grammatically precise sentences. Finally, one may ask how LLMs equipped with CoT can bypass", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "the impossibility results outlined in Theorems 3.1 and 3.2. Actually, this can be understood via the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 473, + 495 + ], + "score": 1.0, + "content": "effective depth of the Transformer circuit. By employing CoT, the effective depth is no longer", + "type": "text" + }, + { + "bbox": [ + 473, + 483, + 482, + 492 + ], + "score": 0.73, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "since", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "the generated outputs are repeatedly looped back to the input. The dependency between output tokens", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "leads to a significantly deeper circuit with depth proportional to the length of the CoT solution. Note", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "that even if the recursive procedure is repeated within a fixed Transformer (or circuit), the expressivity", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 524, + 508, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 199, + 542 + ], + "score": 1.0, + "content": "can still be far beyond", + "type": "text" + }, + { + "bbox": [ + 199, + 525, + 218, + 537 + ], + "score": 0.88, + "content": "{ \\bar { \\mathsf { T } } } { \\mathsf { C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 524, + 508, + 542 + ], + "score": 1.0, + "content": ": as will be shown in Section 4, with a sufficient number of CoT steps,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 538, + 373, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 282, + 551 + ], + "score": 1.0, + "content": "autoregressive Transformers can even solve", + "type": "text" + }, + { + "bbox": [ + 282, + 538, + 290, + 548 + ], + "score": 0.28, + "content": "\\mathsf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 538, + 373, + 551 + ], + "score": 1.0, + "content": "-complete problems.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 352, + 508, + 551 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 440, + 578 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 442, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 442, + 581 + ], + "score": 1.0, + "content": "4 CoT is the Key to Solving General Decision-Making Problems", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "The previous section has delineated the critical role of CoT in solving math problems. In this section,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "we will switch our attention to a more general setting beyond mathematics. Remarkably, we find that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "LLMs with CoT are theoretically capable of emulating a powerful decision-making framework called", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 622, + 491, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 491, + 635 + ], + "score": 1.0, + "content": "Dynamic Programming [6], thus strongly justifying the ability of CoT in solving complex tasks.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 589, + 506, + 635 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 231, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 233, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 233, + 662 + ], + "score": 1.0, + "content": "4.1 Dynamic Programming", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Dynamic programming (DP) is widely regarded as a core technique to solve decision-making", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "problems [55]. 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It can be generally written as", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 223, + 388, + 237 + ], + "lines": [ + { + "bbox": [ + 222, + 223, + 388, + 237 + ], + "spans": [ + { + "bbox": [ + 222, + 223, + 388, + 237 + ], + "score": 0.93, + "content": "\\mathsf { d } \\mathsf { p } ( i ) = T \\left( \\boldsymbol { n } , \\boldsymbol { s } , i , \\left\\{ \\left( j , \\mathsf { d } \\mathsf { p } ( j ) \\right) : j \\prec i \\right\\} \\right) ,", + "type": "interline_equation", + "image_path": "8aab84ceb9393cf915b27ba7ba97e9b400b3c302fc2e3b9dc3b5343ca365e08d.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 222, + 223, + 388, + 237 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 104, + 238, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 134, + 253 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 243, + 141, + 250 + ], + "score": 0.7, + "content": "\\pmb { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 238, + 325, + 253 + ], + "score": 1.0, + "content": "is the concatenation of all input sequences", + "type": "text" + }, + { + "bbox": [ + 325, + 239, + 384, + 252 + ], + "score": 0.91, + "content": "\\pmb { s } ^ { ( 1 ) } , \\cdots , \\pmb { s } ^ { ( N ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 238, + 506, + 253 + ], + "score": 1.0, + "content": ". 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ProblemLongest increasing subsequenceEdit distance
InputA string s of length nTwo strings g(1), g(2) of length n1 = |s(1)|and n2 = |s(2)|, concatenated together
State space{(j,k):j∈[n],k ∈{0,..,j-1}}{0,..·,n1} × {0,..,n2}
Transitionfunction1 ifk=0dp(j,k)= max(dp(j,k-1),ifk>0dp(k,k-1)×I[sj>Sk]+1)dp(j,k)=ak if j=0bj ifk=0min(dp(j,k-1) +a,dp(j-1,k)+b,dp(j-1,k-1) otherwise+[1)≠s2)]）
Aggregationfunctionmaxi∈[n] dp(i,i-1)dp(n1,n2)
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As examples, the longest increasing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 456, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 507, + 469 + ], + "score": 1.0, + "content": "subsequence (LIS) and edit distance (ED) are two well-known DP problems presented in the “Introduc-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "tion to Algorithms” book [17] (see Appendix G.1 for problem descriptions and DP solutions). We list", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 502, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 502, + 491 + ], + "score": 1.0, + "content": "the state space, transition function, and aggregation function of the two problems in the table below.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 443, + 507, + 491 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 495, + 508, + 646 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 495, + 508, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 495, + 508, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 508, + 646 + ], + "score": 0.984, + "html": "
ProblemLongest increasing subsequenceEdit distance
InputA string s of length nTwo strings g(1), g(2) of length n1 = |s(1)|and n2 = |s(2)|, concatenated together
State space{(j,k):j∈[n],k ∈{0,..,j-1}}{0,..·,n1} × {0,..,n2}
Transitionfunction1 ifk=0dp(j,k)= max(dp(j,k-1),ifk>0dp(k,k-1)×I[sj>Sk]+1)dp(j,k)=ak if j=0bj ifk=0min(dp(j,k-1) +a,dp(j-1,k)+b,dp(j-1,k-1) otherwise+[1)≠s2)]）
Aggregationfunctionmaxi∈[n] dp(i,i-1)dp(n1,n2)
", + "type": "table", + "image_path": "b3841f7b303def7a6df23ab8ecc6f006c8ef74e8d10491750ca5abce20cf822f.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 106, + 495, + 508, + 545.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 106, + 545.3333333333334, + 508, + 595.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 106, + 595.6666666666667, + 508, + 646.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 660, + 210, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 211, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 211, + 673 + ], + "score": 1.0, + "content": "4.2 Theoretical results", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 680, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "We begin by investigating whether LLMs with CoT can solve the general DP problems defined above.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "score": 1.0, + "content": "We consider a natural CoT generation process, where the generated sequence has the following form:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 680, + 506, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 140, + 704, + 469, + 721 + ], + "lines": [ + { + "bbox": [ + 141, + 700, + 472, + 725 + ], + "spans": [ + { + "bbox": [ + 141, + 704, + 416, + 720 + ], + "score": 0.71, + "content": "\\begin{array} { r l r l r l r l } { s ^ { ( 1 ) } } & { { } | } & { } & { \\cdots } & { { } | } & { } { s ^ { ( N ) } } & { { } | } & { ( i _ { 1 } , \\mathsf { d p } ( i _ { 1 } ) ) } & { { } \\dots } & { { } ( i _ { | \\mathcal { Z } _ { n } | } , \\mathsf { d p } ( i _ { | \\mathcal { Z } _ { n } | } ) ) } \\end{array}", + "type": "inline_equation", + "image_path": "6460c4c5df7c8e60b67dfb241fc29477d9c77de82a29ed445ef36450cfc91db7.jpg" + }, + { + "bbox": [ + 417, + 700, + 472, + 725 + ], + "score": 1.0, + "content": "final answer", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 141, + 700, + 472, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 104, + 72, + 506, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 72, + 254, + 88 + ], + "score": 1.0, + "content": "Here, the input sequence consists of", + "type": "text" + }, + { + "bbox": [ + 255, + 73, + 265, + 83 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 72, + 435, + 88 + ], + "score": 1.0, + "content": "strings separated by special symbols, and", + "type": "text" + }, + { + "bbox": [ + 436, + 73, + 494, + 85 + ], + "score": 0.9, + "content": "( i _ { 1 } , \\cdots , i _ { | \\mathbb { Z } _ { n } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 72, + 506, + 88 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 298, + 97 + ], + "score": 1.0, + "content": "a feasible topological ordering of the state space", + "type": "text" + }, + { + "bbox": [ + 299, + 84, + 311, + 95 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 82, + 428, + 97 + ], + "score": 1.0, + "content": ". We assume that all domains", + "type": "text" + }, + { + "bbox": [ + 428, + 84, + 436, + 93 + ], + "score": 0.51, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 82, + 439, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 439, + 84, + 449, + 93 + ], + "score": 0.5, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 82, + 465, + 97 + ], + "score": 1.0, + "content": ", Y ,", + "type": "text" + }, + { + "bbox": [ + 465, + 85, + 474, + 94 + ], + "score": 0.66, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "belong", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "to the real vector space so that their elements can be effectively represented and handled by a neural", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 167, + 118 + ], + "score": 1.0, + "content": "network. Each", + "type": "text" + }, + { + "bbox": [ + 167, + 105, + 243, + 118 + ], + "score": 0.92, + "content": "( i , \\mathsf { d p } ( i ) ) \\in \\mathcal { T } \\times \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "above will be represented as a single vector and generated jointly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 277, + 129 + ], + "score": 1.0, + "content": "in the CoT output. We further assume that", + "type": "text" + }, + { + "bbox": [ + 277, + 117, + 285, + 126 + ], + "score": 0.46, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 117, + 311, + 127 + ], + "score": 0.29, + "content": ", \\mathcal { X } , \\mathcal { V } .", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 117, + 314, + 129 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 315, + 117, + 324, + 126 + ], + "score": 0.72, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "are discrete spaces (e.g., integers) so that the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 140 + ], + "score": 1.0, + "content": "elements can be precisely represented using finite precision. To simplify our analysis, we consider", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "a regression setting where each element in the CoT output directly corresponds to the output of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "last Transformer layer (without using a softmax layer for tokenization as in Section 3). Instead, the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "Transformer output is simply projected to the nearest element in the corresponding discrete space", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 128, + 183 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + }, + { + "bbox": [ + 128, + 171, + 156, + 182 + ], + "score": 0.9, + "content": "\\mathcal { T } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 170, + 167, + 183 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 168, + 172, + 177, + 181 + ], + "score": 0.77, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "). Likewise, each generated output is directly looped back to the Transformer input", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "without using an embedding layer. This regression setting is convenient for manipulating numerical", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 192, + 360, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 360, + 205 + ], + "score": 1.0, + "content": "values and has been extensively adopted in prior works [24, 1].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 393, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 394, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 394, + 224 + ], + "score": 1.0, + "content": "Before presenting our main result, we make the following assumptions:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 415, + 236 + ], + "score": 1.0, + "content": "Definition 4.1 (Polynomially-efficient approximation). Given neural network", + "type": "text" + }, + { + "bbox": [ + 415, + 224, + 427, + 234 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "and target function", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 231, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 107, + 234, + 174, + 246 + ], + "score": 0.91, + "content": "f : \\mathcal { X } ^ { \\mathrm { i n } } \\to \\mathcal { X } ^ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 231, + 204, + 248 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 204, + 234, + 251, + 244 + ], + "score": 0.92, + "content": "\\chi ^ { \\mathrm { i n } } \\subset \\mathbb { R } ^ { d _ { \\mathrm { i n } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 231, + 271, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 234, + 323, + 244 + ], + "score": 0.9, + "content": "\\mathcal { X } ^ { \\mathrm { o u t } } \\subset \\mathbb { R } ^ { d _ { \\mathrm { o u t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 231, + 359, + 248 + ], + "score": 1.0, + "content": ", we say", + "type": "text" + }, + { + "bbox": [ + 360, + 235, + 367, + 245 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 231, + 470, + 248 + ], + "score": 1.0, + "content": "can be approximated by", + "type": "text" + }, + { + "bbox": [ + 471, + 235, + 483, + 245 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 231, + 506, + 248 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 244, + 502, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 252, + 257 + ], + "score": 1.0, + "content": "polynomial efficiency if there exist", + "type": "text" + }, + { + "bbox": [ + 252, + 245, + 277, + 256 + ], + "score": 0.88, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 244, + 281, + 257 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 281, + 245, + 308, + 255 + ], + "score": 0.78, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 244, + 403, + 257 + ], + "score": 1.0, + "content": "such that for any error", + "type": "text" + }, + { + "bbox": [ + 403, + 246, + 428, + 255 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 244, + 474, + 257 + ], + "score": 1.0, + "content": "and radius", + "type": "text" + }, + { + "bbox": [ + 474, + 245, + 502, + 255 + ], + "score": 0.9, + "content": "R > 0", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 253, + 502, + 270 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 199, + 270 + ], + "score": 1.0, + "content": "there exists parameter", + "type": "text" + }, + { + "bbox": [ + 199, + 257, + 205, + 266 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 253, + 280, + 270 + ], + "score": 1.0, + "content": "satisfying that (i)", + "type": "text" + }, + { + "bbox": [ + 281, + 256, + 438, + 268 + ], + "score": 0.9, + "content": "\\| f ( \\pmb { x } ) - P _ { \\theta } ( \\pmb { x } + \\pmb { \\delta } ) \\| _ { \\infty } < \\epsilon + \\lambda \\| \\pmb { \\delta } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 253, + 467, + 270 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 467, + 256, + 502, + 266 + ], + "score": 0.89, + "content": "\\pmb { x } \\in \\mathcal { X } ^ { \\mathrm { i n } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 266, + 497, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 267, + 154, + 279 + ], + "score": 0.92, + "content": "\\| { \\pmb x } \\| _ { \\infty } \\le R", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 266, + 184, + 280 + ], + "score": 1.0, + "content": "and all", + "type": "text" + }, + { + "bbox": [ + 185, + 267, + 228, + 279 + ], + "score": 0.93, + "content": "\\| \\delta \\| _ { \\infty } < \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 266, + 351, + 280 + ], + "score": 1.0, + "content": "; (ii) all elements of parameter", + "type": "text" + }, + { + "bbox": [ + 351, + 268, + 357, + 277 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 266, + 423, + 280 + ], + "score": 1.0, + "content": "are bounded by", + "type": "text" + }, + { + "bbox": [ + 423, + 267, + 490, + 279 + ], + "score": 0.92, + "content": "\\dot { O } ( \\mathrm { p o l y } ( R , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 266, + 497, + 280 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 108, + 281, + 503, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "Assumption 4.2. The size of the state space can be polynomially upper bounded by the problem size", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 107, + 294, + 115, + 302 + ], + "score": 0.67, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\pmb { n } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 291, + 136, + 305 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 136, + 291, + 220, + 304 + ], + "score": 0.92, + "content": "\\bar { | } \\mathcal { T } _ { n } | = O ( \\mathrm { p o l y } ( | s | ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 291, + 505, + 305 + ], + "score": 1.0, + "content": ". Similarly, all input elements, DP values, and answers are polynomially", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 303, + 262, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 249, + 316 + ], + "score": 1.0, + "content": "upper bounded by the problem size", + "type": "text" + }, + { + "bbox": [ + 249, + 305, + 257, + 313 + ], + "score": 0.72, + "content": "\\textbf { \\em n }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 303, + 262, + 316 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 503, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 238, + 329 + ], + "score": 1.0, + "content": "Assumption 4.3. Each function", + "type": "text" + }, + { + "bbox": [ + 239, + 317, + 268, + 329 + ], + "score": 0.87, + "content": "f , g , h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 316, + 286, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 319, + 294, + 327 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "in (5) and (6) can be approximated with polynomial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 327, + 375, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 375, + 340 + ], + "score": 1.0, + "content": "efficiency by a perceptron of constant size (with GeLU activation).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 506, + 376 + ], + "lines": [ + { + "bbox": [ + 104, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 240, + 354 + ], + "score": 1.0, + "content": "Assumption 4.4. The function", + "type": "text" + }, + { + "bbox": [ + 240, + 340, + 324, + 352 + ], + "score": 0.92, + "content": "F : \\mathbb { N } ^ { N } \\times \\mathcal { T } \\to \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 339, + 372, + 354 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 372, + 342, + 444, + 354 + ], + "score": 0.93, + "content": "F ( n , i _ { k } ) = i _ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 339, + 462, + 354 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 463, + 341, + 502, + 352 + ], + "score": 0.92, + "content": "\\mathbf { \\Omega } _ { n } \\in \\mathbb { N } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 339, + 506, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 107, + 352, + 164, + 365 + ], + "score": 0.92, + "content": "k \\in [ | \\bar { \\mathcal { T } _ { n } } | - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "can be approximated with polynomial efficiency by a perceptron of constant size (with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 488, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 208, + 377 + ], + "score": 1.0, + "content": "GeLU activation), where", + "type": "text" + }, + { + "bbox": [ + 208, + 363, + 267, + 377 + ], + "score": 0.93, + "content": "( i _ { 1 } , \\cdots , i _ { | \\mathbb { Z } _ { n } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 362, + 472, + 377 + ], + "score": 1.0, + "content": "is a feasible topological ordering of the state space", + "type": "text" + }, + { + "bbox": [ + 472, + 364, + 485, + 375 + ], + "score": 0.89, + "content": "\\mathcal { I } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 362, + 488, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 234, + 393 + ], + "score": 1.0, + "content": "Assumption 4.5. The function", + "type": "text" + }, + { + "bbox": [ + 234, + 378, + 324, + 392 + ], + "score": 0.91, + "content": "F : \\mathbb { N } ^ { N } \\times \\mathcal { T } \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 376, + 368, + 393 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 368, + 379, + 451, + 392 + ], + "score": 0.92, + "content": "F ( \\pmb { n } , i ) = \\mathbb { I } [ i \\in \\mathcal { A } _ { n } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 376, + 506, + 393 + ], + "score": 1.0, + "content": "(see (6)) can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "be approximated with polynomial efficiency by a perceptron of constant size (with GeLU activation).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "Remark 4.6. All assumptions above are mild. Assumption 4.2 is necessary to ensure that the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "state vectors, inputs, and DP values can be represented using log-precision, and Assumptions 4.3", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "to 4.5 guarantee that all basic functions that determine the DP process can be well-approximated", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 450 + ], + "score": 1.0, + "content": "by a composition of finite log-precision Transformer layers of constant size. In Appendix G.1, we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "show these assumptions are satisfied for LIS and ED problems described above as well as the CFG", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 459, + 293, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 293, + 471 + ], + "score": 1.0, + "content": "Membership Testing problem in Theorem 4.8.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 108, + 474, + 503, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "We are now ready to present our main result, which shows that LLMs with CoT can solve all DP", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 484, + 420, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 420, + 497 + ], + "score": 1.0, + "content": "problems satisfying the above assumptions. We give a proof in Appendix G.2.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 476, + 511 + ], + "score": 1.0, + "content": "Theorem 4.7. Consider any DP problem satisfying Assumptions 4.2 to 4.5. For any integer", + "type": "text" + }, + { + "bbox": [ + 477, + 499, + 503, + 509 + ], + "score": 0.88, + "content": "n \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 498, + 506, + 511 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 510, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 357, + 522 + ], + "score": 1.0, + "content": "there exists an autoregressive Transformer with constant depth", + "type": "text" + }, + { + "bbox": [ + 358, + 510, + 366, + 520 + ], + "score": 0.43, + "content": "L _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 510, + 442, + 522 + ], + "score": 1.0, + "content": ", hidden dimension", + "type": "text" + }, + { + "bbox": [ + 442, + 510, + 449, + 519 + ], + "score": 0.39, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 510, + 506, + 522 + ], + "score": 1.0, + "content": "and attention", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 131, + 534 + ], + "score": 1.0, + "content": "heads", + "type": "text" + }, + { + "bbox": [ + 132, + 521, + 142, + 531 + ], + "score": 0.55, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 520, + 205, + 534 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 206, + 524, + 213, + 531 + ], + "score": 0.3, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "), such that the answer generated by the Transformer is correct for all input", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 532, + 499, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 446, + 544 + ], + "score": 1.0, + "content": "sequences of length no more than n. Moreover, all parameter values are bounded by", + "type": "text" + }, + { + "bbox": [ + 446, + 532, + 494, + 543 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 532, + 499, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "To complete the analysis, we next explore whether Transformers can directly predict the answer", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "of DP problems without generating intermediate CoT sequences. We show generally the answer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "is no: many DP problems are intrinsically hard to be solved by a bounded-depth Transformer without", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "CoT. One celebrated example is the Context-Free Grammar (CFG) Membership Testing, which take", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 591, + 135, + 605 + ], + "score": 1.0, + "content": "a CFG", + "type": "text" + }, + { + "bbox": [ + 135, + 592, + 145, + 601 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 591, + 194, + 605 + ], + "score": 1.0, + "content": "and a string", + "type": "text" + }, + { + "bbox": [ + 195, + 594, + 202, + 601 + ], + "score": 0.66, + "content": "\\pmb { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 591, + 308, + 605 + ], + "score": 1.0, + "content": "as input and tests whether", + "type": "text" + }, + { + "bbox": [ + 308, + 594, + 316, + 601 + ], + "score": 0.65, + "content": "\\pmb { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "belongs to the context-free language generated", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 602, + 507, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 119, + 615 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 603, + 128, + 613 + ], + "score": 0.67, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 602, + 507, + 615 + ], + "score": 1.0, + "content": ". A formal definition of this problem and a standard DP solution are given in Appendix G.1.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 613, + 406, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 406, + 626 + ], + "score": 1.0, + "content": "We have the following impossibility result (see Appendix G.3 for a proof):", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 200, + 642 + ], + "score": 1.0, + "content": "Theorem 4.8. Assume", + "type": "text" + }, + { + "bbox": [ + 201, + 627, + 250, + 640 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 626, + 496, + 642 + ], + "score": 1.0, + "content": ". 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To simplify our analysis, we consider", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "a regression setting where each element in the CoT output directly corresponds to the output of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "last Transformer layer (without using a softmax layer for tokenization as in Section 3). Instead, the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "Transformer output is simply projected to the nearest element in the corresponding discrete space", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 128, + 183 + ], + "score": 1.0, + "content": "(e.g.,", + "type": "text" + }, + { + "bbox": [ + 128, + 171, + 156, + 182 + ], + "score": 0.9, + "content": "\\mathcal { T } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 170, + 167, + 183 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 168, + 172, + 177, + 181 + ], + "score": 0.77, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "). 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Given neural network", + "type": "text" + }, + { + "bbox": [ + 415, + 224, + 427, + 234 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "and target function", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 231, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 107, + 234, + 174, + 246 + ], + "score": 0.91, + "content": "f : \\mathcal { X } ^ { \\mathrm { i n } } \\to \\mathcal { X } ^ { \\mathrm { o u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 231, + 204, + 248 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 204, + 234, + 251, + 244 + ], + "score": 0.92, + "content": "\\chi ^ { \\mathrm { i n } } \\subset \\mathbb { R } ^ { d _ { \\mathrm { i n } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 231, + 271, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 234, + 323, + 244 + ], + "score": 0.9, + "content": "\\mathcal { X } ^ { \\mathrm { o u t } } \\subset \\mathbb { R } ^ { d _ { \\mathrm { o u t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 231, + 359, + 248 + ], + "score": 1.0, + "content": ", we say", + "type": "text" + }, + { + "bbox": [ + 360, + 235, + 367, + 245 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 231, + 470, + 248 + ], + "score": 1.0, + "content": "can be approximated by", + "type": "text" + }, + { + "bbox": [ + 471, + 235, + 483, + 245 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 231, + 506, + 248 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 244, + 502, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 252, + 257 + ], + "score": 1.0, + "content": "polynomial efficiency if there exist", + "type": "text" + }, + { + "bbox": [ + 252, + 245, + 277, + 256 + ], + "score": 0.88, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 244, + 281, + 257 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 281, + 245, + 308, + 255 + ], + "score": 0.78, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 244, + 403, + 257 + ], + "score": 1.0, + "content": "such that for any error", + "type": "text" + }, + { + "bbox": [ + 403, + 246, + 428, + 255 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 244, + 474, + 257 + ], + "score": 1.0, + "content": "and radius", + "type": "text" + }, + { + "bbox": [ + 474, + 245, + 502, + 255 + ], + "score": 0.9, + "content": "R > 0", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 253, + 502, + 270 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 199, + 270 + ], + "score": 1.0, + "content": "there exists parameter", + "type": "text" + }, + { + "bbox": [ + 199, + 257, + 205, + 266 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 253, + 280, + 270 + ], + "score": 1.0, + "content": "satisfying that (i)", + "type": "text" + }, + { + "bbox": [ + 281, + 256, + 438, + 268 + ], + "score": 0.9, + "content": "\\| f ( \\pmb { x } ) - P _ { \\theta } ( \\pmb { x } + \\pmb { \\delta } ) \\| _ { \\infty } < \\epsilon + \\lambda \\| \\pmb { \\delta } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 253, + 467, + 270 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 467, + 256, + 502, + 266 + ], + "score": 0.89, + "content": "\\pmb { x } \\in \\mathcal { X } ^ { \\mathrm { i n } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 266, + 497, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 267, + 154, + 279 + ], + "score": 0.92, + "content": "\\| { \\pmb x } \\| _ { \\infty } \\le R", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 266, + 184, + 280 + ], + "score": 1.0, + "content": "and all", + "type": "text" + }, + { + "bbox": [ + 185, + 267, + 228, + 279 + ], + "score": 0.93, + "content": "\\| \\delta \\| _ { \\infty } < \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 266, + 351, + 280 + ], + "score": 1.0, + "content": "; (ii) all elements of parameter", + "type": "text" + }, + { + "bbox": [ + 351, + 268, + 357, + 277 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 266, + 423, + 280 + ], + "score": 1.0, + "content": "are bounded by", + "type": "text" + }, + { + "bbox": [ + 423, + 267, + 490, + 279 + ], + "score": 0.92, + "content": "\\dot { O } ( \\mathrm { p o l y } ( R , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 266, + 497, + 280 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 222, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 281, + 503, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "Assumption 4.2. The size of the state space can be polynomially upper bounded by the problem size", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 107, + 294, + 115, + 302 + ], + "score": 0.67, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\pmb { n } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 291, + 136, + 305 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 136, + 291, + 220, + 304 + ], + "score": 0.92, + "content": "\\bar { | } \\mathcal { T } _ { n } | = O ( \\mathrm { p o l y } ( | s | ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 291, + 505, + 305 + ], + "score": 1.0, + "content": ". Similarly, all input elements, DP values, and answers are polynomially", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 303, + 262, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 249, + 316 + ], + "score": 1.0, + "content": "upper bounded by the problem size", + "type": "text" + }, + { + "bbox": [ + 249, + 305, + 257, + 313 + ], + "score": 0.72, + "content": "\\textbf { \\em n }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 303, + 262, + 316 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 280, + 505, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 503, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 238, + 329 + ], + "score": 1.0, + "content": "Assumption 4.3. Each function", + "type": "text" + }, + { + "bbox": [ + 239, + 317, + 268, + 329 + ], + "score": 0.87, + "content": "f , g , h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 316, + 286, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 319, + 294, + 327 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "in (5) and (6) can be approximated with polynomial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 327, + 375, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 375, + 340 + ], + "score": 1.0, + "content": "efficiency by a perceptron of constant size (with GeLU activation).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 106, + 316, + 505, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 506, + 376 + ], + "lines": [ + { + "bbox": [ + 104, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 240, + 354 + ], + "score": 1.0, + "content": "Assumption 4.4. The function", + "type": "text" + }, + { + "bbox": [ + 240, + 340, + 324, + 352 + ], + "score": 0.92, + "content": "F : \\mathbb { N } ^ { N } \\times \\mathcal { T } \\to \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 339, + 372, + 354 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 372, + 342, + 444, + 354 + ], + "score": 0.93, + "content": "F ( n , i _ { k } ) = i _ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 339, + 462, + 354 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 463, + 341, + 502, + 352 + ], + "score": 0.92, + "content": "\\mathbf { \\Omega } _ { n } \\in \\mathbb { N } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 339, + 506, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 107, + 352, + 164, + 365 + ], + "score": 0.92, + "content": "k \\in [ | \\bar { \\mathcal { T } _ { n } } | - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "can be approximated with polynomial efficiency by a perceptron of constant size (with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 488, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 208, + 377 + ], + "score": 1.0, + "content": "GeLU activation), where", + "type": "text" + }, + { + "bbox": [ + 208, + 363, + 267, + 377 + ], + "score": 0.93, + "content": "( i _ { 1 } , \\cdots , i _ { | \\mathbb { Z } _ { n } | } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 362, + 472, + 377 + ], + "score": 1.0, + "content": "is a feasible topological ordering of the state space", + "type": "text" + }, + { + "bbox": [ + 472, + 364, + 485, + 375 + ], + "score": 0.89, + "content": "\\mathcal { I } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 362, + 488, + 377 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 339, + 506, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 234, + 393 + ], + "score": 1.0, + "content": "Assumption 4.5. The function", + "type": "text" + }, + { + "bbox": [ + 234, + 378, + 324, + 392 + ], + "score": 0.91, + "content": "F : \\mathbb { N } ^ { N } \\times \\mathcal { T } \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 376, + 368, + 393 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 368, + 379, + 451, + 392 + ], + "score": 0.92, + "content": "F ( \\pmb { n } , i ) = \\mathbb { I } [ i \\in \\mathcal { A } _ { n } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 376, + 506, + 393 + ], + "score": 1.0, + "content": "(see (6)) can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "be approximated with polynomial efficiency by a perceptron of constant size (with GeLU activation).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 376, + 506, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "Remark 4.6. All assumptions above are mild. Assumption 4.2 is necessary to ensure that the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "state vectors, inputs, and DP values can be represented using log-precision, and Assumptions 4.3", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "to 4.5 guarantee that all basic functions that determine the DP process can be well-approximated", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 450 + ], + "score": 1.0, + "content": "by a composition of finite log-precision Transformer layers of constant size. In Appendix G.1, we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "show these assumptions are satisfied for LIS and ED problems described above as well as the CFG", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 459, + 293, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 293, + 471 + ], + "score": 1.0, + "content": "Membership Testing problem in Theorem 4.8.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 404, + 506, + 471 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 474, + 503, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "We are now ready to present our main result, which shows that LLMs with CoT can solve all DP", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 484, + 420, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 420, + 497 + ], + "score": 1.0, + "content": "problems satisfying the above assumptions. We give a proof in Appendix G.2.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 473, + 505, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 476, + 511 + ], + "score": 1.0, + "content": "Theorem 4.7. Consider any DP problem satisfying Assumptions 4.2 to 4.5. For any integer", + "type": "text" + }, + { + "bbox": [ + 477, + 499, + 503, + 509 + ], + "score": 0.88, + "content": "n \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 498, + 506, + 511 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 510, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 357, + 522 + ], + "score": 1.0, + "content": "there exists an autoregressive Transformer with constant depth", + "type": "text" + }, + { + "bbox": [ + 358, + 510, + 366, + 520 + ], + "score": 0.43, + "content": "L _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 510, + 442, + 522 + ], + "score": 1.0, + "content": ", hidden dimension", + "type": "text" + }, + { + "bbox": [ + 442, + 510, + 449, + 519 + ], + "score": 0.39, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 510, + 506, + 522 + ], + "score": 1.0, + "content": "and attention", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 131, + 534 + ], + "score": 1.0, + "content": "heads", + "type": "text" + }, + { + "bbox": [ + 132, + 521, + 142, + 531 + ], + "score": 0.55, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 520, + 205, + 534 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 206, + 524, + 213, + 531 + ], + "score": 0.3, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "), such that the answer generated by the Transformer is correct for all input", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 532, + 499, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 446, + 544 + ], + "score": 1.0, + "content": "sequences of length no more than n. Moreover, all parameter values are bounded by", + "type": "text" + }, + { + "bbox": [ + 446, + 532, + 494, + 543 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 532, + 499, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 498, + 506, + 544 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "To complete the analysis, we next explore whether Transformers can directly predict the answer", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "of DP problems without generating intermediate CoT sequences. We show generally the answer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "is no: many DP problems are intrinsically hard to be solved by a bounded-depth Transformer without", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "CoT. One celebrated example is the Context-Free Grammar (CFG) Membership Testing, which take", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 591, + 135, + 605 + ], + "score": 1.0, + "content": "a CFG", + "type": "text" + }, + { + "bbox": [ + 135, + 592, + 145, + 601 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 591, + 194, + 605 + ], + "score": 1.0, + "content": "and a string", + "type": "text" + }, + { + "bbox": [ + 195, + 594, + 202, + 601 + ], + "score": 0.66, + "content": "\\pmb { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 591, + 308, + 605 + ], + "score": 1.0, + "content": "as input and tests whether", + "type": "text" + }, + { + "bbox": [ + 308, + 594, + 316, + 601 + ], + "score": 0.65, + "content": "\\pmb { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "belongs to the context-free language generated", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 602, + 507, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 119, + 615 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 603, + 128, + 613 + ], + "score": 0.67, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 602, + 507, + 615 + ], + "score": 1.0, + "content": ". A formal definition of this problem and a standard DP solution are given in Appendix G.1.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 613, + 406, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 406, + 626 + ], + "score": 1.0, + "content": "We have the following impossibility result (see Appendix G.3 for a proof):", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 548, + 507, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 200, + 642 + ], + "score": 1.0, + "content": "Theorem 4.8. Assume", + "type": "text" + }, + { + "bbox": [ + 201, + 627, + 250, + 640 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 626, + 496, + 642 + ], + "score": 1.0, + "content": ". There exists a context-free language such that for any depth", + "type": "text" + }, + { + "bbox": [ + 496, + 629, + 504, + 639 + ], + "score": 0.36, + "content": "L", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 639, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 188, + 652 + ], + "score": 1.0, + "content": "and any polynomial", + "type": "text" + }, + { + "bbox": [ + 189, + 640, + 197, + 651 + ], + "score": 0.78, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 639, + 322, + 652 + ], + "score": 1.0, + "content": ", there exists a sequence length", + "type": "text" + }, + { + "bbox": [ + 323, + 640, + 349, + 650 + ], + "score": 0.88, + "content": "n \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 639, + 505, + 652 + ], + "score": 1.0, + "content": "where no log-precision autoregressive", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 201, + 663 + ], + "score": 1.0, + "content": "transformer with depth", + "type": "text" + }, + { + "bbox": [ + 201, + 651, + 209, + 660 + ], + "score": 0.61, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 650, + 302, + 663 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 302, + 651, + 343, + 663 + ], + "score": 0.92, + "content": "d \\leq Q ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "can generate the correct answer for the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 661, + 375, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 365, + 675 + ], + "score": 1.0, + "content": "CFG Membership Testing problem for all input strings of length", + "type": "text" + }, + { + "bbox": [ + 365, + 666, + 371, + 671 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 661, + 375, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 626, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "In contrast, enabling CoT substantially improves the expressivity of Transformers: as proved in", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "Jones & Laaser [33], the universal CFG Membership Testing is a celebrated P-complete problem and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "is intrinsically hard to be solved by a well-parallelized computation model. Combined with these", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 711, + 484, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 484, + 723 + ], + "score": 1.0, + "content": "results, we conclude that CoT plays a critical role in tackling tasks that are inherently difficult.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 71, + 191, + 85 + ], + "lines": [ + { + "bbox": [ + 103, + 68, + 193, + 88 + ], + "spans": [ + { + "bbox": [ + 103, + 68, + 193, + 88 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "score": 1.0, + "content": "In previous sections, we proved by construction that LLMs exhibit sufficient expressive power to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 107, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 505, + 119 + ], + "score": 1.0, + "content": "solve mathematical and decision-making tasks. On the other hand, it is still essential to check whether", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "a Transformer model can learn such ability directly from training data. Below, we will complement", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 144 + ], + "score": 1.0, + "content": "our theoretical results with experimental evidence, showing that the model can easily learn underlying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 140, + 366, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 366, + 152 + ], + "score": 1.0, + "content": "task solutions when equipped with CoT training demonstrations.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 164, + 220, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 221, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 221, + 180 + ], + "score": 1.0, + "content": "5.1 Experimental Design", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "Tasks and datasets. We choose four tasks for evaluation: Arithmetic, Equation, LIS, and ED. The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "score": 1.0, + "content": "first two tasks (Arithmetic and Equation) as well as their input/CoT formats have been illustrated in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Figure 1. For the LIS task, the goal is to find the length of the longest increasing subsequence of a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "given integer sequence. For the ED task, the goal is to calculate the minimum cost required (called", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "edit distance) to convert one sequence to another using three basic edit operations: insert, delete and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "replace. All input sequences, CoT demonstrations, and answers in LIS and ED are bounded-range", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "integers and can therefore be tokenized (similar to the first two tasks). We consider two settings:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "score": 1.0, + "content": "(i) CoT datasets, which consist of samples; (ii) Direct datasets, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "are used to train models that directly predict the answer without CoT steps. These datasets are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 412, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 412, + 296 + ], + "score": 1.0, + "content": "constructed by removing all intermediate derivations from the CoT datasets.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "For each task, we construct three datasets with increasing difficulty. For Arithmetic, we build datasets", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 311, + 324 + ], + "score": 1.0, + "content": "with different numbers of operators ranging from", + "type": "text" + }, + { + "bbox": [ + 312, + 311, + 346, + 322 + ], + "score": 0.92, + "content": "\\{ 4 , 5 , 6 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 310, + 506, + 324 + ], + "score": 1.0, + "content": ". For Equation, we build datasets with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 281, + 334 + ], + "score": 1.0, + "content": "different numbers of variables ranging from", + "type": "text" + }, + { + "bbox": [ + 282, + 322, + 316, + 333 + ], + "score": 0.93, + "content": "\\{ 3 , 4 , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 321, + 506, + 334 + ], + "score": 1.0, + "content": ". For LIS, we build datasets with different input", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 230, + 345 + ], + "score": 1.0, + "content": "sequence lengths ranging from", + "type": "text" + }, + { + "bbox": [ + 230, + 333, + 284, + 345 + ], + "score": 0.9, + "content": "\\{ 5 0 , 8 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 332, + 506, + 345 + ], + "score": 1.0, + "content": ". For ED, we build datasets with different string lengths,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 322, + 356 + ], + "score": 1.0, + "content": "where the average length of the two strings ranges from", + "type": "text" + }, + { + "bbox": [ + 322, + 343, + 371, + 356 + ], + "score": 0.91, + "content": "\\{ 1 2 , 1 6 , 2 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 343, + 506, + 356 + ], + "score": 1.0, + "content": ". Creating these datasets allows us", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "to investigate how model performance varies with the increase of problem size. We generate 1M sam-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 235, + 378 + ], + "score": 1.0, + "content": "ples for each training dataset and", + "type": "text" + }, + { + "bbox": [ + 236, + 365, + 258, + 376 + ], + "score": 0.39, + "content": "0 . 1 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "for testing while ensuring that duplicate samples between train-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 507, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 507, + 389 + ], + "score": 1.0, + "content": "ing and testing are removed. More details about the dataset construction can be found in Appendix H.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "Model training and inference. For all experiments, we use standard Transformer models with hidden", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 151, + 416 + ], + "score": 1.0, + "content": "dimension", + "type": "text" + }, + { + "bbox": [ + 151, + 404, + 186, + 414 + ], + "score": 0.88, + "content": "d = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 403, + 215, + 416 + ], + "score": 1.0, + "content": ", heads", + "type": "text" + }, + { + "bbox": [ + 215, + 404, + 244, + 414 + ], + "score": 0.89, + "content": "H = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 403, + 358, + 416 + ], + "score": 1.0, + "content": ", and different model depths", + "type": "text" + }, + { + "bbox": [ + 359, + 404, + 367, + 414 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 403, + 505, + 416 + ], + "score": 1.0, + "content": ". We adopt the AdamW optimizer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 147, + 428 + ], + "score": 1.0, + "content": "[39] with", + "type": "text" + }, + { + "bbox": [ + 147, + 414, + 283, + 426 + ], + "score": 0.84, + "content": "\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 9 , \\mathrm { l r } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 413, + 360, + 428 + ], + "score": 1.0, + "content": ", and weight decay", + "type": "text" + }, + { + "bbox": [ + 360, + 415, + 391, + 425 + ], + "score": 0.8, + "content": "= 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "in all experiments. We use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "a fixed dropout ratio of 0.1 for all experiments to improve generalization. For CoT datasets, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "optimize the negative log-likelihood loss on all tokens in the CoT steps and answers. This is similar to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "the so-called process supervision proposed in a concurrent work [36]. For direct datasets, we optimize", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "the negative log-likelihood loss on answer tokens. All models are trained on 4 V100 GPUs for 100", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "epochs. During inference, models trained on the direct datasets are required to output the answer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "directly, and models trained on CoT datasets will generate the whole CoT process token-by-token", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "(using greedy search) until generating the End-of-Sentence token, where the output in the final", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "step is regarded as the answer. We report the accuracy as the evaluation metric. Please refer to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 513, + 316, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 316, + 526 + ], + "score": 1.0, + "content": "Appendix H for more training configuration details.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 222, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 223, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 223, + 551 + ], + "score": 1.0, + "content": "5.2 Experimental Results", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "Main results. All results are shown in Figure 2, where each subfigure corresponds to a task with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 113, + 579 + ], + "score": 0.45, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 569, + 422, + 582 + ], + "score": 1.0, + "content": "-axis representing the difficulty level and y-axis representing the test accuracy", + "type": "text" + }, + { + "bbox": [ + 422, + 569, + 437, + 580 + ], + "score": 0.76, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ". We repeat each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "experiment five times and report the error bars. In each subfigure, the purple bar and blue bars indicate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 591, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 504, + 604 + ], + "score": 1.0, + "content": "the performance of the model trained on the CoT and direct datasets, respectively. The model depths", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "are specified in the legend. From these results, one can easily see that 3-layer Transformers with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "CoT already achieve near-perfect performance on all tasks for all difficulty levels, which is consistent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "to Theorems 3.3, 3.4 and 4.7 and may further imply that they can learn better solutions with fewer", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "model depth than our constructions. Moreover, it is worth noting that, for some difficult datasets such", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "as Equation of 5 variables, a perfect accuracy would imply that the model can precisely generate", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "a correct CoT with a very long length of roughly 500 for all inputs. In contrast, models trained on", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "direct datasets perform much worse even when using larger depths (particularly on the Equation task).", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "While increasing the depth usually helps the performance of direct prediction (which is consistent", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 507, + 702 + ], + "score": 1.0, + "content": "with our theory), the performance drops significantly when the length of the input sequence grows.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "All these empirical findings verify our theoretical results and clearly demonstrate the benefit of CoT", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 223, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 223, + 723 + ], + "score": 1.0, + "content": "in autoregressive generation.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 45 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 71, + 191, + 85 + ], + "lines": [ + { + "bbox": [ + 103, + 68, + 193, + 88 + ], + "spans": [ + { + "bbox": [ + 103, + 68, + 193, + 88 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 108 + ], + "score": 1.0, + "content": "In previous sections, we proved by construction that LLMs exhibit sufficient expressive power to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 107, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 505, + 119 + ], + "score": 1.0, + "content": "solve mathematical and decision-making tasks. On the other hand, it is still essential to check whether", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "a Transformer model can learn such ability directly from training data. Below, we will complement", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 144 + ], + "score": 1.0, + "content": "our theoretical results with experimental evidence, showing that the model can easily learn underlying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 140, + 366, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 366, + 152 + ], + "score": 1.0, + "content": "task solutions when equipped with CoT training demonstrations.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 96, + 505, + 152 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 164, + 220, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 221, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 221, + 180 + ], + "score": 1.0, + "content": "5.1 Experimental Design", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 185, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "Tasks and datasets. We choose four tasks for evaluation: Arithmetic, Equation, LIS, and ED. The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 208 + ], + "score": 1.0, + "content": "first two tasks (Arithmetic and Equation) as well as their input/CoT formats have been illustrated in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "Figure 1. For the LIS task, the goal is to find the length of the longest increasing subsequence of a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "given integer sequence. For the ED task, the goal is to calculate the minimum cost required (called", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "edit distance) to convert one sequence to another using three basic edit operations: insert, delete and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "replace. All input sequences, CoT demonstrations, and answers in LIS and ED are bounded-range", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "integers and can therefore be tokenized (similar to the first two tasks). We consider two settings:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "score": 1.0, + "content": "(i) CoT datasets, which consist of samples; (ii) Direct datasets, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "are used to train models that directly predict the answer without CoT steps. These datasets are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 412, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 412, + 296 + ], + "score": 1.0, + "content": "constructed by removing all intermediate derivations from the CoT datasets.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 185, + 506, + 296 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "For each task, we construct three datasets with increasing difficulty. For Arithmetic, we build datasets", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 311, + 324 + ], + "score": 1.0, + "content": "with different numbers of operators ranging from", + "type": "text" + }, + { + "bbox": [ + 312, + 311, + 346, + 322 + ], + "score": 0.92, + "content": "\\{ 4 , 5 , 6 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 310, + 506, + 324 + ], + "score": 1.0, + "content": ". For Equation, we build datasets with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 281, + 334 + ], + "score": 1.0, + "content": "different numbers of variables ranging from", + "type": "text" + }, + { + "bbox": [ + 282, + 322, + 316, + 333 + ], + "score": 0.93, + "content": "\\{ 3 , 4 , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 321, + 506, + 334 + ], + "score": 1.0, + "content": ". For LIS, we build datasets with different input", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 230, + 345 + ], + "score": 1.0, + "content": "sequence lengths ranging from", + "type": "text" + }, + { + "bbox": [ + 230, + 333, + 284, + 345 + ], + "score": 0.9, + "content": "\\{ 5 0 , 8 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 332, + 506, + 345 + ], + "score": 1.0, + "content": ". For ED, we build datasets with different string lengths,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 322, + 356 + ], + "score": 1.0, + "content": "where the average length of the two strings ranges from", + "type": "text" + }, + { + "bbox": [ + 322, + 343, + 371, + 356 + ], + "score": 0.91, + "content": "\\{ 1 2 , 1 6 , 2 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 343, + 506, + 356 + ], + "score": 1.0, + "content": ". Creating these datasets allows us", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "to investigate how model performance varies with the increase of problem size. We generate 1M sam-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 235, + 378 + ], + "score": 1.0, + "content": "ples for each training dataset and", + "type": "text" + }, + { + "bbox": [ + 236, + 365, + 258, + 376 + ], + "score": 0.39, + "content": "0 . 1 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 366, + 506, + 378 + ], + "score": 1.0, + "content": "for testing while ensuring that duplicate samples between train-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 507, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 507, + 389 + ], + "score": 1.0, + "content": "ing and testing are removed. More details about the dataset construction can be found in Appendix H.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 300, + 507, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "Model training and inference. For all experiments, we use standard Transformer models with hidden", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 151, + 416 + ], + "score": 1.0, + "content": "dimension", + "type": "text" + }, + { + "bbox": [ + 151, + 404, + 186, + 414 + ], + "score": 0.88, + "content": "d = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 403, + 215, + 416 + ], + "score": 1.0, + "content": ", heads", + "type": "text" + }, + { + "bbox": [ + 215, + 404, + 244, + 414 + ], + "score": 0.89, + "content": "H = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 403, + 358, + 416 + ], + "score": 1.0, + "content": ", and different model depths", + "type": "text" + }, + { + "bbox": [ + 359, + 404, + 367, + 414 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 403, + 505, + 416 + ], + "score": 1.0, + "content": ". We adopt the AdamW optimizer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 147, + 428 + ], + "score": 1.0, + "content": "[39] with", + "type": "text" + }, + { + "bbox": [ + 147, + 414, + 283, + 426 + ], + "score": 0.84, + "content": "\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 9 , \\mathrm { l r } = 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 413, + 360, + 428 + ], + "score": 1.0, + "content": ", and weight decay", + "type": "text" + }, + { + "bbox": [ + 360, + 415, + 391, + 425 + ], + "score": 0.8, + "content": "= 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "in all experiments. We use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "a fixed dropout ratio of 0.1 for all experiments to improve generalization. For CoT datasets, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "optimize the negative log-likelihood loss on all tokens in the CoT steps and answers. This is similar to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "the so-called process supervision proposed in a concurrent work [36]. For direct datasets, we optimize", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "the negative log-likelihood loss on answer tokens. All models are trained on 4 V100 GPUs for 100", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "epochs. During inference, models trained on the direct datasets are required to output the answer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "directly, and models trained on CoT datasets will generate the whole CoT process token-by-token", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "(using greedy search) until generating the End-of-Sentence token, where the output in the final", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "step is regarded as the answer. We report the accuracy as the evaluation metric. Please refer to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 513, + 316, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 316, + 526 + ], + "score": 1.0, + "content": "Appendix H for more training configuration details.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 391, + 506, + 526 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 222, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 223, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 223, + 551 + ], + "score": 1.0, + "content": "5.2 Experimental Results", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "Main results. All results are shown in Figure 2, where each subfigure corresponds to a task with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 113, + 579 + ], + "score": 0.45, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 569, + 422, + 582 + ], + "score": 1.0, + "content": "-axis representing the difficulty level and y-axis representing the test accuracy", + "type": "text" + }, + { + "bbox": [ + 422, + 569, + 437, + 580 + ], + "score": 0.76, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ". We repeat each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "experiment five times and report the error bars. In each subfigure, the purple bar and blue bars indicate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 591, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 504, + 604 + ], + "score": 1.0, + "content": "the performance of the model trained on the CoT and direct datasets, respectively. The model depths", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "are specified in the legend. From these results, one can easily see that 3-layer Transformers with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "CoT already achieve near-perfect performance on all tasks for all difficulty levels, which is consistent", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "to Theorems 3.3, 3.4 and 4.7 and may further imply that they can learn better solutions with fewer", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "model depth than our constructions. Moreover, it is worth noting that, for some difficult datasets such", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "as Equation of 5 variables, a perfect accuracy would imply that the model can precisely generate", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "a correct CoT with a very long length of roughly 500 for all inputs. In contrast, models trained on", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "direct datasets perform much worse even when using larger depths (particularly on the Equation task).", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "While increasing the depth usually helps the performance of direct prediction (which is consistent", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 507, + 702 + ], + "score": 1.0, + "content": "with our theory), the performance drops significantly when the length of the input sequence grows.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "All these empirical findings verify our theoretical results and clearly demonstrate the benefit of CoT", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 223, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 223, + 723 + ], + "score": 1.0, + "content": "in autoregressive generation.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 558, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 70, + 504, + 320 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 70, + 504, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 504, + 320 + ], + "score": 0.976, + "type": "image", + "image_path": "2b6296309d7840ab82ec20c4c4c6f5150d128adb343c79dd6f4c6e265cb69631.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 70, + 504, + 153.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 153.33333333333331, + 504, + 236.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 236.66666666666663, + 504, + 319.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 326, + 506, + 370 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 326, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 507, + 338 + ], + "score": 1.0, + "content": "Figure 2: Model performance on different tasks. For all tasks and various difficulty levels, autore-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "gressive Transformers with CoT consistently outperform Transformers trained on direct datasets. In", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "particular, 3-layer Transformers already succeed in these tasks with almost perfect accuracy, while", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 358, + 417, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 194, + 371 + ], + "score": 1.0, + "content": "deeper Transformers", + "type": "text" + }, + { + "bbox": [ + 194, + 359, + 241, + 370 + ], + "score": 0.88, + "content": "\\mathcal { L } = 3 , 4 , 5", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 358, + 417, + 371 + ], + "score": 1.0, + "content": ") trained on the direct datasets typically fail.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "Robustness to data quality. Unlike the synthetic datasets constructed above, real-world training", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "datasets are not perfect and often involve corruption or miss intermediate steps. This calls into question", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "whether the model can still perform well when training on low-quality datasets. To investigate this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "question, we construct corrupted datasets for the arithmetic task in Appendix I. Surprisingly, our", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 406, + 442 + ], + "score": 1.0, + "content": "results show that the 3-layer Transformer model can still achieve more than", + "type": "text" + }, + { + "bbox": [ + 406, + 429, + 426, + 440 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "accuracy even with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 127, + 451 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "of the data missing an intermediate CoT step and involving a single-token corruption. This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 450, + 412, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 412, + 464 + ], + "score": 1.0, + "content": "clearly demonstrates the robustness of CoT training on low-quality datasets.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 329, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 329, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 329, + 480 + ], + "score": 1.0, + "content": "Length extrapolation. We finally study whether the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 478, + 329, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 329, + 491 + ], + "score": 1.0, + "content": "learned autoregressive models can further extrapolate to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 488, + 329, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 329, + 502 + ], + "score": 1.0, + "content": "data with longer lengths. 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For all tasks and various difficulty levels, autore-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "gressive Transformers with CoT consistently outperform Transformers trained on direct datasets. In", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "particular, 3-layer Transformers already succeed in these tasks with almost perfect accuracy, while", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 358, + 417, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 194, + 371 + ], + "score": 1.0, + "content": "deeper Transformers", + "type": "text" + }, + { + "bbox": [ + 194, + 359, + 241, + 370 + ], + "score": 0.88, + "content": "\\mathcal { L } = 3 , 4 , 5", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 358, + 417, + 371 + ], + "score": 1.0, + "content": ") trained on the direct datasets typically fail.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "Robustness to data quality. 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This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 450, + 412, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 412, + 464 + ], + "score": 1.0, + "content": "clearly demonstrates the robustness of CoT training on low-quality datasets.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 384, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 329, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 329, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 329, + 480 + ], + "score": 1.0, + "content": "Length extrapolation. We finally study whether the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 478, + 329, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 329, + 491 + ], + "score": 1.0, + "content": "learned autoregressive models can further extrapolate to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 488, + 329, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 329, + 502 + ], + "score": 1.0, + "content": "data with longer lengths. 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Initially, researchers primarily focused on exploring the expressive power", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "of Transformers in the context of function approximation. Yun et al. [67] proved that Transformers", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "with sufficient size can universally approximate arbitrary continuous sequence-to-sequence functions", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "on a compact domain. Recently, universality results have been extended to model variants such as", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 469, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 469, + 723 + ], + "score": 1.0, + "content": "Sparse Transformers [68] and Transformers with relative positional encodings (RPE) [41].", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 645, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 507, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 507, + 86 + ], + "score": 1.0, + "content": "More relevant to this paper, another line of works investigated the power of Transformers from a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "computation perspective. Early results have shown that both standard encoder-decoder Transformers", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "[58] and looped Transformer encoders are Turing-complete [49, 47, 20, 8]. However, these results", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "depend on the unreasonable assumption of infinite precision, yielding a quite unrealistic construction", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "that does not match practical scenarios. Recently, Giannou [25] demonstrated that a constant-depth", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "looped Transformer encoder can simulate practical computer programs. Wei et al. [60] showed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "that finite-precision encoder-decoder Transformers can approximately simulate Turing machines", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "with bounded computation time. Liu et al. [37] considered a restricted setting of learning automata,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "for which a shallow non-recursive Transformer provably suffices. Yao et al. [66] demonstrated", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "that Transformers can recognize or generate bounded-depth Dyck language [30], a specific type", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "score": 1.0, + "content": "of context-free language. Besides affirmative results, other works characterized the expressivity", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 193, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 205 + ], + "score": 1.0, + "content": "limitation of Transformers via the perspective of modeling formal languages [26, 7, 62, 66, 14] or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 507, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 217 + ], + "score": 1.0, + "content": "simulating circuits [27, 43, 42]. However, none of these works (except [66]) explored the setting of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "autoregressive Transformers typically adopted in LLMs, which we study in this paper. Moreover, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "consider a more practical setting that targets the emergent ability of LLMs in solving basic reasoning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 236, + 434, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 434, + 249 + ], + "score": 1.0, + "content": "problems via a readable CoT output, which aligns well with real-world scenarios.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "Recently, the power of Transformers has regained attention due to the exceptional in-context learn-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "ability exhibited by LLMs [9]. Garg et al. [24] demonstrated that autoregressive Transformers can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "in-context learn basic function classes (e.g., linear functions, MLPs, and decision trees) via input", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "sample sequences. Subsequent works further revealed that Transformers can implement learning", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "algorithms such as linear regression [1], gradient descent [1, 59, 19], and Bayesian inference [64], and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "a broad class of machine learning algorithms [3]. The works of [23, 45] studied in-context learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "via the concept of “induction heads”. All the above works investigated the power of (autoregressive)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "score": 1.0, + "content": "Transformer models from an expressivity perspective, which shares similarities to this paper. Here,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "score": 1.0, + "content": "we focus on the reasoning capability of Transformers and underscore the key role of CoT in improving", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 351, + 188, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 188, + 363 + ], + "score": 1.0, + "content": "the power of LLMs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 301, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 302, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 302, + 397 + ], + "score": 1.0, + "content": "7 Limitations and Future Directions", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "In this work, from a model-capacity perspective, we theoretically analyze why Chain-of-Thought", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "prompting is essential in solving mathematical and decision-making problems. Focusing on two", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "basic mathematical problems as well as Dynamic Programming, we show that a bounded-depth", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "Transformer without CoT struggles with these tasks unless its size grows prohibitively large. In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "contrast to our negative results, we prove by construction that when equipped with CoT, constant-size", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Transformers are sufficiently capable of addressing these tasks by generating intermediate derivations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "sequentially. Extensive experiments show that models trained on CoT datasets can indeed learn", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "solutions almost perfectly, while direct prediction always fails. We further demonstrate that CoT has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 495, + 353, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 353, + 508 + ], + "score": 1.0, + "content": "the potential to generalize to unseen data with longer lengths.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Several foundational questions remain to be answered. Firstly, while this paper investigates why", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "CoT enhances the expressivity of LLMs, we do not yet answer how the CoT generation process", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "is triggered by specific prompts. Revealing the relation between prompts and outputs is valuable", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "for better harnessing LLMs. Secondly, it has been empirically observed that scaling the model size", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "significantly improves the CoT ability [61]. Theoretically understanding how model size plays a role", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "in CoT would be an interesting research problem. Thirdly, this paper mainly studies the expressivity", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "of LLMs in generating CoT solutions, without theoretically thinking about their generalization ability.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 586, + 507, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 507, + 602 + ], + "score": 1.0, + "content": "Given our experimental results, we believe it is an important future direction for theoretically studying", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 597, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 507, + 613 + ], + "score": 1.0, + "content": "how LLMs can generalize from CoT demonstrations (even in the out-of-distribution setting, e.g.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "length extrapolation (Figure 3)) [60, 13]. Finally, from a practical perspective, it is interesting to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "investigate how models can learn CoT solutions when there are only limited CoT demonstrations in", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "training (or even purely from direct datasets). We would like to leave these questions as future work,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 642, + 441, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 441, + 655 + ], + "score": 1.0, + "content": "which we believe are beneficial to better reveal the power and limitations of LLMs.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Acknowledgement. This work is supported by National Key R&D Program of China", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "(2022ZD0114900) and National Science Foundation of China (NSFC62276005), and is partially", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "funded by Microsoft Research Asia Collaborative Research Project. The authors are grateful to David", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Chiang, who pointed out a mistake regarding the P-completeness of CFG Membership Testing in the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 711, + 484, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 484, + 723 + ], + "score": 1.0, + "content": "early version of this paper. The authors also thank all reviewers for their valuable suggestions.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 507, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 507, + 86 + ], + "score": 1.0, + "content": "More relevant to this paper, another line of works investigated the power of Transformers from a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "computation perspective. Early results have shown that both standard encoder-decoder Transformers", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "[58] and looped Transformer encoders are Turing-complete [49, 47, 20, 8]. However, these results", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "depend on the unreasonable assumption of infinite precision, yielding a quite unrealistic construction", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "that does not match practical scenarios. Recently, Giannou [25] demonstrated that a constant-depth", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "looped Transformer encoder can simulate practical computer programs. Wei et al. [60] showed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "that finite-precision encoder-decoder Transformers can approximately simulate Turing machines", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "with bounded computation time. Liu et al. [37] considered a restricted setting of learning automata,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "for which a shallow non-recursive Transformer provably suffices. Yao et al. [66] demonstrated", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "that Transformers can recognize or generate bounded-depth Dyck language [30], a specific type", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "score": 1.0, + "content": "of context-free language. Besides affirmative results, other works characterized the expressivity", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 193, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 205 + ], + "score": 1.0, + "content": "limitation of Transformers via the perspective of modeling formal languages [26, 7, 62, 66, 14] or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 507, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 217 + ], + "score": 1.0, + "content": "simulating circuits [27, 43, 42]. However, none of these works (except [66]) explored the setting of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "autoregressive Transformers typically adopted in LLMs, which we study in this paper. Moreover, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "consider a more practical setting that targets the emergent ability of LLMs in solving basic reasoning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 236, + 434, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 434, + 249 + ], + "score": 1.0, + "content": "problems via a readable CoT output, which aligns well with real-world scenarios.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 72, + 507, + 249 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "Recently, the power of Transformers has regained attention due to the exceptional in-context learn-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "ability exhibited by LLMs [9]. Garg et al. [24] demonstrated that autoregressive Transformers can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "in-context learn basic function classes (e.g., linear functions, MLPs, and decision trees) via input", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "sample sequences. Subsequent works further revealed that Transformers can implement learning", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "algorithms such as linear regression [1], gradient descent [1, 59, 19], and Bayesian inference [64], and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "a broad class of machine learning algorithms [3]. The works of [23, 45] studied in-context learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "via the concept of “induction heads”. All the above works investigated the power of (autoregressive)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "score": 1.0, + "content": "Transformer models from an expressivity perspective, which shares similarities to this paper. Here,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "score": 1.0, + "content": "we focus on the reasoning capability of Transformers and underscore the key role of CoT in improving", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 351, + 188, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 188, + 363 + ], + "score": 1.0, + "content": "the power of LLMs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 253, + 507, + 363 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 301, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 302, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 302, + 397 + ], + "score": 1.0, + "content": "7 Limitations and Future Directions", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "In this work, from a model-capacity perspective, we theoretically analyze why Chain-of-Thought", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "prompting is essential in solving mathematical and decision-making problems. Focusing on two", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "basic mathematical problems as well as Dynamic Programming, we show that a bounded-depth", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "Transformer without CoT struggles with these tasks unless its size grows prohibitively large. In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "contrast to our negative results, we prove by construction that when equipped with CoT, constant-size", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Transformers are sufficiently capable of addressing these tasks by generating intermediate derivations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "sequentially. Extensive experiments show that models trained on CoT datasets can indeed learn", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "solutions almost perfectly, while direct prediction always fails. We further demonstrate that CoT has", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 495, + 353, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 353, + 508 + ], + "score": 1.0, + "content": "the potential to generalize to unseen data with longer lengths.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 408, + 506, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Several foundational questions remain to be answered. Firstly, while this paper investigates why", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "CoT enhances the expressivity of LLMs, we do not yet answer how the CoT generation process", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "is triggered by specific prompts. Revealing the relation between prompts and outputs is valuable", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "for better harnessing LLMs. Secondly, it has been empirically observed that scaling the model size", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "significantly improves the CoT ability [61]. Theoretically understanding how model size plays a role", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "in CoT would be an interesting research problem. Thirdly, this paper mainly studies the expressivity", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "of LLMs in generating CoT solutions, without theoretically thinking about their generalization ability.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 586, + 507, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 507, + 602 + ], + "score": 1.0, + "content": "Given our experimental results, we believe it is an important future direction for theoretically studying", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 597, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 507, + 613 + ], + "score": 1.0, + "content": "how LLMs can generalize from CoT demonstrations (even in the out-of-distribution setting, e.g.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "length extrapolation (Figure 3)) [60, 13]. Finally, from a practical perspective, it is interesting to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "investigate how models can learn CoT solutions when there are only limited CoT demonstrations in", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "training (or even purely from direct datasets). We would like to leave these questions as future work,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 642, + 441, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 441, + 655 + ], + "score": 1.0, + "content": "which we believe are beneficial to better reveal the power and limitations of LLMs.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 511, + 507, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Acknowledgement. This work is supported by National Key R&D Program of China", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "(2022ZD0114900) and National Science Foundation of China (NSFC62276005), and is partially", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "funded by Microsoft Research Asia Collaborative Research Project. The authors are grateful to David", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Chiang, who pointed out a mistake regarding the P-completeness of CFG Membership Testing in the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 711, + 484, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 484, + 723 + ], + "score": 1.0, + "content": "early version of this paper. The authors also thank all reviewers for their valuable suggestions.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 666, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 218, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 220, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 220, + 87 + ], + "score": 1.0, + "content": "Author Contributions", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 109 + ], + "score": 1.0, + "content": "Guhao Feng proposed to analyze the expressivity of Transformers using circuit complexity and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "proved all impossibility results in this paper, including Theorems 3.1, 3.2 and 4.8. He came up with", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "the initial idea of Dynamic Programming. He and Bohang Zhang formalized the DP framework and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "proved all positive results in this paper, including Theorems 3.3, 3.4 and 4.7. He contributed to the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 254, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 254, + 153 + ], + "score": 1.0, + "content": "paper writing of Appendices C to G.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 156, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 506, + 168 + ], + "score": 1.0, + "content": "Bohang Zhang supervised all undergraduate students. He raised the the problem of linear equation.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 180 + ], + "score": 1.0, + "content": "He and Guhao Feng formalized the DP framework and proved all positive results in this paper,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 178, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 504, + 190 + ], + "score": 1.0, + "content": "including Theorems 3.3, 3.4 and 4.7. In the experimental part, he helped generate datasets with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 202 + ], + "score": 1.0, + "content": "Yuntian Gu, conducted hyper-parameter tuning, and finalized all experiments in Section 5. He was", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 444, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 444, + 212 + ], + "score": 1.0, + "content": "responsible for writing the majority of this paper and checking/correcting all proofs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "Yuntian Gu was responsible for the experimental part. He wrote the entire code, including the model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "details, training pipeline, and evaluation. He created the datasets for all four tasks in Section 5 and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 504, + 251 + ], + "score": 1.0, + "content": "conducted extensive experimental exploration during this project (with the help of Bohang Zhang).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 105, + 254, + 503, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "Haotian Ye participated in regular discussions, raised ideas, and helped check the code in experiments.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 270, + 504, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 285 + ], + "score": 1.0, + "content": "Di He initiated the problem of studying the capability of Transformers in basic tasks like evaluating", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "arithmetic expressions. 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However, both fields contain an infinite number of elements. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 484, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 507, + 496 + ], + "score": 1.0, + "content": "this paper, we consider a class of fields called finite fields, which contain a finite number of elements.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 196, + 507 + ], + "score": 1.0, + "content": "Given a prime number", + "type": "text" + }, + { + "bbox": [ + 196, + 496, + 203, + 506 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 495, + 262, + 507 + ], + "score": 1.0, + "content": ", the finite field", + "type": "text" + }, + { + "bbox": [ + 262, + 495, + 275, + 507 + ], + "score": 0.9, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 495, + 371, + 507 + ], + "score": 1.0, + "content": "is the field consisting of", + "type": "text" + }, + { + "bbox": [ + 371, + 496, + 378, + 506 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "elements, which can be denoted", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 118, + 520 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 506, + 178, + 517 + ], + "score": 0.91, + "content": "0 , 1 , \\cdots , p - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 505, + 194, + 520 + ], + "score": 1.0, + "content": ". In", + "type": "text" + }, + { + "bbox": [ + 194, + 506, + 206, + 518 + ], + "score": 0.88, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 505, + 506, + 520 + ], + "score": 1.0, + "content": ", both addition and multiplication are defined by simply adding/multiplying", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 342, + 529 + ], + "score": 1.0, + "content": "two input integers and then taking the remainder modulo", + "type": "text" + }, + { + "bbox": [ + 342, + 519, + 348, + 528 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 516, + 505, + 529 + ], + "score": 1.0, + "content": ". It can be easily checked that the two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "operations satisfy the six properties described above. Thus, subtraction and division can be defined", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 459, + 551 + ], + "score": 1.0, + "content": "accordingly. 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Both tasks of evaluating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 614, + 483, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 483, + 627 + ], + "score": 1.0, + "content": "arithmetic expressions and solving linear equations (Section 3.1) are well-defined in this field.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 212, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 213, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 213, + 655 + ], + "score": 1.0, + "content": "A.2 Circuit complexity", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "In circuit complexity theory, there are several fundamental complexity classes that capture different", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 672, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 684 + ], + "score": 1.0, + "content": "levels of computation power. Below, we provide a brief overview of these classes; however, for a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 682, + 428, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 428, + 695 + ], + "score": 1.0, + "content": "comprehensive introduction, we recommend readers refer to Arora & Barak [2].", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 384, + 713 + ], + "score": 1.0, + "content": "The basic complexity classes we will discuss in this subsection are", + "type": "text" + }, + { + "bbox": [ + 385, + 698, + 473, + 711 + ], + "score": 0.37, + "content": "\\mathsf { N C } ^ { 0 } , \\mathsf { A C } ^ { 0 } , \\mathsf { T C } ^ { 0 } , \\mathsf { N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 696, + 506, + 713 + ], + "score": 1.0, + "content": ", and P.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "These classes represent increasing levels of computation complexity. The relationships between these", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 742, + 311, + 750 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 93, + 192, + 115 + ], + "lines": [ + { + "bbox": [ + 103, + 91, + 195, + 117 + ], + "spans": [ + { + "bbox": [ + 103, + 91, + 195, + 117 + ], + "score": 1.0, + "content": "Appendix", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 506, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "The Appendix is organized as follows. Appendix A introduces additional mathematical background", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "and useful notations. Appendix B presents formal definitions and CoT solutions of the arithmetic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "expression task and the linear equation task. Appendix C gives several technical lemmas, which", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "will be frequently used in our subsequent proofs. 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It can be easily checked that the two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "operations satisfy the six properties described above. Thus, subtraction and division can be defined", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 459, + 551 + ], + "score": 1.0, + "content": "accordingly. Remarkably, a key result in abstract algebra shows that all finite fields with", + "type": "text" + }, + { + "bbox": [ + 460, + 540, + 466, + 550 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "elements", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "are isomorphic, which means that the above definitions of addition, subtraction, multiplication, and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 560, + 271, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 271, + 573 + ], + "score": 1.0, + "content": "division are unique (up to isomorphism).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 462, + 507, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 503, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 266, + 589 + ], + "score": 1.0, + "content": "As an example, consider the finite field", + "type": "text" + }, + { + "bbox": [ + 266, + 577, + 279, + 588 + ], + "score": 0.88, + "content": "\\mathbb { Z } _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 576, + 338, + 589 + ], + "score": 1.0, + "content": ". We have that", + "type": "text" + }, + { + "bbox": [ + 338, + 577, + 361, + 587 + ], + "score": 0.87, + "content": "2 + 3", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 576, + 424, + 589 + ], + "score": 1.0, + "content": "equals 0, since", + "type": "text" + }, + { + "bbox": [ + 425, + 577, + 455, + 589 + ], + "score": 0.89, + "content": "( 2 + 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 576, + 478, + 589 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 479, + 577, + 503, + 587 + ], + "score": 0.86, + "content": "5 = 0", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 353, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 147, + 600 + ], + "score": 1.0, + "content": "Similarly,", + "type": "text" + }, + { + "bbox": [ + 148, + 588, + 172, + 598 + ], + "score": 0.88, + "content": "2 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 587, + 200, + 600 + ], + "score": 1.0, + "content": "equals", + "type": "text" + }, + { + "bbox": [ + 200, + 588, + 235, + 599 + ], + "score": 0.41, + "content": "1 ; 2 - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 587, + 290, + 600 + ], + "score": 1.0, + "content": "equals 4; and", + "type": "text" + }, + { + "bbox": [ + 291, + 588, + 314, + 598 + ], + "score": 0.86, + "content": "2 \\div 3", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 587, + 353, + 600 + ], + "score": 1.0, + "content": "equals 4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 576, + 503, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 504, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 237, + 618 + ], + "score": 1.0, + "content": "In Section 3, we utilize the field", + "type": "text" + }, + { + "bbox": [ + 237, + 604, + 250, + 616 + ], + "score": 0.89, + "content": "\\mathbb { Z } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 602, + 506, + 618 + ], + "score": 1.0, + "content": "to address the issue of infinite tokens. Both tasks of evaluating", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 614, + 483, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 483, + 627 + ], + "score": 1.0, + "content": "arithmetic expressions and solving linear equations (Section 3.1) are well-defined in this field.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 602, + 506, + 627 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 212, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 213, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 213, + 655 + ], + "score": 1.0, + "content": "A.2 Circuit complexity", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "In circuit complexity theory, there are several fundamental complexity classes that capture different", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 672, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 684 + ], + "score": 1.0, + "content": "levels of computation power. Below, we provide a brief overview of these classes; however, for a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 682, + 428, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 428, + 695 + ], + "score": 1.0, + "content": "comprehensive introduction, we recommend readers refer to Arora & Barak [2].", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 660, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 384, + 713 + ], + "score": 1.0, + "content": "The basic complexity classes we will discuss in this subsection are", + "type": "text" + }, + { + "bbox": [ + 385, + 698, + 473, + 711 + ], + "score": 0.37, + "content": "\\mathsf { N C } ^ { 0 } , \\mathsf { A C } ^ { 0 } , \\mathsf { T C } ^ { 0 } , \\mathsf { N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 696, + 506, + 713 + ], + "score": 1.0, + "content": ", and P.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "These classes represent increasing levels of computation complexity. The relationships between these", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 696, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 73, + 261, + 83 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 262, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 262, + 85 + ], + "score": 1.0, + "content": "classes can be summarized as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 87, + 372, + 102 + ], + "lines": [ + { + "bbox": [ + 239, + 87, + 372, + 102 + ], + "spans": [ + { + "bbox": [ + 239, + 87, + 372, + 102 + ], + "score": 0.91, + "content": "{ \\mathsf { N C } } ^ { 0 } \\subsetneq { \\mathsf { A C } } ^ { 0 } \\subsetneq { \\mathsf { T C } } ^ { 0 } \\subset { \\mathsf { N C } } ^ { 1 } \\subset { \\mathsf { P } }", + "type": "interline_equation", + "image_path": "f402ffd96403bbdc008172fdec1fb3e89d844268b16305d35441a2bf22c0792b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 239, + 87, + 372, + 102 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 504, + 118 + ], + "score": 1.0, + "content": "Moreover, in the field of computational theory, it is widely conjectured that all subset relations in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "the hierarchy are proper subset relations. This means that each class is believed to capture a strictly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "larger set of computational problems than its predecessor in the hierarchy. However, proving some", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "of these subset relations to be proper remains a critical open question in computational complexity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 194, + 163 + ], + "score": 1.0, + "content": "theory. For example,", + "type": "text" + }, + { + "bbox": [ + 194, + 150, + 233, + 161 + ], + "score": 0.92, + "content": "{ \\mathsf { N C } } ^ { 1 } = { \\mathsf { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 150, + 279, + 163 + ], + "score": 1.0, + "content": "will imply", + "type": "text" + }, + { + "bbox": [ + 279, + 150, + 313, + 162 + ], + "score": 0.89, + "content": "{ \\mathsf { P } } = { \\mathsf { N P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 150, + 505, + 163 + ], + "score": 1.0, + "content": ", which is widely regarded as impossible but is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 317, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 317, + 173 + ], + "score": 1.0, + "content": "still a celebrated open question in computer science.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "To formally define these classes, we first introduce the concept of Boolean circuits. A Boolean circuit", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 126, + 201 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 191, + 134, + 199 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "input bits is a directed acyclic graph (DAG), in which every node is either an input bit or an", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "internal node representing one bit (also called a gate). The value of each internal node depends on its", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "score": 1.0, + "content": "direct predecessors. Furthermore, several internal nodes are designated as output nodes, representing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "the output of the Boolean circuit. The in-degree of a node is called its fan-in number, and the input", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 233, + 201, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 201, + 244 + ], + "score": 1.0, + "content": "nodes have zero fan-in.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 108, + 249, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "A Boolean circuit can only simulate a computation problem of a fixed number of input bits. When the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "input length varies, a series of distinct Boolean circuits will be required, each designed to process a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 507, + 285 + ], + "score": 1.0, + "content": "specific length. In this case, circuit complexity studies how the circuit size (e.g., depth, fan-in number,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "width) increases with respect to the input length for a given computation problem. We now describe", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 243, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 243, + 306 + ], + "score": 1.0, + "content": "each complexity class as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 127, + 312, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 126, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 126, + 309, + 136, + 324 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 309, + 155, + 322 + ], + "score": 0.72, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "is the class of constant-depth, constant-fan-in, polynomial-sized circuits consisting of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 135, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 135, + 322, + 252, + 336 + ], + "score": 1.0, + "content": "AND, OR, and NOT gates.", + "type": "text" + }, + { + "bbox": [ + 252, + 322, + 272, + 334 + ], + "score": 0.76, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "circuits is the weakest class in the above hierarchy with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 135, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 135, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "limited expressive power because they cannot express functions that depend on a growing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 135, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 135, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "number of inputs as the input size increases. For example, the basic logical-AND function", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 135, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 135, + 357, + 315, + 369 + ], + "score": 1.0, + "content": "with an arbitrary number of input bits is not in", + "type": "text" + }, + { + "bbox": [ + 316, + 356, + 335, + 368 + ], + "score": 0.81, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 357, + 505, + 369 + ], + "score": 1.0, + "content": ". In [22], the authors considered a restricted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 135, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 135, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "version of the Transformer model with constant depth and a constant-degree sparse selection", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 135, + 379, + 402, + 392 + ], + "spans": [ + { + "bbox": [ + 135, + 379, + 402, + 392 + ], + "score": 1.0, + "content": "construction, which can be characterized by this complexity class.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 126, + 391, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 126, + 391, + 136, + 407 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 391, + 155, + 404 + ], + "score": 0.8, + "content": "\\mathsf { A C } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 391, + 505, + 407 + ], + "score": 1.0, + "content": "is the class of constant-depth, unbounded-fan-in, polynomial-sized circuits consisting", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 135, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 135, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "of AND, OR, and NOT gates, with NOT gates allowed only at the inputs. It is strictly more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 134, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 134, + 415, + 192, + 430 + ], + "score": 1.0, + "content": "powerful than", + "type": "text" + }, + { + "bbox": [ + 193, + 415, + 212, + 427 + ], + "score": 0.8, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "mainly because the fan-in number can (polynomially) depend on the input", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 136, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 136, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "length. However, there are still several fundamental Boolean functions that are not in this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 136, + 438, + 464, + 451 + ], + "spans": [ + { + "bbox": [ + 136, + 438, + 464, + 451 + ], + "score": 1.0, + "content": "complexity class, such as the parity function or the majority function (see below).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 136, + 450, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 136, + 450, + 155, + 462 + ], + "score": 0.8, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 450, + 228, + 465 + ], + "score": 1.0, + "content": "is an extension of", + "type": "text" + }, + { + "bbox": [ + 228, + 450, + 247, + 462 + ], + "score": 0.84, + "content": "\\mathsf { A C } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 450, + 506, + 465 + ], + "score": 1.0, + "content": "that introduces an additional gate called MAJ (i.e., the majority).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 135, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 135, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "The MAJ gate takes an arbitrary number of input bits and evaluates to false when half or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 136, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 136, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "more of the input bits are false, and true otherwise. Previous work [42, 43] showed that the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 135, + 485, + 304, + 497 + ], + "spans": [ + { + "bbox": [ + 135, + 485, + 304, + 497 + ], + "score": 1.0, + "content": "log-precision Transformer is in this class.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 131, + 497, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 131, + 497, + 136, + 512 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 497, + 155, + 509 + ], + "score": 0.84, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 497, + 506, + 512 + ], + "score": 1.0, + "content": "is a complexity class that consists of constant-fan-in, polynomial-sized circuits with a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 135, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 135, + 510, + 220, + 523 + ], + "score": 1.0, + "content": "logarithmic depth of", + "type": "text" + }, + { + "bbox": [ + 220, + 510, + 257, + 522 + ], + "score": 0.92, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 510, + 288, + 523 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 288, + 512, + 295, + 520 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 510, + 415, + 523 + ], + "score": 1.0, + "content": "is the input length. Similar to", + "type": "text" + }, + { + "bbox": [ + 415, + 509, + 434, + 521 + ], + "score": 0.86, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ", the basic logical", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 135, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 135, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "gates are AND, OR, and NOT. Allowing the number of layers to depend on the input length", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 135, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 135, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "significantly increases the expressiveness of the circuit. On the other hand, the logarithmic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 135, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 135, + 543, + 376, + 558 + ], + "score": 1.0, + "content": "dependency still enables a descent parallelizability. Indeed,", + "type": "text" + }, + { + "bbox": [ + 376, + 543, + 395, + 555 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "is widely recognized as an", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 135, + 556, + 446, + 568 + ], + "spans": [ + { + "bbox": [ + 135, + 556, + 446, + 568 + ], + "score": 1.0, + "content": "important complexity class that captures efficiently parallelizable algorithms.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 129, + 567, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 129, + 567, + 505, + 582 + ], + "score": 1.0, + "content": "• P is the complexity class that contains problems that can be solved by a Turing machine in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 135, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 135, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "polynomial time. It contains a set of problems that do not have an efficient parallel algorithm.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 135, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 135, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "For example, the Context-Free-Grammar Membership Testing is in this class and is proved to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 135, + 600, + 218, + 614 + ], + "spans": [ + { + "bbox": [ + 135, + 600, + 218, + 614 + ], + "score": 1.0, + "content": "be P-complete [33].", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 625, + 191, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 193, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 193, + 639 + ], + "score": 1.0, + "content": "A.3 Log-precision", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "In this work, we focus on Transformers whose neuron values are restricted to be floating-point", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 656, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 504, + 669 + ], + "score": 1.0, + "content": "numbers of finite precision, and all computations operated on floating-point numbers will be finally", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "truncated, similar to how a computer processes real numbers. In practice, the two most common", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "formats to store real numbers are the fixed-point format and floating-point format (e.g., the IEEE-754", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "standard [31]). Likewise, there are several popular truncation approaches (also called rounding), such", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 104, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "as round-to-the-nearest, round-to-zero, round-up, and round-down. Our results in this paper hold for", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 710, + 304, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 304, + 723 + ], + "score": 1.0, + "content": "both formats and all these truncation approaches.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 312, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 312, + 754 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 73, + 261, + 83 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 262, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 262, + 85 + ], + "score": 1.0, + "content": "classes can be summarized as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 72, + 262, + 85 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 87, + 372, + 102 + ], + "lines": [ + { + "bbox": [ + 239, + 87, + 372, + 102 + ], + "spans": [ + { + "bbox": [ + 239, + 87, + 372, + 102 + ], + "score": 0.91, + "content": "{ \\mathsf { N C } } ^ { 0 } \\subsetneq { \\mathsf { A C } } ^ { 0 } \\subsetneq { \\mathsf { T C } } ^ { 0 } \\subset { \\mathsf { N C } } ^ { 1 } \\subset { \\mathsf { P } }", + "type": "interline_equation", + "image_path": "f402ffd96403bbdc008172fdec1fb3e89d844268b16305d35441a2bf22c0792b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 239, + 87, + 372, + 102 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 504, + 118 + ], + "score": 1.0, + "content": "Moreover, in the field of computational theory, it is widely conjectured that all subset relations in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "the hierarchy are proper subset relations. This means that each class is believed to capture a strictly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "larger set of computational problems than its predecessor in the hierarchy. However, proving some", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "of these subset relations to be proper remains a critical open question in computational complexity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 194, + 163 + ], + "score": 1.0, + "content": "theory. For example,", + "type": "text" + }, + { + "bbox": [ + 194, + 150, + 233, + 161 + ], + "score": 0.92, + "content": "{ \\mathsf { N C } } ^ { 1 } = { \\mathsf { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 150, + 279, + 163 + ], + "score": 1.0, + "content": "will imply", + "type": "text" + }, + { + "bbox": [ + 279, + 150, + 313, + 162 + ], + "score": 0.89, + "content": "{ \\mathsf { P } } = { \\mathsf { N P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 150, + 505, + 163 + ], + "score": 1.0, + "content": ", which is widely regarded as impossible but is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 317, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 317, + 173 + ], + "score": 1.0, + "content": "still a celebrated open question in computer science.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 107, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "To formally define these classes, we first introduce the concept of Boolean circuits. A Boolean circuit", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 126, + 201 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 191, + 134, + 199 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "input bits is a directed acyclic graph (DAG), in which every node is either an input bit or an", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "internal node representing one bit (also called a gate). The value of each internal node depends on its", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "score": 1.0, + "content": "direct predecessors. Furthermore, several internal nodes are designated as output nodes, representing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "the output of the Boolean circuit. The in-degree of a node is called its fan-in number, and the input", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 233, + 201, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 201, + 244 + ], + "score": 1.0, + "content": "nodes have zero fan-in.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 178, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 249, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "A Boolean circuit can only simulate a computation problem of a fixed number of input bits. When the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "input length varies, a series of distinct Boolean circuits will be required, each designed to process a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 507, + 285 + ], + "score": 1.0, + "content": "specific length. In this case, circuit complexity studies how the circuit size (e.g., depth, fan-in number,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "width) increases with respect to the input length for a given computation problem. We now describe", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 243, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 243, + 306 + ], + "score": 1.0, + "content": "each complexity class as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 249, + 507, + 306 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 312, + 506, + 613 + ], + "lines": [ + { + "bbox": [ + 126, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 126, + 309, + 136, + 324 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 309, + 155, + 322 + ], + "score": 0.72, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "is the class of constant-depth, constant-fan-in, polynomial-sized circuits consisting of", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 135, + 322, + 252, + 336 + ], + "score": 1.0, + "content": "AND, OR, and NOT gates.", + "type": "text" + }, + { + "bbox": [ + 252, + 322, + 272, + 334 + ], + "score": 0.76, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "circuits is the weakest class in the above hierarchy with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 135, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 135, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "limited expressive power because they cannot express functions that depend on a growing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 135, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 135, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "number of inputs as the input size increases. For example, the basic logical-AND function", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 135, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 135, + 357, + 315, + 369 + ], + "score": 1.0, + "content": "with an arbitrary number of input bits is not in", + "type": "text" + }, + { + "bbox": [ + 316, + 356, + 335, + 368 + ], + "score": 0.81, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 357, + 505, + 369 + ], + "score": 1.0, + "content": ". In [22], the authors considered a restricted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 135, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 135, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "version of the Transformer model with constant depth and a constant-degree sparse selection", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 135, + 379, + 402, + 392 + ], + "spans": [ + { + "bbox": [ + 135, + 379, + 402, + 392 + ], + "score": 1.0, + "content": "construction, which can be characterized by this complexity class.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 391, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 126, + 391, + 136, + 407 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 391, + 155, + 404 + ], + "score": 0.8, + "content": "\\mathsf { A C } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 391, + 505, + 407 + ], + "score": 1.0, + "content": "is the class of constant-depth, unbounded-fan-in, polynomial-sized circuits consisting", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 135, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "of AND, OR, and NOT gates, with NOT gates allowed only at the inputs. It is strictly more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 134, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 134, + 415, + 192, + 430 + ], + "score": 1.0, + "content": "powerful than", + "type": "text" + }, + { + "bbox": [ + 193, + 415, + 212, + 427 + ], + "score": 0.8, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "mainly because the fan-in number can (polynomially) depend on the input", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 136, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 136, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "length. However, there are still several fundamental Boolean functions that are not in this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 136, + 438, + 464, + 451 + ], + "spans": [ + { + "bbox": [ + 136, + 438, + 464, + 451 + ], + "score": 1.0, + "content": "complexity class, such as the parity function or the majority function (see below).", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 136, + 450, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 136, + 450, + 155, + 462 + ], + "score": 0.8, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 450, + 228, + 465 + ], + "score": 1.0, + "content": "is an extension of", + "type": "text" + }, + { + "bbox": [ + 228, + 450, + 247, + 462 + ], + "score": 0.84, + "content": "\\mathsf { A C } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 450, + 506, + 465 + ], + "score": 1.0, + "content": "that introduces an additional gate called MAJ (i.e., the majority).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 135, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 135, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "The MAJ gate takes an arbitrary number of input bits and evaluates to false when half or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 136, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 136, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "more of the input bits are false, and true otherwise. Previous work [42, 43] showed that the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 135, + 485, + 304, + 497 + ], + "spans": [ + { + "bbox": [ + 135, + 485, + 304, + 497 + ], + "score": 1.0, + "content": "log-precision Transformer is in this class.", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 497, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 131, + 497, + 136, + 512 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 497, + 155, + 509 + ], + "score": 0.84, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 497, + 506, + 512 + ], + "score": 1.0, + "content": "is a complexity class that consists of constant-fan-in, polynomial-sized circuits with a", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 135, + 510, + 220, + 523 + ], + "score": 1.0, + "content": "logarithmic depth of", + "type": "text" + }, + { + "bbox": [ + 220, + 510, + 257, + 522 + ], + "score": 0.92, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 510, + 288, + 523 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 288, + 512, + 295, + 520 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 510, + 415, + 523 + ], + "score": 1.0, + "content": "is the input length. Similar to", + "type": "text" + }, + { + "bbox": [ + 415, + 509, + 434, + 521 + ], + "score": 0.86, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 510, + 505, + 523 + ], + "score": 1.0, + "content": ", the basic logical", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 135, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 135, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "gates are AND, OR, and NOT. Allowing the number of layers to depend on the input length", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 135, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 135, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "significantly increases the expressiveness of the circuit. On the other hand, the logarithmic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 135, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 135, + 543, + 376, + 558 + ], + "score": 1.0, + "content": "dependency still enables a descent parallelizability. Indeed,", + "type": "text" + }, + { + "bbox": [ + 376, + 543, + 395, + 555 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "is widely recognized as an", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 135, + 556, + 446, + 568 + ], + "spans": [ + { + "bbox": [ + 135, + 556, + 446, + 568 + ], + "score": 1.0, + "content": "important complexity class that captures efficiently parallelizable algorithms.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 567, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 129, + 567, + 505, + 582 + ], + "score": 1.0, + "content": "• P is the complexity class that contains problems that can be solved by a Turing machine in", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 135, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "polynomial time. It contains a set of problems that do not have an efficient parallel algorithm.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 135, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 135, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "For example, the Context-Free-Grammar Membership Testing is in this class and is proved to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 135, + 600, + 218, + 614 + ], + "spans": [ + { + "bbox": [ + 135, + 600, + 218, + 614 + ], + "score": 1.0, + "content": "be P-complete [33].", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + } + ], + "index": 31.5, + "bbox_fs": [ + 126, + 309, + 506, + 614 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 625, + 191, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 193, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 193, + 639 + ], + "score": 1.0, + "content": "A.3 Log-precision", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "In this work, we focus on Transformers whose neuron values are restricted to be floating-point", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 656, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 504, + 669 + ], + "score": 1.0, + "content": "numbers of finite precision, and all computations operated on floating-point numbers will be finally", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "truncated, similar to how a computer processes real numbers. In practice, the two most common", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "formats to store real numbers are the fixed-point format and floating-point format (e.g., the IEEE-754", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "standard [31]). Likewise, there are several popular truncation approaches (also called rounding), such", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 104, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "as round-to-the-nearest, round-to-zero, round-up, and round-down. Our results in this paper hold for", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 710, + 304, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 304, + 723 + ], + "score": 1.0, + "content": "both formats and all these truncation approaches.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49, + "bbox_fs": [ + 104, + 645, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 71, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 369, + 86 + ], + "score": 1.0, + "content": "Specifically, the log-precision assumption means that we can use", + "type": "text" + }, + { + "bbox": [ + 369, + 73, + 412, + 85 + ], + "score": 0.92, + "content": "O ( \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "bits to represent a real", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 368, + 97 + ], + "score": 1.0, + "content": "number, where the length of the input sequence is bounded by", + "type": "text" + }, + { + "bbox": [ + 369, + 86, + 375, + 93 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 83, + 506, + 97 + ], + "score": 1.0, + "content": ". For any floating-point format", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 191, + 108 + ], + "score": 1.0, + "content": "described above with", + "type": "text" + }, + { + "bbox": [ + 192, + 94, + 234, + 106 + ], + "score": 0.92, + "content": "{ \\bar { O ( } } \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "bits, an important property is that it can represent all real numbers of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 151, + 119 + ], + "score": 1.0, + "content": "magnitude", + "type": "text" + }, + { + "bbox": [ + 151, + 105, + 200, + 118 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 106, + 231, + 119 + ], + "score": 1.0, + "content": "within", + "type": "text" + }, + { + "bbox": [ + 231, + 105, + 289, + 118 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( 1 / n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "truncation error. We next analyze how the truncation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "error will propagate and magnify in a log-precision Transformer from the input to the output layer.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "Note that since the functions represented by Transformers are continuous, the approximation error in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "a hidden neuron will smoothly influence the approximation error of subsequent neurons in deeper", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "layers. This impact can be bounded by the Lipschitz constant of the Transformer, which depends", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "on its basic layers. In particular, the softmax function (in attention) is 1-Lipschitz, the GeLU", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "activation is 2-Lipschitz, and the Lipschitz constant of a linear layer depends on the scale of its", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "weight parameters. Combining these together leads to the following result: given a bounded-depth", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "score": 1.0, + "content": "log-precision Transformer of polynomial size, when all parameter values of the Transformer are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 185, + 217 + ], + "score": 1.0, + "content": "further bounded by", + "type": "text" + }, + { + "bbox": [ + 185, + 203, + 234, + 216 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 203, + 362, + 217 + ], + "score": 1.0, + "content": ", all neuron values only yield an", + "type": "text" + }, + { + "bbox": [ + 363, + 203, + 421, + 216 + ], + "score": 0.93, + "content": "O ( \\mathrm { p o l y } ( 1 / n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "approximation error", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "compared to the infinite-precision counterpart. Therefore, if a problem can be solved by a bounded-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "depth polynomial-size infinite-precision Transformer with polynomially-bounded parameters, it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "can also be solved by a log-precision Transformer of the same size. This finding is helpful for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 274, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 274, + 259 + ], + "score": 1.0, + "content": "understanding Theorems 3.3, 3.4 and 4.7.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "Finally, we point out that a key property of log-precision Transformer is that each neuron can only", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 507, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 126, + 287 + ], + "score": 1.0, + "content": "hold", + "type": "text" + }, + { + "bbox": [ + 127, + 275, + 169, + 286 + ], + "score": 0.92, + "content": "O ( \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 273, + 507, + 287 + ], + "score": 1.0, + "content": "-bit information and thus cannot store the full information of the entire input sequence.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "Therefore, the log-precision assumption captures the idea that the computation must be somehow", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "distributed on each token, which well-resembles practical situations and the way Transformers work.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 328, + 325, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 325, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 325, + 344 + ], + "score": 1.0, + "content": "B Formal Definitions of CoT in Section 3", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 357, + 504, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "In this section, we will formally define the CoT derivation formats for the two math problems", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 368, + 200, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 200, + 380 + ], + "score": 1.0, + "content": "described in Section 3.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "Arithmetic expression. In an arithmetic expression that contains operators, there exists at least one", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 395, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 507, + 408 + ], + "score": 1.0, + "content": "pair of neighboring numbers connected by an operator that can be calculated, which we refer to as a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "handle. More precisely, one can represent an arithmetic expression into a (binary) syntax tree where", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "each number is a leaf node and each operator is an internal node that has two children. In this case, a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "pair of neighboring numbers is a handle if they share the same parent in the syntax tree. For instance,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 439, + 480, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 234, + 451 + ], + "score": 1.0, + "content": "consider the arithmetic formula", + "type": "text" + }, + { + "bbox": [ + 235, + 439, + 317, + 451 + ], + "score": 0.91, + "content": "7 \\times ( 6 + 5 + 4 \\times 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 440, + 346, + 451 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 347, + 440, + 370, + 450 + ], + "score": 0.87, + "content": "6 + 5", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 440, + 388, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 388, + 440, + 412, + 450 + ], + "score": 0.89, + "content": "4 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 440, + 480, + 451 + ], + "score": 1.0, + "content": "are two handles.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 426, + 468 + ], + "score": 1.0, + "content": "An important observation is that we can determine whether a pair of numbers", + "type": "text" + }, + { + "bbox": [ + 426, + 458, + 433, + 466 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 456, + 451, + 468 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 452, + 456, + 457, + 466 + ], + "score": 0.71, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "can form a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "score": 1.0, + "content": "handle with the operator op by examining the token before", + "type": "text" + }, + { + "bbox": [ + 337, + 469, + 344, + 477 + ], + "score": 0.77, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 467, + 419, + 479 + ], + "score": 1.0, + "content": "and the token after", + "type": "text" + }, + { + "bbox": [ + 420, + 467, + 425, + 477 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 467, + 505, + 479 + ], + "score": 1.0, + "content": ", where these tokens", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 507, + 491 + ], + "score": 1.0, + "content": "are either operators, brackets, or empty (i.e., approaching the beginning/ending of the sequence,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 207, + 502 + ], + "score": 1.0, + "content": "including the equal sign", + "type": "text" + }, + { + "bbox": [ + 208, + 489, + 221, + 499 + ], + "score": 0.65, + "content": "\\cdot _ { = } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 488, + 361, + 502 + ], + "score": 1.0, + "content": "). Specifically, given subsequence", + "type": "text" + }, + { + "bbox": [ + 361, + 490, + 380, + 500 + ], + "score": 0.84, + "content": "s _ { 1 } \\ a", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 488, + 397, + 502 + ], + "score": 1.0, + "content": "op", + "type": "text" + }, + { + "bbox": [ + 398, + 489, + 415, + 500 + ], + "score": 0.86, + "content": "b s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 488, + 473, + 502 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 474, + 491, + 481, + 498 + ], + "score": 0.7, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 488, + 498, + 502 + ], + "score": 1.0, + "content": "op", + "type": "text" + }, + { + "bbox": [ + 498, + 489, + 504, + 498 + ], + "score": 0.73, + "content": "b", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 334, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 334, + 511 + ], + "score": 1.0, + "content": "forms a handle iff one of the following conditions holds:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "In the proposed chain of thought (CoT), an autoregressive Transformer calculates one handle at each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 445, + 571 + ], + "score": 1.0, + "content": "step. If there are multiple handles, the leftmost handle is selected. The subsequence", + "type": "text" + }, + { + "bbox": [ + 445, + 561, + 452, + 569 + ], + "score": 0.59, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 559, + 469, + 571 + ], + "score": 1.0, + "content": "op", + "type": "text" + }, + { + "bbox": [ + 469, + 559, + 475, + 569 + ], + "score": 0.69, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "is then", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 298, + 582 + ], + "score": 1.0, + "content": "replaced by the calculated result. For the case of", + "type": "text" + }, + { + "bbox": [ + 298, + 570, + 326, + 581 + ], + "score": 0.86, + "content": "s _ { 1 } = (", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 569, + 344, + 582 + ], + "score": 1.0, + "content": "( and", + "type": "text" + }, + { + "bbox": [ + 345, + 570, + 372, + 581 + ], + "score": 0.89, + "content": "s _ { 2 } = )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 569, + 506, + 582 + ], + "score": 1.0, + "content": ", there will be a pair of redundant", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "brackets and thus the two tokens are removed. It is easy to see that the resulting sequence is still a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "valid arithmetic expression. By following this process, each CoT step reduces one operator and the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 602, + 475, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 475, + 615 + ], + "score": 1.0, + "content": "formula is gradually simplified until there is only one number left, yielding the final answer.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 366, + 631 + ], + "score": 1.0, + "content": "System of linear equations. Assume that we have a system of", + "type": "text" + }, + { + "bbox": [ + 366, + 621, + 376, + 628 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "linear equations with variables", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 631, + 168, + 641 + ], + "score": 0.84, + "content": "x _ { 1 } , x _ { 2 } , \\ldots , x _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 629, + 190, + 643 + ], + "score": 1.0, + "content": ". The", + "type": "text" + }, + { + "bbox": [ + 190, + 631, + 195, + 640 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 629, + 437, + 643 + ], + "score": 1.0, + "content": "-th equation in the input sequence is grammatically written as", + "type": "text" + }, + { + "bbox": [ + 437, + 631, + 506, + 641 + ], + "score": 0.89, + "content": "a _ { i 1 } x _ { 1 } + a _ { i 2 } x _ { 2 } +", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 107, + 641, + 181, + 652 + ], + "score": 0.89, + "content": "\\cdots + a _ { i m } x _ { m } = b _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 641, + 212, + 653 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 212, + 640, + 298, + 653 + ], + "score": 0.93, + "content": "\\bar { a } _ { i j } \\in \\{ 0 , \\cdots , \\bar { p ^ { - } } 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 641, + 317, + 653 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 640, + 399, + 652 + ], + "score": 0.92, + "content": "b _ { i } \\in \\bar { \\{ 0 , \\cdot \\cdot \\cdot , p - \\bar { 1 } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 641, + 506, + 653 + ], + "score": 1.0, + "content": ". For simplicity, we do not", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 651, + 382, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 166, + 665 + ], + "score": 1.0, + "content": "omit the token", + "type": "text" + }, + { + "bbox": [ + 167, + 654, + 180, + 664 + ], + "score": 0.86, + "content": "a _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 651, + 192, + 665 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 192, + 654, + 215, + 664 + ], + "score": 0.9, + "content": "a _ { i j } x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 651, + 327, + 665 + ], + "score": 1.0, + "content": "in the input sequence when", + "type": "text" + }, + { + "bbox": [ + 328, + 652, + 377, + 664 + ], + "score": 0.91, + "content": "a _ { i j } \\in \\{ 1 , 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 651, + 382, + 665 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 106, + 667, + 504, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 241, + 680 + ], + "score": 1.0, + "content": "We can construct the following", + "type": "text" + }, + { + "bbox": [ + 241, + 668, + 262, + 678 + ], + "score": 0.29, + "content": "\\mathrm { C o T }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "to solve the equations by using the Gaussian elimination", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 678, + 468, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 202, + 691 + ], + "score": 1.0, + "content": "algorithm. At each step", + "type": "text" + }, + { + "bbox": [ + 203, + 680, + 207, + 689 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 678, + 299, + 691 + ], + "score": 1.0, + "content": ", we select an equation", + "type": "text" + }, + { + "bbox": [ + 299, + 680, + 306, + 688 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 678, + 468, + 691 + ], + "score": 1.0, + "content": "satisfying the following two conditions:", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 127, + 695, + 317, + 723 + ], + "lines": [ + { + "bbox": [ + 127, + 693, + 267, + 708 + ], + "spans": [ + { + "bbox": [ + 127, + 693, + 209, + 708 + ], + "score": 1.0, + "content": "• The coefficient of", + "type": "text" + }, + { + "bbox": [ + 209, + 697, + 219, + 706 + ], + "score": 0.84, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 693, + 267, + 708 + ], + "score": 1.0, + "content": "is nonzero.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 126, + 708, + 318, + 725 + ], + "spans": [ + { + "bbox": [ + 126, + 708, + 213, + 725 + ], + "score": 1.0, + "content": "• The coefficients of", + "type": "text" + }, + { + "bbox": [ + 213, + 713, + 267, + 722 + ], + "score": 0.87, + "content": "x _ { 1 } , \\cdots , x _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 708, + 318, + 725 + ], + "score": 1.0, + "content": "are all zero.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 71, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 369, + 86 + ], + "score": 1.0, + "content": "Specifically, the log-precision assumption means that we can use", + "type": "text" + }, + { + "bbox": [ + 369, + 73, + 412, + 85 + ], + "score": 0.92, + "content": "O ( \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "bits to represent a real", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 368, + 97 + ], + "score": 1.0, + "content": "number, where the length of the input sequence is bounded by", + "type": "text" + }, + { + "bbox": [ + 369, + 86, + 375, + 93 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 83, + 506, + 97 + ], + "score": 1.0, + "content": ". For any floating-point format", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 191, + 108 + ], + "score": 1.0, + "content": "described above with", + "type": "text" + }, + { + "bbox": [ + 192, + 94, + 234, + 106 + ], + "score": 0.92, + "content": "{ \\bar { O ( } } \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "bits, an important property is that it can represent all real numbers of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 151, + 119 + ], + "score": 1.0, + "content": "magnitude", + "type": "text" + }, + { + "bbox": [ + 151, + 105, + 200, + 118 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 106, + 231, + 119 + ], + "score": 1.0, + "content": "within", + "type": "text" + }, + { + "bbox": [ + 231, + 105, + 289, + 118 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( 1 / n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "truncation error. We next analyze how the truncation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "error will propagate and magnify in a log-precision Transformer from the input to the output layer.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "Note that since the functions represented by Transformers are continuous, the approximation error in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "a hidden neuron will smoothly influence the approximation error of subsequent neurons in deeper", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "layers. This impact can be bounded by the Lipschitz constant of the Transformer, which depends", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "on its basic layers. In particular, the softmax function (in attention) is 1-Lipschitz, the GeLU", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "activation is 2-Lipschitz, and the Lipschitz constant of a linear layer depends on the scale of its", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "weight parameters. Combining these together leads to the following result: given a bounded-depth", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "score": 1.0, + "content": "log-precision Transformer of polynomial size, when all parameter values of the Transformer are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 185, + 217 + ], + "score": 1.0, + "content": "further bounded by", + "type": "text" + }, + { + "bbox": [ + 185, + 203, + 234, + 216 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 203, + 362, + 217 + ], + "score": 1.0, + "content": ", all neuron values only yield an", + "type": "text" + }, + { + "bbox": [ + 363, + 203, + 421, + 216 + ], + "score": 0.93, + "content": "O ( \\mathrm { p o l y } ( 1 / n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "approximation error", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "compared to the infinite-precision counterpart. Therefore, if a problem can be solved by a bounded-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "depth polynomial-size infinite-precision Transformer with polynomially-bounded parameters, it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "can also be solved by a log-precision Transformer of the same size. This finding is helpful for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 248, + 274, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 274, + 259 + ], + "score": 1.0, + "content": "understanding Theorems 3.3, 3.4 and 4.7.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 73, + 506, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "Finally, we point out that a key property of log-precision Transformer is that each neuron can only", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 507, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 126, + 287 + ], + "score": 1.0, + "content": "hold", + "type": "text" + }, + { + "bbox": [ + 127, + 275, + 169, + 286 + ], + "score": 0.92, + "content": "O ( \\log ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 273, + 507, + 287 + ], + "score": 1.0, + "content": "-bit information and thus cannot store the full information of the entire input sequence.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "Therefore, the log-precision assumption captures the idea that the computation must be somehow", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "distributed on each token, which well-resembles practical situations and the way Transformers work.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 263, + 507, + 308 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 328, + 325, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 325, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 325, + 344 + ], + "score": 1.0, + "content": "B Formal Definitions of CoT in Section 3", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 357, + 504, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "In this section, we will formally define the CoT derivation formats for the two math problems", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 368, + 200, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 200, + 380 + ], + "score": 1.0, + "content": "described in Section 3.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 356, + 505, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "Arithmetic expression. In an arithmetic expression that contains operators, there exists at least one", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 395, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 507, + 408 + ], + "score": 1.0, + "content": "pair of neighboring numbers connected by an operator that can be calculated, which we refer to as a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "handle. More precisely, one can represent an arithmetic expression into a (binary) syntax tree where", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "each number is a leaf node and each operator is an internal node that has two children. In this case, a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "pair of neighboring numbers is a handle if they share the same parent in the syntax tree. For instance,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 439, + 480, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 234, + 451 + ], + "score": 1.0, + "content": "consider the arithmetic formula", + "type": "text" + }, + { + "bbox": [ + 235, + 439, + 317, + 451 + ], + "score": 0.91, + "content": "7 \\times ( 6 + 5 + 4 \\times 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 440, + 346, + 451 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 347, + 440, + 370, + 450 + ], + "score": 0.87, + "content": "6 + 5", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 440, + 388, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 388, + 440, + 412, + 450 + ], + "score": 0.89, + "content": "4 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 440, + 480, + 451 + ], + "score": 1.0, + "content": "are two handles.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 384, + 507, + 451 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 426, + 468 + ], + "score": 1.0, + "content": "An important observation is that we can determine whether a pair of numbers", + "type": "text" + }, + { + "bbox": [ + 426, + 458, + 433, + 466 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 456, + 451, + 468 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 452, + 456, + 457, + 466 + ], + "score": 0.71, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "can form a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 337, + 479 + ], + "score": 1.0, + "content": "handle with the operator op by examining the token before", + "type": "text" + }, + { + "bbox": [ + 337, + 469, + 344, + 477 + ], + "score": 0.77, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 467, + 419, + 479 + ], + "score": 1.0, + "content": "and the token after", + "type": "text" + }, + { + "bbox": [ + 420, + 467, + 425, + 477 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 467, + 505, + 479 + ], + "score": 1.0, + "content": ", where these tokens", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 507, + 491 + ], + "score": 1.0, + "content": "are either operators, brackets, or empty (i.e., approaching the beginning/ending of the sequence,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 207, + 502 + ], + "score": 1.0, + "content": "including the equal sign", + "type": "text" + }, + { + "bbox": [ + 208, + 489, + 221, + 499 + ], + "score": 0.65, + "content": "\\cdot _ { = } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 488, + 361, + 502 + ], + "score": 1.0, + "content": "). Specifically, given subsequence", + "type": "text" + }, + { + "bbox": [ + 361, + 490, + 380, + 500 + ], + "score": 0.84, + "content": "s _ { 1 } \\ a", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 488, + 397, + 502 + ], + "score": 1.0, + "content": "op", + "type": "text" + }, + { + "bbox": [ + 398, + 489, + 415, + 500 + ], + "score": 0.86, + "content": "b s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 488, + 473, + 502 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 474, + 491, + 481, + 498 + ], + "score": 0.7, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 488, + 498, + 502 + ], + "score": 1.0, + "content": "op", + "type": "text" + }, + { + "bbox": [ + 498, + 489, + 504, + 498 + ], + "score": 0.73, + "content": "b", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 334, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 334, + 511 + ], + "score": 1.0, + "content": "forms a handle iff one of the following conditions holds:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 456, + 507, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "In the proposed chain of thought (CoT), an autoregressive Transformer calculates one handle at each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 445, + 571 + ], + "score": 1.0, + "content": "step. If there are multiple handles, the leftmost handle is selected. The subsequence", + "type": "text" + }, + { + "bbox": [ + 445, + 561, + 452, + 569 + ], + "score": 0.59, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 559, + 469, + 571 + ], + "score": 1.0, + "content": "op", + "type": "text" + }, + { + "bbox": [ + 469, + 559, + 475, + 569 + ], + "score": 0.69, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "is then", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 298, + 582 + ], + "score": 1.0, + "content": "replaced by the calculated result. For the case of", + "type": "text" + }, + { + "bbox": [ + 298, + 570, + 326, + 581 + ], + "score": 0.86, + "content": "s _ { 1 } = (", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 569, + 344, + 582 + ], + "score": 1.0, + "content": "( and", + "type": "text" + }, + { + "bbox": [ + 345, + 570, + 372, + 581 + ], + "score": 0.89, + "content": "s _ { 2 } = )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 569, + 506, + 582 + ], + "score": 1.0, + "content": ", there will be a pair of redundant", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "brackets and thus the two tokens are removed. It is easy to see that the resulting sequence is still a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "valid arithmetic expression. By following this process, each CoT step reduces one operator and the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 602, + 475, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 475, + 615 + ], + "score": 1.0, + "content": "formula is gradually simplified until there is only one number left, yielding the final answer.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 547, + 506, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 366, + 631 + ], + "score": 1.0, + "content": "System of linear equations. Assume that we have a system of", + "type": "text" + }, + { + "bbox": [ + 366, + 621, + 376, + 628 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "linear equations with variables", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 631, + 168, + 641 + ], + "score": 0.84, + "content": "x _ { 1 } , x _ { 2 } , \\ldots , x _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 629, + 190, + 643 + ], + "score": 1.0, + "content": ". The", + "type": "text" + }, + { + "bbox": [ + 190, + 631, + 195, + 640 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 629, + 437, + 643 + ], + "score": 1.0, + "content": "-th equation in the input sequence is grammatically written as", + "type": "text" + }, + { + "bbox": [ + 437, + 631, + 506, + 641 + ], + "score": 0.89, + "content": "a _ { i 1 } x _ { 1 } + a _ { i 2 } x _ { 2 } +", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 107, + 641, + 181, + 652 + ], + "score": 0.89, + "content": "\\cdots + a _ { i m } x _ { m } = b _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 641, + 212, + 653 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 212, + 640, + 298, + 653 + ], + "score": 0.93, + "content": "\\bar { a } _ { i j } \\in \\{ 0 , \\cdots , \\bar { p ^ { - } } 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 641, + 317, + 653 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 640, + 399, + 652 + ], + "score": 0.92, + "content": "b _ { i } \\in \\bar { \\{ 0 , \\cdot \\cdot \\cdot , p - \\bar { 1 } \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 641, + 506, + 653 + ], + "score": 1.0, + "content": ". For simplicity, we do not", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 651, + 382, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 166, + 665 + ], + "score": 1.0, + "content": "omit the token", + "type": "text" + }, + { + "bbox": [ + 167, + 654, + 180, + 664 + ], + "score": 0.86, + "content": "a _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 651, + 192, + 665 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 192, + 654, + 215, + 664 + ], + "score": 0.9, + "content": "a _ { i j } x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 651, + 327, + 665 + ], + "score": 1.0, + "content": "in the input sequence when", + "type": "text" + }, + { + "bbox": [ + 328, + 652, + 377, + 664 + ], + "score": 0.91, + "content": "a _ { i j } \\in \\{ 1 , 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 651, + 382, + 665 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 618, + 506, + 665 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 667, + 504, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 241, + 680 + ], + "score": 1.0, + "content": "We can construct the following", + "type": "text" + }, + { + "bbox": [ + 241, + 668, + 262, + 678 + ], + "score": 0.29, + "content": "\\mathrm { C o T }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "to solve the equations by using the Gaussian elimination", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 678, + 468, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 202, + 691 + ], + "score": 1.0, + "content": "algorithm. At each step", + "type": "text" + }, + { + "bbox": [ + 203, + 680, + 207, + 689 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 678, + 299, + 691 + ], + "score": 1.0, + "content": ", we select an equation", + "type": "text" + }, + { + "bbox": [ + 299, + 680, + 306, + 688 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 678, + 468, + 691 + ], + "score": 1.0, + "content": "satisfying the following two conditions:", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 667, + 505, + 691 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 695, + 317, + 723 + ], + "lines": [ + { + "bbox": [ + 127, + 693, + 267, + 708 + ], + "spans": [ + { + "bbox": [ + 127, + 693, + 209, + 708 + ], + "score": 1.0, + "content": "• The coefficient of", + "type": "text" + }, + { + "bbox": [ + 209, + 697, + 219, + 706 + ], + "score": 0.84, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 693, + 267, + 708 + ], + "score": 1.0, + "content": "is nonzero.", + "type": "text" + } + ], + "index": 47, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 708, + 318, + 725 + ], + "spans": [ + { + "bbox": [ + 126, + 708, + 213, + 725 + ], + "score": 1.0, + "content": "• The coefficients of", + "type": "text" + }, + { + "bbox": [ + 213, + 713, + 267, + 722 + ], + "score": 0.87, + "content": "x _ { 1 } , \\cdots , x _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 708, + 318, + 725 + ], + "score": 1.0, + "content": "are all zero.", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 47.5, + "bbox_fs": [ + 126, + 693, + 318, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Such an equation must exist, otherwise the solution is not unique or does not exist. If there are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 376, + 95 + ], + "score": 1.0, + "content": "multiple equations satisfying the above conditions, we choose the", + "type": "text" + }, + { + "bbox": [ + 376, + 84, + 383, + 93 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "-th equation with the smallest", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 131, + 108 + ], + "score": 1.0, + "content": "index", + "type": "text" + }, + { + "bbox": [ + 131, + 95, + 137, + 104 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 93, + 265, + 108 + ], + "score": 1.0, + "content": ". We then swap it with equation", + "type": "text" + }, + { + "bbox": [ + 266, + 95, + 270, + 104 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 93, + 317, + 108 + ], + "score": 1.0, + "content": ", so that the", + "type": "text" + }, + { + "bbox": [ + 317, + 96, + 322, + 104 + ], + "score": 0.79, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "-th equation now satisfy the above conditions.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 231, + 124 + ], + "score": 1.0, + "content": "We then eliminate the variable", + "type": "text" + }, + { + "bbox": [ + 231, + 113, + 241, + 122 + ], + "score": 0.85, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 110, + 425, + 124 + ], + "score": 1.0, + "content": "in all other equations by leveraging equation", + "type": "text" + }, + { + "bbox": [ + 425, + 113, + 430, + 121 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 110, + 505, + 124 + ], + "score": 1.0, + "content": ". Formally, denote", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 121, + 246, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 121, + 135 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 123, + 126, + 132 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 121, + 199, + 135 + ], + "score": 1.0, + "content": "-th equation at the", + "type": "text" + }, + { + "bbox": [ + 200, + 123, + 204, + 131 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 121, + 246, + 135 + ], + "score": 1.0, + "content": "-th step as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 135, + 390, + 152 + ], + "lines": [ + { + "bbox": [ + 220, + 135, + 390, + 152 + ], + "spans": [ + { + "bbox": [ + 220, + 135, + 390, + 152 + ], + "score": 0.92, + "content": "a _ { i i } ^ { ( i ) } x _ { i } + a _ { i , i + 1 } ^ { ( i ) } x _ { i + 1 } + \\cdot \\cdot \\cdot + a _ { i m } ^ { ( i ) } x _ { m } = b _ { i } ,", + "type": "interline_equation", + "image_path": "524977f895ba1e986548aff07ba4b928b61c179f48e01526d8acc8b1f1aa9007.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 220, + 135, + 390, + 152 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 155, + 506, + 222 + ], + "lines": [ + { + "bbox": [ + 103, + 153, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 103, + 153, + 405, + 172 + ], + "score": 1.0, + "content": "and denote the coefficient of xi in the j-th equation (j ̸= i) as a(i)ji .", + "type": "text" + }, + { + "bbox": [ + 401, + 156, + 505, + 171 + ], + "score": 1.0, + "content": "We can multiply (7) by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 169, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 163, + 187 + ], + "score": 0.94, + "content": "- ( a _ { i i } ^ { ( i ) } ) ^ { - 1 } a _ { j i } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 172, + 322, + 185 + ], + "score": 1.0, + "content": "and add the resulting equation to the", + "type": "text" + }, + { + "bbox": [ + 323, + 173, + 329, + 185 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "-th equation. This will eliminate the term", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 117, + 199 + ], + "score": 0.83, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 183, + 146, + 206 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 146, + 189, + 152, + 200 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 183, + 339, + 206 + ], + "score": 1.0, + "content": "-th equation. We further normalize equation", + "type": "text" + }, + { + "bbox": [ + 339, + 189, + 344, + 198 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 183, + 439, + 206 + ], + "score": 1.0, + "content": "so that the coefficient", + "type": "text" + }, + { + "bbox": [ + 439, + 186, + 455, + 201 + ], + "score": 0.92, + "content": "a _ { i i } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 183, + 506, + 206 + ], + "score": 1.0, + "content": "becomes 1.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 204, + 213 + ], + "score": 1.0, + "content": "Depending on whether", + "type": "text" + }, + { + "bbox": [ + 205, + 200, + 231, + 211 + ], + "score": 0.91, + "content": "j \\ \\leq \\ i", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 198, + 245, + 213 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 245, + 200, + 271, + 211 + ], + "score": 0.92, + "content": "j > i", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 198, + 506, + 213 + ], + "score": 1.0, + "content": ", the resulting equation in the CoT output will have the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 225, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 225, + 223 + ], + "score": 1.0, + "content": "following grammatical form:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 506, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 256, + 272 + ], + "score": 1.0, + "content": "Note that we remove all zero terms", + "type": "text" + }, + { + "bbox": [ + 256, + 260, + 282, + 271 + ], + "score": 0.92, + "content": "\\tilde { a } _ { j k } x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 258, + 299, + 272 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 299, + 259, + 356, + 271 + ], + "score": 0.92, + "content": "k \\leq i , k \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "in the CoT output and also remove", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 187, + 282 + ], + "score": 1.0, + "content": "the coefficient 1 in", + "type": "text" + }, + { + "bbox": [ + 187, + 271, + 214, + 281 + ], + "score": 0.91, + "content": "\\tilde { a } _ { k k } x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 270, + 231, + 282 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 231, + 271, + 257, + 281 + ], + "score": 0.9, + "content": "k \\leq i", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 270, + 505, + 282 + ], + "score": 1.0, + "content": ", similar to how human write solutions (see Figure 1 for an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 476, + 294 + ], + "score": 1.0, + "content": "illustration). However, to simplify our proof, we reserve the coefficient 0 or 1 (i.e., outputting", + "type": "text" + }, + { + "bbox": [ + 477, + 281, + 493, + 292 + ], + "score": 0.89, + "content": "0 x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 123, + 303 + ], + "score": 0.86, + "content": "1 \\boldsymbol { x } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 291, + 151, + 305 + ], + "score": 1.0, + "content": ") when", + "type": "text" + }, + { + "bbox": [ + 152, + 292, + 175, + 302 + ], + "score": 0.9, + "content": "k > i", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "since it cannot be determined easily before computing the coefficient. The above", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 436, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 198, + 316 + ], + "score": 1.0, + "content": "process is repeated for", + "type": "text" + }, + { + "bbox": [ + 198, + 303, + 226, + 313 + ], + "score": 0.9, + "content": "m - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 303, + 436, + 316 + ], + "score": 1.0, + "content": "steps, and after the final step we obtain the solution.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 328, + 224, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 226, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 226, + 345 + ], + "score": 1.0, + "content": "C Technical Lemmas", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 352, + 248, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 249, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 249, + 365 + ], + "score": 1.0, + "content": "C.1 Technical lemmas for MLP", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 387 + ], + "score": 1.0, + "content": "In this subsection, we will demonstrate the representation efficiency of two-layer MLPs in performing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "several basic operations, such as multiplication, linear transformation, conditional selection, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 394, + 470, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 470, + 407 + ], + "score": 1.0, + "content": "look-up table. These operations will serve as building blocks in performing complex tasks.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 410, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "We first show that a two-layer MLP with GeLU activation can efficiently approximate the scalar", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 422, + 483, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 283, + 435 + ], + "score": 1.0, + "content": "multiplication, with all weights bounded by", + "type": "text" + }, + { + "bbox": [ + 283, + 422, + 339, + 434 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 422, + 368, + 435 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 368, + 424, + 374, + 432 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 422, + 483, + 435 + ], + "score": 1.0, + "content": "is the approximation error.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 178, + 448 + ], + "score": 1.0, + "content": "Lemma C.1. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 434, + 228, + 447 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "be a two-layer MLP with GeLU activation, and the hidden dimension", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 182, + 459 + ], + "score": 1.0, + "content": "is 4. Then, for any", + "type": "text" + }, + { + "bbox": [ + 182, + 447, + 206, + 457 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 446, + 225, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 447, + 255, + 457 + ], + "score": 0.89, + "content": "M > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 446, + 392, + 459 + ], + "score": 1.0, + "content": ", there exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 392, + 447, + 405, + 457 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "norm upper bounded by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 457, + 414, + 470 + ], + "spans": [ + { + "bbox": [ + 107, + 457, + 177, + 469 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 457, + 219, + 470 + ], + "score": 1.0, + "content": ") such that", + "type": "text" + }, + { + "bbox": [ + 219, + 457, + 291, + 469 + ], + "score": 0.91, + "content": "| f ( a , b ) - a b | \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 457, + 344, + 470 + ], + "score": 1.0, + "content": "holds for all", + "type": "text" + }, + { + "bbox": [ + 345, + 457, + 410, + 469 + ], + "score": 0.92, + "content": "a , b \\in [ - M , M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 457, + 414, + 470 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 295, + 493 + ], + "score": 1.0, + "content": "Proof. Denote the input vector to the MLP as", + "type": "text" + }, + { + "bbox": [ + 296, + 480, + 342, + 492 + ], + "score": 0.92, + "content": "( a , b ) \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 478, + 506, + 493 + ], + "score": 1.0, + "content": ". After the first linear layer, it is easy to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 325, + 504 + ], + "score": 1.0, + "content": "construct a weight matrix such that the hidden vector is", + "type": "text" + }, + { + "bbox": [ + 325, + 491, + 475, + 504 + ], + "score": 0.9, + "content": "\\underline { { 1 } } ( a + b , - a - b , a - b , - a + b ) \\in \\mathbb { R } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 491, + 505, + 504 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 503, + 114, + 512 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 501, + 249, + 515 + ], + "score": 1.0, + "content": "is an arbitrary scaling factor. Let", + "type": "text" + }, + { + "bbox": [ + 249, + 504, + 257, + 512 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "be the GeLU activation. We can similarly construct a weight", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 293, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 293, + 525 + ], + "score": 1.0, + "content": "vector such that the final output of the MLP is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 526, + 472, + 555 + ], + "lines": [ + { + "bbox": [ + 138, + 526, + 472, + 555 + ], + "spans": [ + { + "bbox": [ + 138, + 526, + 472, + 555 + ], + "score": 0.92, + "content": "f ( a , b ) = \\frac { \\sqrt { 2 \\pi } \\lambda ^ { 2 } } { 8 } \\left( \\sigma \\left( \\frac { a + b } { \\lambda } \\right) + \\sigma \\left( \\frac { - a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { - a + b } { \\lambda } \\right) \\right) .", + "type": "interline_equation", + "image_path": "df7d705ab6a45c8ec4e901b9995b48ac897c2abec6099a770838dfeee705d761.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 138, + 526, + 472, + 535.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 138, + 535.6666666666666, + 472, + 545.3333333333333 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 138, + 545.3333333333333, + 472, + 554.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 438, + 568 + ], + "score": 1.0, + "content": "We will prove that the above MLP satisfies the theorem by picking an appropriate", + "type": "text" + }, + { + "bbox": [ + 438, + 556, + 445, + 566 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 555, + 505, + 568 + ], + "score": 1.0, + "content": ". By definition", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 191, + 579 + ], + "score": 1.0, + "content": "of GeLU activation,", + "type": "text" + }, + { + "bbox": [ + 191, + 567, + 253, + 579 + ], + "score": 0.93, + "content": "\\sigma ( x ) = x \\Phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 567, + 282, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 282, + 567, + 304, + 578 + ], + "score": 0.91, + "content": "\\Phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "is the standard Gaussian cumulative distribution", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 201, + 592 + ], + "score": 1.0, + "content": "function. We thus have", + "type": "text" + }, + { + "bbox": [ + 202, + 579, + 250, + 591 + ], + "score": 0.92, + "content": "\\sigma ^ { \\prime } ( 0 ) = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 578, + 269, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 578, + 333, + 592 + ], + "score": 0.93, + "content": "\\sigma ^ { \\prime \\prime } ( 0 ) = \\sqrt { 2 / \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 578, + 505, + 592 + ], + "score": 1.0, + "content": ". Applying Taylor’s formula and assuming", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 589, + 182, + 603 + ], + "spans": [ + { + "bbox": [ + 107, + 590, + 142, + 601 + ], + "score": 0.9, + "content": "\\lambda > 2 M", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 589, + 182, + 603 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 601, + 472, + 722 + ], + "lines": [ + { + "bbox": [ + 137, + 601, + 472, + 722 + ], + "spans": [ + { + "bbox": [ + 137, + 601, + 472, + 722 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\left| \\sigma \\left( \\frac { a + b } { \\lambda } \\right) + \\sigma \\left( \\frac { - a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { - a + b } { \\lambda } \\right) - \\frac { 8 a b } { \\sqrt { 2 \\pi } \\lambda ^ { 2 } } \\right| } \\\\ & { \\le \\left| \\frac { 1 } { 2 } \\sqrt { \\frac { 2 } { \\pi } } \\left( \\left( \\frac { a + b } { \\lambda } \\right) ^ { 2 } + \\left( \\frac { - a - b } { \\lambda } \\right) ^ { 2 } - \\left( \\frac { a - b } { \\lambda } \\right) ^ { 2 } - \\left( \\frac { - a + b } { \\lambda } \\right) ^ { 2 } \\right) - \\frac { 8 a b } { \\sqrt { 2 \\pi } \\lambda ^ { 2 } } \\right| } \\\\ & { \\qquad + \\frac { 4 } { 3 ! } \\frac { ( 2 M ) ^ { 3 } } { \\lambda ^ { 3 } } \\left| \\underset { x \\in [ - 1 , 1 ] } { \\operatorname* { m a x } } \\sigma ^ { ( 3 ) } ( x ) \\right| } \\\\ & { = \\frac { 1 6 M ^ { 3 } } { 3 \\lambda ^ { 3 } } \\underset { x \\in [ - 1 , 1 ] } { \\operatorname* { m a x } } \\frac { 1 } { \\sqrt { 2 \\pi } } ( x ^ { 3 } - 4 x ) \\exp ( - \\frac { x ^ { 2 } } { 2 } ) < \\frac { 8 0 M ^ { 3 } } { 3 \\sqrt { 2 \\pi } \\lambda ^ { 3 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "9cb10dc7db3665ebc6645f7e379dfa7ddaab2403721a14cca497f1a045e39b82.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 137, + 601, + 472, + 641.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 137, + 641.3333333333334, + 472, + 681.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 137, + 681.6666666666667, + 472, + 722.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 312, + 755 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 312, + 755 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Such an equation must exist, otherwise the solution is not unique or does not exist. If there are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 376, + 95 + ], + "score": 1.0, + "content": "multiple equations satisfying the above conditions, we choose the", + "type": "text" + }, + { + "bbox": [ + 376, + 84, + 383, + 93 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "-th equation with the smallest", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 131, + 108 + ], + "score": 1.0, + "content": "index", + "type": "text" + }, + { + "bbox": [ + 131, + 95, + 137, + 104 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 93, + 265, + 108 + ], + "score": 1.0, + "content": ". 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This will eliminate the term", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 117, + 199 + ], + "score": 0.83, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 183, + 146, + 206 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 146, + 189, + 152, + 200 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 183, + 339, + 206 + ], + "score": 1.0, + "content": "-th equation. We further normalize equation", + "type": "text" + }, + { + "bbox": [ + 339, + 189, + 344, + 198 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 183, + 439, + 206 + ], + "score": 1.0, + "content": "so that the coefficient", + "type": "text" + }, + { + "bbox": [ + 439, + 186, + 455, + 201 + ], + "score": 0.92, + "content": "a _ { i i } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 183, + 506, + 206 + ], + "score": 1.0, + "content": "becomes 1.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 204, + 213 + ], + "score": 1.0, + "content": "Depending on whether", + "type": "text" + }, + { + "bbox": [ + 205, + 200, + 231, + 211 + ], + "score": 0.91, + "content": "j \\ \\leq \\ i", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 198, + 245, + 213 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 245, + 200, + 271, + 211 + ], + "score": 0.92, + "content": "j > i", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 198, + 506, + 213 + ], + "score": 1.0, + "content": ", the resulting equation in the CoT output will have the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 225, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 225, + 223 + ], + "score": 1.0, + "content": "following grammatical form:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 103, + 153, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 506, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 256, + 272 + ], + "score": 1.0, + "content": "Note that we remove all zero terms", + "type": "text" + }, + { + "bbox": [ + 256, + 260, + 282, + 271 + ], + "score": 0.92, + "content": "\\tilde { a } _ { j k } x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 258, + 299, + 272 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 299, + 259, + 356, + 271 + ], + "score": 0.92, + "content": "k \\leq i , k \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "in the CoT output and also remove", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 187, + 282 + ], + "score": 1.0, + "content": "the coefficient 1 in", + "type": "text" + }, + { + "bbox": [ + 187, + 271, + 214, + 281 + ], + "score": 0.91, + "content": "\\tilde { a } _ { k k } x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 270, + 231, + 282 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 231, + 271, + 257, + 281 + ], + "score": 0.9, + "content": "k \\leq i", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 270, + 505, + 282 + ], + "score": 1.0, + "content": ", similar to how human write solutions (see Figure 1 for an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 476, + 294 + ], + "score": 1.0, + "content": "illustration). However, to simplify our proof, we reserve the coefficient 0 or 1 (i.e., outputting", + "type": "text" + }, + { + "bbox": [ + 477, + 281, + 493, + 292 + ], + "score": 0.89, + "content": "0 x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 123, + 303 + ], + "score": 0.86, + "content": "1 \\boldsymbol { x } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 291, + 151, + 305 + ], + "score": 1.0, + "content": ") when", + "type": "text" + }, + { + "bbox": [ + 152, + 292, + 175, + 302 + ], + "score": 0.9, + "content": "k > i", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "since it cannot be determined easily before computing the coefficient. The above", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 436, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 198, + 316 + ], + "score": 1.0, + "content": "process is repeated for", + "type": "text" + }, + { + "bbox": [ + 198, + 303, + 226, + 313 + ], + "score": 0.9, + "content": "m - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 303, + 436, + 316 + ], + "score": 1.0, + "content": "steps, and after the final step we obtain the solution.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 258, + 506, + 316 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 328, + 224, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 226, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 226, + 345 + ], + "score": 1.0, + "content": "C Technical Lemmas", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 352, + 248, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 249, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 249, + 365 + ], + "score": 1.0, + "content": "C.1 Technical lemmas for MLP", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 387 + ], + "score": 1.0, + "content": "In this subsection, we will demonstrate the representation efficiency of two-layer MLPs in performing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "several basic operations, such as multiplication, linear transformation, conditional selection, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 394, + 470, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 470, + 407 + ], + "score": 1.0, + "content": "look-up table. These operations will serve as building blocks in performing complex tasks.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 371, + 505, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 410, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "We first show that a two-layer MLP with GeLU activation can efficiently approximate the scalar", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 422, + 483, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 283, + 435 + ], + "score": 1.0, + "content": "multiplication, with all weights bounded by", + "type": "text" + }, + { + "bbox": [ + 283, + 422, + 339, + 434 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 422, + 368, + 435 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 368, + 424, + 374, + 432 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 422, + 483, + 435 + ], + "score": 1.0, + "content": "is the approximation error.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 106, + 411, + 505, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 178, + 448 + ], + "score": 1.0, + "content": "Lemma C.1. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 434, + 228, + 447 + ], + "score": 0.91, + "content": "f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "be a two-layer MLP with GeLU activation, and the hidden dimension", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 182, + 459 + ], + "score": 1.0, + "content": "is 4. Then, for any", + "type": "text" + }, + { + "bbox": [ + 182, + 447, + 206, + 457 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 446, + 225, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 447, + 255, + 457 + ], + "score": 0.89, + "content": "M > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 446, + 392, + 459 + ], + "score": 1.0, + "content": ", there exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 392, + 447, + 405, + 457 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "norm upper bounded by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 457, + 414, + 470 + ], + "spans": [ + { + "bbox": [ + 107, + 457, + 177, + 469 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 457, + 219, + 470 + ], + "score": 1.0, + "content": ") such that", + "type": "text" + }, + { + "bbox": [ + 219, + 457, + 291, + 469 + ], + "score": 0.91, + "content": "| f ( a , b ) - a b | \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 457, + 344, + 470 + ], + "score": 1.0, + "content": "holds for all", + "type": "text" + }, + { + "bbox": [ + 345, + 457, + 410, + 469 + ], + "score": 0.92, + "content": "a , b \\in [ - M , M ]", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 457, + 414, + 470 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 434, + 505, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 295, + 493 + ], + "score": 1.0, + "content": "Proof. Denote the input vector to the MLP as", + "type": "text" + }, + { + "bbox": [ + 296, + 480, + 342, + 492 + ], + "score": 0.92, + "content": "( a , b ) \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 478, + 506, + 493 + ], + "score": 1.0, + "content": ". After the first linear layer, it is easy to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 325, + 504 + ], + "score": 1.0, + "content": "construct a weight matrix such that the hidden vector is", + "type": "text" + }, + { + "bbox": [ + 325, + 491, + 475, + 504 + ], + "score": 0.9, + "content": "\\underline { { 1 } } ( a + b , - a - b , a - b , - a + b ) \\in \\mathbb { R } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 491, + 505, + 504 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 107, + 503, + 114, + 512 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 501, + 249, + 515 + ], + "score": 1.0, + "content": "is an arbitrary scaling factor. Let", + "type": "text" + }, + { + "bbox": [ + 249, + 504, + 257, + 512 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "be the GeLU activation. We can similarly construct a weight", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 293, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 293, + 525 + ], + "score": 1.0, + "content": "vector such that the final output of the MLP is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 478, + 506, + 525 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 526, + 472, + 555 + ], + "lines": [ + { + "bbox": [ + 138, + 526, + 472, + 555 + ], + "spans": [ + { + "bbox": [ + 138, + 526, + 472, + 555 + ], + "score": 0.92, + "content": "f ( a , b ) = \\frac { \\sqrt { 2 \\pi } \\lambda ^ { 2 } } { 8 } \\left( \\sigma \\left( \\frac { a + b } { \\lambda } \\right) + \\sigma \\left( \\frac { - a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { - a + b } { \\lambda } \\right) \\right) .", + "type": "interline_equation", + "image_path": "df7d705ab6a45c8ec4e901b9995b48ac897c2abec6099a770838dfeee705d761.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 138, + 526, + 472, + 535.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 138, + 535.6666666666666, + 472, + 545.3333333333333 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 138, + 545.3333333333333, + 472, + 554.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 438, + 568 + ], + "score": 1.0, + "content": "We will prove that the above MLP satisfies the theorem by picking an appropriate", + "type": "text" + }, + { + "bbox": [ + 438, + 556, + 445, + 566 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 555, + 505, + 568 + ], + "score": 1.0, + "content": ". By definition", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 191, + 579 + ], + "score": 1.0, + "content": "of GeLU activation,", + "type": "text" + }, + { + "bbox": [ + 191, + 567, + 253, + 579 + ], + "score": 0.93, + "content": "\\sigma ( x ) = x \\Phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 567, + 282, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 282, + 567, + 304, + 578 + ], + "score": 0.91, + "content": "\\Phi ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "is the standard Gaussian cumulative distribution", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 201, + 592 + ], + "score": 1.0, + "content": "function. We thus have", + "type": "text" + }, + { + "bbox": [ + 202, + 579, + 250, + 591 + ], + "score": 0.92, + "content": "\\sigma ^ { \\prime } ( 0 ) = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 578, + 269, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 578, + 333, + 592 + ], + "score": 0.93, + "content": "\\sigma ^ { \\prime \\prime } ( 0 ) = \\sqrt { 2 / \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 578, + 505, + 592 + ], + "score": 1.0, + "content": ". Applying Taylor’s formula and assuming", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 589, + 182, + 603 + ], + "spans": [ + { + "bbox": [ + 107, + 590, + 142, + 601 + ], + "score": 0.9, + "content": "\\lambda > 2 M", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 589, + 182, + 603 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 555, + 505, + 603 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 601, + 472, + 722 + ], + "lines": [ + { + "bbox": [ + 137, + 601, + 472, + 722 + ], + "spans": [ + { + "bbox": [ + 137, + 601, + 472, + 722 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\left| \\sigma \\left( \\frac { a + b } { \\lambda } \\right) + \\sigma \\left( \\frac { - a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { a - b } { \\lambda } \\right) - \\sigma \\left( \\frac { - a + b } { \\lambda } \\right) - \\frac { 8 a b } { \\sqrt { 2 \\pi } \\lambda ^ { 2 } } \\right| } \\\\ & { \\le \\left| \\frac { 1 } { 2 } \\sqrt { \\frac { 2 } { \\pi } } \\left( \\left( \\frac { a + b } { \\lambda } \\right) ^ { 2 } + \\left( \\frac { - a - b } { \\lambda } \\right) ^ { 2 } - \\left( \\frac { a - b } { \\lambda } \\right) ^ { 2 } - \\left( \\frac { - a + b } { \\lambda } \\right) ^ { 2 } \\right) - \\frac { 8 a b } { \\sqrt { 2 \\pi } \\lambda ^ { 2 } } \\right| } \\\\ & { \\qquad + \\frac { 4 } { 3 ! } \\frac { ( 2 M ) ^ { 3 } } { \\lambda ^ { 3 } } \\left| \\underset { x \\in [ - 1 , 1 ] } { \\operatorname* { m a x } } \\sigma ^ { ( 3 ) } ( x ) \\right| } \\\\ & { = \\frac { 1 6 M ^ { 3 } } { 3 \\lambda ^ { 3 } } \\underset { x \\in [ - 1 , 1 ] } { \\operatorname* { m a x } } \\frac { 1 } { \\sqrt { 2 \\pi } } ( x ^ { 3 } - 4 x ) \\exp ( - \\frac { x ^ { 2 } } { 2 } ) < \\frac { 8 0 M ^ { 3 } } { 3 \\sqrt { 2 \\pi } \\lambda ^ { 3 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "9cb10dc7db3665ebc6645f7e379dfa7ddaab2403721a14cca497f1a045e39b82.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 137, + 601, + 472, + 641.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 137, + 641.3333333333334, + 472, + 681.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 137, + 681.6666666666667, + 472, + 722.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 70, + 507, + 97 + ], + "lines": [ + { + "bbox": [ + 105, + 69, + 507, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 147, + 88 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 148, + 70, + 235, + 86 + ], + "score": 0.92, + "content": "\\begin{array} { r } { | f ( a , b ) - a b | < \\frac { 1 0 M ^ { 3 } } { 3 \\lambda } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 69, + 253, + 88 + ], + "score": 1.0, + "content": ". Set", + "type": "text" + }, + { + "bbox": [ + 254, + 70, + 296, + 86 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\lambda \\ge \\frac { 1 0 M ^ { 3 } } { 3 \\epsilon } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 69, + 390, + 88 + ], + "score": 1.0, + "content": ", and then we can obtain", + "type": "text" + }, + { + "bbox": [ + 390, + 73, + 460, + 85 + ], + "score": 0.92, + "content": "| f ( a , b ) - a b | < \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 69, + 507, + 88 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 317, + 98 + ], + "score": 1.0, + "content": "each weight element in the MLP is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 317, + 85, + 344, + 97 + ], + "score": 0.92, + "content": "O ( \\lambda ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 84, + 411, + 98 + ], + "score": 1.0, + "content": ", which is clearly", + "type": "text" + }, + { + "bbox": [ + 412, + 85, + 482, + 97 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 84, + 505, + 98 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 109, + 506, + 143 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 506, + 123 + ], + "score": 1.0, + "content": "Next, we will demonstrate that a two-layer MLP with GeLU activation can efficiently approximate a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 398, + 134 + ], + "score": 1.0, + "content": "two-layer MLP with ReLU activation, with all weights upper bounded by", + "type": "text" + }, + { + "bbox": [ + 399, + 120, + 455, + 133 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 119, + 506, + 134 + ], + "score": 1.0, + "content": ". This result", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 269, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 269, + 145 + ], + "score": 1.0, + "content": "is useful in proving subsequent lemmas.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 181, + 159 + ], + "score": 1.0, + "content": "Lemma C.2. Let", + "type": "text" + }, + { + "bbox": [ + 181, + 145, + 246, + 158 + ], + "score": 0.91, + "content": "\\pmb { g } : \\mathbb { R } ^ { d _ { 1 } } \\mathbb { R } ^ { d _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 145, + 505, + 159 + ], + "score": 1.0, + "content": "be a two-layer MLP with ReLU activation, and all parameter", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 504, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 221, + 170 + ], + "score": 1.0, + "content": "values are upper bounded by", + "type": "text" + }, + { + "bbox": [ + 221, + 158, + 232, + 167 + ], + "score": 0.63, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 158, + 291, + 170 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 291, + 158, + 315, + 168 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 158, + 433, + 170 + ], + "score": 1.0, + "content": ", there exists a two-layer MLP", + "type": "text" + }, + { + "bbox": [ + 433, + 158, + 441, + 169 + ], + "score": 0.74, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 158, + 504, + 170 + ], + "score": 1.0, + "content": "of the same size", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 342, + 181 + ], + "score": 1.0, + "content": "with GeLU activation and parameters upper bounded by", + "type": "text" + }, + { + "bbox": [ + 343, + 168, + 414, + 180 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 168, + 442, + 181 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 442, + 169, + 455, + 180 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "norm, such", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 320, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 151, + 194 + ], + "score": 1.0, + "content": "that for all", + "type": "text" + }, + { + "bbox": [ + 152, + 180, + 186, + 191 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 179, + 226, + 194 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 226, + 180, + 317, + 192 + ], + "score": 0.93, + "content": "\\| f ( \\pmb { x } ) - \\pmb { g } ( \\pmb { x } ) \\| _ { \\infty } \\leq \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 179, + 320, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 504, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 152, + 218 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 204, + 264, + 217 + ], + "score": 0.92, + "content": "\\pmb { g } ( \\pmb { x } ) = \\pmb { W } _ { 2 } \\cdot \\mathrm { R e L U } ( \\pmb { W } _ { 1 } \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 203, + 325, + 218 + ], + "score": 1.0, + "content": ". We construct", + "type": "text" + }, + { + "bbox": [ + 325, + 204, + 450, + 217 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\pmb { f } ( \\pmb { x } ) = \\frac { 1 } { \\lambda } \\pmb { W } _ { 2 } \\cdot \\mathrm { G e L U } ( \\lambda \\pmb { W } _ { 1 } \\pmb { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 203, + 479, + 218 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 479, + 205, + 504, + 215 + ], + "score": 0.88, + "content": "\\lambda > 0", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 292, + 231 + ], + "score": 1.0, + "content": "is a sufficiently large constant. To prove that", + "type": "text" + }, + { + "bbox": [ + 292, + 217, + 383, + 230 + ], + "score": 0.9, + "content": "\\| f ( \\pmb { x } ) - \\pmb { g } ( \\pmb { x } ) \\| _ { \\infty } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 216, + 412, + 231 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 412, + 217, + 448, + 228 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 216, + 506, + 231 + ], + "score": 1.0, + "content": ", it suffices to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 149, + 242 + ], + "score": 1.0, + "content": "prove that", + "type": "text" + }, + { + "bbox": [ + 149, + 228, + 312, + 243 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W _ { 2 } ( \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) ) \\| _ { \\infty } \\le \\epsilon } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 229, + 340, + 242 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 340, + 229, + 371, + 240 + ], + "score": 0.9, + "content": "z \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 229, + 399, + 242 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 399, + 230, + 406, + 239 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "is the hidden size. Since", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 246, + 474, + 305 + ], + "lines": [ + { + "bbox": [ + 135, + 246, + 474, + 305 + ], + "spans": [ + { + "bbox": [ + 135, + 246, + 474, + 305 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left\\| { \\cal W } _ { 2 } \\left( \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) \\right) \\right\\| _ { \\infty } \\leq \\| { \\cal W } _ { 2 } \\| _ { \\infty } \\left\\| \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) ) \\right\\| _ { \\infty } } \\\\ & { \\qquad \\leq M d \\left\\| \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) ) \\right\\| _ { \\infty } , } \\end{array}", + "type": "interline_equation", + "image_path": "549461fe6aa7298c5cf33b6cd7388a4179f33568ffc571e7c3096233cb0acd5a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 135, + 246, + 474, + 265.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 135, + 265.6666666666667, + 474, + 285.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 135, + 285.33333333333337, + 474, + 305.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 309, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 329, + 324 + ], + "score": 1.0, + "content": "it suffices to consider the scalar setting and prove that", + "type": "text" + }, + { + "bbox": [ + 329, + 309, + 477, + 324 + ], + "score": 0.91, + "content": "| \\textstyle { \\frac { 1 } { \\lambda } } \\mathrm { G e L U } ( \\lambda y ) - \\mathrm { R e L U } ( y ) | \\leq \\epsilon / M d", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 321, + 314, + 335 + ], + "spans": [ + { + "bbox": [ + 107, + 322, + 132, + 333 + ], + "score": 0.9, + "content": "y \\in \\mathbb R", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 321, + 314, + 335 + ], + "score": 1.0, + "content": ". 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\\infty } ^ { + \\infty } { \\frac { \\lambda y } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t - \\int _ { - \\infty } ^ { \\lambda y } { \\frac { \\lambda y } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t \\right| = { \\frac { 1 } { \\lambda } } \\int _ { \\lambda y } ^ { + \\infty } { \\frac { \\lambda y } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t .", + "type": "interline_equation", + "image_path": "9018a8d82f3f7f6ca1964d74133ee26115ff609499f208a017c10957b3861e5e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 113, + 394, + 497, + 405.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 113, + 405.0, + 497, + 416.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 113, + 416.0, + 497, + 427.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 379, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 379, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 216, + 446 + ], + "score": 1.0, + "content": "Combined with the case of", + "type": "text" + }, + { + "bbox": [ + 216, + 432, + 240, + 443 + ], + "score": 0.91, + "content": "y < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 429, + 379, + 446 + ], + "score": 1.0, + "content": ", (8) can be consistently written as", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 448, + 482, + 537 + ], + "lines": [ + { + "bbox": [ + 128, + 448, + 482, + 537 + ], + "spans": [ + { + "bbox": [ + 128, + 448, + 482, + 537 + ], + "score": 0.95, + "content": "{ \\begin{array} { r l } { \\left| { \\frac { 1 } { \\lambda } } \\mathrm { G e L U } ( \\lambda y ) - \\operatorname { R e L U } ( y ) \\right| = { \\cfrac { 1 } { \\lambda } } \\int _ { \\lambda \\mid y \\mid } ^ { + \\infty } { \\frac { \\lambda \\mid y \\mid } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t } & { } \\\\ { \\leq { \\cfrac { 1 } { { \\sqrt { 2 \\pi } } \\lambda } } \\int _ { \\lambda \\mid y \\mid } ^ { + \\infty } t \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t = { \\frac { 1 } { { \\sqrt { 2 \\pi } } \\lambda } } \\exp \\left( - { \\frac { \\lambda ^ { 2 } y ^ { 2 } } { 2 } } \\right) } & { } \\\\ { \\leq { \\frac { 1 } { { \\sqrt { 2 \\pi } } \\lambda } } . } \\end{array} }", + "type": "interline_equation", + "image_path": "70230d3e8bd22d18afc2bd5dd765a89b1778eb5fdba954eed71e5de0aa0d287b.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 128, + 448, + 482, + 477.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 128, + 477.6666666666667, + 482, + 507.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 128, + 507.33333333333337, + 482, + 537.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 379, + 556 + ], + "lines": [ + { + "bbox": [ + 104, + 538, + 380, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 184, + 556 + ], + "score": 1.0, + "content": "Picking λ = √Md", + "type": "text" + }, + { + "bbox": [ + 181, + 540, + 380, + 558 + ], + "score": 1.0, + "content": "yields the desired result and completes the proof.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 568, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "Equipped with the above result, we now prove that a two-layer MLP with GeLU activation can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 332, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 332, + 592 + ], + "score": 1.0, + "content": "perform linear transformation and conditional selection.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 198, + 607 + ], + "score": 1.0, + "content": "Proposition C.3. Let", + "type": "text" + }, + { + "bbox": [ + 198, + 593, + 261, + 605 + ], + "score": 0.89, + "content": "\\pmb { f } : \\mathbb { R } ^ { d _ { 1 } } \\mathbb { R } ^ { d _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 592, + 506, + 607 + ], + "score": 1.0, + "content": "be a two-layer MLP with GeLU activation, and the hidden", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 161, + 619 + ], + "score": 1.0, + "content": "dimension is", + "type": "text" + }, + { + "bbox": [ + 161, + 606, + 176, + 617 + ], + "score": 0.87, + "content": "2 d _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 603, + 198, + 619 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 199, + 605, + 256, + 616 + ], + "score": 0.9, + "content": "W \\in \\mathbb { R } ^ { d _ { 2 } \\times d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 603, + 366, + 619 + ], + "score": 1.0, + "content": "be any matrix and denote", + "type": "text" + }, + { + "bbox": [ + 366, + 606, + 441, + 617 + ], + "score": 0.9, + "content": "M = \\operatorname* { m a x } _ { i j } | W _ { i j } |", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 603, + 506, + 619 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 107, + 617, + 130, + 627 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 616, + 269, + 630 + ], + "score": 1.0, + "content": ", there exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 269, + 617, + 283, + 627 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 616, + 358, + 630 + ], + "score": 1.0, + "content": "norm bounded by", + "type": "text" + }, + { + "bbox": [ + 358, + 617, + 429, + 628 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , \\bar { 1 / \\epsilon } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 616, + 505, + 630 + ], + "score": 1.0, + "content": ", such that for any", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 626, + 274, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 141, + 639 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 626, + 181, + 642 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 181, + 628, + 271, + 640 + ], + "score": 0.91, + "content": "\\| f ( \\pmb { x } ) - \\pmb { W } \\pmb { x } \\| _ { \\infty } \\le \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 626, + 274, + 642 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 504, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 415, + 666 + ], + "score": 1.0, + "content": "Proof. We can use a two-layer MLP with ReLU activation to implement", + "type": "text" + }, + { + "bbox": [ + 416, + 653, + 474, + 665 + ], + "score": 0.91, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { g } } } ( \\mathbf { \\boldsymbol { x } } ) \\mathbf { \\nabla } = \\mathbf { \\nabla } W \\mathbf { \\boldsymbol { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 651, + 505, + 666 + ], + "score": 1.0, + "content": "by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 663, + 203, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 203, + 676 + ], + "score": 1.0, + "content": "following construction:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 680, + 385, + 694 + ], + "lines": [ + { + "bbox": [ + 226, + 680, + 385, + 694 + ], + "spans": [ + { + "bbox": [ + 226, + 680, + 385, + 694 + ], + "score": 0.92, + "content": "W x = { \\mathrm { R e L U } } ( W x ) + { \\mathrm { R e L U } } ( - W x )", + "type": "interline_equation", + "image_path": "09528bafb1b7f990efbc052736c0dd9b9e3cf6817717d77ed2e02e39e806885e.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 226, + 680, + 385, + 694 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 699, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 313, + 713 + ], + "score": 1.0, + "content": "Combined with Lemma C.2, we can also implement", + "type": "text" + }, + { + "bbox": [ + 313, + 700, + 334, + 712 + ], + "score": 0.92, + "content": "{ \\pmb g } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 699, + 507, + 713 + ], + "score": 1.0, + "content": "by a two-layer MLP with GeLU activation.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 494, + 542, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 495, + 543, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 495, + 543, + 505, + 553 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 70, + 507, + 97 + ], + "lines": [ + { + "bbox": [ + 105, + 69, + 507, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 147, + 88 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 148, + 70, + 235, + 86 + ], + "score": 0.92, + "content": "\\begin{array} { r } { | f ( a , b ) - a b | < \\frac { 1 0 M ^ { 3 } } { 3 \\lambda } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 69, + 253, + 88 + ], + "score": 1.0, + "content": ". Set", + "type": "text" + }, + { + "bbox": [ + 254, + 70, + 296, + 86 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\lambda \\ge \\frac { 1 0 M ^ { 3 } } { 3 \\epsilon } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 69, + 390, + 88 + ], + "score": 1.0, + "content": ", and then we can obtain", + "type": "text" + }, + { + "bbox": [ + 390, + 73, + 460, + 85 + ], + "score": 0.92, + "content": "| f ( a , b ) - a b | < \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 69, + 507, + 88 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 317, + 98 + ], + "score": 1.0, + "content": "each weight element in the MLP is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 317, + 85, + 344, + 97 + ], + "score": 0.92, + "content": "O ( \\lambda ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 84, + 411, + 98 + ], + "score": 1.0, + "content": ", which is clearly", + "type": "text" + }, + { + "bbox": [ + 412, + 85, + 482, + 97 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 84, + 505, + 98 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 69, + 507, + 98 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 109, + 506, + 143 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 506, + 123 + ], + "score": 1.0, + "content": "Next, we will demonstrate that a two-layer MLP with GeLU activation can efficiently approximate a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 398, + 134 + ], + "score": 1.0, + "content": "two-layer MLP with ReLU activation, with all weights upper bounded by", + "type": "text" + }, + { + "bbox": [ + 399, + 120, + 455, + 133 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 119, + 506, + 134 + ], + "score": 1.0, + "content": ". This result", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 269, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 269, + 145 + ], + "score": 1.0, + "content": "is useful in proving subsequent lemmas.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 108, + 506, + 145 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 181, + 159 + ], + "score": 1.0, + "content": "Lemma C.2. Let", + "type": "text" + }, + { + "bbox": [ + 181, + 145, + 246, + 158 + ], + "score": 0.91, + "content": "\\pmb { g } : \\mathbb { R } ^ { d _ { 1 } } \\mathbb { R } ^ { d _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 145, + 505, + 159 + ], + "score": 1.0, + "content": "be a two-layer MLP with ReLU activation, and all parameter", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 504, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 221, + 170 + ], + "score": 1.0, + "content": "values are upper bounded by", + "type": "text" + }, + { + "bbox": [ + 221, + 158, + 232, + 167 + ], + "score": 0.63, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 158, + 291, + 170 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 291, + 158, + 315, + 168 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 158, + 433, + 170 + ], + "score": 1.0, + "content": ", there exists a two-layer MLP", + "type": "text" + }, + { + "bbox": [ + 433, + 158, + 441, + 169 + ], + "score": 0.74, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 158, + 504, + 170 + ], + "score": 1.0, + "content": "of the same size", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 342, + 181 + ], + "score": 1.0, + "content": "with GeLU activation and parameters upper bounded by", + "type": "text" + }, + { + "bbox": [ + 343, + 168, + 414, + 180 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 168, + 442, + 181 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 442, + 169, + 455, + 180 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "norm, such", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 320, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 151, + 194 + ], + "score": 1.0, + "content": "that for all", + "type": "text" + }, + { + "bbox": [ + 152, + 180, + 186, + 191 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 179, + 226, + 194 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 226, + 180, + 317, + 192 + ], + "score": 0.93, + "content": "\\| f ( \\pmb { x } ) - \\pmb { g } ( \\pmb { x } ) \\| _ { \\infty } \\leq \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 179, + 320, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 145, + 505, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 504, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 152, + 218 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 204, + 264, + 217 + ], + "score": 0.92, + "content": "\\pmb { g } ( \\pmb { x } ) = \\pmb { W } _ { 2 } \\cdot \\mathrm { R e L U } ( \\pmb { W } _ { 1 } \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 203, + 325, + 218 + ], + "score": 1.0, + "content": ". We construct", + "type": "text" + }, + { + "bbox": [ + 325, + 204, + 450, + 217 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\pmb { f } ( \\pmb { x } ) = \\frac { 1 } { \\lambda } \\pmb { W } _ { 2 } \\cdot \\mathrm { G e L U } ( \\lambda \\pmb { W } _ { 1 } \\pmb { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 203, + 479, + 218 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 479, + 205, + 504, + 215 + ], + "score": 0.88, + "content": "\\lambda > 0", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 292, + 231 + ], + "score": 1.0, + "content": "is a sufficiently large constant. To prove that", + "type": "text" + }, + { + "bbox": [ + 292, + 217, + 383, + 230 + ], + "score": 0.9, + "content": "\\| f ( \\pmb { x } ) - \\pmb { g } ( \\pmb { x } ) \\| _ { \\infty } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 216, + 412, + 231 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 412, + 217, + 448, + 228 + ], + "score": 0.9, + "content": "\\pmb { x } \\in \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 216, + 506, + 231 + ], + "score": 1.0, + "content": ", it suffices to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 149, + 242 + ], + "score": 1.0, + "content": "prove that", + "type": "text" + }, + { + "bbox": [ + 149, + 228, + 312, + 243 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\| W _ { 2 } ( \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) ) \\| _ { \\infty } \\le \\epsilon } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 229, + 340, + 242 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 340, + 229, + 371, + 240 + ], + "score": 0.9, + "content": "z \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 229, + 399, + 242 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 399, + 230, + 406, + 239 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "is the hidden size. Since", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 203, + 506, + 243 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 246, + 474, + 305 + ], + "lines": [ + { + "bbox": [ + 135, + 246, + 474, + 305 + ], + "spans": [ + { + "bbox": [ + 135, + 246, + 474, + 305 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left\\| { \\cal W } _ { 2 } \\left( \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) \\right) \\right\\| _ { \\infty } \\leq \\| { \\cal W } _ { 2 } \\| _ { \\infty } \\left\\| \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) ) \\right\\| _ { \\infty } } \\\\ & { \\qquad \\leq M d \\left\\| \\mathrm { R e L U } ( z ) - \\frac { 1 } { \\lambda } \\mathrm { G e L U } ( \\lambda z ) ) \\right\\| _ { \\infty } , } \\end{array}", + "type": "interline_equation", + "image_path": "549461fe6aa7298c5cf33b6cd7388a4179f33568ffc571e7c3096233cb0acd5a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 135, + 246, + 474, + 265.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 135, + 265.6666666666667, + 474, + 285.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 135, + 285.33333333333337, + 474, + 305.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 309, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 329, + 324 + ], + "score": 1.0, + "content": "it suffices to consider the scalar setting and prove that", + "type": "text" + }, + { + "bbox": [ + 329, + 309, + 477, + 324 + ], + "score": 0.91, + "content": "| \\textstyle { \\frac { 1 } { \\lambda } } \\mathrm { G e L U } ( \\lambda y ) - \\mathrm { R e L U } ( y ) | \\leq \\epsilon / M d", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 321, + 314, + 335 + ], + "spans": [ + { + "bbox": [ + 107, + 322, + 132, + 333 + ], + "score": 0.9, + "content": "y \\in \\mathbb R", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 321, + 314, + 335 + ], + "score": 1.0, + "content": ". 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\\infty } ^ { + \\infty } { \\frac { \\lambda y } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t - \\int _ { - \\infty } ^ { \\lambda y } { \\frac { \\lambda y } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t \\right| = { \\frac { 1 } { \\lambda } } \\int _ { \\lambda y } ^ { + \\infty } { \\frac { \\lambda y } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t .", + "type": "interline_equation", + "image_path": "9018a8d82f3f7f6ca1964d74133ee26115ff609499f208a017c10957b3861e5e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 113, + 394, + 497, + 405.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 113, + 405.0, + 497, + 416.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 113, + 416.0, + 497, + 427.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 379, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 379, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 216, + 446 + ], + "score": 1.0, + "content": "Combined with the case of", + "type": "text" + }, + { + "bbox": [ + 216, + 432, + 240, + 443 + ], + "score": 0.91, + "content": "y < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 429, + 379, + 446 + ], + "score": 1.0, + "content": ", (8) can be consistently written as", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 429, + 379, + 446 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 448, + 482, + 537 + ], + "lines": [ + { + "bbox": [ + 128, + 448, + 482, + 537 + ], + "spans": [ + { + "bbox": [ + 128, + 448, + 482, + 537 + ], + "score": 0.95, + "content": "{ \\begin{array} { r l } { \\left| { \\frac { 1 } { \\lambda } } \\mathrm { G e L U } ( \\lambda y ) - \\operatorname { R e L U } ( y ) \\right| = { \\cfrac { 1 } { \\lambda } } \\int _ { \\lambda \\mid y \\mid } ^ { + \\infty } { \\frac { \\lambda \\mid y \\mid } { { \\sqrt { 2 \\pi } } } } \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t } & { } \\\\ { \\leq { \\cfrac { 1 } { { \\sqrt { 2 \\pi } } \\lambda } } \\int _ { \\lambda \\mid y \\mid } ^ { + \\infty } t \\exp \\left( - { \\frac { t ^ { 2 } } { 2 } } \\right) \\mathrm { d } t = { \\frac { 1 } { { \\sqrt { 2 \\pi } } \\lambda } } \\exp \\left( - { \\frac { \\lambda ^ { 2 } y ^ { 2 } } { 2 } } \\right) } & { } \\\\ { \\leq { \\frac { 1 } { { \\sqrt { 2 \\pi } } \\lambda } } . } \\end{array} }", + "type": "interline_equation", + "image_path": "70230d3e8bd22d18afc2bd5dd765a89b1778eb5fdba954eed71e5de0aa0d287b.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 128, + 448, + 482, + 477.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 128, + 477.6666666666667, + 482, + 507.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 128, + 507.33333333333337, + 482, + 537.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 379, + 556 + ], + "lines": [ + { + "bbox": [ + 104, + 538, + 380, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 184, + 556 + ], + "score": 1.0, + "content": "Picking λ = √Md", + "type": "text" + }, + { + "bbox": [ + 181, + 540, + 380, + 558 + ], + "score": 1.0, + "content": "yields the desired result and completes the proof.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 538, + 380, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 568, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "Equipped with the above result, we now prove that a two-layer MLP with GeLU activation can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 332, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 332, + 592 + ], + "score": 1.0, + "content": "perform linear transformation and conditional selection.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 568, + 505, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 198, + 607 + ], + "score": 1.0, + "content": "Proposition C.3. Let", + "type": "text" + }, + { + "bbox": [ + 198, + 593, + 261, + 605 + ], + "score": 0.89, + "content": "\\pmb { f } : \\mathbb { R } ^ { d _ { 1 } } \\mathbb { R } ^ { d _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 592, + 506, + 607 + ], + "score": 1.0, + "content": "be a two-layer MLP with GeLU activation, and the hidden", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 161, + 619 + ], + "score": 1.0, + "content": "dimension is", + "type": "text" + }, + { + "bbox": [ + 161, + 606, + 176, + 617 + ], + "score": 0.87, + "content": "2 d _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 603, + 198, + 619 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 199, + 605, + 256, + 616 + ], + "score": 0.9, + "content": "W \\in \\mathbb { R } ^ { d _ { 2 } \\times d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 603, + 366, + 619 + ], + "score": 1.0, + "content": "be any matrix and denote", + "type": "text" + }, + { + "bbox": [ + 366, + 606, + 441, + 617 + ], + "score": 0.9, + "content": "M = \\operatorname* { m a x } _ { i j } | W _ { i j } |", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 603, + 506, + 619 + ], + "score": 1.0, + "content": ". 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We can use a two-layer MLP with ReLU activation to implement", + "type": "text" + }, + { + "bbox": [ + 416, + 653, + 474, + 665 + ], + "score": 0.91, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { g } } } ( \\mathbf { \\boldsymbol { x } } ) \\mathbf { \\nabla } = \\mathbf { \\nabla } W \\mathbf { \\boldsymbol { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 651, + 505, + 666 + ], + "score": 1.0, + "content": "by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 663, + 203, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 203, + 676 + ], + "score": 1.0, + "content": "following construction:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 651, + 505, + 676 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 680, + 385, + 694 + ], + "lines": [ + { + "bbox": [ + 226, + 680, + 385, + 694 + ], + "spans": [ + { + "bbox": [ + 226, + 680, + 385, + 694 + ], + "score": 0.92, + "content": "W x = { \\mathrm { R e L U } } ( W x ) + { \\mathrm { R e L U } } ( - W x )", + "type": "interline_equation", + "image_path": "09528bafb1b7f990efbc052736c0dd9b9e3cf6817717d77ed2e02e39e806885e.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 226, + 680, + 385, + 694 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 699, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 313, + 713 + ], + "score": 1.0, + "content": "Combined with Lemma C.2, we can also implement", + "type": "text" + }, + { + "bbox": [ + 313, + 700, + 334, + 712 + ], + "score": 0.92, + "content": "{ \\pmb g } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 699, + 507, + 713 + ], + "score": 1.0, + "content": "by a two-layer MLP with GeLU activation.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 699, + 507, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 71, + 424, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 426, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 280, + 87 + ], + "score": 1.0, + "content": "Lemma C.4. Define the selection function", + "type": "text" + }, + { + "bbox": [ + 280, + 72, + 378, + 84 + ], + "score": 0.9, + "content": "\\pmb { g } : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } \\times \\mathbb { R } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 70, + 426, + 87 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 87, + 361, + 113 + ], + "lines": [ + { + "bbox": [ + 244, + 87, + 361, + 113 + ], + "spans": [ + { + "bbox": [ + 244, + 87, + 361, + 113 + ], + "score": 0.93, + "content": "\\begin{array} { r } { g ( x , y , t ) = \\left\\{ \\begin{array} { l l } { x } & { i f t \\ge 0 , } \\\\ { y } & { i f t < 0 . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "8b14c37af241f6cdc2a371322c426306d910a7c50ed070bb48419a97812603c7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 244, + 87, + 361, + 113 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 115, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 122, + 128 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 115, + 221, + 127 + ], + "score": 0.9, + "content": "\\pmb { f } : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } \\times \\mathbb { R } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "be a two-layer MLP with GeLU activation, and the hidden dimension", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 117, + 140 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 128, + 147, + 137 + ], + "score": 0.89, + "content": "2 d + 2", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 125, + 217, + 140 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 218, + 127, + 246, + 137 + ], + "score": 0.81, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 125, + 251, + 140 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 252, + 127, + 282, + 137 + ], + "score": 0.71, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 125, + 307, + 140 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 307, + 127, + 342, + 137 + ], + "score": 0.9, + "content": "M \\ > \\ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 125, + 489, + 140 + ], + "score": 1.0, + "content": ", there exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 490, + 127, + 504, + 138 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 182, + 151 + ], + "score": 1.0, + "content": "norm bounded by", + "type": "text" + }, + { + "bbox": [ + 182, + 138, + 272, + 149 + ], + "score": 0.89, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\alpha , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 136, + 348, + 151 + ], + "score": 1.0, + "content": "), such that for all", + "type": "text" + }, + { + "bbox": [ + 348, + 137, + 412, + 149 + ], + "score": 0.84, + "content": "\\pmb { x } \\in [ - M , M ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 136, + 417, + 151 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 418, + 137, + 482, + 150 + ], + "score": 0.91, + "content": "\\mathbf { \\dot { y } } \\in [ - M , M ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 136, + 506, + 151 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 381, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 210, + 161 + ], + "score": 0.91, + "content": "t \\in [ - \\infty , - \\alpha ] \\cup [ \\alpha , + \\infty ]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 149, + 250, + 162 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 250, + 149, + 377, + 161 + ], + "score": 0.89, + "content": "\\| f ( x , y , t ) - g ( x , y , t ) \\| _ { \\infty } \\leq \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 149, + 381, + 162 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 428, + 186 + ], + "score": 1.0, + "content": "Proof. We can simply use a two-layer MLP with ReLU activation to implement", + "type": "text" + }, + { + "bbox": [ + 428, + 174, + 435, + 183 + ], + "score": 0.79, + "content": "\\textbf { { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 169, + 506, + 186 + ], + "score": 1.0, + "content": "by the following", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 183, + 161, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 161, + 195 + ], + "score": 1.0, + "content": "construction:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 194, + 499, + 225 + ], + "lines": [ + { + "bbox": [ + 111, + 194, + 499, + 225 + ], + "spans": [ + { + "bbox": [ + 111, + 194, + 499, + 225 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { h ( x , y , t ) = ( h _ { 1 } , h _ { 2 } , h _ { 3 } , h _ { 4 } ) : = ( x + \\alpha ^ { - 1 } M t { \\bf 1 } _ { d } , y - \\alpha ^ { - 1 } M t { \\bf 1 } _ { d } , \\alpha ^ { - 1 } M t , - \\alpha ^ { - 1 } M t ) \\in \\mathbb { R } ^ { 2 d + 2 } , } \\\\ & { f ( x , y , t ) = \\mathrm { R e L U } ( h _ { 1 } ) - \\mathrm { R e L U } ( h _ { 3 } ) { \\bf 1 } _ { d } + \\mathrm { R e L U } ( h _ { 2 } ) - \\mathrm { R e L U } ( h _ { 4 } ) { \\bf 1 } _ { d } , } \\end{array}", + "type": "interline_equation", + "image_path": "5e828fdac3625640557cdec66b4c278b677449ebe81775a182de863a710f6a82.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 194, + 499, + 204.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 204.33333333333334, + 499, + 214.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 214.66666666666669, + 499, + 225.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 133, + 239 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 227, + 145, + 238 + ], + "score": 0.86, + "content": "{ \\bf 1 } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 226, + 240, + 239 + ], + "score": 1.0, + "content": "is the all-one vector of", + "type": "text" + }, + { + "bbox": [ + 240, + 227, + 247, + 237 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 226, + 418, + 239 + ], + "score": 1.0, + "content": "dimension. It is easy to check that, for all", + "type": "text" + }, + { + "bbox": [ + 419, + 226, + 505, + 239 + ], + "score": 0.92, + "content": "\\pmb { x } \\in [ - M , M ] ^ { d _ { 1 } } , \\pmb { y } \\in", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 237, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 107, + 239, + 154, + 251 + ], + "score": 0.92, + "content": "[ - M , M ] ^ { d _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 237, + 176, + 253 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 176, + 239, + 279, + 250 + ], + "score": 0.91, + "content": "t \\in [ - \\infty , - \\alpha ] \\cup [ \\alpha , + \\infty ]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 237, + 318, + 253 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 319, + 239, + 411, + 251 + ], + "score": 0.92, + "content": "f ( \\pmb { x } , \\pmb { y } , t ) = \\pmb { g } ( \\pmb { x } , \\pmb { y } , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 237, + 507, + 253 + ], + "score": 1.0, + "content": ". Moreover, all parame-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 189, + 262 + ], + "score": 1.0, + "content": "ters are bounded by", + "type": "text" + }, + { + "bbox": [ + 189, + 250, + 227, + 262 + ], + "score": 0.91, + "content": "O ( M / \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 250, + 462, + 262 + ], + "score": 1.0, + "content": ". Therefore, by using Lemma C.2, we can also implement", + "type": "text" + }, + { + "bbox": [ + 463, + 250, + 484, + 261 + ], + "score": 0.91, + "content": "\\pmb { g } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "by a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 390, + 273 + ], + "score": 1.0, + "content": "two-layer MLP with GeLU activation and all parameters are bounded by", + "type": "text" + }, + { + "bbox": [ + 391, + 261, + 482, + 273 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\alpha , 1 / \\acute { \\epsilon } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 260, + 494, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 261, + 505, + 271 + ], + "score": 0.28, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 506, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 507, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 495, + 296 + ], + "score": 1.0, + "content": "We final show that a two-layer MLP can efficiently represent a look-up table. Consider a", + "type": "text" + }, + { + "bbox": [ + 495, + 284, + 502, + 294 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 283, + 507, + 296 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 507, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 213, + 307 + ], + "score": 1.0, + "content": "dimensional table of size", + "type": "text" + }, + { + "bbox": [ + 214, + 294, + 225, + 305 + ], + "score": 0.86, + "content": "d ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 293, + 496, + 307 + ], + "score": 1.0, + "content": ", where each element in the table is an integer ranging from 1 to", + "type": "text" + }, + { + "bbox": [ + 497, + 295, + 503, + 304 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 293, + 507, + 307 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 166, + 318 + ], + "score": 1.0, + "content": "Denote the set", + "type": "text" + }, + { + "bbox": [ + 166, + 306, + 241, + 318 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ e _ { i } : i \\in [ d ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 306, + 272, + 318 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 272, + 307, + 282, + 317 + ], + "score": 0.86, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 306, + 299, + 318 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 300, + 306, + 306, + 316 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 306, + 454, + 318 + ], + "score": 1.0, + "content": "-dimensional one-hot vector with the", + "type": "text" + }, + { + "bbox": [ + 454, + 307, + 459, + 316 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "-th element", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 430, + 330 + ], + "score": 1.0, + "content": "being 1. The above look-up table can thus be represented as a discrete function", + "type": "text" + }, + { + "bbox": [ + 430, + 317, + 482, + 329 + ], + "score": 0.92, + "content": "g : { \\mathcal { D } } ^ { k } { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 316, + 506, + 330 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 487, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 221, + 340 + ], + "score": 1.0, + "content": "following lemma shows that", + "type": "text" + }, + { + "bbox": [ + 222, + 330, + 229, + 340 + ], + "score": 0.79, + "content": "\\textbf { { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 329, + 487, + 340 + ], + "score": 1.0, + "content": "can be implemented by a two-layer MLP with GeLU activation.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 104, + 338, + 507, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 177, + 356 + ], + "score": 1.0, + "content": "Lemma C.5. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 341, + 229, + 353 + ], + "score": 0.9, + "content": "g : { \\mathcal { D } } ^ { k } { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 338, + 380, + 356 + ], + "score": 1.0, + "content": "be any function defined above, and let", + "type": "text" + }, + { + "bbox": [ + 380, + 341, + 446, + 353 + ], + "score": 0.9, + "content": "\\pmb { f } : \\mathbb { R } ^ { k \\times d } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 338, + 507, + 356 + ], + "score": 1.0, + "content": "be a two-layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 379, + 366 + ], + "score": 1.0, + "content": "MLP with GeLU activation and bias, and the hidden dimension is", + "type": "text" + }, + { + "bbox": [ + 379, + 353, + 390, + 363 + ], + "score": 0.84, + "content": "d ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 351, + 453, + 366 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 453, + 353, + 478, + 363 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 362, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 214, + 376 + ], + "score": 1.0, + "content": "exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 214, + 364, + 228, + 375 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 362, + 300, + 376 + ], + "score": 1.0, + "content": "norm bounded by", + "type": "text" + }, + { + "bbox": [ + 300, + 364, + 366, + 375 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( k , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 362, + 435, + 376 + ], + "score": 1.0, + "content": ", such that for all", + "type": "text" + }, + { + "bbox": [ + 435, + 363, + 504, + 374 + ], + "score": 0.91, + "content": "\\pmb { x } \\in \\mathcal { D } ^ { k } \\subset \\mathbb { R } ^ { k \\times d }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 372, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 189, + 389 + ], + "score": 1.0, + "content": "and all perturbation", + "type": "text" + }, + { + "bbox": [ + 190, + 375, + 281, + 388 + ], + "score": 0.91, + "content": "\\pmb { \\delta } \\in [ - 1 / 2 k , 1 / 2 k ] ^ { k \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 372, + 320, + 389 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 320, + 375, + 475, + 388 + ], + "score": 0.91, + "content": "\\| f ( \\pmb { x } + \\pmb { \\delta } ) - \\pmb { g } ( \\pmb { x } ) \\| _ { \\infty } \\le \\epsilon + 2 k \\| \\pmb { \\delta } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 372, + 506, + 389 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 387, + 362, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 387, + 131, + 399 + ], + "score": 0.91, + "content": "\\| \\delta \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 387, + 184, + 399 + ], + "score": 1.0, + "content": "is the vector", + "type": "text" + }, + { + "bbox": [ + 185, + 387, + 198, + 398 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 387, + 351, + 399 + ], + "score": 1.0, + "content": "-norm applied to the flattended matrix", + "type": "text" + }, + { + "bbox": [ + 352, + 388, + 358, + 397 + ], + "score": 0.63, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 387, + 362, + 399 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 428, + 424 + ], + "score": 1.0, + "content": "Proof. We can simply use a two-layer MLP with ReLU activation to implement", + "type": "text" + }, + { + "bbox": [ + 428, + 412, + 435, + 421 + ], + "score": 0.77, + "content": "\\textbf { { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 407, + 505, + 424 + ], + "score": 1.0, + "content": "by the following", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 340, + 433 + ], + "score": 1.0, + "content": "construction. Denote the index of the MLP hidden layer as", + "type": "text" + }, + { + "bbox": [ + 340, + 420, + 416, + 433 + ], + "score": 0.94, + "content": "( i _ { 1 } , \\cdot \\cdot \\cdot , i _ { k } ) \\bar { \\in [ d ] ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 420, + 505, + 433 + ], + "score": 1.0, + "content": ". We can construct the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 431, + 267, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 267, + 443 + ], + "score": 1.0, + "content": "weights of the first MLP layer such that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 444, + 426, + 459 + ], + "lines": [ + { + "bbox": [ + 183, + 444, + 426, + 459 + ], + "spans": [ + { + "bbox": [ + 183, + 444, + 426, + 459 + ], + "score": 0.87, + "content": "h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\pmb x } ) = 2 ( x _ { i _ { 1 } } + x _ { d + i _ { 2 } } \\cdot \\cdot \\cdot + x _ { ( k - 1 ) d + i _ { k } } ) - 2 k + 1 .", + "type": "interline_equation", + "image_path": "f8f6e9a8ccdabc8cb213eac96e4cd2b1d5372a22aa7aa1f466097e6d62aff25e.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 183, + 444, + 426, + 459 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 479, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 482, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 482, + 473 + ], + "score": 1.0, + "content": "We can then construct the weights of the second layer such that the final output of the MLP is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 473, + 401, + 503 + ], + "lines": [ + { + "bbox": [ + 209, + 473, + 401, + 503 + ], + "spans": [ + { + "bbox": [ + 209, + 473, + 401, + 503 + ], + "score": 0.92, + "content": "f _ { j } ( \\pmb { x } ) = \\sum _ { g _ { j } ( e _ { i _ { 1 } } , \\cdots , e _ { i _ { k } } ) = 1 } { \\mathrm { R e L U } } ( h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( \\pmb { x } ) ) .", + "type": "interline_equation", + "image_path": "087e1e60efa04bd95b1f3bb934765df4ac7853fb6d8f5a167bed2a999a1820b8.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 473, + 401, + 488.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 209, + 488.0, + 401, + 503.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 455, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 457, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 185, + 520 + ], + "score": 1.0, + "content": "One can check that", + "type": "text" + }, + { + "bbox": [ + 186, + 506, + 241, + 518 + ], + "score": 0.93, + "content": "\\pmb { f } ( \\pmb { x } ) = \\pmb { g } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 503, + 293, + 520 + ], + "score": 1.0, + "content": "holds for all", + "type": "text" + }, + { + "bbox": [ + 293, + 505, + 361, + 516 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathcal { D } ^ { k } \\subset \\mathbb { R } ^ { d \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 503, + 457, + 520 + ], + "score": 1.0, + "content": ". Furthermore, we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 519, + 510, + 596 + ], + "lines": [ + { + "bbox": [ + 111, + 519, + 510, + 596 + ], + "spans": [ + { + "bbox": [ + 111, + 519, + 510, + 596 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { f ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) - g ( { \\boldsymbol x } ) \\| _ { \\infty } = \\| f ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) - f ( { \\boldsymbol x } ) \\| _ { \\infty } } & { } \\\\ & { \\qquad = \\underset { j \\in [ d ] } { \\operatorname* { m a x } } \\left| \\sum _ { \\substack { \\boldsymbol y _ { j } ( e _ { i _ { 1 } } , \\ldots , e _ { i _ { k } } ) = 1 } } \\left( \\operatorname { R e L U } ( h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) ) - \\operatorname { R e L U } ( h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } ) ) \\right) \\right| } \\\\ & { \\qquad \\le \\underset { ( i _ { 1 } , \\cdots , i _ { k } ) \\in [ d ] ^ { k } } { \\operatorname* { m a x } } \\left| h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) - h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } ) \\right| \\le 2 k \\| { \\boldsymbol \\delta } \\| _ { \\infty } } \\end{array}", + "type": "interline_equation", + "image_path": "74e9f6364147cc961657b35441ba13ee92951f57f7b13101ddf75b200a9fa4a2.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 109, + 519, + 510, + 544.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 109, + 544.6666666666666, + 510, + 570.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 109, + 570.3333333333333, + 510, + 595.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 598, + 503, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 186, + 613 + ], + "score": 1.0, + "content": "for all perturbations", + "type": "text" + }, + { + "bbox": [ + 187, + 598, + 279, + 612 + ], + "score": 0.92, + "content": "\\pmb { \\delta } \\in [ - 1 / 2 k , 1 / 2 k ] ^ { k \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 596, + 483, + 613 + ], + "score": 1.0, + "content": ". Thus by using Lemma C.2, we can also implement", + "type": "text" + }, + { + "bbox": [ + 484, + 599, + 505, + 611 + ], + "score": 0.9, + "content": "\\pmb { g } ( \\pmb { x } )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 416, + 623 + ], + "score": 1.0, + "content": "by a two-layer MLP with GeLU activation and all parameters are bounded by", + "type": "text" + }, + { + "bbox": [ + 416, + 610, + 482, + 622 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( k , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 609, + 506, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "title", + "bbox": [ + 106, + 634, + 304, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 305, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 305, + 648 + ], + "score": 1.0, + "content": "C.2 Technical lemmas for the attention layer", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "In this subsection, we will introduce two special operations that can be performed by the attention", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 663, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 256, + 678 + ], + "score": 1.0, + "content": "layer (with causal mask). Below, let", + "type": "text" + }, + { + "bbox": [ + 256, + 666, + 283, + 675 + ], + "score": 0.89, + "content": "n \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 663, + 370, + 678 + ], + "score": 1.0, + "content": "be an integer and let", + "type": "text" + }, + { + "bbox": [ + 370, + 667, + 433, + 676 + ], + "score": 0.89, + "content": "{ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\cdots , { \\pmb x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 663, + 506, + 678 + ], + "score": 1.0, + "content": "be a sequence of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 673, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 673, + 166, + 690 + ], + "score": 1.0, + "content": "vectors where", + "type": "text" + }, + { + "bbox": [ + 166, + 676, + 298, + 688 + ], + "score": 0.86, + "content": "\\pmb { x } _ { i } = ( \\tilde { \\pmb { x } } _ { i } , r _ { i } , 1 ) \\in [ - M , M ] ^ { d + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 673, + 302, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 676, + 339, + 687 + ], + "score": 0.76, + "content": "\\tilde { \\mathbf { x } } _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 673, + 343, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 343, + 676, + 373, + 687 + ], + "score": 0.82, + "content": "r _ { i } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 673, + 394, + 690 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 395, + 677, + 407, + 686 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 673, + 507, + 690 + ], + "score": 1.0, + "content": "is a large constant. Let", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 685, + 508, + 704 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 198, + 700 + ], + "score": 0.9, + "content": "K , Q , V \\in \\mathbb { R } ^ { d ^ { \\prime } \\times ( d + 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 685, + 284, + 704 + ], + "score": 1.0, + "content": "be any matrices with", + "type": "text" + }, + { + "bbox": [ + 284, + 689, + 330, + 700 + ], + "score": 0.94, + "content": "\\| \\ b { V } \\| _ { \\infty } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 685, + 363, + 704 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 363, + 689, + 421, + 700 + ], + "score": 0.91, + "content": "0 < \\rho , \\delta < M", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 685, + 508, + 704 + ], + "score": 1.0, + "content": "be any real numbers.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 137, + 713 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 137, + 700, + 179, + 712 + ], + "score": 0.83, + "content": "\\pmb q _ { i } = \\pmb Q x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 699, + 183, + 713 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 183, + 700, + 228, + 712 + ], + "score": 0.79, + "content": "k _ { j } = K x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 699, + 232, + 713 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 232, + 700, + 276, + 712 + ], + "score": 0.87, + "content": "{ \\pmb v } _ { j } = { \\pmb V } { \\pmb x } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 699, + 389, + 713 + ], + "score": 1.0, + "content": ", and define the matching set", + "type": "text" + }, + { + "bbox": [ + 390, + 700, + 503, + 712 + ], + "score": 0.92, + "content": "S _ { i } = \\{ j \\leq i : | \\pmb { q } _ { i } \\cdot \\pmb { k } _ { j } | \\leq \\rho \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 699, + 507, + 713 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 402, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 402, + 723 + ], + "score": 1.0, + "content": "Equipped with these notations, we define two basic operations as follows:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 310, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 312, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 312, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 71, + 424, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 426, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 280, + 87 + ], + "score": 1.0, + "content": "Lemma C.4. Define the selection function", + "type": "text" + }, + { + "bbox": [ + 280, + 72, + 378, + 84 + ], + "score": 0.9, + "content": "\\pmb { g } : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } \\times \\mathbb { R } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 70, + 426, + 87 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 70, + 426, + 87 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 87, + 361, + 113 + ], + "lines": [ + { + "bbox": [ + 244, + 87, + 361, + 113 + ], + "spans": [ + { + "bbox": [ + 244, + 87, + 361, + 113 + ], + "score": 0.93, + "content": "\\begin{array} { r } { g ( x , y , t ) = \\left\\{ \\begin{array} { l l } { x } & { i f t \\ge 0 , } \\\\ { y } & { i f t < 0 . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "8b14c37af241f6cdc2a371322c426306d910a7c50ed070bb48419a97812603c7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 244, + 87, + 361, + 113 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 115, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 122, + 128 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 115, + 221, + 127 + ], + "score": 0.9, + "content": "\\pmb { f } : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } \\times \\mathbb { R } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "be a two-layer MLP with GeLU activation, and the hidden dimension", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 117, + 140 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 128, + 147, + 137 + ], + "score": 0.89, + "content": "2 d + 2", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 125, + 217, + 140 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 218, + 127, + 246, + 137 + ], + "score": 0.81, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 125, + 251, + 140 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 252, + 127, + 282, + 137 + ], + "score": 0.71, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 125, + 307, + 140 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 307, + 127, + 342, + 137 + ], + "score": 0.9, + "content": "M \\ > \\ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 125, + 489, + 140 + ], + "score": 1.0, + "content": ", there exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 490, + 127, + 504, + 138 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 182, + 151 + ], + "score": 1.0, + "content": "norm bounded by", + "type": "text" + }, + { + "bbox": [ + 182, + 138, + 272, + 149 + ], + "score": 0.89, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\alpha , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 136, + 348, + 151 + ], + "score": 1.0, + "content": "), such that for all", + "type": "text" + }, + { + "bbox": [ + 348, + 137, + 412, + 149 + ], + "score": 0.84, + "content": "\\pmb { x } \\in [ - M , M ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 136, + 417, + 151 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 418, + 137, + 482, + 150 + ], + "score": 0.91, + "content": "\\mathbf { \\dot { y } } \\in [ - M , M ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 136, + 506, + 151 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 381, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 210, + 161 + ], + "score": 0.91, + "content": "t \\in [ - \\infty , - \\alpha ] \\cup [ \\alpha , + \\infty ]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 149, + 250, + 162 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 250, + 149, + 377, + 161 + ], + "score": 0.89, + "content": "\\| f ( x , y , t ) - g ( x , y , t ) \\| _ { \\infty } \\leq \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 149, + 381, + 162 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 114, + 506, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 428, + 186 + ], + "score": 1.0, + "content": "Proof. We can simply use a two-layer MLP with ReLU activation to implement", + "type": "text" + }, + { + "bbox": [ + 428, + 174, + 435, + 183 + ], + "score": 0.79, + "content": "\\textbf { { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 169, + 506, + 186 + ], + "score": 1.0, + "content": "by the following", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 183, + 161, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 161, + 195 + ], + "score": 1.0, + "content": "construction:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 169, + 506, + 195 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 194, + 499, + 225 + ], + "lines": [ + { + "bbox": [ + 111, + 194, + 499, + 225 + ], + "spans": [ + { + "bbox": [ + 111, + 194, + 499, + 225 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { h ( x , y , t ) = ( h _ { 1 } , h _ { 2 } , h _ { 3 } , h _ { 4 } ) : = ( x + \\alpha ^ { - 1 } M t { \\bf 1 } _ { d } , y - \\alpha ^ { - 1 } M t { \\bf 1 } _ { d } , \\alpha ^ { - 1 } M t , - \\alpha ^ { - 1 } M t ) \\in \\mathbb { R } ^ { 2 d + 2 } , } \\\\ & { f ( x , y , t ) = \\mathrm { R e L U } ( h _ { 1 } ) - \\mathrm { R e L U } ( h _ { 3 } ) { \\bf 1 } _ { d } + \\mathrm { R e L U } ( h _ { 2 } ) - \\mathrm { R e L U } ( h _ { 4 } ) { \\bf 1 } _ { d } , } \\end{array}", + "type": "interline_equation", + "image_path": "5e828fdac3625640557cdec66b4c278b677449ebe81775a182de863a710f6a82.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 194, + 499, + 204.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 204.33333333333334, + 499, + 214.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 214.66666666666669, + 499, + 225.00000000000003 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 133, + 239 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 227, + 145, + 238 + ], + "score": 0.86, + "content": "{ \\bf 1 } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 226, + 240, + 239 + ], + "score": 1.0, + "content": "is the all-one vector of", + "type": "text" + }, + { + "bbox": [ + 240, + 227, + 247, + 237 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 226, + 418, + 239 + ], + "score": 1.0, + "content": "dimension. It is easy to check that, for all", + "type": "text" + }, + { + "bbox": [ + 419, + 226, + 505, + 239 + ], + "score": 0.92, + "content": "\\pmb { x } \\in [ - M , M ] ^ { d _ { 1 } } , \\pmb { y } \\in", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 237, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 107, + 239, + 154, + 251 + ], + "score": 0.92, + "content": "[ - M , M ] ^ { d _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 237, + 176, + 253 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 176, + 239, + 279, + 250 + ], + "score": 0.91, + "content": "t \\in [ - \\infty , - \\alpha ] \\cup [ \\alpha , + \\infty ]", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 237, + 318, + 253 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 319, + 239, + 411, + 251 + ], + "score": 0.92, + "content": "f ( \\pmb { x } , \\pmb { y } , t ) = \\pmb { g } ( \\pmb { x } , \\pmb { y } , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 237, + 507, + 253 + ], + "score": 1.0, + "content": ". Moreover, all parame-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 189, + 262 + ], + "score": 1.0, + "content": "ters are bounded by", + "type": "text" + }, + { + "bbox": [ + 189, + 250, + 227, + 262 + ], + "score": 0.91, + "content": "O ( M / \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 250, + 462, + 262 + ], + "score": 1.0, + "content": ". Therefore, by using Lemma C.2, we can also implement", + "type": "text" + }, + { + "bbox": [ + 463, + 250, + 484, + 261 + ], + "score": 0.91, + "content": "\\pmb { g } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "by a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 390, + 273 + ], + "score": 1.0, + "content": "two-layer MLP with GeLU activation and all parameters are bounded by", + "type": "text" + }, + { + "bbox": [ + 391, + 261, + 482, + 273 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( M , 1 / \\alpha , 1 / \\acute { \\epsilon } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 260, + 494, + 273 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 261, + 505, + 271 + ], + "score": 0.28, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 226, + 507, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 506, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 507, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 495, + 296 + ], + "score": 1.0, + "content": "We final show that a two-layer MLP can efficiently represent a look-up table. Consider a", + "type": "text" + }, + { + "bbox": [ + 495, + 284, + 502, + 294 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 283, + 507, + 296 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 507, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 213, + 307 + ], + "score": 1.0, + "content": "dimensional table of size", + "type": "text" + }, + { + "bbox": [ + 214, + 294, + 225, + 305 + ], + "score": 0.86, + "content": "d ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 293, + 496, + 307 + ], + "score": 1.0, + "content": ", where each element in the table is an integer ranging from 1 to", + "type": "text" + }, + { + "bbox": [ + 497, + 295, + 503, + 304 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 293, + 507, + 307 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 166, + 318 + ], + "score": 1.0, + "content": "Denote the set", + "type": "text" + }, + { + "bbox": [ + 166, + 306, + 241, + 318 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ e _ { i } : i \\in [ d ] \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 306, + 272, + 318 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 272, + 307, + 282, + 317 + ], + "score": 0.86, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 306, + 299, + 318 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 300, + 306, + 306, + 316 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 306, + 454, + 318 + ], + "score": 1.0, + "content": "-dimensional one-hot vector with the", + "type": "text" + }, + { + "bbox": [ + 454, + 307, + 459, + 316 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "-th element", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 430, + 330 + ], + "score": 1.0, + "content": "being 1. The above look-up table can thus be represented as a discrete function", + "type": "text" + }, + { + "bbox": [ + 430, + 317, + 482, + 329 + ], + "score": 0.92, + "content": "g : { \\mathcal { D } } ^ { k } { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 316, + 506, + 330 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 487, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 221, + 340 + ], + "score": 1.0, + "content": "following lemma shows that", + "type": "text" + }, + { + "bbox": [ + 222, + 330, + 229, + 340 + ], + "score": 0.79, + "content": "\\textbf { { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 329, + 487, + 340 + ], + "score": 1.0, + "content": "can be implemented by a two-layer MLP with GeLU activation.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 283, + 507, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 104, + 338, + 507, + 356 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 177, + 356 + ], + "score": 1.0, + "content": "Lemma C.5. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 341, + 229, + 353 + ], + "score": 0.9, + "content": "g : { \\mathcal { D } } ^ { k } { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 338, + 380, + 356 + ], + "score": 1.0, + "content": "be any function defined above, and let", + "type": "text" + }, + { + "bbox": [ + 380, + 341, + 446, + 353 + ], + "score": 0.9, + "content": "\\pmb { f } : \\mathbb { R } ^ { k \\times d } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 338, + 507, + 356 + ], + "score": 1.0, + "content": "be a two-layer", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 379, + 366 + ], + "score": 1.0, + "content": "MLP with GeLU activation and bias, and the hidden dimension is", + "type": "text" + }, + { + "bbox": [ + 379, + 353, + 390, + 363 + ], + "score": 0.84, + "content": "d ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 351, + 453, + 366 + ], + "score": 1.0, + "content": ". Then, for any", + "type": "text" + }, + { + "bbox": [ + 453, + 353, + 478, + 363 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 362, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 214, + 376 + ], + "score": 1.0, + "content": "exist MLP parameters with", + "type": "text" + }, + { + "bbox": [ + 214, + 364, + 228, + 375 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 362, + 300, + 376 + ], + "score": 1.0, + "content": "norm bounded by", + "type": "text" + }, + { + "bbox": [ + 300, + 364, + 366, + 375 + ], + "score": 0.91, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( k , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 362, + 435, + 376 + ], + "score": 1.0, + "content": ", such that for all", + "type": "text" + }, + { + "bbox": [ + 435, + 363, + 504, + 374 + ], + "score": 0.91, + "content": "\\pmb { x } \\in \\mathcal { D } ^ { k } \\subset \\mathbb { R } ^ { k \\times d }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 372, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 189, + 389 + ], + "score": 1.0, + "content": "and all perturbation", + "type": "text" + }, + { + "bbox": [ + 190, + 375, + 281, + 388 + ], + "score": 0.91, + "content": "\\pmb { \\delta } \\in [ - 1 / 2 k , 1 / 2 k ] ^ { k \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 372, + 320, + 389 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 320, + 375, + 475, + 388 + ], + "score": 0.91, + "content": "\\| f ( \\pmb { x } + \\pmb { \\delta } ) - \\pmb { g } ( \\pmb { x } ) \\| _ { \\infty } \\le \\epsilon + 2 k \\| \\pmb { \\delta } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 372, + 506, + 389 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 387, + 362, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 387, + 131, + 399 + ], + "score": 0.91, + "content": "\\| \\delta \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 387, + 184, + 399 + ], + "score": 1.0, + "content": "is the vector", + "type": "text" + }, + { + "bbox": [ + 185, + 387, + 198, + 398 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 387, + 351, + 399 + ], + "score": 1.0, + "content": "-norm applied to the flattended matrix", + "type": "text" + }, + { + "bbox": [ + 352, + 388, + 358, + 397 + ], + "score": 0.63, + "content": "\\pmb { \\delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 387, + 362, + 399 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 338, + 507, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 428, + 424 + ], + "score": 1.0, + "content": "Proof. We can simply use a two-layer MLP with ReLU activation to implement", + "type": "text" + }, + { + "bbox": [ + 428, + 412, + 435, + 421 + ], + "score": 0.77, + "content": "\\textbf { { g } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 407, + 505, + 424 + ], + "score": 1.0, + "content": "by the following", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 340, + 433 + ], + "score": 1.0, + "content": "construction. Denote the index of the MLP hidden layer as", + "type": "text" + }, + { + "bbox": [ + 340, + 420, + 416, + 433 + ], + "score": 0.94, + "content": "( i _ { 1 } , \\cdot \\cdot \\cdot , i _ { k } ) \\bar { \\in [ d ] ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 420, + 505, + 433 + ], + "score": 1.0, + "content": ". We can construct the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 431, + 267, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 267, + 443 + ], + "score": 1.0, + "content": "weights of the first MLP layer such that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 407, + 505, + 443 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 444, + 426, + 459 + ], + "lines": [ + { + "bbox": [ + 183, + 444, + 426, + 459 + ], + "spans": [ + { + "bbox": [ + 183, + 444, + 426, + 459 + ], + "score": 0.87, + "content": "h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\pmb x } ) = 2 ( x _ { i _ { 1 } } + x _ { d + i _ { 2 } } \\cdot \\cdot \\cdot + x _ { ( k - 1 ) d + i _ { k } } ) - 2 k + 1 .", + "type": "interline_equation", + "image_path": "f8f6e9a8ccdabc8cb213eac96e4cd2b1d5372a22aa7aa1f466097e6d62aff25e.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 183, + 444, + 426, + 459 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 479, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 482, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 482, + 473 + ], + "score": 1.0, + "content": "We can then construct the weights of the second layer such that the final output of the MLP is", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 459, + 482, + 473 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 473, + 401, + 503 + ], + "lines": [ + { + "bbox": [ + 209, + 473, + 401, + 503 + ], + "spans": [ + { + "bbox": [ + 209, + 473, + 401, + 503 + ], + "score": 0.92, + "content": "f _ { j } ( \\pmb { x } ) = \\sum _ { g _ { j } ( e _ { i _ { 1 } } , \\cdots , e _ { i _ { k } } ) = 1 } { \\mathrm { R e L U } } ( h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( \\pmb { x } ) ) .", + "type": "interline_equation", + "image_path": "087e1e60efa04bd95b1f3bb934765df4ac7853fb6d8f5a167bed2a999a1820b8.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 473, + 401, + 488.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 209, + 488.0, + 401, + 503.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 455, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 503, + 457, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 185, + 520 + ], + "score": 1.0, + "content": "One can check that", + "type": "text" + }, + { + "bbox": [ + 186, + 506, + 241, + 518 + ], + "score": 0.93, + "content": "\\pmb { f } ( \\pmb { x } ) = \\pmb { g } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 503, + 293, + 520 + ], + "score": 1.0, + "content": "holds for all", + "type": "text" + }, + { + "bbox": [ + 293, + 505, + 361, + 516 + ], + "score": 0.92, + "content": "\\pmb { x } \\in \\mathcal { D } ^ { k } \\subset \\mathbb { R } ^ { d \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 503, + 457, + 520 + ], + "score": 1.0, + "content": ". Furthermore, we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 503, + 457, + 520 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 519, + 510, + 596 + ], + "lines": [ + { + "bbox": [ + 111, + 519, + 510, + 596 + ], + "spans": [ + { + "bbox": [ + 111, + 519, + 510, + 596 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { f ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) - g ( { \\boldsymbol x } ) \\| _ { \\infty } = \\| f ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) - f ( { \\boldsymbol x } ) \\| _ { \\infty } } & { } \\\\ & { \\qquad = \\underset { j \\in [ d ] } { \\operatorname* { m a x } } \\left| \\sum _ { \\substack { \\boldsymbol y _ { j } ( e _ { i _ { 1 } } , \\ldots , e _ { i _ { k } } ) = 1 } } \\left( \\operatorname { R e L U } ( h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) ) - \\operatorname { R e L U } ( h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } ) ) \\right) \\right| } \\\\ & { \\qquad \\le \\underset { ( i _ { 1 } , \\cdots , i _ { k } ) \\in [ d ] ^ { k } } { \\operatorname* { m a x } } \\left| h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } + { \\boldsymbol \\delta } ) - h _ { ( i _ { 1 } , \\cdots , i _ { k } ) } ( { \\boldsymbol x } ) \\right| \\le 2 k \\| { \\boldsymbol \\delta } \\| _ { \\infty } } \\end{array}", + "type": "interline_equation", + "image_path": "74e9f6364147cc961657b35441ba13ee92951f57f7b13101ddf75b200a9fa4a2.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 109, + 519, + 510, + 544.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 109, + 544.6666666666666, + 510, + 570.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 109, + 570.3333333333333, + 510, + 595.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 598, + 503, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 186, + 613 + ], + "score": 1.0, + "content": "for all perturbations", + "type": "text" + }, + { + "bbox": [ + 187, + 598, + 279, + 612 + ], + "score": 0.92, + "content": "\\pmb { \\delta } \\in [ - 1 / 2 k , 1 / 2 k ] ^ { k \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 596, + 483, + 613 + ], + "score": 1.0, + "content": ". Thus by using Lemma C.2, we can also implement", + "type": "text" + }, + { + "bbox": [ + 484, + 599, + 505, + 611 + ], + "score": 0.9, + "content": "\\pmb { g } ( \\pmb { x } )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 416, + 623 + ], + "score": 1.0, + "content": "by a two-layer MLP with GeLU activation and all parameters are bounded by", + "type": "text" + }, + { + "bbox": [ + 416, + 610, + 482, + 622 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( k , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 609, + 506, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 596, + 506, + 623 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 634, + 304, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 305, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 305, + 648 + ], + "score": 1.0, + "content": "C.2 Technical lemmas for the attention layer", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "In this subsection, we will introduce two special operations that can be performed by the attention", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 663, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 256, + 678 + ], + "score": 1.0, + "content": "layer (with causal mask). Below, let", + "type": "text" + }, + { + "bbox": [ + 256, + 666, + 283, + 675 + ], + "score": 0.89, + "content": "n \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 663, + 370, + 678 + ], + "score": 1.0, + "content": "be an integer and let", + "type": "text" + }, + { + "bbox": [ + 370, + 667, + 433, + 676 + ], + "score": 0.89, + "content": "{ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\cdots , { \\pmb x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 663, + 506, + 678 + ], + "score": 1.0, + "content": "be a sequence of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 673, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 673, + 166, + 690 + ], + "score": 1.0, + "content": "vectors where", + "type": "text" + }, + { + "bbox": [ + 166, + 676, + 298, + 688 + ], + "score": 0.86, + "content": "\\pmb { x } _ { i } = ( \\tilde { \\pmb { x } } _ { i } , r _ { i } , 1 ) \\in [ - M , M ] ^ { d + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 673, + 302, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 676, + 339, + 687 + ], + "score": 0.76, + "content": "\\tilde { \\mathbf { x } } _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 673, + 343, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 343, + 676, + 373, + 687 + ], + "score": 0.82, + "content": "r _ { i } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 673, + 394, + 690 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 395, + 677, + 407, + 686 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 673, + 507, + 690 + ], + "score": 1.0, + "content": "is a large constant. Let", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 685, + 508, + 704 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 198, + 700 + ], + "score": 0.9, + "content": "K , Q , V \\in \\mathbb { R } ^ { d ^ { \\prime } \\times ( d + 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 685, + 284, + 704 + ], + "score": 1.0, + "content": "be any matrices with", + "type": "text" + }, + { + "bbox": [ + 284, + 689, + 330, + 700 + ], + "score": 0.94, + "content": "\\| \\ b { V } \\| _ { \\infty } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 685, + 363, + 704 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 363, + 689, + 421, + 700 + ], + "score": 0.91, + "content": "0 < \\rho , \\delta < M", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 685, + 508, + 704 + ], + "score": 1.0, + "content": "be any real numbers.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 137, + 713 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 137, + 700, + 179, + 712 + ], + "score": 0.83, + "content": "\\pmb q _ { i } = \\pmb Q x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 699, + 183, + 713 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 183, + 700, + 228, + 712 + ], + "score": 0.79, + "content": "k _ { j } = K x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 699, + 232, + 713 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 232, + 700, + 276, + 712 + ], + "score": 0.87, + "content": "{ \\pmb v } _ { j } = { \\pmb V } { \\pmb x } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 699, + 389, + 713 + ], + "score": 1.0, + "content": ", and define the matching set", + "type": "text" + }, + { + "bbox": [ + 390, + 700, + 503, + 712 + ], + "score": 0.92, + "content": "S _ { i } = \\{ j \\leq i : | \\pmb { q } _ { i } \\cdot \\pmb { k } _ { j } | \\leq \\rho \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 699, + 507, + 713 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 402, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 402, + 723 + ], + "score": 1.0, + "content": "Equipped with these notations, we define two basic operations as follows:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 653, + 508, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 127, + 72, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 126, + 70, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 126, + 70, + 313, + 87 + ], + "score": 1.0, + "content": "• COPY: The output is a sequence of vectors", + "type": "text" + }, + { + "bbox": [ + 313, + 75, + 361, + 84 + ], + "score": 0.9, + "content": "\\mathbf { } u _ { 1 } , \\cdots , \\mathbf { } u _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 70, + 383, + 87 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 384, + 75, + 434, + 86 + ], + "score": 0.91, + "content": "{ \\mathbf { } } u _ { i } = { \\mathbf { } } v _ { \\mathrm { p o s } ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 70, + 466, + 87 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 466, + 73, + 506, + 85 + ], + "score": 0.88, + "content": "\\mathrm { p o s } ( i ) =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 136, + 83, + 203, + 100 + ], + "spans": [ + { + "bbox": [ + 136, + 86, + 198, + 97 + ], + "score": 0.78, + "content": "\\operatorname { a r g m a x } _ { j \\in S _ { i } } r _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 83, + 203, + 100 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 126, + 93, + 465, + 111 + ], + "spans": [ + { + "bbox": [ + 126, + 93, + 314, + 111 + ], + "score": 1.0, + "content": "• MEAN: The output is a sequence of vectors", + "type": "text" + }, + { + "bbox": [ + 314, + 98, + 362, + 108 + ], + "score": 0.9, + "content": "\\mathbf { } u _ { 1 } , \\cdots , \\mathbf { } u _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 93, + 384, + 111 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 384, + 98, + 461, + 109 + ], + "score": 0.9, + "content": "\\mathbf { \\Delta } \\mathbf { u } _ { i } = \\mathrm { m e a n } _ { j \\in S _ { i } } v _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 93, + 465, + 111 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 115, + 111, + 481, + 124 + ], + "lines": [ + { + "bbox": [ + 112, + 110, + 480, + 127 + ], + "spans": [ + { 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We next make the following regularity assumption:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 105, + 126, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 237, + 140 + ], + "score": 1.0, + "content": "Assumption C.6. The matrices", + "type": "text" + }, + { + "bbox": [ + 238, + 126, + 276, + 138 + ], + "score": 0.91, + "content": "Q , K , V", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 125, + 325, + 140 + ], + "score": 1.0, + "content": "and scalars", + "type": "text" + }, + { + "bbox": [ + 325, + 127, + 340, + 138 + ], + "score": 0.9, + "content": "\\rho , \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "satisfy that for all considered sequences", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 138, + 252, + 149 + ], + "spans": [ + { + "bbox": [ + 107, + 139, + 170, + 149 + ], + "score": 0.9, + "content": "{ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\cdots , { \\pmb x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 138, + 252, + 149 + ], + "score": 1.0, + "content": ", the following hold:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 127, + 153, + 355, + 183 + ], + "lines": [ + { + "bbox": [ + 127, + 151, + 356, + 167 + ], + "spans": [ + { + "bbox": [ + 127, + 151, + 169, + 167 + ], + "score": 1.0, + "content": "• For any", + "type": "text" + }, + { + "bbox": [ + 169, + 153, + 207, + 165 + ], + "score": 0.92, + "content": "i , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 151, + 236, + 167 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 236, + 153, + 287, + 166 + ], + "score": 0.92, + "content": "| \\boldsymbol { q } _ { i } \\cdot \\boldsymbol { k } _ { j } | \\le \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 151, + 299, + 167 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 299, + 153, + 352, + 166 + ], + "score": 0.91, + "content": "\\pmb q _ { i } \\cdot \\pmb k _ { j } \\le - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 151, + 356, + 167 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 127, + 168, + 327, + 183 + ], + "spans": [ + { + "bbox": [ + 127, + 168, + 169, + 183 + ], + "score": 1.0, + "content": "• For any", + "type": "text" + }, + { + "bbox": [ + 169, + 170, + 207, + 182 + ], + "score": 0.92, + "content": "i , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 168, + 235, + 183 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 236, + 171, + 258, + 182 + ], + "score": 0.89, + "content": "i = j", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 168, + 270, + 183 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 271, + 169, + 324, + 182 + ], + "score": 0.91, + "content": "| r _ { i } - r _ { j } | \\ge \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 168, + 327, + 183 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 506, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 456, + 202 + ], + "score": 1.0, + "content": "Assumption C.6 says that there are sufficient gaps between the attended position (e.g.,", + "type": "text" + }, + { + "bbox": [ + 456, + 189, + 486, + 201 + ], + "score": 0.84, + "content": "\\mathrm { p o s } ( i ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 506, + 212 + ], + "score": 1.0, + "content": "other positions. The two lemmas below show that the attention layer with casual mask can implement", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 329, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 329, + 224 + ], + "score": 1.0, + "content": "both COPY operation and MEAN operation efficiently.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 506, + 294 + ], + "lines": [ + { + "bbox": [ + 103, + 222, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 103, + 222, + 362, + 243 + ], + "score": 1.0, + "content": "Lemma C.7. Assume Assumption C.6 holds with ρ ≤ δ28M .", + "type": "text" + }, + { + "bbox": [ + 359, + 225, + 397, + 239 + ], + "score": 1.0, + "content": "For any", + "type": "text" + }, + { + "bbox": [ + 397, + 227, + 424, + 237 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 225, + 506, + 239 + ], + "score": 1.0, + "content": ", there exists an at-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 248, + 251 + ], + "score": 1.0, + "content": "tention layer with embedding size", + "type": "text" + }, + { + "bbox": [ + 248, + 238, + 270, + 250 + ], + "score": 0.89, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "and one causal attention head that can approximate the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 507, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 441, + 263 + ], + "score": 1.0, + "content": "COPY operation defined above. Formally, for any considered sequence of vectors", + "type": "text" + }, + { + "bbox": [ + 441, + 251, + 503, + 260 + ], + "score": 0.86, + "content": "{ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\ldots , { \\pmb x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 247, + 507, + 263 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 298, + 273 + ], + "score": 1.0, + "content": "denote the corresponding attention output as", + "type": "text" + }, + { + "bbox": [ + 298, + 262, + 357, + 271 + ], + "score": 0.88, + "content": "o _ { 1 } , o _ { 2 } , \\ldots , o _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 259, + 430, + 273 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + }, + { + "bbox": [ + 430, + 260, + 505, + 272 + ], + "score": 0.89, + "content": "\\| \\pmb { o } _ { i } - \\pmb { u } _ { i } \\| _ { \\infty } \\le \\epsilon", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 136, + 284 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 137, + 271, + 171, + 282 + ], + "score": 0.89, + "content": "i \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 271, + 194, + 284 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 195, + 271, + 229, + 282 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\boldsymbol { S } _ { i } \\neq \\boldsymbol { \\emptyset } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 271, + 299, + 284 + ], + "score": 1.0, + "content": ". Moreover, the", + "type": "text" + }, + { + "bbox": [ + 299, + 271, + 313, + 282 + ], + "score": 0.87, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "norm of attention parameters is bounded by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 253, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 247, + 294 + ], + "score": 0.85, + "content": "O ( \\mathrm { p o l y } ( M , \\bar { 1 } / \\bar { \\delta } , \\log ( n ) , \\log ( 1 / \\epsilon ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 281, + 253, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "Proof. The purpose of the attention head is to focus only on the vector that needs to be copied. To", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 313, + 507, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 507, + 330 + ], + "score": 1.0, + "content": "achieve this, we construct the key, query, and value vectors as follows (by assigning suitable key,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 327, + 330, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 330, + 338 + ], + "score": 1.0, + "content": "query, and value weight matrices in the attention head):", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 127, + 342, + 235, + 389 + ], + "lines": [ + { + "bbox": [ + 127, + 342, + 235, + 355 + ], + "spans": [ + { + "bbox": [ + 127, + 342, + 167, + 354 + ], + "score": 1.0, + "content": "• Query:", + "type": "text" + }, + { + "bbox": [ + 168, + 342, + 235, + 355 + ], + "score": 0.81, + "content": "( \\lambda \\pmb q _ { i } , \\mu ) \\in \\mathbb { R } ^ { d + 1 }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 126, + 358, + 223, + 372 + ], + "spans": [ + { + "bbox": [ + 126, + 358, + 158, + 372 + ], + "score": 1.0, + "content": "• Key:", + "type": "text" + }, + { + "bbox": [ + 159, + 359, + 223, + 372 + ], + "score": 0.52, + "content": "( k _ { i } , r _ { i } ) \\in \\mathbb { R } ^ { d + 1 }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 127, + 376, + 200, + 389 + ], + "spans": [ + { + "bbox": [ + 127, + 376, + 200, + 389 + ], + "score": 0.34, + "content": "\\bullet \\mathrm { ~ V a l u e : ~ } v _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation", + "image_path": "5d4b0dfb8ba7098b6a21aad4087d32367abf3aafe20fd916800cc6ece4e47c27.jpg" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 103, + 393, + 477, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 478, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 133, + 407 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 394, + 140, + 404 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 392, + 158, + 407 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 396, + 165, + 405 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 392, + 354, + 407 + ], + "score": 1.0, + "content": "are constants that will be defined later. Denote", + "type": "text" + }, + { + "bbox": [ + 354, + 396, + 367, + 407 + ], + "score": 0.88, + "content": "a _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 392, + 478, + 407 + ], + "score": 1.0, + "content": "as the attention score, then", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 408, + 459, + 438 + ], + "lines": [ + { + "bbox": [ + 151, + 408, + 459, + 438 + ], + "spans": [ + { + "bbox": [ + 151, + 408, + 459, + 438 + ], + "score": 0.94, + "content": "a _ { i , j } = { \\frac { \\exp ( \\lambda ( q _ { i } \\cdot k _ { j } ) + \\mu r _ { j } ) } { \\sum _ { j ^ { \\prime } } \\exp ( \\lambda ( q _ { i } \\cdot k _ { j ^ { \\prime } } ) + \\mu r _ { j ^ { \\prime } } ) } } = { \\frac { \\exp ( \\lambda ( q _ { i } \\cdot k _ { j } ) ) } { \\sum _ { j ^ { \\prime } } \\exp ( \\lambda ( q _ { i } \\cdot k _ { j ^ { \\prime } } ) + \\mu ( r _ { j ^ { \\prime } } - r _ { j } ) ) } } .", + "type": "interline_equation", + "image_path": "7bf0c8a13aaa0fae83d6ce1de0a7799b3e013676e98a23bdd8e09b5845de7b9e.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 151, + 408, + 459, + 418.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 151, + 418.0, + 459, + 428.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 151, + 428.0, + 459, + 438.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 441, + 502, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 503, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 173, + 459 + ], + "score": 1.0, + "content": "Since ρ ≤ δ 8M", + "type": "text" + }, + { + "bbox": [ + 167, + 438, + 186, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 443, + 218, + 455 + ], + "score": 0.89, + "content": "M \\geq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 438, + 258, + 456 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 258, + 443, + 308, + 456 + ], + "score": 0.89, + "content": "\\delta - \\rho \\ge \\frac { 7 } { 8 } \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 438, + 358, + 456 + ], + "score": 1.0, + "content": ". 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We next make the following regularity assumption:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 112, + 110, + 480, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 126, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 237, + 140 + ], + "score": 1.0, + "content": "Assumption C.6. The matrices", + "type": "text" + }, + { + "bbox": [ + 238, + 126, + 276, + 138 + ], + "score": 0.91, + "content": "Q , K , V", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 125, + 325, + 140 + ], + "score": 1.0, + "content": "and scalars", + "type": "text" + }, + { + "bbox": [ + 325, + 127, + 340, + 138 + ], + "score": 0.9, + "content": "\\rho , \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "satisfy that for all considered sequences", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 138, + 252, + 149 + ], + "spans": [ + { + "bbox": [ + 107, + 139, + 170, + 149 + ], + "score": 0.9, + "content": "{ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\cdots , { \\pmb x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 138, + 252, + 149 + ], + "score": 1.0, + "content": ", the following hold:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 125, + 505, + 149 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 153, + 355, + 183 + ], + "lines": [ + { + "bbox": [ + 127, + 151, + 356, + 167 + ], + "spans": [ + { + "bbox": [ + 127, + 151, + 169, + 167 + ], + "score": 1.0, + "content": "• For any", + "type": "text" + }, + { + "bbox": [ + 169, + 153, + 207, + 165 + ], + "score": 0.92, + "content": "i , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 151, + 236, + 167 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 236, + 153, + 287, + 166 + ], + "score": 0.92, + "content": "| \\boldsymbol { q } _ { i } \\cdot \\boldsymbol { k } _ { j } | \\le \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 151, + 299, + 167 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 299, + 153, + 352, + 166 + ], + "score": 0.91, + "content": "\\pmb q _ { i } \\cdot \\pmb k _ { j } \\le - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 151, + 356, + 167 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6, + "is_list_end_line": true + }, + { + "bbox": [ + 127, + 168, + 327, + 183 + ], + "spans": [ + { + "bbox": [ + 127, + 168, + 169, + 183 + ], + "score": 1.0, + "content": "• For any", + "type": "text" + }, + { + "bbox": [ + 169, + 170, + 207, + 182 + ], + "score": 0.92, + "content": "i , j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 168, + 235, + 183 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 236, + 171, + 258, + 182 + ], + "score": 0.89, + "content": "i = j", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 168, + 270, + 183 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 271, + 169, + 324, + 182 + ], + "score": 0.91, + "content": "| r _ { i } - r _ { j } | \\ge \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 168, + 327, + 183 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 6.5, + "bbox_fs": [ + 127, + 151, + 356, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 506, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 456, + 202 + ], + "score": 1.0, + "content": "Assumption C.6 says that there are sufficient gaps between the attended position (e.g.,", + "type": "text" + }, + { + "bbox": [ + 456, + 189, + 486, + 201 + ], + "score": 0.84, + "content": "\\mathrm { p o s } ( i ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 506, + 212 + ], + "score": 1.0, + "content": "other positions. The two lemmas below show that the attention layer with casual mask can implement", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 329, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 329, + 224 + ], + "score": 1.0, + "content": "both COPY operation and MEAN operation efficiently.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 188, + 506, + 224 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 506, + 294 + ], + "lines": [ + { + "bbox": [ + 103, + 222, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 103, + 222, + 362, + 243 + ], + "score": 1.0, + "content": "Lemma C.7. Assume Assumption C.6 holds with ρ ≤ δ28M .", + "type": "text" + }, + { + "bbox": [ + 359, + 225, + 397, + 239 + ], + "score": 1.0, + "content": "For any", + "type": "text" + }, + { + "bbox": [ + 397, + 227, + 424, + 237 + ], + "score": 0.87, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 225, + 506, + 239 + ], + "score": 1.0, + "content": ", there exists an at-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 248, + 251 + ], + "score": 1.0, + "content": "tention layer with embedding size", + "type": "text" + }, + { + "bbox": [ + 248, + 238, + 270, + 250 + ], + "score": 0.89, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "and one causal attention head that can approximate the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 507, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 441, + 263 + ], + "score": 1.0, + "content": "COPY operation defined above. Formally, for any considered sequence of vectors", + "type": "text" + }, + { + "bbox": [ + 441, + 251, + 503, + 260 + ], + "score": 0.86, + "content": "{ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , \\ldots , { \\pmb x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 247, + 507, + 263 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 298, + 273 + ], + "score": 1.0, + "content": "denote the corresponding attention output as", + "type": "text" + }, + { + "bbox": [ + 298, + 262, + 357, + 271 + ], + "score": 0.88, + "content": "o _ { 1 } , o _ { 2 } , \\ldots , o _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 259, + 430, + 273 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + }, + { + "bbox": [ + 430, + 260, + 505, + 272 + ], + "score": 0.89, + "content": "\\| \\pmb { o } _ { i } - \\pmb { u } _ { i } \\| _ { \\infty } \\le \\epsilon", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 136, + 284 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 137, + 271, + 171, + 282 + ], + "score": 0.89, + "content": "i \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 271, + 194, + 284 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 195, + 271, + 229, + 282 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\boldsymbol { S } _ { i } \\neq \\boldsymbol { \\emptyset } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 271, + 299, + 284 + ], + "score": 1.0, + "content": ". Moreover, the", + "type": "text" + }, + { + "bbox": [ + 299, + 271, + 313, + 282 + ], + "score": 0.87, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "norm of attention parameters is bounded by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 253, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 247, + 294 + ], + "score": 0.85, + "content": "O ( \\mathrm { p o l y } ( M , \\bar { 1 } / \\bar { \\delta } , \\log ( n ) , \\log ( 1 / \\epsilon ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 281, + 253, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 103, + 222, + 507, + 294 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "Proof. The purpose of the attention head is to focus only on the vector that needs to be copied. To", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 313, + 507, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 507, + 330 + ], + "score": 1.0, + "content": "achieve this, we construct the key, query, and value vectors as follows (by assigning suitable key,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 327, + 330, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 330, + 338 + ], + "score": 1.0, + "content": "query, and value weight matrices in the attention head):", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 304, + 507, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 127, + 342, + 235, + 389 + ], + "lines": [ + { + "bbox": [ + 127, + 342, + 235, + 355 + ], + "spans": [ + { + "bbox": [ + 127, + 342, + 167, + 354 + ], + "score": 1.0, + "content": "• Query:", + "type": "text" + }, + { + "bbox": [ + 168, + 342, + 235, + 355 + ], + "score": 0.81, + "content": "( \\lambda \\pmb q _ { i } , \\mu ) \\in \\mathbb { R } ^ { d + 1 }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 126, + 358, + 223, + 372 + ], + "spans": [ + { + "bbox": [ + 126, + 358, + 158, + 372 + ], + "score": 1.0, + "content": "• Key:", + "type": "text" + }, + { + "bbox": [ + 159, + 359, + 223, + 372 + ], + "score": 0.52, + "content": "( k _ { i } , r _ { i } ) \\in \\mathbb { R } ^ { d + 1 }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 127, + 376, + 200, + 389 + ], + "spans": [ + { + "bbox": [ + 127, + 376, + 200, + 389 + ], + "score": 0.34, + "content": "\\bullet \\mathrm { ~ V a l u e : ~ } v _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation", + "image_path": "5d4b0dfb8ba7098b6a21aad4087d32367abf3aafe20fd916800cc6ece4e47c27.jpg" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 126, + 342, + 235, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 393, + 477, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 478, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 133, + 407 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 394, + 140, + 404 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 392, + 158, + 407 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 396, + 165, + 405 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 392, + 354, + 407 + ], + "score": 1.0, + "content": "are constants that will be defined later. Denote", + "type": "text" + }, + { + "bbox": [ + 354, + 396, + 367, + 407 + ], + "score": 0.88, + "content": "a _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 392, + 478, + 407 + ], + "score": 1.0, + "content": "as the attention score, then", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 392, + 478, + 407 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 408, + 459, + 438 + ], + "lines": [ + { + "bbox": [ + 151, + 408, + 459, + 438 + ], + "spans": [ + { + "bbox": [ + 151, + 408, + 459, + 438 + ], + "score": 0.94, + "content": "a _ { i , j } = { \\frac { \\exp ( \\lambda ( q _ { i } \\cdot k _ { j } ) + \\mu r _ { j } ) } { \\sum _ { j ^ { \\prime } } \\exp ( \\lambda ( q _ { i } \\cdot k _ { j ^ { \\prime } } ) + \\mu r _ { j ^ { \\prime } } ) } } = { \\frac { \\exp ( \\lambda ( q _ { i } \\cdot k _ { j } ) ) } { \\sum _ { j ^ { \\prime } } \\exp ( \\lambda ( q _ { i } \\cdot k _ { j ^ { \\prime } } ) + \\mu ( r _ { j ^ { \\prime } } - r _ { j } ) ) } } .", + "type": "interline_equation", + "image_path": "7bf0c8a13aaa0fae83d6ce1de0a7799b3e013676e98a23bdd8e09b5845de7b9e.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 151, + 408, + 459, + 418.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 151, + 418.0, + 459, + 428.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 151, + 428.0, + 459, + 438.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 441, + 502, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 503, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 173, + 459 + ], + "score": 1.0, + "content": "Since ρ ≤ δ 8M", + "type": "text" + }, + { + "bbox": [ + 167, + 438, + 186, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 443, + 218, + 455 + ], + "score": 0.89, + "content": "M \\geq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 438, + 258, + 456 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 258, + 443, + 308, + 456 + ], + "score": 0.89, + "content": "\\delta - \\rho \\ge \\frac { 7 } { 8 } \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 438, + 358, + 456 + ], + "score": 1.0, + "content": ". By setting", + "type": "text" + }, + { + "bbox": [ + 358, + 439, + 425, + 457 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\lambda = \\frac { 8 M \\ln ( \\frac { 2 n M } { \\epsilon } ) } { \\delta ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 438, + 444, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 439, + 503, + 457 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mu = \\frac { 3 \\ln ( \\frac { 2 n M } { \\epsilon } ) } { \\delta } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 455, + 385, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 200, + 467 + ], + "score": 1.0, + "content": "(which are bounded by", + "type": "text" + }, + { + "bbox": [ + 200, + 455, + 343, + 467 + ], + "score": 0.9, + "content": "O ( \\mathrm { p o l y } ( M , 1 / \\delta , \\log ( n ) , \\log ( 1 / \\epsilon ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 455, + 385, + 467 + ], + "score": 1.0, + "content": "), we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 438, + 503, + 467 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 470, + 444, + 615 + ], + "lines": [ + { + "bbox": [ + 167, + 470, + 444, + 615 + ], + "spans": [ + { + "bbox": [ + 167, + 470, + 444, + 615 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { a _ { i , \\mathrm { p o s } ( i ) } \\geq \\frac { \\exp ( - \\lambda \\rho ) } { \\exp ( - \\lambda \\rho ) + ( n - 1 ) \\exp ( \\operatorname* { m a x } ( - \\lambda \\delta ) + 2 M \\mu , \\lambda \\rho - \\mu \\delta ) ) } } \\\\ & { = \\frac { 1 } { 1 + ( n - 1 ) \\exp ( \\operatorname* { m a x } ( - \\lambda ( \\delta - \\rho ) + 2 M \\mu , 2 \\lambda \\rho - \\mu \\delta ) ) } } \\\\ & { \\geq 1 - n \\exp ( \\operatorname* { m a x } ( - \\lambda ( \\delta - \\rho ) + 2 M \\mu , 2 \\lambda \\rho - \\mu \\delta ) ) } \\\\ & { \\geq 1 - n \\exp \\bigg ( \\operatorname* { m a x } \\bigg ( - \\frac { M } { \\delta } \\ln \\bigg ( \\frac { 2 n M } { \\epsilon } \\bigg ) , - \\ln \\bigg ( \\frac { 2 n M } { \\epsilon } \\bigg ) \\bigg ) \\bigg ) } \\\\ & { \\geq 1 - n \\exp \\bigg ( - \\ln \\bigg ( \\frac { 2 n M } { \\epsilon } \\bigg ) \\bigg ) } \\\\ & { = 1 - \\frac { \\epsilon } { 2 M } , } \\end{array}", + "type": "interline_equation", + "image_path": "e996fb288b31f39bcf4129ac9a2f2706c9974aea5755cb31280d79de4b42fc5f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 167, + 470, + 444, + 518.3333333333334 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 167, + 518.3333333333334, + 444, + 566.6666666666667 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 167, + 566.6666666666667, + 444, + 615.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 504, + 653 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 372, + 630 + ], + "score": 1.0, + "content": "where in (10) we use Assumption C.6, which implies that whenever", + "type": "text" + }, + { + "bbox": [ + 373, + 617, + 420, + 629 + ], + "score": 0.93, + "content": "j ^ { \\prime } \\ne \\mathrm { p o s } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 615, + 449, + 630 + ], + "score": 1.0, + "content": ", either", + "type": "text" + }, + { + "bbox": [ + 449, + 617, + 504, + 629 + ], + "score": 0.9, + "content": "\\mathbf { \\boldsymbol { q } } _ { i } \\cdot \\mathbf { \\boldsymbol { k } } _ { j ^ { \\prime } } \\leq - \\delta", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 628, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 119, + 642 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 119, + 630, + 169, + 642 + ], + "score": 0.91, + "content": "( \\pmb q _ { i } \\cdot \\pmb k _ { j ^ { \\prime } } \\leq \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 628, + 186, + 642 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 186, + 630, + 248, + 642 + ], + "score": 0.91, + "content": "r _ { j ^ { \\prime } } - r _ { j } \\leq - \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 628, + 367, + 642 + ], + "score": 1.0, + "content": "; in (11) we use the inequality", + "type": "text" + }, + { + "bbox": [ + 367, + 629, + 420, + 643 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { 1 + x } \\geq 1 - x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 629, + 448, + 642 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 448, + 630, + 473, + 640 + ], + "score": 0.88, + "content": "x \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "; in (12)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 641, + 276, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 185, + 654 + ], + "score": 1.0, + "content": "we use the fact that", + "type": "text" + }, + { + "bbox": [ + 186, + 642, + 216, + 653 + ], + "score": 0.9, + "content": "M \\geq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 641, + 276, + 654 + ], + "score": 1.0, + "content": ". We thus have", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 615, + 505, + 654 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 654, + 478, + 711 + ], + "lines": [ + { + "bbox": [ + 132, + 654, + 478, + 711 + ], + "spans": [ + { + "bbox": [ + 132, + 654, + 478, + 711 + ], + "score": 0.93, + "content": "\\| \\pmb { o } _ { i } - \\pmb { u } _ { i } \\| _ { \\infty } = \\left\\| \\sum _ { j } a _ { i j } \\pmb { v } _ { j } - \\pmb { v } _ { \\mathrm { p o s } ( i ) } \\right\\| _ { \\infty } \\le M \\| V \\| _ { \\infty } \\cdot \\left( 1 - a _ { i , \\mathrm { p o s } ( i ) } + \\sum _ { j \\neq \\mathrm { p o s } ( i ) } a _ { i , j } \\right)", + "type": "interline_equation", + "image_path": "bb16c95909bf08a758dd2162dc897a3f87c0b68aaf004e5f6ededdef2b2db5ec.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 132, + 654, + 478, + 673.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 132, + 673.0, + 478, + 692.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 132, + 692.0, + 478, + 711.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 711, + 216, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 216, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 216, + 723 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 106, + 710, + 216, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 71, + 506, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 507, + 89 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 302, + 88 + ], + "score": 1.0, + "content": "Lemma C.8. Assume Assumption C.6 holds with", + "type": "text" + }, + { + "bbox": [ + 316, + 70, + 507, + 89 + ], + "score": 1.0, + "content": "δϵ16M ln( 4Mn ) . For any 0 < ϵ ≤ M , there exists", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 506, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 261, + 101 + ], + "score": 1.0, + "content": "an attention layer with embedding size", + "type": "text" + }, + { + "bbox": [ + 261, + 88, + 282, + 100 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 87, + 506, + 101 + ], + "score": 1.0, + "content": "and one causal attention head that can approximate the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 97, + 507, + 113 + ], + "spans": [ + { + "bbox": [ + 104, + 97, + 441, + 113 + ], + "score": 1.0, + "content": "MEAN operation defined above. Formally, for any considered sequence of vectors", + "type": "text" + }, + { + "bbox": [ + 443, + 101, + 502, + 110 + ], + "score": 0.83, + "content": "\\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 97, + 507, + 113 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 223, + 123 + ], + "score": 1.0, + "content": "denote the attention output as", + "type": "text" + }, + { + "bbox": [ + 224, + 111, + 282, + 121 + ], + "score": 0.86, + "content": "o _ { 1 } , o _ { 2 } , \\ldots , o _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 109, + 344, + 123 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + }, + { + "bbox": [ + 344, + 110, + 408, + 121 + ], + "score": 0.89, + "content": "\\| \\pmb { O } _ { i } - \\pmb { u } _ { i } \\| _ { \\infty } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 109, + 435, + 123 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 435, + 110, + 463, + 121 + ], + "score": 0.88, + "content": "i \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 109, + 482, + 123 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 483, + 110, + 505, + 121 + ], + "score": 0.88, + "content": "\\quad S _ { i } \\neq ", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 107, + 120, + 113, + 131 + ], + "score": 0.47, + "content": "\\varnothing .", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 119, + 172, + 133 + ], + "score": 1.0, + "content": ". Moreover, the", + "type": "text" + }, + { + "bbox": [ + 173, + 121, + 187, + 132 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 119, + 360, + 133 + ], + "score": 1.0, + "content": "norm of attention parameters is bounded by", + "type": "text" + }, + { + "bbox": [ + 361, + 121, + 502, + 132 + ], + "score": 0.88, + "content": "O ( \\mathrm { p o l y } ( M , 1 / \\delta , \\log ( n ) , \\log ( 1 / \\epsilon ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 119, + 506, + 133 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 104, + 142, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "Proof. The purpose of the attention head is to average across all tokens that satisfy the condition", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 154, + 452, + 167 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 152, + 167 + ], + "score": 0.92, + "content": "\\pmb q _ { i } \\cdot \\pmb k _ { j } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 154, + 452, + 167 + ], + "score": 1.0, + "content": ". To achieve this, we construct the key, query, and value vectors as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 170, + 205, + 217 + ], + "lines": [ + { + "bbox": [ + 125, + 170, + 205, + 217 + ], + "spans": [ + { + "bbox": [ + 125, + 170, + 205, + 217 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\bullet \\mathrm { ~ Q u e r y } \\colon \\lambda \\pmb { q } _ { i } \\in \\mathbb { R } ^ { d } } \\\\ & { \\bullet \\mathrm { ~ K e y } \\colon \\pmb { k } _ { i } \\in \\mathbb { R } ^ { d } } \\\\ & { \\bullet \\mathrm { ~ V a l u e } \\colon \\pmb { v } _ { i } \\in \\mathbb { R } ^ { d } } \\end{array}", + "type": "interline_equation", + "image_path": "d1a6fe2cee731df83cb152d51c30c5ce72d0c2c7e0b3386a657c311ac00cbcc2.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 125, + 170, + 205, + 193.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 125, + 193.5, + 205, + 217.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 460, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 220, + 459, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 133, + 234 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 222, + 140, + 231 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 220, + 335, + 234 + ], + "score": 1.0, + "content": "is a constant which will be defined later. Denote", + "type": "text" + }, + { + "bbox": [ + 336, + 224, + 349, + 234 + ], + "score": 0.88, + "content": "a _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 220, + 459, + 234 + ], + "score": 1.0, + "content": "as the attention score, then", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 235, + 361, + 264 + ], + "lines": [ + { + "bbox": [ + 249, + 235, + 361, + 264 + ], + "spans": [ + { + "bbox": [ + 249, + 235, + 361, + 264 + ], + "score": 0.94, + "content": "a _ { i , j } = \\frac { \\exp ( \\lambda \\pmb q _ { i } \\cdot \\pmb k _ { j } ) } { \\sum _ { j ^ { \\prime } } \\exp ( \\lambda \\pmb q _ { i } \\cdot \\pmb k _ { j ^ { \\prime } } ) } .", + "type": "interline_equation", + "image_path": "4d0429d415a481dcc6e69a1ca7f72c4a2f92a9236affe6862e4114100a9ef8e2.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 235, + 361, + 249.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 249, + 249.5, + 361, + 264.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 267, + 492, + 281 + ], + "lines": [ + { + "bbox": [ + 103, + 264, + 496, + 283 + ], + "spans": [ + { + "bbox": [ + 103, + 264, + 150, + 283 + ], + "score": 1.0, + "content": "By setting", + "type": "text" + }, + { + "bbox": [ + 150, + 267, + 216, + 281 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\lambda = \\frac { 1 } { \\delta } \\ln \\left( \\frac { 4 M n } { \\epsilon } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 264, + 306, + 283 + ], + "score": 1.0, + "content": "(which is bounded by", + "type": "text" + }, + { + "bbox": [ + 307, + 268, + 450, + 280 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( M , 1 / \\delta , \\log ( n ) , \\log ( 1 / \\epsilon ) ) ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 264, + 496, + 283 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 283, + 406, + 413 + ], + "lines": [ + { + "bbox": [ + 203, + 283, + 406, + 413 + ], + "spans": [ + { + "bbox": [ + 203, + 283, + 406, + 413 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\displaystyle \\sum _ { j \\notin { \\mathcal S } _ { i } } a _ { i j } \\leq \\frac { ( n - | S _ { i } | ) \\exp ( - \\lambda \\delta ) } { ( n - | S _ { i } | ) \\exp ( - \\lambda \\delta ) + | S _ { i } | \\exp ( - \\lambda \\rho ) } } & { } \\\\ { \\displaystyle } & { = \\frac { | S _ { i } | } { 1 + \\frac { | S _ { i } | } { n - | S _ { i } | } \\exp ( - \\lambda ( \\rho - \\delta ) ) } } \\\\ { \\displaystyle } & { < n \\exp ( \\lambda \\rho ) \\exp ( - \\lambda \\delta ) ) } \\\\ { \\displaystyle } & { \\leq n \\exp \\left( \\frac { \\epsilon } { 1 6 M } \\right) \\exp \\left( - \\ln \\left( \\frac { 4 M n } { \\epsilon } \\right) \\right) } \\\\ { \\displaystyle } & { < \\frac { \\epsilon } { 3 M } , } \\end{array}", + "type": "interline_equation", + "image_path": "49a50f61b24de86f34cf7b67ca08bd405ddb1ab62599ab36809ed81be0c6957d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 203, + 283, + 406, + 297.44444444444446 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 203, + 297.44444444444446, + 406, + 311.8888888888889 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 203, + 311.8888888888889, + 406, + 326.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 203, + 326.33333333333337, + 406, + 340.7777777777778 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 340.7777777777778, + 406, + 355.2222222222223 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 203, + 355.2222222222223, + 406, + 369.66666666666674 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 203, + 369.66666666666674, + 406, + 384.1111111111112 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 203, + 384.1111111111112, + 406, + 398.55555555555566 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 203, + 398.55555555555566, + 406, + 413.0000000000001 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 504, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 331, + 427 + ], + "score": 1.0, + "content": "where in (13) we use Assumption C.6, which implies that", + "type": "text" + }, + { + "bbox": [ + 331, + 414, + 381, + 426 + ], + "score": 0.92, + "content": "\\mathbf { \\boldsymbol { q } } _ { i } \\cdot \\mathbf { \\boldsymbol { k } } _ { j } \\le - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 411, + 408, + 427 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 408, + 414, + 435, + 426 + ], + "score": 0.92, + "content": "j \\notin S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 411, + 453, + 427 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 453, + 414, + 504, + 426 + ], + "score": 0.92, + "content": "q _ { i } \\cdot k _ { j } \\geq - \\rho", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 420, + 509, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 132, + 441 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 132, + 426, + 159, + 438 + ], + "score": 0.91, + "content": "j \\in { \\mathcal { S } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 426, + 276, + 441 + ], + "score": 1.0, + "content": "; in (14) we use the inequality", + "type": "text" + }, + { + "bbox": [ + 277, + 425, + 314, + 439 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 1 } { 1 + x } } < { \\frac { 1 } { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 420, + 341, + 446 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 341, + 427, + 366, + 437 + ], + "score": 0.88, + "content": "x > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 420, + 509, + 446 + ], + "score": 1.0, + "content": "; in (15) we use that the assumption", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 297, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 124, + 451 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 439, + 153, + 450 + ], + "score": 0.9, + "content": "\\epsilon \\leq M", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 438, + 221, + 451 + ], + "score": 1.0, + "content": "and the fact that", + "type": "text" + }, + { + "bbox": [ + 221, + 439, + 293, + 451 + ], + "score": 0.92, + "content": "\\exp ( 1 / 1 6 ) < 4 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 438, + 297, + 451 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 245, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 245, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 178, + 468 + ], + "score": 1.0, + "content": "Similarly, for any", + "type": "text" + }, + { + "bbox": [ + 179, + 455, + 206, + 467 + ], + "score": 0.91, + "content": "j \\in S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 454, + 245, + 468 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 469, + 483, + 631 + ], + "lines": [ + { + "bbox": [ + 110, + 469, + 483, + 631 + ], + "spans": [ + { + "bbox": [ + 110, + 469, + 483, + 631 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\displaystyle a _ { i \\ j } - \\frac { 1 } { | \\mathcal S _ { i } | } \\Big | \\leq \\operatorname* { m a x } ( \\frac { 1 } { | \\mathcal S _ { i } | } - \\frac { \\exp ( - \\rho \\lambda ) } { | \\mathcal S _ { i } | \\exp ( \\rho \\lambda ) + ( n - | \\mathcal S _ { i } | ) \\exp ( - \\lambda \\delta ) } , \\frac { \\exp ( \\rho \\lambda ) } { | \\mathcal S _ { i } | \\exp ( - \\rho \\lambda ) } - \\frac { 1 } { | \\mathcal S _ { i } | } ) } & { } \\\\ { \\displaystyle \\leq \\frac { 1 } { | \\mathcal S _ { i } | } \\operatorname* { m a x } ( 1 - \\frac { 1 } { \\exp ( 2 \\rho \\lambda ) + n \\exp ( - \\lambda ( \\delta - \\rho ) ) } , \\exp ( 2 \\rho \\lambda ) - 1 ) } & { } \\\\ { \\displaystyle \\leq \\frac { 1 } { | \\mathcal S _ { i } | } \\operatorname* { m a x } ( \\exp ( 2 \\rho \\lambda ) - 1 + n \\exp ( - \\lambda ( \\delta - \\rho ) ) , \\exp ( 2 \\rho \\lambda ) - 1 ) } & { } \\\\ { \\displaystyle = \\frac { 1 } { | \\mathcal S _ { i } | } ( \\exp ( 2 \\rho \\lambda ) - 1 + n \\exp ( - \\lambda ( \\delta - \\rho ) ) ) } & { } \\\\ { \\displaystyle \\leq \\frac { 1 } { | \\mathcal S _ { i } | } ( \\exp ( \\frac { \\epsilon } { 8 \\lambda M } ) - 1 + \\frac { \\epsilon } { 3 M } ) } & { } \\\\ { \\displaystyle \\leq \\frac { 2 \\epsilon } { | \\mathcal S _ { i } | } \\exp ( - \\frac { \\epsilon } { 8 \\lambda M } ) - 1 + \\frac { \\epsilon } { 3 M } ) } & { } \\\\ { \\displaystyle \\leq \\frac { 2 \\epsilon } { 3 M | \\mathcal S _ { i } | } } \\end{array}", + "type": "interline_equation", + "image_path": "6ed37e266ba2264a4147f5d2d4b519843a4a3740dbc71330876a16be10e85e9e.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 110, + 469, + 483, + 523.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 110, + 523.0, + 483, + 577.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 110, + 577.0, + 483, + 631.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 133, + 658 + ], + "score": 1.0, + "content": "wherefor all", + "type": "text" + }, + { + "bbox": [ + 133, + 645, + 158, + 655 + ], + "score": 0.88, + "content": "x > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 632, + 446, + 658 + ], + "score": 1.0, + "content": "we use Assumption C.6 similarly as before; in (17) we use the inequality ; in (18) we use the inequality previously derived in (15); in (19) we use", + "type": "text" + }, + { + "bbox": [ + 446, + 632, + 505, + 646 + ], + "score": 0.91, + "content": "\\begin{array} { r } { 1 - \\frac { 1 } { x } \\le x - 1 } \\end{array}", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 655, + 316, + 667 + ], + "spans": [ + { + "bbox": [ + 107, + 655, + 178, + 667 + ], + "score": 0.91, + "content": "\\exp ( x ) \\leq 1 + 2 x", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 655, + 205, + 667 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 205, + 656, + 249, + 666 + ], + "score": 0.91, + "content": "0 \\leq x \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 655, + 316, + 667 + ], + "score": 1.0, + "content": ". We thus obtain", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 668, + 501, + 709 + ], + "lines": [ + { + "bbox": [ + 111, + 668, + 501, + 709 + ], + "spans": [ + { + "bbox": [ + 111, + 668, + 501, + 709 + ], + "score": 0.93, + "content": "\\| \\sigma _ { i } - u _ { i } \\| _ { \\infty } = \\left\\| \\sum _ { j } a _ { i j } v _ { j } - \\frac { 1 } { | S _ { i } | } \\sum _ { j \\in S _ { i } } v _ { j } \\right\\| _ { \\infty } \\leq M \\| V \\| _ { \\infty } \\cdot \\left( \\sum _ { j \\notin S _ { i } } a _ { i j } + \\sum _ { j \\in S _ { i } } \\left| a _ { i j } - \\frac { 1 } { | S _ { i } | } \\right| \\right) \\leq \\epsilon ,", + "type": "interline_equation", + "image_path": "7e036fa051b82f132fbf4b5feef59cf92b6797b80a9511af6577cc604e7f0edc.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 111, + 668, + 501, + 681.6666666666666 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 111, + 681.6666666666666, + 501, + 695.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 111, + 695.3333333333333, + 501, + 708.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 711, + 216, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 216, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 216, + 723 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 494, + 711, + 505, + 721 + ], + "lines": [ + { + "bbox": [ + 495, + 712, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 495, + 712, + 505, + 722 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 71, + 506, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 507, + 89 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 302, + 88 + ], + "score": 1.0, + "content": "Lemma C.8. Assume Assumption C.6 holds with", + "type": "text" + }, + { + "bbox": [ + 316, + 70, + 507, + 89 + ], + "score": 1.0, + "content": "δϵ16M ln( 4Mn ) . For any 0 < ϵ ≤ M , there exists", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 506, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 261, + 101 + ], + "score": 1.0, + "content": "an attention layer with embedding size", + "type": "text" + }, + { + "bbox": [ + 261, + 88, + 282, + 100 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 87, + 506, + 101 + ], + "score": 1.0, + "content": "and one causal attention head that can approximate the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 97, + 507, + 113 + ], + "spans": [ + { + "bbox": [ + 104, + 97, + 441, + 113 + ], + "score": 1.0, + "content": "MEAN operation defined above. Formally, for any considered sequence of vectors", + "type": "text" + }, + { + "bbox": [ + 443, + 101, + 502, + 110 + ], + "score": 0.83, + "content": "\\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } , \\ldots , \\pmb { x } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 97, + 507, + 113 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 109, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 223, + 123 + ], + "score": 1.0, + "content": "denote the attention output as", + "type": "text" + }, + { + "bbox": [ + 224, + 111, + 282, + 121 + ], + "score": 0.86, + "content": "o _ { 1 } , o _ { 2 } , \\ldots , o _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 109, + 344, + 123 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + }, + { + "bbox": [ + 344, + 110, + 408, + 121 + ], + "score": 0.89, + "content": "\\| \\pmb { O } _ { i } - \\pmb { u } _ { i } \\| _ { \\infty } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 109, + 435, + 123 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 435, + 110, + 463, + 121 + ], + "score": 0.88, + "content": "i \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 109, + 482, + 123 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 483, + 110, + 505, + 121 + ], + "score": 0.88, + "content": "\\quad S _ { i } \\neq ", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 107, + 120, + 113, + 131 + ], + "score": 0.47, + "content": "\\varnothing .", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 119, + 172, + 133 + ], + "score": 1.0, + "content": ". Moreover, the", + "type": "text" + }, + { + "bbox": [ + 173, + 121, + 187, + 132 + ], + "score": 0.88, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 119, + 360, + 133 + ], + "score": 1.0, + "content": "norm of attention parameters is bounded by", + "type": "text" + }, + { + "bbox": [ + 361, + 121, + 502, + 132 + ], + "score": 0.88, + "content": "O ( \\mathrm { p o l y } ( M , 1 / \\delta , \\log ( n ) , \\log ( 1 / \\epsilon ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 119, + 506, + 133 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 70, + 507, + 133 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 142, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "Proof. The purpose of the attention head is to average across all tokens that satisfy the condition", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 154, + 452, + 167 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 152, + 167 + ], + "score": 0.92, + "content": "\\pmb q _ { i } \\cdot \\pmb k _ { j } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 154, + 452, + 167 + ], + "score": 1.0, + "content": ". To achieve this, we construct the key, query, and value vectors as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 143, + 505, + 167 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 170, + 205, + 217 + ], + "lines": [ + { + "bbox": [ + 125, + 170, + 205, + 217 + ], + "spans": [ + { + "bbox": [ + 125, + 170, + 205, + 217 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\bullet \\mathrm { ~ Q u e r y } \\colon \\lambda \\pmb { q } _ { i } \\in \\mathbb { R } ^ { d } } \\\\ & { \\bullet \\mathrm { ~ K e y } \\colon \\pmb { k } _ { i } \\in \\mathbb { R } ^ { d } } \\\\ & { \\bullet \\mathrm { ~ V a l u e } \\colon \\pmb { v } _ { i } \\in \\mathbb { R } ^ { d } } \\end{array}", + "type": "interline_equation", + "image_path": "d1a6fe2cee731df83cb152d51c30c5ce72d0c2c7e0b3386a657c311ac00cbcc2.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 125, + 170, + 205, + 193.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 125, + 193.5, + 205, + 217.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 221, + 460, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 220, + 459, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 133, + 234 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 222, + 140, + 231 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 220, + 335, + 234 + ], + "score": 1.0, + "content": "is a constant which will be defined later. Denote", + "type": "text" + }, + { + "bbox": [ + 336, + 224, + 349, + 234 + ], + "score": 0.88, + "content": "a _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 220, + 459, + 234 + ], + "score": 1.0, + "content": "as the attention score, then", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 220, + 459, + 234 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 235, + 361, + 264 + ], + "lines": [ + { + "bbox": [ + 249, + 235, + 361, + 264 + ], + "spans": [ + { + "bbox": [ + 249, + 235, + 361, + 264 + ], + "score": 0.94, + "content": "a _ { i , j } = \\frac { \\exp ( \\lambda \\pmb q _ { i } \\cdot \\pmb k _ { j } ) } { \\sum _ { j ^ { \\prime } } \\exp ( \\lambda \\pmb q _ { i } \\cdot \\pmb k _ { j ^ { \\prime } } ) } .", + "type": "interline_equation", + "image_path": "4d0429d415a481dcc6e69a1ca7f72c4a2f92a9236affe6862e4114100a9ef8e2.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 235, + 361, + 249.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 249, + 249.5, + 361, + 264.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 267, + 492, + 281 + ], + "lines": [ + { + "bbox": [ + 103, + 264, + 496, + 283 + ], + "spans": [ + { + "bbox": [ + 103, + 264, + 150, + 283 + ], + "score": 1.0, + "content": "By setting", + "type": "text" + }, + { + "bbox": [ + 150, + 267, + 216, + 281 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\lambda = \\frac { 1 } { \\delta } \\ln \\left( \\frac { 4 M n } { \\epsilon } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 264, + 306, + 283 + ], + "score": 1.0, + "content": "(which is bounded by", + "type": "text" + }, + { + "bbox": [ + 307, + 268, + 450, + 280 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( M , 1 / \\delta , \\log ( n ) , \\log ( 1 / \\epsilon ) ) ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 264, + 496, + 283 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 103, + 264, + 496, + 283 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 283, + 406, + 413 + ], + "lines": [ + { + "bbox": [ + 203, + 283, + 406, + 413 + ], + "spans": [ + { + "bbox": [ + 203, + 283, + 406, + 413 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\displaystyle \\sum _ { j \\notin { \\mathcal S } _ { i } } a _ { i j } \\leq \\frac { ( n - | S _ { i } | ) \\exp ( - \\lambda \\delta ) } { ( n - | S _ { i } | ) \\exp ( - \\lambda \\delta ) + | S _ { i } | \\exp ( - \\lambda \\rho ) } } & { } \\\\ { \\displaystyle } & { = \\frac { | S _ { i } | } { 1 + \\frac { | S _ { i } | } { n - | S _ { i } | } \\exp ( - \\lambda ( \\rho - \\delta ) ) } } \\\\ { \\displaystyle } & { < n \\exp ( \\lambda \\rho ) \\exp ( - \\lambda \\delta ) ) } \\\\ { \\displaystyle } & { \\leq n \\exp \\left( \\frac { \\epsilon } { 1 6 M } \\right) \\exp \\left( - \\ln \\left( \\frac { 4 M n } { \\epsilon } \\right) \\right) } \\\\ { \\displaystyle } & { < \\frac { \\epsilon } { 3 M } , } \\end{array}", + "type": "interline_equation", + "image_path": "49a50f61b24de86f34cf7b67ca08bd405ddb1ab62599ab36809ed81be0c6957d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 203, + 283, + 406, + 297.44444444444446 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 203, + 297.44444444444446, + 406, + 311.8888888888889 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 203, + 311.8888888888889, + 406, + 326.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 203, + 326.33333333333337, + 406, + 340.7777777777778 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 340.7777777777778, + 406, + 355.2222222222223 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 203, + 355.2222222222223, + 406, + 369.66666666666674 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 203, + 369.66666666666674, + 406, + 384.1111111111112 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 203, + 384.1111111111112, + 406, + 398.55555555555566 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 203, + 398.55555555555566, + 406, + 413.0000000000001 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 504, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 331, + 427 + ], + "score": 1.0, + "content": "where in (13) we use Assumption C.6, which implies that", + "type": "text" + }, + { + "bbox": [ + 331, + 414, + 381, + 426 + ], + "score": 0.92, + "content": "\\mathbf { \\boldsymbol { q } } _ { i } \\cdot \\mathbf { \\boldsymbol { k } } _ { j } \\le - 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\\frac { 1 } { | \\mathcal S _ { i } | } \\Big | \\leq \\operatorname* { m a x } ( \\frac { 1 } { | \\mathcal S _ { i } | } - \\frac { \\exp ( - \\rho \\lambda ) } { | \\mathcal S _ { i } | \\exp ( \\rho \\lambda ) + ( n - | \\mathcal S _ { i } | ) \\exp ( - \\lambda \\delta ) } , \\frac { \\exp ( \\rho \\lambda ) } { | \\mathcal S _ { i } | \\exp ( - \\rho \\lambda ) } - \\frac { 1 } { | \\mathcal S _ { i } | } ) } & { } \\\\ { \\displaystyle \\leq \\frac { 1 } { | \\mathcal S _ { i } | } \\operatorname* { m a x } ( 1 - \\frac { 1 } { \\exp ( 2 \\rho \\lambda ) + n \\exp ( - \\lambda ( \\delta - \\rho ) ) } , \\exp ( 2 \\rho \\lambda ) - 1 ) } & { } \\\\ { \\displaystyle \\leq \\frac { 1 } { | \\mathcal S _ { i } | } \\operatorname* { m a x } ( \\exp ( 2 \\rho \\lambda ) - 1 + n \\exp ( - \\lambda ( \\delta - \\rho ) ) , \\exp ( 2 \\rho \\lambda ) - 1 ) } & { } \\\\ { \\displaystyle = \\frac { 1 } { | \\mathcal S _ { i } | } ( \\exp ( 2 \\rho \\lambda ) - 1 + n \\exp ( - \\lambda ( \\delta - \\rho ) ) ) } & { } \\\\ { \\displaystyle \\leq \\frac { 1 } { | \\mathcal S _ { i } | } ( \\exp ( \\frac { \\epsilon } { 8 \\lambda M } ) - 1 + \\frac { \\epsilon } { 3 M } ) } & { } \\\\ { \\displaystyle \\leq \\frac { 2 \\epsilon } { | \\mathcal S _ { i } | } \\exp ( - \\frac { \\epsilon } { 8 \\lambda M } ) - 1 + \\frac { \\epsilon } { 3 M } ) } & { } \\\\ { \\displaystyle \\leq \\frac { 2 \\epsilon } { 3 M | \\mathcal S _ { i } | } } \\end{array}", + "type": "interline_equation", + "image_path": "6ed37e266ba2264a4147f5d2d4b519843a4a3740dbc71330876a16be10e85e9e.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 110, + 469, + 483, + 523.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 110, + 523.0, + 483, + 577.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 110, + 577.0, + 483, + 631.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 133, + 658 + ], + "score": 1.0, + "content": "wherefor all", + "type": "text" + }, + { + "bbox": [ + 133, + 645, + 158, + 655 + ], + "score": 0.88, + "content": "x > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 632, + 446, + 658 + ], + "score": 1.0, + "content": "we use Assumption C.6 similarly as before; in (17) we use the inequality ; in (18) we use the inequality previously derived in (15); in (19) we use", + "type": "text" + }, + { + "bbox": [ + 446, + 632, + 505, + 646 + ], + "score": 0.91, + "content": "\\begin{array} { r } { 1 - 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We thus obtain", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 632, + 505, + 667 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 668, + 501, + 709 + ], + "lines": [ + { + "bbox": [ + 111, + 668, + 501, + 709 + ], + "spans": [ + { + "bbox": [ + 111, + 668, + 501, + 709 + ], + "score": 0.93, + "content": "\\| \\sigma _ { i } - u _ { i } \\| _ { \\infty } = \\left\\| \\sum _ { j } a _ { i j } v _ { j } - \\frac { 1 } { | S _ { i } | } \\sum _ { j \\in S _ { i } } v _ { j } \\right\\| _ { \\infty } \\leq M \\| V \\| _ { \\infty } \\cdot \\left( \\sum _ { j \\notin S _ { i } } a _ { i j } + \\sum _ { j \\in S _ { i } } \\left| a _ { i j } - \\frac { 1 } { | S _ { i } | } \\right| \\right) \\leq \\epsilon ,", + "type": "interline_equation", + "image_path": "7e036fa051b82f132fbf4b5feef59cf92b6797b80a9511af6577cc604e7f0edc.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 111, + 668, + 501, + 681.6666666666666 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 111, + 681.6666666666666, + 501, + 695.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 111, + 695.3333333333333, + 501, + 708.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 711, + 216, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 216, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 216, + 723 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 710, + 216, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 232, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 232, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 232, + 87 + ], + "score": 1.0, + "content": "D Arithmetic Formula", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 109 + ], + "score": 1.0, + "content": "In this section, we prove that the autoregressive Transformer can evaluate arithmetic expressions", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 107, + 412, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 412, + 119 + ], + "score": 1.0, + "content": "when equipped with CoT, whereas the it cannot solve this task without CoT.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 223, + 144 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 224, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 224, + 145 + ], + "score": 1.0, + "content": "D.1 Proof of Theorem 3.3", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "Before proving this theorem, there is one point that needs to be clarified: all residual connections", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "in the attention/MLP layers can be replaced by concatenation, in the sense that both architectures", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "have the same expressive power. Formally, consider an MLP (or an attention layer) denoted as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 184, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 107, + 185, + 162, + 198 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 184, + 196, + 199 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 196, + 186, + 238, + 197 + ], + "score": 0.92, + "content": "\\pmb { y } = \\pmb { f } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 184, + 506, + 199 + ], + "score": 1.0, + "content": ". It is easy to see that we can construct another MLP (or attention", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 195, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 175, + 212 + ], + "score": 1.0, + "content": "layer) denoted as", + "type": "text" + }, + { + "bbox": [ + 176, + 198, + 237, + 209 + ], + "score": 0.92, + "content": "\\pmb { g } : \\mathbb { R } ^ { 2 d } \\mathbb { R } ^ { 2 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 195, + 276, + 212 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 276, + 197, + 454, + 210 + ], + "score": 0.94, + "content": "g ( \\pmb { x } , \\mathbf { 0 } ) + ( \\pmb { x } , \\mathbf { 0 } ) = ( \\mathbf { 0 } , \\pmb { y } ) + ( \\pmb { x } , \\mathbf { 0 } ) = ( \\pmb { x } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 195, + 506, + 212 + ], + "score": 1.0, + "content": ", namely, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "residual connection can implement concatenation. Conversely, concatenation can implement residual", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "connection by using a linear projection. Based on the equivalence, we can use the concatenation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 507, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 507, + 243 + ], + "score": 1.0, + "content": "operation instead of residual connection in all subsequent proofs presented in Appendices D, E and G.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Similarly, the output of multi-head attention can be replaced by the concatenation of the output of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 252, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 103, + 253, + 347, + 270 + ], + "score": 1.0, + "content": "each head (instead of performing aggregation via matrices", + "type": "text" + }, + { + "bbox": [ + 347, + 252, + 377, + 267 + ], + "score": 0.93, + "content": "{ W } _ { O } ^ { ( l , h ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 253, + 506, + 270 + ], + "score": 1.0, + "content": "defined in (2)). For clarity, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "further omit the unnecessary parts in the concatenated outputs and only retain the outputs that are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 213, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 213, + 289 + ], + "score": 1.0, + "content": "used in subsequent layers.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 485, + 305 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 485, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 485, + 306 + ], + "score": 1.0, + "content": "We now present the proof of Theorem 3.3. For ease of reading, we restate Theorem 3.3 below:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 225, + 321 + ], + "score": 1.0, + "content": "Theorem D.1. For any prime", + "type": "text" + }, + { + "bbox": [ + 225, + 311, + 231, + 320 + ], + "score": 0.44, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 307, + 278, + 321 + ], + "score": 1.0, + "content": "and integer", + "type": "text" + }, + { + "bbox": [ + 279, + 309, + 303, + 319 + ], + "score": 0.87, + "content": "n > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 307, + 506, + 321 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer defined", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 222, + 332 + ], + "score": 1.0, + "content": "in Section 2 with hidden size", + "type": "text" + }, + { + "bbox": [ + 222, + 320, + 288, + 331 + ], + "score": 0.91, + "content": "d = { \\cal O } ( \\mathrm { p o l y } ( p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 319, + 355, + 332 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 355, + 322, + 362, + 330 + ], + "score": 0.57, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 319, + 393, + 332 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 393, + 320, + 419, + 330 + ], + "score": 0.89, + "content": "L = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 319, + 506, + 332 + ], + "score": 1.0, + "content": ", and 5 heads in each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 359, + 343 + ], + "score": 1.0, + "content": "layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 359, + 331, + 367, + 340 + ], + "score": 0.49, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 330, + 479, + 343 + ], + "score": 1.0, + "content": "for all inputs in Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 330, + 502, + 342 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 428, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 375, + 354 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 341, + 423, + 353 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 339, + 428, + 354 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "Proof sketch. The intuition behind our construction is that when the CoT output proceeds to a certain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "position, the Transformer can read the context related to this position and determine whether it should", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "copy a token or perform a calculation. Remarkably, the context only contains a fixed number of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "score": 1.0, + "content": "tokens (as discussed in Appendix B). Based on the key observation, we can construct our five-layer", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "transformer as follows. The first layer collects important positional information. The second and third", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "layers determine whether to perform a calculation by examining the context related to the current", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "token, which contains five tokens. The fourth layer and the fifth layers are used to generate the output", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 439, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 450 + ], + "score": 1.0, + "content": "via three cases: before/at/after the position that performs a calculation. For the first and the last cases,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "the output simply copies a previous token with position computed by the two layers. For the middle", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 481, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 430, + 473 + ], + "score": 1.0, + "content": "case, the outcome is computed via a look-up table that stores the arithemtic rules", + "type": "text" + }, + { + "bbox": [ + 431, + 461, + 477, + 472 + ], + "score": 0.9, + "content": "( + , - , \\times , \\div )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 460, + 481, + 473 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 277, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 279, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 279, + 500 + ], + "score": 1.0, + "content": "Proof. We construct each layer as follows.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 364, + 516 + ], + "score": 1.0, + "content": "Token Embeddings. Assume that we have a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 365, + 505, + 405, + 514 + ], + "score": 0.9, + "content": "s _ { 1 } , \\ldots , s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "and we want to generate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 512, + 507, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 164, + 527 + ], + "score": 1.0, + "content": "the next token", + "type": "text" + }, + { + "bbox": [ + 165, + 515, + 184, + 525 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 512, + 221, + 527 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 221, + 513, + 250, + 525 + ], + "score": 0.92, + "content": "j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 512, + 266, + 527 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 266, + 514, + 293, + 526 + ], + "score": 0.82, + "content": "\\operatorname { i d } ( s _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 512, + 379, + 527 + ], + "score": 1.0, + "content": "be the index of token", + "type": "text" + }, + { + "bbox": [ + 379, + 515, + 390, + 526 + ], + "score": 0.86, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 512, + 507, + 527 + ], + "score": 1.0, + "content": "in the embedding dictionary,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 103, + 523, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 103, + 523, + 489, + 544 + ], + "score": 1.0, + "content": "with values ranging from 1 to the number of tokens. We can embed the token sj by x(0)j", + "type": "text" + }, + { + "bbox": [ + 472, + 525, + 505, + 541 + ], + "score": 0.92, + "content": "\\pmb { x } _ { j } ^ { ( 0 ) } =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 537, + 508, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 222, + 554 + ], + "score": 0.91, + "content": "( e _ { \\mathrm { i d } ( s _ { j } ) } , j , 1 ) \\in \\mathbb { R } ^ { \\mathrm { n u m \\_ t o k e n s + 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 537, + 253, + 556 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 253, + 543, + 264, + 554 + ], + "score": 0.87, + "content": "e _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 537, + 376, + 556 + ], + "score": 1.0, + "content": "is a one-hot vector with the", + "type": "text" + }, + { + "bbox": [ + 376, + 542, + 382, + 553 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 537, + 462, + 556 + ], + "score": 1.0, + "content": "-th element being 1,", + "type": "text" + }, + { + "bbox": [ + 462, + 542, + 495, + 553 + ], + "score": 0.86, + "content": "j \\in \\mathbb { N } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 537, + 508, + 556 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 551, + 423, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 423, + 565 + ], + "score": 1.0, + "content": "the positional embedding, and the constant embedding 1 is used as a bias term.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "Layer 1. The first layer of the autoregressive Transformer uses two attention heads to perform the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 173, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 173, + 593 + ], + "score": 1.0, + "content": "following tasks:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 120, + 595, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 123, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 123, + 594, + 267, + 609 + ], + "score": 1.0, + "content": "1. Count the number of equal signs", + "type": "text" + }, + { + "bbox": [ + 267, + 596, + 286, + 607 + ], + "score": 0.65, + "content": "( \\ ' = \\ ' )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 594, + 408, + 609 + ], + "score": 1.0, + "content": "in previous tokens, denoted as", + "type": "text" + }, + { + "bbox": [ + 409, + 597, + 422, + 608 + ], + "score": 0.89, + "content": "n _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 594, + 443, + 609 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 443, + 595, + 502, + 608 + ], + "score": 0.91, + "content": "n _ { i } ^ { = } = | \\{ j \\leq i", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 594, + 506, + 609 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 136, + 606, + 184, + 620 + ], + "spans": [ + { + "bbox": [ + 136, + 607, + 178, + 620 + ], + "score": 0.92, + "content": "s _ { j } = \\cdot { = } ^ { , } \\} |", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 606, + 184, + 620 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 122, + 621, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 122, + 621, + 341, + 637 + ], + "score": 1.0, + "content": "2. Copy the position of the last equal sign, denoted as", + "type": "text" + }, + { + "bbox": [ + 342, + 624, + 354, + 635 + ], + "score": 0.88, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 621, + 375, + 637 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 375, + 623, + 503, + 636 + ], + "score": 0.91, + "content": "p _ { i } ^ { = } = \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathit { \\iota } ^ { : } = \\mathit { \\iota } ^ { : } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 621, + 506, + 637 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 134, + 632, + 360, + 647 + ], + "spans": [ + { + "bbox": [ + 134, + 632, + 174, + 647 + ], + "score": 1.0, + "content": "If the set", + "type": "text" + }, + { + "bbox": [ + 174, + 633, + 258, + 646 + ], + "score": 0.92, + "content": "\\{ j : j \\leq i , s _ { j } = \\stackrel { } { = } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 632, + 324, + 647 + ], + "score": 1.0, + "content": "is empty, define", + "type": "text" + }, + { + "bbox": [ + 325, + 635, + 356, + 646 + ], + "score": 0.92, + "content": "p _ { i } ^ { = } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 632, + 360, + 647 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 122, + 650, + 189, + 664 + ], + "spans": [ + { + "bbox": [ + 122, + 650, + 175, + 664 + ], + "score": 1.0, + "content": "3. Compute", + "type": "text" + }, + { + "bbox": [ + 176, + 651, + 184, + 662 + ], + "score": 0.85, + "content": "i ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 650, + 189, + 664 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 109, + 668, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 108, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 108, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Based on Appendix C.2 (Lemma C.8), we can use the first attention head to perform the MEAN", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 438, + 691 + ], + "score": 1.0, + "content": "operation that counts the percentage of equal signs in the preceding sentences (i.e.,", + "type": "text" + }, + { + "bbox": [ + 439, + 679, + 461, + 691 + ], + "score": 0.9, + "content": "n _ { i } ^ { = } / i", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "). This can", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 690, + 489, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 197, + 704 + ], + "score": 1.0, + "content": "be achieved by setting", + "type": "text" + }, + { + "bbox": [ + 198, + 691, + 226, + 703 + ], + "score": 0.79, + "content": "Q = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 690, + 230, + 704 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 691, + 261, + 702 + ], + "score": 0.78, + "content": "\\pmb { K } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 690, + 264, + 704 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 265, + 690, + 348, + 704 + ], + "score": 0.9, + "content": "V = ( e _ { \\mathrm { i d } ( ^ { \\cdot } = ^ { \\cdot } ) } , 0 , 0 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 690, + 489, + 704 + ], + "score": 1.0, + "content": "(defined in Appendix C.2), so that", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 709, + 444, + 725 + ], + "lines": [ + { + "bbox": [ + 166, + 709, + 444, + 725 + ], + "spans": [ + { + "bbox": [ + 166, + 709, + 444, + 725 + ], + "score": 0.87, + "content": "\\begin{array} { r } { q _ { i } = \\mathbf { 0 } , \\quad k _ { j } = \\mathbf { 0 } , \\quad v _ { j } = e _ { \\mathrm { i d } ( \\cdot = \\cdot ) } \\cdot e _ { \\mathrm { i d } ( s _ { j } ) } = \\mathbb { I } [ s _ { j } = \\cdot = \\cdot ] , \\quad S _ { i } = [ i ] . } \\end{array}", + "type": "interline_equation", + "image_path": "abe0ab9d625c6831c23a4a18573d66575d65569261c57f31e4f5a268b8b99161.jpg" + } + ] + } + ], + "index": 47, + "virtual_lines": [ + { + "bbox": [ + 166, + 709, + 444, + 725 + ], + "spans": [], + "index": 47 + } + ] + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 232, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 232, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 232, + 87 + ], + "score": 1.0, + "content": "D Arithmetic Formula", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 109 + ], + "score": 1.0, + "content": "In this section, we prove that the autoregressive Transformer can evaluate arithmetic expressions", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 107, + 412, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 412, + 119 + ], + "score": 1.0, + "content": "when equipped with CoT, whereas the it cannot solve this task without CoT.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 95, + 505, + 119 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 223, + 144 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 224, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 224, + 145 + ], + "score": 1.0, + "content": "D.1 Proof of Theorem 3.3", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "Before proving this theorem, there is one point that needs to be clarified: all residual connections", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "in the attention/MLP layers can be replaced by concatenation, in the sense that both architectures", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "have the same expressive power. Formally, consider an MLP (or an attention layer) denoted as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 184, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 107, + 185, + 162, + 198 + ], + "score": 0.92, + "content": "f : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 184, + 196, + 199 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 196, + 186, + 238, + 197 + ], + "score": 0.92, + "content": "\\pmb { y } = \\pmb { f } ( \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 184, + 506, + 199 + ], + "score": 1.0, + "content": ". It is easy to see that we can construct another MLP (or attention", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 195, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 175, + 212 + ], + "score": 1.0, + "content": "layer) denoted as", + "type": "text" + }, + { + "bbox": [ + 176, + 198, + 237, + 209 + ], + "score": 0.92, + "content": "\\pmb { g } : \\mathbb { R } ^ { 2 d } \\mathbb { R } ^ { 2 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 195, + 276, + 212 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 276, + 197, + 454, + 210 + ], + "score": 0.94, + "content": "g ( \\pmb { x } , \\mathbf { 0 } ) + ( \\pmb { x } , \\mathbf { 0 } ) = ( \\mathbf { 0 } , \\pmb { y } ) + ( \\pmb { x } , \\mathbf { 0 } ) = ( \\pmb { x } , \\pmb { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 195, + 506, + 212 + ], + "score": 1.0, + "content": ", namely, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "residual connection can implement concatenation. Conversely, concatenation can implement residual", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "connection by using a linear projection. Based on the equivalence, we can use the concatenation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 507, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 507, + 243 + ], + "score": 1.0, + "content": "operation instead of residual connection in all subsequent proofs presented in Appendices D, E and G.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Similarly, the output of multi-head attention can be replaced by the concatenation of the output of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 252, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 103, + 253, + 347, + 270 + ], + "score": 1.0, + "content": "each head (instead of performing aggregation via matrices", + "type": "text" + }, + { + "bbox": [ + 347, + 252, + 377, + 267 + ], + "score": 0.93, + "content": "{ W } _ { O } ^ { ( l , h ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 253, + 506, + 270 + ], + "score": 1.0, + "content": "defined in (2)). For clarity, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "further omit the unnecessary parts in the concatenated outputs and only retain the outputs that are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 213, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 213, + 289 + ], + "score": 1.0, + "content": "used in subsequent layers.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 153, + 507, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 485, + 305 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 485, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 485, + 306 + ], + "score": 1.0, + "content": "We now present the proof of Theorem 3.3. For ease of reading, we restate Theorem 3.3 below:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 291, + 485, + 306 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 225, + 321 + ], + "score": 1.0, + "content": "Theorem D.1. For any prime", + "type": "text" + }, + { + "bbox": [ + 225, + 311, + 231, + 320 + ], + "score": 0.44, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 307, + 278, + 321 + ], + "score": 1.0, + "content": "and integer", + "type": "text" + }, + { + "bbox": [ + 279, + 309, + 303, + 319 + ], + "score": 0.87, + "content": "n > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 307, + 506, + 321 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer defined", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 222, + 332 + ], + "score": 1.0, + "content": "in Section 2 with hidden size", + "type": "text" + }, + { + "bbox": [ + 222, + 320, + 288, + 331 + ], + "score": 0.91, + "content": "d = { \\cal O } ( \\mathrm { p o l y } ( p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 319, + 355, + 332 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 355, + 322, + 362, + 330 + ], + "score": 0.57, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 319, + 393, + 332 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 393, + 320, + 419, + 330 + ], + "score": 0.89, + "content": "L = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 319, + 506, + 332 + ], + "score": 1.0, + "content": ", and 5 heads in each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 359, + 343 + ], + "score": 1.0, + "content": "layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 359, + 331, + 367, + 340 + ], + "score": 0.49, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 330, + 479, + 343 + ], + "score": 1.0, + "content": "for all inputs in Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 330, + 502, + 342 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 428, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 375, + 354 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 341, + 423, + 353 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 339, + 428, + 354 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 307, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "Proof sketch. The intuition behind our construction is that when the CoT output proceeds to a certain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "position, the Transformer can read the context related to this position and determine whether it should", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "copy a token or perform a calculation. Remarkably, the context only contains a fixed number of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 407 + ], + "score": 1.0, + "content": "tokens (as discussed in Appendix B). Based on the key observation, we can construct our five-layer", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "transformer as follows. The first layer collects important positional information. The second and third", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "layers determine whether to perform a calculation by examining the context related to the current", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "token, which contains five tokens. The fourth layer and the fifth layers are used to generate the output", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 439, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 450 + ], + "score": 1.0, + "content": "via three cases: before/at/after the position that performs a calculation. For the first and the last cases,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "the output simply copies a previous token with position computed by the two layers. For the middle", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 481, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 430, + 473 + ], + "score": 1.0, + "content": "case, the outcome is computed via a look-up table that stores the arithemtic rules", + "type": "text" + }, + { + "bbox": [ + 431, + 461, + 477, + 472 + ], + "score": 0.9, + "content": "( + , - , \\times , \\div )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 460, + 481, + 473 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 361, + 506, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 277, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 279, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 279, + 500 + ], + "score": 1.0, + "content": "Proof. We construct each layer as follows.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 484, + 279, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 364, + 516 + ], + "score": 1.0, + "content": "Token Embeddings. Assume that we have a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 365, + 505, + 405, + 514 + ], + "score": 0.9, + "content": "s _ { 1 } , \\ldots , s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "and we want to generate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 512, + 507, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 164, + 527 + ], + "score": 1.0, + "content": "the next token", + "type": "text" + }, + { + "bbox": [ + 165, + 515, + 184, + 525 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 512, + 221, + 527 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 221, + 513, + 250, + 525 + ], + "score": 0.92, + "content": "j \\in [ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 512, + 266, + 527 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 266, + 514, + 293, + 526 + ], + "score": 0.82, + "content": "\\operatorname { i d } ( s _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 512, + 379, + 527 + ], + "score": 1.0, + "content": "be the index of token", + "type": "text" + }, + { + "bbox": [ + 379, + 515, + 390, + 526 + ], + "score": 0.86, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 512, + 507, + 527 + ], + "score": 1.0, + "content": "in the embedding dictionary,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 103, + 523, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 103, + 523, + 489, + 544 + ], + "score": 1.0, + "content": "with values ranging from 1 to the number of tokens. We can embed the token sj by x(0)j", + "type": "text" + }, + { + "bbox": [ + 472, + 525, + 505, + 541 + ], + "score": 0.92, + "content": "\\pmb { x } _ { j } ^ { ( 0 ) } =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 537, + 508, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 222, + 554 + ], + "score": 0.91, + "content": "( e _ { \\mathrm { i d } ( s _ { j } ) } , j , 1 ) \\in \\mathbb { R } ^ { \\mathrm { n u m \\_ t o k e n s + 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 537, + 253, + 556 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 253, + 543, + 264, + 554 + ], + "score": 0.87, + "content": "e _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 537, + 376, + 556 + ], + "score": 1.0, + "content": "is a one-hot vector with the", + "type": "text" + }, + { + "bbox": [ + 376, + 542, + 382, + 553 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 537, + 462, + 556 + ], + "score": 1.0, + "content": "-th element being 1,", + "type": "text" + }, + { + "bbox": [ + 462, + 542, + 495, + 553 + ], + "score": 0.86, + "content": "j \\in \\mathbb { N } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 537, + 508, + 556 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 551, + 423, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 423, + 565 + ], + "score": 1.0, + "content": "the positional embedding, and the constant embedding 1 is used as a bias term.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 103, + 501, + 508, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "Layer 1. The first layer of the autoregressive Transformer uses two attention heads to perform the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 173, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 173, + 593 + ], + "score": 1.0, + "content": "following tasks:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 567, + 505, + 593 + ] + }, + { + "type": "list", + "bbox": [ + 120, + 595, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 123, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 123, + 594, + 267, + 609 + ], + "score": 1.0, + "content": "1. Count the number of equal signs", + "type": "text" + }, + { + "bbox": [ + 267, + 596, + 286, + 607 + ], + "score": 0.65, + "content": "( \\ ' = \\ ' )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 594, + 408, + 609 + ], + "score": 1.0, + "content": "in previous tokens, denoted as", + "type": "text" + }, + { + "bbox": [ + 409, + 597, + 422, + 608 + ], + "score": 0.89, + "content": "n _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 594, + 443, + 609 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 443, + 595, + 502, + 608 + ], + "score": 0.91, + "content": "n _ { i } ^ { = } = | \\{ j \\leq i", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 594, + 506, + 609 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 606, + 184, + 620 + ], + "spans": [ + { + "bbox": [ + 136, + 607, + 178, + 620 + ], + "score": 0.92, + "content": "s _ { j } = \\cdot { = } ^ { , } \\} |", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 606, + 184, + 620 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 122, + 621, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 122, + 621, + 341, + 637 + ], + "score": 1.0, + "content": "2. Copy the position of the last equal sign, denoted as", + "type": "text" + }, + { + "bbox": [ + 342, + 624, + 354, + 635 + ], + "score": 0.88, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 621, + 375, + 637 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 375, + 623, + 503, + 636 + ], + "score": 0.91, + "content": "p _ { i } ^ { = } = \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathit { \\iota } ^ { : } = \\mathit { \\iota } ^ { : } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 621, + 506, + 637 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 134, + 632, + 360, + 647 + ], + "spans": [ + { + "bbox": [ + 134, + 632, + 174, + 647 + ], + "score": 1.0, + "content": "If the set", + "type": "text" + }, + { + "bbox": [ + 174, + 633, + 258, + 646 + ], + "score": 0.92, + "content": "\\{ j : j \\leq i , s _ { j } = \\stackrel { } { = } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 632, + 324, + 647 + ], + "score": 1.0, + "content": "is empty, define", + "type": "text" + }, + { + "bbox": [ + 325, + 635, + 356, + 646 + ], + "score": 0.92, + "content": "p _ { i } ^ { = } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 632, + 360, + 647 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + }, + { + "bbox": [ + 122, + 650, + 189, + 664 + ], + "spans": [ + { + "bbox": [ + 122, + 650, + 175, + 664 + ], + "score": 1.0, + "content": "3. Compute", + "type": "text" + }, + { + "bbox": [ + 176, + 651, + 184, + 662 + ], + "score": 0.85, + "content": "i ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 650, + 189, + 664 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 41, + "bbox_fs": [ + 122, + 594, + 506, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 668, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 108, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 108, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Based on Appendix C.2 (Lemma C.8), we can use the first attention head to perform the MEAN", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 438, + 691 + ], + "score": 1.0, + "content": "operation that counts the percentage of equal signs in the preceding sentences (i.e.,", + "type": "text" + }, + { + "bbox": [ + 439, + 679, + 461, + 691 + ], + "score": 0.9, + "content": "n _ { i } ^ { = } / i", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "). This can", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 690, + 489, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 197, + 704 + ], + "score": 1.0, + "content": "be achieved by setting", + "type": "text" + }, + { + "bbox": [ + 198, + 691, + 226, + 703 + ], + "score": 0.79, + "content": "Q = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 690, + 230, + 704 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 691, + 261, + 702 + ], + "score": 0.78, + "content": "\\pmb { K } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 690, + 264, + 704 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 265, + 690, + 348, + 704 + ], + "score": 0.9, + "content": "V = ( e _ { \\mathrm { i d } ( ^ { \\cdot } = ^ { \\cdot } ) } , 0 , 0 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 690, + 489, + 704 + ], + "score": 1.0, + "content": "(defined in Appendix C.2), so that", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 106, + 667, + 505, + 704 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 709, + 444, + 725 + ], + "lines": [ + { + "bbox": [ + 166, + 709, + 444, + 725 + ], + "spans": [ + { + "bbox": [ + 166, + 709, + 444, + 725 + ], + "score": 0.87, + "content": "\\begin{array} { r } { q _ { i } = \\mathbf { 0 } , \\quad k _ { j } = \\mathbf { 0 } , \\quad v _ { j } = e _ { \\mathrm { i d } ( \\cdot = \\cdot ) } \\cdot e _ { \\mathrm { i d } ( s _ { j } ) } = \\mathbb { I } [ s _ { j } = \\cdot = \\cdot ] , \\quad S _ { i } = [ i ] . } \\end{array}", + "type": "interline_equation", + "image_path": "abe0ab9d625c6831c23a4a18573d66575d65569261c57f31e4f5a268b8b99161.jpg" + } + ] + } + ], + "index": 47, + "virtual_lines": [ + { + "bbox": [ + 166, + 709, + 444, + 725 + ], + "spans": [], + "index": 47 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 108 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "Similarly, we can use the second attention head to perform a COPY operation that copies the position", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 437, + 97 + ], + "score": 1.0, + "content": "index of the last equal sign (by Lemma C.7). This can be achieved by setting", + "type": "text" + }, + { + "bbox": [ + 437, + 83, + 502, + 96 + ], + "score": 0.93, + "content": "\\dot { \\boldsymbol { Q } } = ( \\dot { \\mathbf { 0 } } , 0 , 1 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 83, + 506, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 94, + 439, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 199, + 109 + ], + "score": 0.87, + "content": "\\pmb { K } = ( e _ { \\mathrm { i d } ( \\cdot = \\cdot ) } , 0 , - 1 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 94, + 204, + 109 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 204, + 95, + 266, + 108 + ], + "score": 0.86, + "content": "V = ( \\mathbf { 0 } , 1 , 0 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 94, + 270, + 109 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 270, + 95, + 298, + 108 + ], + "score": 0.9, + "content": "r _ { j } = j", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 94, + 439, + 109 + ], + "score": 1.0, + "content": "(defined in Appendix C.2), so that", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 113, + 446, + 127 + ], + "lines": [ + { + "bbox": [ + 164, + 113, + 446, + 127 + ], + "spans": [ + { + "bbox": [ + 164, + 113, + 446, + 127 + ], + "score": 0.78, + "content": "q _ { i } = 1 , \\quad k _ { j } = \\mathbb { I } [ s _ { j } = \\cdot = \\cdot = 1 , \\quad v _ { j } = j , \\quad S _ { i } = \\{ j \\leq i : s _ { j } = \\cdot = \\cdot = \\} .", + "type": "interline_equation", + "image_path": "707bc5f7566a155ec219f7741731b0b5d9ce758d24ecb8fe2bf158eb8c63fe71.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 164, + 113, + 446, + 127 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 104, + 130, + 503, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 130, + 323, + 145 + ], + "score": 1.0, + "content": "It is easy to check that the above construction outputs", + "type": "text" + }, + { + "bbox": [ + 324, + 131, + 448, + 144 + ], + "score": 0.91, + "content": "u _ { i } = \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathit { \\iota } ^ { \\circ } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 130, + 474, + 145 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 474, + 132, + 503, + 144 + ], + "score": 0.93, + "content": "\\boldsymbol { S } _ { i } \\neq \\boldsymbol { \\emptyset }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 298, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 145, + 155 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 145, + 156, + 154 + ], + "score": 0.85, + "content": "u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 142, + 226, + 155 + ], + "score": 1.0, + "content": "may not equal to", + "type": "text" + }, + { + "bbox": [ + 226, + 144, + 239, + 155 + ], + "score": 0.89, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 142, + 264, + 155 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 265, + 143, + 293, + 154 + ], + "score": 0.91, + "content": "\\boldsymbol { S } _ { i } = \\boldsymbol { \\emptyset }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 142, + 298, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 506, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "Using the residual connection to perform concatenation, the output of the attention layer has the form", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 286, + 183 + ], + "score": 0.75, + "content": "( e _ { \\mathrm { i d } ( s _ { i } ) } , i , 1 , n _ { i } ^ { = } / i , \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathrm { \\iota } ^ { \\circ } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 170, + 447, + 183 + ], + "score": 1.0, + "content": "). We can then use an MLP to multiply", + "type": "text" + }, + { + "bbox": [ + 448, + 170, + 470, + 182 + ], + "score": 0.93, + "content": "n _ { i } ^ { = } / i", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 170, + 488, + 183 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 489, + 171, + 493, + 180 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 135, + 196 + ], + "score": 1.0, + "content": "obtain", + "type": "text" + }, + { + "bbox": [ + 135, + 184, + 148, + 195 + ], + "score": 0.88, + "content": "n _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 181, + 288, + 196 + ], + "score": 1.0, + "content": "and use another MLP to compute", + "type": "text" + }, + { + "bbox": [ + 288, + 182, + 298, + 193 + ], + "score": 0.87, + "content": "i ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "according to Lemma C.1; Simultaneously, we can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 298, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 182, + 207 + ], + "score": 1.0, + "content": "compute the value", + "type": "text" + }, + { + "bbox": [ + 182, + 195, + 195, + 206 + ], + "score": 0.9, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 194, + 298, + 207 + ], + "score": 1.0, + "content": "using the following way:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 210, + 408, + 238 + ], + "lines": [ + { + "bbox": [ + 196, + 210, + 408, + 238 + ], + "spans": [ + { + "bbox": [ + 196, + 210, + 408, + 238 + ], + "score": 0.92, + "content": "p _ { i } ^ { = } = \\left\\{ \\begin{array} { l l } { \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathit { \\iota } ^ { * } = ^ { \\prime } \\} } & { \\mathrm { i f ~ } n _ { i } ^ { = } / i \\geq 1 / n , } \\\\ { 0 } & { \\mathrm { i f ~ } n _ { i } ^ { = } / i = 0 , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "36c991f806b77a5881771d35879575a9e653624ff2704116d0f78a17ecd5e82a.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 196, + 210, + 408, + 238 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "which is a conditional selection operation and can be implemented by an MLP (Lemma C.4). Also", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 253, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 244, + 266 + ], + "score": 1.0, + "content": "note that the gap in Lemma C.4 is", + "type": "text" + }, + { + "bbox": [ + 245, + 253, + 286, + 266 + ], + "score": 0.92, + "content": "\\alpha = 1 / 2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 253, + 504, + 266 + ], + "score": 1.0, + "content": ", which can be implemented within log-precision. The", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 261, + 407, + 284 + ], + "spans": [ + { + "bbox": [ + 103, + 261, + 273, + 284 + ], + "score": 1.0, + "content": "final output of the first layer has the form", + "type": "text" + }, + { + "bbox": [ + 273, + 265, + 402, + 280 + ], + "score": 0.93, + "content": "\\pmb { x } _ { i } ^ { ( 1 ) } = ( e _ { \\mathrm { i d } ( s _ { i } ) } , i , i ^ { 2 } , n _ { i } ^ { = } , p _ { i } ^ { = } , 1 ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 261, + 407, + 284 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 504, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 298 + ], + "score": 1.0, + "content": "Layer 2. The second layer of the Transformer does some tricky preparation work for the next layer.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 121, + 301, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 123, + 300, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 123, + 300, + 409, + 315 + ], + "score": 1.0, + "content": "1. Compute the distance to the nearest and the last equal sign, denoted as", + "type": "text" + }, + { + "bbox": [ + 410, + 301, + 423, + 313 + ], + "score": 0.87, + "content": "d _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 300, + 439, + 315 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 300, + 452, + 314 + ], + "score": 0.9, + "content": "\\hat { d } _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 300, + 506, + 315 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 135, + 313, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 135, + 314, + 178, + 328 + ], + "score": 1.0, + "content": "Formally,", + "type": "text" + }, + { + "bbox": [ + 178, + 314, + 325, + 327 + ], + "score": 0.9, + "content": "d _ { i } ^ { = } = i - \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\ ' = \\ ' \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 314, + 329, + 328 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 329, + 313, + 476, + 327 + ], + "score": 0.88, + "content": "\\hat { d } _ { i } ^ { = } = i - \\operatorname* { m a x } \\{ j : j < i , s _ { j } = { \\stackrel { . } { \\mathbf { \\ i } } } = { \\stackrel { . } { \\mathbf { \\ j } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 314, + 506, + 328 + ], + "score": 1.0, + "content": ". If the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 135, + 326, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 135, + 326, + 311, + 342 + ], + "score": 1.0, + "content": "nearest/last equal sign does not exist, define", + "type": "text" + }, + { + "bbox": [ + 312, + 329, + 341, + 340 + ], + "score": 0.9, + "content": "d _ { i } ^ { = } = i", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 326, + 353, + 342 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 353, + 327, + 383, + 340 + ], + "score": 0.91, + "content": "\\hat { d } _ { i } ^ { = } = i", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 326, + 473, + 342 + ], + "score": 1.0, + "content": ". The relation between", + "type": "text" + }, + { + "bbox": [ + 473, + 327, + 502, + 340 + ], + "score": 0.26, + "content": "d _ { i } ^ { = } , { \\hat { d } } _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 326, + 506, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 135, + 340, + 370, + 356 + ], + "spans": [ + { + "bbox": [ + 135, + 340, + 153, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 342, + 166, + 354 + ], + "score": 0.89, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 340, + 248, + 356 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + }, + { + "bbox": [ + 248, + 340, + 366, + 354 + ], + "score": 0.73, + "content": "d _ { i } ^ { = } = i - p _ { i } ^ { = } , \\hat { d } _ { i } ^ { = } = i - p _ { i - 1 } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 340, + 370, + 356 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 122, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 122, + 356, + 419, + 371 + ], + "score": 1.0, + "content": "2. Count the number of equal signs in strictly previous tokens, denoted as", + "type": "text" + }, + { + "bbox": [ + 419, + 358, + 433, + 370 + ], + "score": 0.9, + "content": "\\hat { n } _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 356, + 453, + 371 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 454, + 357, + 505, + 370 + ], + "score": 0.92, + "content": "\\hat { n } _ { i } ^ { \\equiv } = | \\{ j <", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 137, + 368, + 196, + 382 + ], + "spans": [ + { + "bbox": [ + 137, + 369, + 191, + 381 + ], + "score": 0.91, + "content": "i : s _ { j } = \\operatorname { } \\operatorname { \\overrightarrow { \\partial } } \\} |", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 368, + 196, + 382 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 122, + 385, + 309, + 400 + ], + "spans": [ + { + "bbox": [ + 122, + 386, + 176, + 400 + ], + "score": 1.0, + "content": "3. Compute", + "type": "text" + }, + { + "bbox": [ + 176, + 387, + 260, + 400 + ], + "score": 0.6, + "content": "( n _ { i } ^ { = } ) ^ { 2 } , ( \\hat { n } _ { i } ^ { = } ) ^ { 2 } , ( d _ { i } ^ { = } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 386, + 281, + 400 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 281, + 385, + 306, + 400 + ], + "score": 0.91, + "content": "( \\hat { d } _ { i } ^ { = } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 386, + 309, + 400 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 426, + 416 + ], + "lines": [ + { + "bbox": [ + 104, + 401, + 427, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 427, + 419 + ], + "score": 1.0, + "content": "The first and the second tasks can be done using the COPY operation by setting", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 420, + 510, + 438 + ], + "lines": [ + { + "bbox": [ + 111, + 420, + 510, + 438 + ], + "spans": [ + { + "bbox": [ + 111, + 420, + 510, + 438 + ], + "score": 0.9, + "content": "t _ { i } = Q x _ { i } ^ { ( 1 ) } = ( ( i - 1 ) ^ { 2 } , i - 1 , 1 ) , \\quad k _ { j } = K x _ { j } ^ { ( 1 ) } = ( - 1 , 2 j , - j ^ { 2 } ) , \\quad r _ { j } = 0 , \\quad v _ { j } = ( n _ { j } , j - p _ { j } ^ { = } + 1 ) .", + "type": "interline_equation", + "image_path": "8faf5dad76c703e2865d92f6ffe2d3730fa445da9fae1555bf5850eae0ef29cd.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 111, + 420, + 510, + 438 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 507, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 269, + 458 + ], + "score": 1.0, + "content": "Under the above construction, we have", + "type": "text" + }, + { + "bbox": [ + 269, + 443, + 370, + 457 + ], + "score": 0.91, + "content": "\\pmb { q } _ { i } \\cdot \\pmb { k } _ { j } = - ( i - j - 1 ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 442, + 411, + 458 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 412, + 444, + 467, + 456 + ], + "score": 0.94, + "content": "S _ { i } = \\{ i - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 442, + 506, + 458 + ], + "score": 1.0, + "content": ", namely,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 160, + 471 + ], + "score": 1.0, + "content": "the output is", + "type": "text" + }, + { + "bbox": [ + 160, + 456, + 194, + 470 + ], + "score": 0.93, + "content": "( \\hat { n } _ { i } , \\hat { d } _ { i } ^ { = } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 456, + 335, + 471 + ], + "score": 1.0, + "content": ". We then use an MLP to calculate", + "type": "text" + }, + { + "bbox": [ + 336, + 456, + 419, + 470 + ], + "score": 0.47, + "content": "( d _ { i } ^ { = } ) ^ { 2 } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , ( n _ { i } ^ { = } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 456, + 441, + 471 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 441, + 457, + 466, + 470 + ], + "score": 0.91, + "content": "( \\hat { n } _ { i } ^ { = } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "by using", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 468, + 293, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 293, + 481 + ], + "score": 1.0, + "content": "Lemma C.1. The output of the second layer is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 485, + 440, + 502 + ], + "lines": [ + { + "bbox": [ + 171, + 485, + 440, + 502 + ], + "spans": [ + { + "bbox": [ + 171, + 485, + 440, + 502 + ], + "score": 0.9, + "content": "\\pmb { x } _ { i } ^ { ( 2 ) } = ( \\pmb { e } _ { \\mathrm { i d } ( s _ { i } ) } , i , n _ { i } ^ { = } , \\hat { n } _ { i } ^ { = } , d _ { i } ^ { = } , \\hat { d } _ { i } ^ { = } , ( n _ { i } ^ { = } ) ^ { 2 } , ( \\hat { n } _ { i } ^ { = } ) ^ { 2 } , ( d _ { i } ^ { = } ) ^ { 2 } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , 1 ) .", + "type": "interline_equation", + "image_path": "edc7488e78fc128437cf225061fb5bd758b3bd01b2613af3b60c043b05ce5fbb.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 171, + 485, + 440, + 502 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "Layer 3. 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If it can be evaluated, compute the result and determine how much", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 135, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 135, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "sentence length will be reduced after this calculation (see Appendix B for details on how", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 135, + 623, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 135, + 623, + 455, + 639 + ], + "score": 1.0, + "content": "the reduced sentence length depends on brackets); otherwise, keep the token", + "type": "text" + }, + { + "bbox": [ + 455, + 626, + 482, + 638 + ], + "score": 0.91, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 623, + 506, + 639 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 136, + 637, + 271, + 653 + ], + "spans": [ + { + "bbox": [ + 136, + 638, + 194, + 653 + ], + "score": 0.9, + "content": "\\hat { n } _ { j } ^ { = } = n _ { i } ^ { = } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 638, + 212, + 652 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 637, + 267, + 653 + ], + "score": 0.94, + "content": "\\hat { d } _ { j } ^ { = } = d _ { i } ^ { = } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 638, + 271, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 104, + 655, + 506, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 497, + 668 + ], + "score": 1.0, + "content": "We can use the multi-head attention to perform the COPY operation five times in parallel. For each", + "type": "text" + }, + { + "bbox": [ + 497, + 657, + 502, + 665 + ], + "score": 0.59, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 653, + 507, + 668 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 384, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 210, + 678 + ], + "score": 1.0, + "content": "we construct the matrices", + "type": "text" + }, + { + "bbox": [ + 210, + 666, + 250, + 677 + ], + "score": 0.34, + "content": "Q , K , V", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 666, + 384, + 678 + ], + "score": 1.0, + "content": "of the COPY operation such that", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 683, + 514, + 716 + ], + "lines": [ + { + "bbox": [ + 111, + 683, + 514, + 716 + ], + "spans": [ + { + "bbox": [ + 111, + 683, + 514, + 716 + ], + "score": 0.83, + "content": "\\begin{array} { l } { { { \\displaystyle t } _ { i } = Q x _ { i } ^ { ( 2 ) } = [ ( n _ { i } ^ { - } ) ^ { 2 } - 2 n _ { i } ^ { - } + 1 , \\qquad 1 , \\qquad n _ { i } ^ { - } - 1 , \\quad ( d _ { i } ^ { - } ) ^ { 2 } + 2 t d _ { i } ^ { - } + t ^ { 2 } , \\qquad 1 , \\qquad d _ { i } ^ { - } - t ] ^ { \\top } } } \\\\ { { \\displaystyle z _ { j } = K x _ { j } ^ { ( 2 ) } = [ \\qquad - 1 , \\qquad - ( \\hat { n } _ { j } ^ { - } ) ^ { 2 } , \\quad \\quad 2 \\hat { n } _ { j } ^ { - } , \\qquad - 1 , \\qquad - ( \\hat { d } _ { j } ^ { - } ) ^ { 2 } , \\quad \\quad 2 \\hat { d } _ { j } ^ { - } { \\bf \\sigma } ] ^ { \\top } } } \\end{array}", + "type": "interline_equation", + "image_path": "dab5d34e72c317329c1c611d914f72e09882a2d77327e2b423039996a52cb90c.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 111, + 683, + 514, + 694.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 111, + 694.0, + 514, + 705.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 111, + 705.0, + 514, + 716.0 + ], + "spans": [], + "index": 43 + } + ] + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 755 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 108 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "Similarly, we can use the second attention head to perform a COPY operation that copies the position", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 437, + 97 + ], + "score": 1.0, + "content": "index of the last equal sign (by Lemma C.7). This can be achieved by setting", + "type": "text" + }, + { + "bbox": [ + 437, + 83, + 502, + 96 + ], + "score": 0.93, + "content": "\\dot { \\boldsymbol { Q } } = ( \\dot { \\mathbf { 0 } } , 0 , 1 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 83, + 506, + 97 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 94, + 439, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 199, + 109 + ], + "score": 0.87, + "content": "\\pmb { K } = ( e _ { \\mathrm { i d } ( \\cdot = \\cdot ) } , 0 , - 1 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 94, + 204, + 109 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 204, + 95, + 266, + 108 + ], + "score": 0.86, + "content": "V = ( \\mathbf { 0 } , 1 , 0 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 94, + 270, + 109 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 270, + 95, + 298, + 108 + ], + "score": 0.9, + "content": "r _ { j } = j", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 94, + 439, + 109 + ], + "score": 1.0, + "content": "(defined in Appendix C.2), so that", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 72, + 506, + 109 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 113, + 446, + 127 + ], + "lines": [ + { + "bbox": [ + 164, + 113, + 446, + 127 + ], + "spans": [ + { + "bbox": [ + 164, + 113, + 446, + 127 + ], + "score": 0.78, + "content": "q _ { i } = 1 , \\quad k _ { j } = \\mathbb { I } [ s _ { j } = \\cdot = \\cdot = 1 , \\quad v _ { j } = j , \\quad S _ { i } = \\{ j \\leq i : s _ { j } = \\cdot = \\cdot = \\} .", + "type": "interline_equation", + "image_path": "707bc5f7566a155ec219f7741731b0b5d9ce758d24ecb8fe2bf158eb8c63fe71.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 164, + 113, + 446, + 127 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 104, + 130, + 503, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 130, + 323, + 145 + ], + "score": 1.0, + "content": "It is easy to check that the above construction outputs", + "type": "text" + }, + { + "bbox": [ + 324, + 131, + 448, + 144 + ], + "score": 0.91, + "content": "u _ { i } = \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathit { \\iota } ^ { \\circ } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 130, + 474, + 145 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 474, + 132, + 503, + 144 + ], + "score": 0.93, + "content": "\\boldsymbol { S } _ { i } \\neq \\boldsymbol { \\emptyset }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 298, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 145, + 155 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 145, + 156, + 154 + ], + "score": 0.85, + "content": "u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 142, + 226, + 155 + ], + "score": 1.0, + "content": "may not equal to", + "type": "text" + }, + { + "bbox": [ + 226, + 144, + 239, + 155 + ], + "score": 0.89, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 142, + 264, + 155 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 265, + 143, + 293, + 154 + ], + "score": 0.91, + "content": "\\boldsymbol { S } _ { i } = \\boldsymbol { \\emptyset }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 142, + 298, + 155 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 130, + 503, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 506, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "Using the residual connection to perform concatenation, the output of the attention layer has the form", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 286, + 183 + ], + "score": 0.75, + "content": "( e _ { \\mathrm { i d } ( s _ { i } ) } , i , 1 , n _ { i } ^ { = } / i , \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathrm { \\iota } ^ { \\circ } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 170, + 447, + 183 + ], + "score": 1.0, + "content": "). We can then use an MLP to multiply", + "type": "text" + }, + { + "bbox": [ + 448, + 170, + 470, + 182 + ], + "score": 0.93, + "content": "n _ { i } ^ { = } / i", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 170, + 488, + 183 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 489, + 171, + 493, + 180 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 135, + 196 + ], + "score": 1.0, + "content": "obtain", + "type": "text" + }, + { + "bbox": [ + 135, + 184, + 148, + 195 + ], + "score": 0.88, + "content": "n _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 181, + 288, + 196 + ], + "score": 1.0, + "content": "and use another MLP to compute", + "type": "text" + }, + { + "bbox": [ + 288, + 182, + 298, + 193 + ], + "score": 0.87, + "content": "i ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "according to Lemma C.1; Simultaneously, we can", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 298, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 182, + 207 + ], + "score": 1.0, + "content": "compute the value", + "type": "text" + }, + { + "bbox": [ + 182, + 195, + 195, + 206 + ], + "score": 0.9, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 194, + 298, + 207 + ], + "score": 1.0, + "content": "using the following way:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 158, + 506, + 207 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 210, + 408, + 238 + ], + "lines": [ + { + "bbox": [ + 196, + 210, + 408, + 238 + ], + "spans": [ + { + "bbox": [ + 196, + 210, + 408, + 238 + ], + "score": 0.92, + "content": "p _ { i } ^ { = } = \\left\\{ \\begin{array} { l l } { \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\mathit { \\iota } ^ { * } = ^ { \\prime } \\} } & { \\mathrm { i f ~ } n _ { i } ^ { = } / i \\geq 1 / n , } \\\\ { 0 } & { \\mathrm { i f ~ } n _ { i } ^ { = } / i = 0 , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "36c991f806b77a5881771d35879575a9e653624ff2704116d0f78a17ecd5e82a.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 196, + 210, + 408, + 238 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "which is a conditional selection operation and can be implemented by an MLP (Lemma C.4). Also", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 253, + 504, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 244, + 266 + ], + "score": 1.0, + "content": "note that the gap in Lemma C.4 is", + "type": "text" + }, + { + "bbox": [ + 245, + 253, + 286, + 266 + ], + "score": 0.92, + "content": "\\alpha = 1 / 2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 253, + 504, + 266 + ], + "score": 1.0, + "content": ", which can be implemented within log-precision. The", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 261, + 407, + 284 + ], + "spans": [ + { + "bbox": [ + 103, + 261, + 273, + 284 + ], + "score": 1.0, + "content": "final output of the first layer has the form", + "type": "text" + }, + { + "bbox": [ + 273, + 265, + 402, + 280 + ], + "score": 0.93, + "content": "\\pmb { x } _ { i } ^ { ( 1 ) } = ( e _ { \\mathrm { i d } ( s _ { i } ) } , i , i ^ { 2 } , n _ { i } ^ { = } , p _ { i } ^ { = } , 1 ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 261, + 407, + 284 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 103, + 242, + 505, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 504, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 298 + ], + "score": 1.0, + "content": "Layer 2. The second layer of the Transformer does some tricky preparation work for the next layer.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 281, + 506, + 298 + ] + }, + { + "type": "list", + "bbox": [ + 121, + 301, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 123, + 300, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 123, + 300, + 409, + 315 + ], + "score": 1.0, + "content": "1. Compute the distance to the nearest and the last equal sign, denoted as", + "type": "text" + }, + { + "bbox": [ + 410, + 301, + 423, + 313 + ], + "score": 0.87, + "content": "d _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 300, + 439, + 315 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 439, + 300, + 452, + 314 + ], + "score": 0.9, + "content": "\\hat { d } _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 300, + 506, + 315 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 313, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 135, + 314, + 178, + 328 + ], + "score": 1.0, + "content": "Formally,", + "type": "text" + }, + { + "bbox": [ + 178, + 314, + 325, + 327 + ], + "score": 0.9, + "content": "d _ { i } ^ { = } = i - \\operatorname* { m a x } \\{ j : j \\leq i , s _ { j } = \\ ' = \\ ' \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 314, + 329, + 328 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 329, + 313, + 476, + 327 + ], + "score": 0.88, + "content": "\\hat { d } _ { i } ^ { = } = i - \\operatorname* { m a x } \\{ j : j < i , s _ { j } = { \\stackrel { . } { \\mathbf { \\ i } } } = { \\stackrel { . } { \\mathbf { \\ j } } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 314, + 506, + 328 + ], + "score": 1.0, + "content": ". If the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 135, + 326, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 135, + 326, + 311, + 342 + ], + "score": 1.0, + "content": "nearest/last equal sign does not exist, define", + "type": "text" + }, + { + "bbox": [ + 312, + 329, + 341, + 340 + ], + "score": 0.9, + "content": "d _ { i } ^ { = } = i", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 326, + 353, + 342 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 353, + 327, + 383, + 340 + ], + "score": 0.91, + "content": "\\hat { d } _ { i } ^ { = } = i", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 326, + 473, + 342 + ], + "score": 1.0, + "content": ". The relation between", + "type": "text" + }, + { + "bbox": [ + 473, + 327, + 502, + 340 + ], + "score": 0.26, + "content": "d _ { i } ^ { = } , { \\hat { d } } _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 326, + 506, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 135, + 340, + 370, + 356 + ], + "spans": [ + { + "bbox": [ + 135, + 340, + 153, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 342, + 166, + 354 + ], + "score": 0.89, + "content": "p _ { i } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 340, + 248, + 356 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + }, + { + "bbox": [ + 248, + 340, + 366, + 354 + ], + "score": 0.73, + "content": "d _ { i } ^ { = } = i - p _ { i } ^ { = } , \\hat { d } _ { i } ^ { = } = i - p _ { i - 1 } ^ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 340, + 370, + 356 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 122, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 122, + 356, + 419, + 371 + ], + "score": 1.0, + "content": "2. 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Note that whether the current position can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 366, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 329, + 381 + ], + "score": 1.0, + "content": "calculated or not depends on the following six tokens", + "type": "text" + }, + { + "bbox": [ + 330, + 367, + 451, + 380 + ], + "score": 0.9, + "content": "( e _ { \\mathrm { i d } ( s _ { i } ) } , e _ { j _ { 1 } } , e _ { j _ { 2 } } , e _ { j _ { 3 } } , e _ { j _ { 4 } } , e _ { j _ { 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 366, + 507, + 381 + ], + "score": 1.0, + "content": ". Concretely,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 377, + 200, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 200, + 389 + ], + "score": 1.0, + "content": "there are several cases:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 127, + 394, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 127, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 127, + 394, + 136, + 407 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 396, + 163, + 407 + ], + "score": 0.88, + "content": "e _ { \\mathrm { i d } ( s _ { i } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "corresponds to the embedding of a number or a right bracket. In this case, the current", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 135, + 405, + 437, + 418 + ], + "spans": [ + { + "bbox": [ + 135, + 405, + 257, + 418 + ], + "score": 1.0, + "content": "position should simply output", + "type": "text" + }, + { + "bbox": [ + 258, + 406, + 272, + 417 + ], + "score": 0.89, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 405, + 437, + 418 + ], + "score": 1.0, + "content": "(which is an operator or a right bracket).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 127, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 127, + 422, + 136, + 433 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 424, + 163, + 434 + ], + "score": 0.89, + "content": "e _ { \\mathrm { i d } ( s _ { i } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 422, + 476, + 433 + ], + "score": 1.0, + "content": "corresponds to the embedding of a left bracket, an operator, or the equal sign", + "type": "text" + }, + { + "bbox": [ + 477, + 422, + 489, + 432 + ], + "score": 0.45, + "content": "\\cdot _ { = } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 422, + 505, + 433 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 135, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 135, + 432, + 177, + 446 + ], + "score": 1.0, + "content": "this case,", + "type": "text" + }, + { + "bbox": [ + 177, + 434, + 191, + 444 + ], + "score": 0.88, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "corresponds to the embedding of a number or a left bracket. There are two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 136, + 444, + 176, + 454 + ], + "spans": [ + { + "bbox": [ + 136, + 444, + 176, + 454 + ], + "score": 1.0, + "content": "subcases:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 154, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 154, + 459, + 165, + 473 + ], + "score": 1.0, + "content": "–", + "type": "text" + }, + { + "bbox": [ + 166, + 461, + 180, + 471 + ], + "score": 0.87, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "corresponds to the embedding of a number. In this case, whether the current posi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 164, + 469, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 164, + 469, + 298, + 486 + ], + "score": 1.0, + "content": "tion can be evaluated depends on", + "type": "text" + }, + { + "bbox": [ + 298, + 471, + 402, + 483 + ], + "score": 0.89, + "content": "( e _ { \\mathrm { i d } ( s _ { i } ) } , e _ { j _ { 1 } } , e _ { j _ { 2 } } , e _ { j _ { 3 } } , e _ { j _ { 4 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 469, + 507, + 486 + ], + "score": 1.0, + "content": "according to Appendix B.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 154, + 481, + 507, + 495 + ], + "spans": [ + { + "bbox": [ + 154, + 481, + 165, + 495 + ], + "score": 1.0, + "content": "–", + "type": "text" + }, + { + "bbox": [ + 166, + 483, + 180, + 495 + ], + "score": 0.83, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 481, + 507, + 495 + ], + "score": 1.0, + "content": "corresponds to the embedding of a left bracket. In this case, whether the cur-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 165, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 165, + 493, + 410, + 507 + ], + "score": 1.0, + "content": "rent position can be evaluated simply depends on whether", + "type": "text" + }, + { + "bbox": [ + 410, + 495, + 424, + 505 + ], + "score": 0.88, + "content": "e _ { j _ { 5 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "corresponds to the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 166, + 504, + 286, + 516 + ], + "spans": [ + { + "bbox": [ + 166, + 504, + 286, + 516 + ], + "score": 1.0, + "content": "embedding of a right bracket.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 104, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 198, + 533 + ], + "score": 1.0, + "content": "When all embeddings", + "type": "text" + }, + { + "bbox": [ + 198, + 522, + 212, + 532 + ], + "score": 0.88, + "content": "e _ { j _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "are one-hot vectors, whether the expression at the current position can", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "be calculated or not forms a look-up table. Therefore, it can be implemented by a two-layer MLP", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 203, + 554 + ], + "score": 1.0, + "content": "with hidden dimension", + "type": "text" + }, + { + "bbox": [ + 203, + 541, + 230, + 554 + ], + "score": 0.93, + "content": "O ( p ^ { 6 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "according to Lemma C.5. Similarly, the computed result and how", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 505, + 566 + ], + "score": 1.0, + "content": "much sentence length will be reduced after this calculation can also be implemented as look-up", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 278, + 577 + ], + "score": 1.0, + "content": "tables. 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In this case, we can similarly implement extra look-up tables that take a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "fewer number of inputs, and the result of which lookup table will be used depends on the position", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "of the equal sign. This corresponds to a multivariate conditional selection operation with multiple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 606, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 186, + 622 + ], + "score": 1.0, + "content": "Boolean conditions", + "type": "text" + }, + { + "bbox": [ + 186, + 608, + 262, + 620 + ], + "score": 0.92, + "content": "\\mathbb { I } [ e _ { j _ { t } } \\cdot e _ { \\mathrm { i d } ( \\cdot = \\cdot ) } = 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 606, + 281, + 622 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 281, + 607, + 353, + 619 + ], + "score": 0.9, + "content": "t \\in \\{ 1 , 2 , 3 , 4 , 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 606, + 506, + 622 + ], + "score": 1.0, + "content": ", which can be similarly implemented", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "by an MLP by extending Lemma C.4. Moreover, we note that the composition of look-up tables", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "and the multivariate conditional selection operation can be merged in just one MLP by following the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 384, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 384, + 653 + ], + "score": 1.0, + "content": "construction in Lemmas C.4 and C.5 (we omit the details for clarity).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 311, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 312, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 312, + 672 + ], + "score": 1.0, + "content": "The final output of the third layer is represented by", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 671, + 448, + 686 + ], + "lines": [ + { + "bbox": [ + 162, + 671, + 448, + 686 + ], + "spans": [ + { + "bbox": [ + 162, + 671, + 448, + 686 + ], + "score": 0.88, + "content": "\\pmb { x } _ { i } ^ { ( 3 ) } = ( \\pmb { e } _ { j _ { 1 } } , \\boldsymbol { n } _ { i } ^ { = } , ( \\boldsymbol { n } _ { i } ^ { = } ) ^ { 2 } , d _ { i } ^ { = } , \\hat { n } _ { i } ^ { = } , ( \\hat { n } _ { i } ^ { = } ) ^ { 2 } , \\hat { d } _ { i } ^ { = } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , f _ { i } , e _ { i } ^ { \\mathrm { o u t c o m e } } , \\boldsymbol { n } _ { i } ^ { \\mathrm { r e d u c e } } ) .", + "type": "interline_equation", + "image_path": "ab4bd7dd3d8dddb5c0efa8f32d66e59263c873e4870d76a09fa6c58fb22adc3d.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 162, + 671, + 448, + 686 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 132, + 700 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 132, + 688, + 141, + 699 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 687, + 412, + 700 + ], + "score": 1.0, + "content": "is a Boolean value recording whether the next output at position", + "type": "text" + }, + { + "bbox": [ + 412, + 689, + 417, + 698 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "is a computed value,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 695, + 508, + 713 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 137, + 711 + ], + "score": 0.87, + "content": "e _ { i } ^ { \\mathrm { { o u t c o m e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 695, + 335, + 713 + ], + "score": 1.0, + "content": "is the one-hot embedding of the outcome when", + "type": "text" + }, + { + "bbox": [ + 335, + 699, + 344, + 710 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 695, + 394, + 713 + ], + "score": 1.0, + "content": "is true, and", + "type": "text" + }, + { + "bbox": [ + 395, + 699, + 421, + 711 + ], + "score": 0.91, + "content": "n _ { i } ^ { \\mathrm { r e d u c e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 695, + 508, + 713 + ], + "score": 1.0, + "content": "records the reduced", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 709, + 471, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 225, + 723 + ], + "score": 1.0, + "content": "length after calculation when", + "type": "text" + }, + { + "bbox": [ + 225, + 711, + 234, + 722 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 709, + 292, + 723 + ], + "score": 1.0, + "content": "is true. 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The output of the attention layer can be written as", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 276, + 506, + 310 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 312, + 453, + 327 + ], + "lines": [ + { + "bbox": [ + 156, + 312, + 453, + 327 + ], + "spans": [ + { + "bbox": [ + 156, + 312, + 453, + 327 + ], + "score": 0.86, + "content": "\\begin{array} { r } { ( e _ { \\mathrm { i d } ( s _ { i } ) } , e _ { j _ { 1 } } , e _ { j _ { 2 } } , e _ { j _ { 3 } } , e _ { j _ { 4 } } , e _ { j _ { 5 } } , i , n _ { i } ^ { = } , ( n _ { i } ^ { = } ) ^ { 2 } , d _ { i } ^ { = } , \\hat { n } _ { i } ^ { = } , ( \\hat { n } _ { i } ^ { = } ) ^ { 2 } , \\hat { d } _ { i } ^ { = } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , 1 ) , } \\end{array}", + "type": "interline_equation", + "image_path": "66a589a0fcb15d28ac04e70b7b2319b81ac10ffdb31b3c3891e6655fbff809f9.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 156, + 312, + 453, + 327 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 329, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 268, + 340 + ], + "score": 1.0, + "content": "where we slightly abuse notation and use", + "type": "text" + }, + { + "bbox": [ + 269, + 331, + 282, + 340 + ], + "score": 0.88, + "content": "e _ { j _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 327, + 451, + 340 + ], + "score": 1.0, + "content": "to denote the embedding we copied by the", + "type": "text" + }, + { + "bbox": [ + 452, + 330, + 456, + 338 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "-th attention", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 136, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 136, + 352 + ], + "score": 1.0, + "content": "heads.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 327, + 505, + 352 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 506, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 367 + ], + "score": 1.0, + "content": "We can then use an MLP to perform the second task. Note that whether the current position can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 366, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 329, + 381 + ], + "score": 1.0, + "content": "calculated or not depends on the following six tokens", + "type": "text" + }, + { + "bbox": [ + 330, + 367, + 451, + 380 + ], + "score": 0.9, + "content": "( e _ { \\mathrm { i d } ( s _ { i } ) } , e _ { j _ { 1 } } , e _ { j _ { 2 } } , e _ { j _ { 3 } } , e _ { j _ { 4 } } , e _ { j _ { 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 366, + 507, + 381 + ], + "score": 1.0, + "content": ". Concretely,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 377, + 200, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 200, + 389 + ], + "score": 1.0, + "content": "there are several cases:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 356, + 507, + 389 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 394, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 127, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 127, + 394, + 136, + 407 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 396, + 163, + 407 + ], + "score": 0.88, + "content": "e _ { \\mathrm { i d } ( s _ { i } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "corresponds to the embedding of a number or a right bracket. In this case, the current", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 405, + 437, + 418 + ], + "spans": [ + { + "bbox": [ + 135, + 405, + 257, + 418 + ], + "score": 1.0, + "content": "position should simply output", + "type": "text" + }, + { + "bbox": [ + 258, + 406, + 272, + 417 + ], + "score": 0.89, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 405, + 437, + 418 + ], + "score": 1.0, + "content": "(which is an operator or a right bracket).", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 127, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 127, + 422, + 136, + 433 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 424, + 163, + 434 + ], + "score": 0.89, + "content": "e _ { \\mathrm { i d } ( s _ { i } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 422, + 476, + 433 + ], + "score": 1.0, + "content": "corresponds to the embedding of a left bracket, an operator, or the equal sign", + "type": "text" + }, + { + "bbox": [ + 477, + 422, + 489, + 432 + ], + "score": 0.45, + "content": "\\cdot _ { = } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 422, + 505, + 433 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 135, + 432, + 177, + 446 + ], + "score": 1.0, + "content": "this case,", + "type": "text" + }, + { + "bbox": [ + 177, + 434, + 191, + 444 + ], + "score": 0.88, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "corresponds to the embedding of a number or a left bracket. There are two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 136, + 444, + 176, + 454 + ], + "spans": [ + { + "bbox": [ + 136, + 444, + 176, + 454 + ], + "score": 1.0, + "content": "subcases:", + "type": "text" + } + ], + "index": 26, + "is_list_end_line": true + }, + { + "bbox": [ + 154, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 154, + 459, + 165, + 473 + ], + "score": 1.0, + "content": "–", + "type": "text" + }, + { + "bbox": [ + 166, + 461, + 180, + 471 + ], + "score": 0.87, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "corresponds to the embedding of a number. In this case, whether the current posi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 164, + 469, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 164, + 469, + 298, + 486 + ], + "score": 1.0, + "content": "tion can be evaluated depends on", + "type": "text" + }, + { + "bbox": [ + 298, + 471, + 402, + 483 + ], + "score": 0.89, + "content": "( e _ { \\mathrm { i d } ( s _ { i } ) } , e _ { j _ { 1 } } , e _ { j _ { 2 } } , e _ { j _ { 3 } } , e _ { j _ { 4 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 469, + 507, + 486 + ], + "score": 1.0, + "content": "according to Appendix B.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 154, + 481, + 507, + 495 + ], + "spans": [ + { + "bbox": [ + 154, + 481, + 165, + 495 + ], + "score": 1.0, + "content": "–", + "type": "text" + }, + { + "bbox": [ + 166, + 483, + 180, + 495 + ], + "score": 0.83, + "content": "e _ { j _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 481, + 507, + 495 + ], + "score": 1.0, + "content": "corresponds to the embedding of a left bracket. In this case, whether the cur-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 165, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 165, + 493, + 410, + 507 + ], + "score": 1.0, + "content": "rent position can be evaluated simply depends on whether", + "type": "text" + }, + { + "bbox": [ + 410, + 495, + 424, + 505 + ], + "score": 0.88, + "content": "e _ { j _ { 5 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "corresponds to the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 166, + 504, + 286, + 516 + ], + "spans": [ + { + "bbox": [ + 166, + 504, + 286, + 516 + ], + "score": 1.0, + "content": "embedding of a right bracket.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + } + ], + "index": 26.5, + "bbox_fs": [ + 127, + 394, + 507, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 104, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 198, + 533 + ], + "score": 1.0, + "content": "When all embeddings", + "type": "text" + }, + { + "bbox": [ + 198, + 522, + 212, + 532 + ], + "score": 0.88, + "content": "e _ { j _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "are one-hot vectors, whether the expression at the current position can", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "be calculated or not forms a look-up table. Therefore, it can be implemented by a two-layer MLP", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 203, + 554 + ], + "score": 1.0, + "content": "with hidden dimension", + "type": "text" + }, + { + "bbox": [ + 203, + 541, + 230, + 554 + ], + "score": 0.93, + "content": "O ( p ^ { 6 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "according to Lemma C.5. Similarly, the computed result and how", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 505, + 566 + ], + "score": 1.0, + "content": "much sentence length will be reduced after this calculation can also be implemented as look-up", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 278, + 577 + ], + "score": 1.0, + "content": "tables. However, some of the embeddings", + "type": "text" + }, + { + "bbox": [ + 279, + 565, + 292, + 576 + ], + "score": 0.88, + "content": "e _ { j _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "may not be one-hot vectors when reaching an equal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "sign (as discussed above). In this case, we can similarly implement extra look-up tables that take a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "fewer number of inputs, and the result of which lookup table will be used depends on the position", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "of the equal sign. This corresponds to a multivariate conditional selection operation with multiple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 606, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 186, + 622 + ], + "score": 1.0, + "content": "Boolean conditions", + "type": "text" + }, + { + "bbox": [ + 186, + 608, + 262, + 620 + ], + "score": 0.92, + "content": "\\mathbb { I } [ e _ { j _ { t } } \\cdot e _ { \\mathrm { i d } ( \\cdot = \\cdot ) } = 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 606, + 281, + 622 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 281, + 607, + 353, + 619 + ], + "score": 0.9, + "content": "t \\in \\{ 1 , 2 , 3 , 4 , 5 \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 606, + 506, + 622 + ], + "score": 1.0, + "content": ", which can be similarly implemented", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "by an MLP by extending Lemma C.4. Moreover, we note that the composition of look-up tables", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "and the multivariate conditional selection operation can be merged in just one MLP by following the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 384, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 384, + 653 + ], + "score": 1.0, + "content": "construction in Lemmas C.4 and C.5 (we omit the details for clarity).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 519, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 311, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 312, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 312, + 672 + ], + "score": 1.0, + "content": "The final output of the third layer is represented by", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 653, + 312, + 672 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 671, + 448, + 686 + ], + "lines": [ + { + "bbox": [ + 162, + 671, + 448, + 686 + ], + "spans": [ + { + "bbox": [ + 162, + 671, + 448, + 686 + ], + "score": 0.88, + "content": "\\pmb { x } _ { i } ^ { ( 3 ) } = ( \\pmb { e } _ { j _ { 1 } } , \\boldsymbol { n } _ { i } ^ { = } , ( \\boldsymbol { n } _ { i } ^ { = } ) ^ { 2 } , d _ { i } ^ { = } , \\hat { n } _ { i } ^ { = } , ( \\hat { n } _ { i } ^ { = } ) ^ { 2 } , \\hat { d } _ { i } ^ { = } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , f _ { i } , e _ { i } ^ { \\mathrm { o u t c o m e } } , \\boldsymbol { n } _ { i } ^ { \\mathrm { r e d u c e } } ) .", + "type": "interline_equation", + "image_path": "ab4bd7dd3d8dddb5c0efa8f32d66e59263c873e4870d76a09fa6c58fb22adc3d.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 162, + 671, + 448, + 686 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 132, + 700 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 132, + 688, + 141, + 699 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 687, + 412, + 700 + ], + "score": 1.0, + "content": "is a Boolean value recording whether the next output at position", + "type": "text" + }, + { + "bbox": [ + 412, + 689, + 417, + 698 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "is a computed value,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 695, + 508, + 713 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 137, + 711 + ], + "score": 0.87, + "content": "e _ { i } ^ { \\mathrm { { o u t c o m e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 695, + 335, + 713 + ], + "score": 1.0, + "content": "is the one-hot embedding of the outcome when", + "type": "text" + }, + { + "bbox": [ + 335, + 699, + 344, + 710 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 695, + 394, + 713 + ], + "score": 1.0, + "content": "is true, and", + "type": "text" + }, + { + "bbox": [ + 395, + 699, + 421, + 711 + ], + "score": 0.91, + "content": "n _ { i } ^ { \\mathrm { r e d u c e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 695, + 508, + 713 + ], + "score": 1.0, + "content": "records the reduced", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 709, + 471, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 225, + 723 + ], + "score": 1.0, + "content": "length after calculation when", + "type": "text" + }, + { + "bbox": [ + 225, + 711, + 234, + 722 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 709, + 292, + 723 + ], + "score": 1.0, + "content": "is true. When", + "type": "text" + }, + { + "bbox": [ + 292, + 711, + 301, + 722 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 709, + 335, + 723 + ], + "score": 1.0, + "content": "is false,", + "type": "text" + }, + { + "bbox": [ + 336, + 711, + 367, + 723 + ], + "score": 0.52, + "content": "e _ { i } ^ { \\mathrm { { o u t c o m e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 709, + 385, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 711, + 411, + 723 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { r e d u c e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 709, + 471, + 723 + ], + "score": 1.0, + "content": "are undefined.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 687, + 508, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "Layer 4. Note that in an arithmetic expression there can be multiple expressions that can be calculated", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "(or handles defined in Appendix B), all of which are processed in the last layer. Therefore, the fourth", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "layer of the Transformer should keep only the leftmost calculation and discard other calculations", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 372, + 118 + ], + "score": 1.0, + "content": "(according to Appendix B). 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We can simultaneously obtain", + "type": "text" + }, + { + "bbox": [ + 350, + 466, + 360, + 480 + ], + "score": 0.9, + "content": "\\hat { f } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 464, + 421, + 485 + ], + "score": 1.0, + "content": "by multiplying", + "type": "text" + }, + { + "bbox": [ + 422, + 465, + 444, + 484 + ], + "score": 0.94, + "content": "\\frac { \\hat { f } _ { i } } { d _ { i } ^ { = } + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 464, + 466, + 485 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 466, + 468, + 503, + 481 + ], + "score": 0.9, + "content": "( d _ { i } ^ { = } + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 464, + 506, + 485 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 480, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 480, + 176, + 496 + ], + "score": 1.0, + "content": "We also compute", + "type": "text" + }, + { + "bbox": [ + 177, + 482, + 238, + 496 + ], + "score": 0.93, + "content": "( d _ { i } ^ { = } + \\hat { n } _ { i } ^ { \\mathrm { r e d u c e } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 480, + 506, + 496 + ], + "score": 1.0, + "content": ", which will be used in the next layer. The final output of the fourth", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 199, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 199, + 507 + ], + "score": 1.0, + "content": "layer is represented by", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 513, + 479, + 529 + ], + "lines": [ + { + "bbox": [ + 132, + 513, + 479, + 529 + ], + "spans": [ + { + "bbox": [ + 132, + 513, + 479, + 529 + ], + "score": 0.89, + "content": "\\boldsymbol x _ { i } ^ { ( 4 ) } = ( \\tilde { e } _ { i } ^ { \\mathrm { o u t c o m e } } , f _ { i } , \\boldsymbol n _ { i } ^ { = } , ( \\boldsymbol n _ { i } ^ { = } ) ^ { 2 } , \\hat { \\boldsymbol n } _ { i } ^ { = } , ( \\hat { \\boldsymbol n } _ { i } ^ { = } ) ^ { 2 } , \\hat { d } _ { i } ^ { = } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , \\hat { f } _ { i } , \\hat { \\boldsymbol n } _ { i } ^ { \\mathrm { r e d u c e } } , d _ { i } ^ { = } , ( d _ { i } ^ { = } + \\hat { n } _ { i } ^ { \\mathrm { r e d u c e } } ) ^ { 2 } ) ,", + "type": "interline_equation", + "image_path": "e14629501d8872943602ad105f309d3a1f0cb858047c0939aa4f755e25575fe0.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 132, + 513, + 479, + 529 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 260, + 549 + ], + "lines": [ + { + "bbox": [ + 133, + 531, + 261, + 554 + ], + "spans": [ + { + "bbox": [ + 133, + 536, + 164, + 549 + ], + "score": 0.86, + "content": "\\tilde { e } _ { i } ^ { \\mathrm { o u t c o m e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 531, + 200, + 554 + ], + "score": 1.0, + "content": "is either", + "type": "text" + }, + { + "bbox": [ + 200, + 536, + 231, + 549 + ], + "score": 0.81, + "content": "e _ { i } ^ { \\mathrm { { o u t c o m e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 531, + 237, + 554 + ], + "score": 1.0, + "content": "o", + "type": "text" + }, + { + "bbox": [ + 231, + 537, + 261, + 550 + ], + "score": 1.0, + "content": "r ej1 .", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "Layer 5. The final layer of the Transformer uses one attention head to copy the corresponding", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 563, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 256, + 580 + ], + "score": 1.0, + "content": "token for generating the output when", + "type": "text" + }, + { + "bbox": [ + 256, + 564, + 284, + 577 + ], + "score": 0.93, + "content": "\\hat { f } _ { i } \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 563, + 302, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 565, + 312, + 577 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 563, + 505, + 580 + ], + "score": 1.0, + "content": "is false. Similar to previous layers, we can copy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 577, + 507, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 169, + 592 + ], + "score": 1.0, + "content": "the embedding", + "type": "text" + }, + { + "bbox": [ + 170, + 579, + 197, + 591 + ], + "score": 0.91, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 577, + 277, + 592 + ], + "score": 1.0, + "content": "located at position", + "type": "text" + }, + { + "bbox": [ + 277, + 579, + 284, + 590 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 577, + 325, + 592 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 578, + 384, + 591 + ], + "score": 0.93, + "content": "\\hat { n } _ { j } ^ { \\mp } = n _ { i } ^ { \\mp } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 577, + 403, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 403, + 577, + 480, + 592 + ], + "score": 0.93, + "content": "\\hat { d } _ { j } ^ { = } = d _ { i } ^ { = } + \\hat { n } _ { i } ^ { \\mathrm { r e d u c e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 577, + 507, + 592 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 590, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 266, + 608 + ], + "score": 1.0, + "content": "output of the attention layer is (e˜outcomei ,", + "type": "text" + }, + { + "bbox": [ + 267, + 591, + 323, + 606 + ], + "score": 0.79, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) } , f _ { i } , { \\hat { f } } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 590, + 507, + 608 + ], + "score": 1.0, + "content": ". We then use an MLP to obtain the output: if", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 605, + 507, + 621 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 152, + 619 + ], + "score": 0.92, + "content": "\\hat { f } _ { i } - f _ { i } \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 605, + 202, + 621 + ], + "score": 1.0, + "content": ", then output", + "type": "text" + }, + { + "bbox": [ + 202, + 609, + 229, + 620 + ], + "score": 0.91, + "content": "e _ { \\operatorname { i d } \\left( s _ { j } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 605, + 301, + 621 + ], + "score": 1.0, + "content": "; otherwise output", + "type": "text" + }, + { + "bbox": [ + 301, + 607, + 332, + 619 + ], + "score": 0.75, + "content": "\\tilde { e } _ { i } ^ { \\mathrm { o u t c o m e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 605, + 507, + 621 + ], + "score": 1.0, + "content": ". This corresponds to a conditional selection", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "operation and can be implemented by an MLP according to Lemma C.4. Finally, we pass the output", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 628, + 333, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 309, + 643 + ], + "score": 1.0, + "content": "through a softmax layer to generate the next token", + "type": "text" + }, + { + "bbox": [ + 309, + 631, + 328, + 641 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 628, + 333, + 643 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Now it remains to conduct an error analysis and determine the scale of parameters. Note that we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 157, + 669 + ], + "score": 1.0, + "content": "can tolerate", + "type": "text" + }, + { + "bbox": [ + 157, + 656, + 179, + 668 + ], + "score": 0.89, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 655, + 456, + 669 + ], + "score": 1.0, + "content": "error of the final layer output in the sense that the generated token", + "type": "text" + }, + { + "bbox": [ + 456, + 658, + 476, + 668 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "is still", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 666, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 666, + 506, + 681 + ], + "score": 1.0, + "content": "correct. Based on Lemmas C.1, C.4, C.5, C.7 and C.8, we can guarantee that when all parameters", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 252, + 691 + ], + "score": 1.0, + "content": "of the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 252, + 678, + 317, + 690 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( n , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 677, + 505, + 691 + ], + "score": 1.0, + "content": ", all intermediate neurons will only induce an", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 155, + 701 + ], + "score": 1.0, + "content": "error below", + "type": "text" + }, + { + "bbox": [ + 156, + 691, + 161, + 699 + ], + "score": 0.37, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 689, + 464, + 701 + ], + "score": 1.0, + "content": ". (Also note that Assumption C.6 in Lemmas C.7 and C.8 is satisfied when", + "type": "text" + }, + { + "bbox": [ + 465, + 691, + 470, + 699 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "is small", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 288, + 713 + ], + "score": 1.0, + "content": "enough.) 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Note that in an arithmetic expression there can be multiple expressions that can be calculated", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "(or handles defined in Appendix B), all of which are processed in the last layer. Therefore, the fourth", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "layer of the Transformer should keep only the leftmost calculation and discard other calculations", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 372, + 118 + ], + "score": 1.0, + "content": "(according to Appendix B). 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Formally, we need two attention heads to perform the following tasks:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 73, + 506, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 120, + 143, + 506, + 224 + ], + "lines": [ + { + "bbox": [ + 123, + 141, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 123, + 141, + 267, + 158 + ], + "score": 1.0, + "content": "1. 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The final output of the fourth", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 199, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 199, + 507 + ], + "score": 1.0, + "content": "layer is represented by", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 454, + 506, + 507 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 513, + 479, + 529 + ], + "lines": [ + { + "bbox": [ + 132, + 513, + 479, + 529 + ], + "spans": [ + { + "bbox": [ + 132, + 513, + 479, + 529 + ], + "score": 0.89, + "content": "\\boldsymbol x _ { i } ^ { ( 4 ) } = ( \\tilde { e } _ { i } ^ { \\mathrm { o u t c o m e } } , f _ { i } , \\boldsymbol n _ { i } ^ { = } , ( \\boldsymbol n _ { i } ^ { = } ) ^ { 2 } , \\hat { \\boldsymbol n } _ { i } ^ { = } , ( \\hat { \\boldsymbol n } _ { i } ^ { = } ) ^ { 2 } , \\hat { d } _ { i } ^ { = } , ( \\hat { d } _ { i } ^ { = } ) ^ { 2 } , \\hat { f } _ { i } , \\hat { \\boldsymbol n } _ { i } ^ { \\mathrm { r e d u c e } } , d _ { i } ^ { = } , ( d _ { i } ^ { = } + \\hat { n } _ { i } ^ { \\mathrm { r e d u c e } } ) ^ { 2 } ) ,", + "type": "interline_equation", + "image_path": "e14629501d8872943602ad105f309d3a1f0cb858047c0939aa4f755e25575fe0.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 132, + 513, + 479, + 529 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 260, + 549 + ], + "lines": [ + { + "bbox": [ + 133, + 531, + 261, + 554 + ], + "spans": [ + { + "bbox": [ + 133, + 536, + 164, + 549 + ], + "score": 0.86, + "content": "\\tilde { e } _ { i } ^ { \\mathrm { o u t c o m e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 531, + 200, + 554 + ], + "score": 1.0, + "content": "is either", + "type": "text" + }, + { + "bbox": [ + 200, + 536, + 231, + 549 + ], + "score": 0.81, + "content": "e _ { i } ^ { \\mathrm { { o u t c o m e } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 531, + 237, + 554 + ], + "score": 1.0, + "content": "o", + "type": "text" + }, + { + "bbox": [ + 231, + 537, + 261, + 550 + ], + "score": 1.0, + "content": "r ej1 .", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 133, + 531, + 261, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "Layer 5. The final layer of the Transformer uses one attention head to copy the corresponding", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 563, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 256, + 580 + ], + "score": 1.0, + "content": "token for generating the output when", + "type": "text" + }, + { + "bbox": [ + 256, + 564, + 284, + 577 + ], + "score": 0.93, + "content": "\\hat { f } _ { i } \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 563, + 302, + 580 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 565, + 312, + 577 + ], + "score": 0.88, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 563, + 505, + 580 + ], + "score": 1.0, + "content": "is false. Similar to previous layers, we can copy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 577, + 507, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 169, + 592 + ], + "score": 1.0, + "content": "the embedding", + "type": "text" + }, + { + "bbox": [ + 170, + 579, + 197, + 591 + ], + "score": 0.91, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 577, + 277, + 592 + ], + "score": 1.0, + "content": "located at position", + "type": "text" + }, + { + "bbox": [ + 277, + 579, + 284, + 590 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 577, + 325, + 592 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 578, + 384, + 591 + ], + "score": 0.93, + "content": "\\hat { n } _ { j } ^ { \\mp } = n _ { i } ^ { \\mp } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 577, + 403, + 592 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 403, + 577, + 480, + 592 + ], + "score": 0.93, + "content": "\\hat { d } _ { j } ^ { = } = d _ { i } ^ { = } + \\hat { n } _ { i } ^ { \\mathrm { r e d u c e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 577, + 507, + 592 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 590, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 266, + 608 + ], + "score": 1.0, + "content": "output of the attention layer is (e˜outcomei ,", + "type": "text" + }, + { + "bbox": [ + 267, + 591, + 323, + 606 + ], + "score": 0.79, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) } , f _ { i } , { \\hat { f } } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 590, + 507, + 608 + ], + "score": 1.0, + "content": ". We then use an MLP to obtain the output: if", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 605, + 507, + 621 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 152, + 619 + ], + "score": 0.92, + "content": "\\hat { f } _ { i } - f _ { i } \\ge 1", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 605, + 202, + 621 + ], + "score": 1.0, + "content": ", then output", + "type": "text" + }, + { + "bbox": [ + 202, + 609, + 229, + 620 + ], + "score": 0.91, + "content": "e _ { \\operatorname { i d } \\left( s _ { j } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 605, + 301, + 621 + ], + "score": 1.0, + "content": "; otherwise output", + "type": "text" + }, + { + "bbox": [ + 301, + 607, + 332, + 619 + ], + "score": 0.75, + "content": "\\tilde { e } _ { i } ^ { \\mathrm { o u t c o m e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 605, + 507, + 621 + ], + "score": 1.0, + "content": ". This corresponds to a conditional selection", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "operation and can be implemented by an MLP according to Lemma C.4. Finally, we pass the output", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 628, + 333, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 309, + 643 + ], + "score": 1.0, + "content": "through a softmax layer to generate the next token", + "type": "text" + }, + { + "bbox": [ + 309, + 631, + 328, + 641 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 628, + 333, + 643 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 552, + 507, + 643 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Now it remains to conduct an error analysis and determine the scale of parameters. Note that we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 157, + 669 + ], + "score": 1.0, + "content": "can tolerate", + "type": "text" + }, + { + "bbox": [ + 157, + 656, + 179, + 668 + ], + "score": 0.89, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 655, + 456, + 669 + ], + "score": 1.0, + "content": "error of the final layer output in the sense that the generated token", + "type": "text" + }, + { + "bbox": [ + 456, + 658, + 476, + 668 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "is still", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 666, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 666, + 506, + 681 + ], + "score": 1.0, + "content": "correct. Based on Lemmas C.1, C.4, C.5, C.7 and C.8, we can guarantee that when all parameters", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 252, + 691 + ], + "score": 1.0, + "content": "of the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 252, + 678, + 317, + 690 + ], + "score": 0.92, + "content": "{ \\cal O } ( \\mathrm { p o l y } ( n , 1 / \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 677, + 505, + 691 + ], + "score": 1.0, + "content": ", all intermediate neurons will only induce an", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 155, + 701 + ], + "score": 1.0, + "content": "error below", + "type": "text" + }, + { + "bbox": [ + 156, + 691, + 161, + 699 + ], + "score": 0.37, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 689, + 464, + 701 + ], + "score": 1.0, + "content": ". (Also note that Assumption C.6 in Lemmas C.7 and C.8 is satisfied when", + "type": "text" + }, + { + "bbox": [ + 465, + 691, + 470, + 699 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "is small", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 288, + 713 + ], + "score": 1.0, + "content": "enough.) Therefore, by picking a fixed small", + "type": "text" + }, + { + "bbox": [ + 288, + 700, + 326, + 712 + ], + "score": 0.92, + "content": "\\epsilon = \\Theta ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ", all parameter values in the Transformer are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 156, + 723 + ], + "score": 1.0, + "content": "bounded by", + "type": "text" + }, + { + "bbox": [ + 156, + 711, + 205, + 723 + ], + "score": 0.92, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 711, + 209, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 711, + 505, + 722 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 645, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 72, + 222, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 224, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 224, + 86 + ], + "score": 1.0, + "content": "D.2 Proof of Theorem 3.1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 92, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 92, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 506, + 105 + ], + "score": 1.0, + "content": "We now prove that evaluating arithmetic expressions without CoT is extremely difficult for bounded-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 455, + 117 + ], + "score": 1.0, + "content": "depth autoregressive Transformers. We will make the widely-believed assumption that", + "type": "text" + }, + { + "bbox": [ + 455, + 103, + 504, + 116 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "(see Appendix A.2 for definitions of these complexity classes). We further need the notion of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "uniformity: informally, this condition says that there exists an efficient algorithm to construct the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 398, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 398, + 150 + ], + "score": 1.0, + "content": "circuits. For a rigorous definition, we refer readers to Arora & Barak [2].", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 504, + 177 + ], + "lines": [ + { + "bbox": [ + 104, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 353, + 168 + ], + "score": 1.0, + "content": "We first present some lemmas on the expressive power of the", + "type": "text" + }, + { + "bbox": [ + 353, + 154, + 372, + 165 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "circuits, based on which we will", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 166, + 265, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 265, + 179 + ], + "score": 1.0, + "content": "give a reduction proof of Theorem 3.1.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 104, + 178, + 504, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 302, + 196 + ], + "score": 1.0, + "content": "Lemma D.2. For any n n-bits binary integers", + "type": "text" + }, + { + "bbox": [ + 302, + 184, + 362, + 193 + ], + "score": 0.83, + "content": "a _ { 1 } , a _ { 2 } , \\cdots , a _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 178, + 380, + 196 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 380, + 180, + 483, + 194 + ], + "score": 0.92, + "content": "f : \\{ 0 , 1 \\} ^ { n ^ { 2 } } \\{ 0 , 1 \\} ^ { 2 n }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 178, + 497, + 196 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 497, + 183, + 504, + 192 + ], + "score": 0.39, + "content": "a", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 190, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 220, + 209 + ], + "score": 1.0, + "content": "boolean function such that", + "type": "text" + }, + { + "bbox": [ + 221, + 193, + 348, + 207 + ], + "score": 0.9, + "content": "\\begin{array} { r } { f ( a _ { 1 } , a _ { 2 } , \\cdot \\cdot \\cdot , a _ { n } ) = \\sum _ { i \\in [ n ] } a _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 190, + 381, + 209 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 382, + 194, + 389, + 205 + ], + "score": 0.8, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 190, + 506, + 209 + ], + "score": 1.0, + "content": "can be implemented by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 196, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 140, + 221 + ], + "score": 1.0, + "content": "uniform", + "type": "text" + }, + { + "bbox": [ + 141, + 206, + 159, + 218 + ], + "score": 0.78, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 205, + 196, + 221 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 318, + 239 + ], + "score": 1.0, + "content": "This lemma demonstrates that computing the sum of", + "type": "text" + }, + { + "bbox": [ + 318, + 229, + 334, + 237 + ], + "score": 0.45, + "content": "_ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "-bit integers (known as iterated addition or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 237, + 252 + ], + "score": 1.0, + "content": "simply summation) is in uniform", + "type": "text" + }, + { + "bbox": [ + 238, + 238, + 256, + 250 + ], + "score": 0.84, + "content": "{ \\mathsf { T } } { \\bar { \\mathsf { C } } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 239, + 505, + 252 + ], + "score": 1.0, + "content": ". The detailed proof of this lemma can be found in the previous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 169, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 169, + 263 + ], + "score": 1.0, + "content": "works [29, 15].", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 506, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 244, + 277 + ], + "score": 1.0, + "content": "Lemma D.3. Consider any string", + "type": "text" + }, + { + "bbox": [ + 244, + 266, + 307, + 276 + ], + "score": 0.89, + "content": "s = s _ { 1 } s _ { 2 } \\cdot \\cdot \\cdot s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 263, + 436, + 277 + ], + "score": 1.0, + "content": "of length n containing brackets", + "type": "text" + }, + { + "bbox": [ + 436, + 264, + 460, + 276 + ], + "score": 0.85, + "content": "^ { \\ast } ( ^ { \\prime } , \\ ^ { \\prime } ) ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 263, + 506, + 277 + ], + "score": 1.0, + "content": ", and other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 296, + 288 + ], + "score": 1.0, + "content": "characters, and all brackets in s are paired. Let", + "type": "text" + }, + { + "bbox": [ + 297, + 276, + 304, + 287 + ], + "score": 0.76, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "be a boolean function taking s as input and output", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 259, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 259, + 298 + ], + "score": 1.0, + "content": "n pairs of integers defined as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 302, + 486, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 486, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 486, + 344 + ], + "score": 0.34, + "content": "f _ { i } ( s ) = \\left\\{ \\begin{array} { l l } { ( - 1 , j ) } & { i f s _ { i } i s a l e f t b r a c k e t a n d s _ { i } , s _ { j } a r e p a i r e d . } \\\\ { ( j , - 1 ) } & { i f s _ { i } i s a r i g h t b r a c k e t a n d s _ { i } , s _ { j } a r e p a i r e d . } \\\\ { ( j , k ) } & { i f s _ { i } i s n o t a b r a c k e t , a n d s _ { j } , s _ { k } i s t h e n e a r e s t p a i r o f m a t c h i n g b r a c k e t s c a l o r b a r a m e t a n d s _ { i } , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "58d9864e319bef12f9de8bbdaaaff5bc8424ad01aa0178ed31de1b4bce90834f.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 106, + 302, + 486, + 316.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 106, + 316.0, + 486, + 330.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 106, + 330.0, + 486, + 344.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 300, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 300, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 128, + 361 + ], + "score": 1.0, + "content": "Then", + "type": "text" + }, + { + "bbox": [ + 128, + 349, + 136, + 360 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 346, + 246, + 361 + ], + "score": 1.0, + "content": "can be implemented by the", + "type": "text" + }, + { + "bbox": [ + 246, + 347, + 265, + 359 + ], + "score": 0.72, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 346, + 300, + 361 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 506, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "Note that the input characters and the output integers are all encoded in binary. This lemma demon-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 376, + 309, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 286, + 394 + ], + "score": 1.0, + "content": "strates that the task of bracket matching is in", + "type": "text" + }, + { + "bbox": [ + 287, + 379, + 305, + 390 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 376, + 309, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 223, + 415 + ], + "score": 1.0, + "content": "Proof. Given the input string", + "type": "text" + }, + { + "bbox": [ + 224, + 406, + 230, + 413 + ], + "score": 0.69, + "content": "\\pmb { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 403, + 317, + 415 + ], + "score": 1.0, + "content": ", we first get the index", + "type": "text" + }, + { + "bbox": [ + 317, + 405, + 322, + 413 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 403, + 392, + 415 + ], + "score": 1.0, + "content": "of each character", + "type": "text" + }, + { + "bbox": [ + 392, + 406, + 401, + 414 + ], + "score": 0.85, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 403, + 505, + 415 + ], + "score": 1.0, + "content": ", which can be hard-coded", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 414, + 415, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 271, + 426 + ], + "score": 1.0, + "content": "into the circuits. Then for each character", + "type": "text" + }, + { + "bbox": [ + 271, + 416, + 280, + 425 + ], + "score": 0.85, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 414, + 415, + 426 + ], + "score": 1.0, + "content": ", we calculate the following term:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 429, + 409, + 457 + ], + "lines": [ + { + "bbox": [ + 201, + 429, + 409, + 457 + ], + "spans": [ + { + "bbox": [ + 201, + 429, + 409, + 457 + ], + "score": 0.91, + "content": "r _ { i } = \\sum _ { j < i } \\mathbb { I } [ s _ { j } = { ^ { * } ( \\cdot ( \\cdot ) } - \\sum _ { j < i } \\mathbb { I } [ s _ { j } = { ^ { * } } ) ^ { * } ] + \\mathbb { I } [ s _ { i } = { ^ { * } ( \\cdot ( \\cdot ) } ] .", + "type": "interline_equation", + "image_path": "b8275307ebe52b542c70abf524567630220c93ec9a0a77b373dfd14e59c5ad83.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 201, + 429, + 409, + 457 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 358, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 359, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 185, + 474 + ], + "score": 1.0, + "content": "Then, the output of", + "type": "text" + }, + { + "bbox": [ + 185, + 460, + 207, + 472 + ], + "score": 0.92, + "content": "f _ { i } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 459, + 359, + 474 + ], + "score": 1.0, + "content": "can be written as the following form:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 476, + 471, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 471, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 471, + 519 + ], + "score": 0.9, + "content": "f _ { i } ( s ) = \\left\\{ \\begin{array} { l l } { ( - 1 , \\operatorname* { m i n } \\{ j > i : s _ { j } = { ^ { \\circ } } \\} , r _ { j } = r _ { i } \\} ) } & { \\mathrm { i f ~ } s _ { i } = { ^ { \\circ } } , } \\\\ { ( \\operatorname* { m a x } \\{ j < i : s _ { j } = { ^ { \\circ } } ( { ^ { \\circ } } , r _ { j } = r _ { i } \\} , - 1 ) } & { \\mathrm { i f ~ } s _ { i } = { ^ { \\circ } } \\} , } \\\\ { ( \\operatorname* { m a x } \\{ j < i : s _ { j } = { ^ { \\circ } } ( { ^ { \\circ } } , r _ { j } = r _ { i } \\} , \\operatorname* { m i n } \\{ j > i : s _ { j } = { ^ { \\circ } } \\} , r _ { j } = r _ { i } \\} ) } & { \\mathrm { i f ~ } s _ { i } \\mathrm { ~ i s ~ n o t ~ a ~ } \\} , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "13c6d0284903e092d54dd87a057c65acdc60d3835004bbfff136c2d86a4ff019.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 106, + 476, + 471, + 490.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 106, + 490.3333333333333, + 471, + 504.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 106, + 504.66666666666663, + 471, + 519.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 422, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 423, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 368, + 536 + ], + "score": 1.0, + "content": "It is simple to verify that the construction can be implemented by", + "type": "text" + }, + { + "bbox": [ + 369, + 521, + 388, + 534 + ], + "score": 0.79, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 521, + 423, + 536 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 201, + 555 + ], + "score": 1.0, + "content": "Theorem D.4. Assume", + "type": "text" + }, + { + "bbox": [ + 202, + 540, + 285, + 554 + ], + "score": 0.73, + "content": "\\mathsf { T C } ^ { 0 } \\neq u n i f o r m \\mathsf { N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 540, + 381, + 555 + ], + "score": 1.0, + "content": ". For any prime number", + "type": "text" + }, + { + "bbox": [ + 381, + 544, + 387, + 553 + ], + "score": 0.45, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 540, + 438, + 555 + ], + "score": 1.0, + "content": ", any integer", + "type": "text" + }, + { + "bbox": [ + 438, + 542, + 446, + 552 + ], + "score": 0.56, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 540, + 506, + 555 + ], + "score": 1.0, + "content": ", and any poly-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 136, + 565 + ], + "score": 1.0, + "content": "nomial", + "type": "text" + }, + { + "bbox": [ + 136, + 553, + 145, + 564 + ], + "score": 0.79, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 553, + 253, + 565 + ], + "score": 1.0, + "content": ", there exists a problem size", + "type": "text" + }, + { + "bbox": [ + 253, + 555, + 260, + 563 + ], + "score": 0.41, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "such that no log-precision autoregressive Transformer defined", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 563, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 200, + 577 + ], + "score": 1.0, + "content": "in Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 200, + 564, + 208, + 574 + ], + "score": 0.62, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 563, + 300, + 577 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 300, + 564, + 341, + 576 + ], + "score": 0.93, + "content": "d \\leq Q ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 563, + 477, + 577 + ], + "score": 1.0, + "content": "can solve the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 477, + 564, + 500, + 576 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 563, + 504, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 295, + 600 + ], + "score": 1.0, + "content": "Proof. Our proof is based on leveraging the", + "type": "text" + }, + { + "bbox": [ + 295, + 587, + 314, + 599 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "-completeness of a classic problem: Boolean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "Formula Evaluation. According to the Buss reduction [12], calculating whether a Boolean formula is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 249, + 624 + ], + "score": 1.0, + "content": "true or false is complete for uniform", + "type": "text" + }, + { + "bbox": [ + 250, + 610, + 268, + 622 + ], + "score": 0.82, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 610, + 505, + 624 + ], + "score": 1.0, + "content": ". Based on this theorem, it suffices to prove that the Boolean", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "Formula Evaluation problem can be reduced to evaluating the arithmetic expression. This will yield", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "the conclusion by using the result that bounded-depth log-precision Transformers with polynomial", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 642, + 401, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 149, + 657 + ], + "score": 1.0, + "content": "size are in", + "type": "text" + }, + { + "bbox": [ + 150, + 644, + 191, + 657 + ], + "score": 0.89, + "content": "{ \\mathsf { T C } } ^ { 0 } { \\mathsf { \\Pi } } [ 4 2 ] ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 642, + 313, + 657 + ], + "score": 1.0, + "content": "as well as the assumption that", + "type": "text" + }, + { + "bbox": [ + 314, + 644, + 398, + 657 + ], + "score": 0.73, + "content": "\\mathsf { T C } ^ { 0 } \\ne \\mathsf { u n i f o r m } \\mathsf { N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 642, + 401, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 162, + 675 + ], + "score": 1.0, + "content": "Formally, let", + "type": "text" + }, + { + "bbox": [ + 162, + 662, + 260, + 674 + ], + "score": 0.93, + "content": "\\Sigma = \\{ 0 , 1 , \\land , \\lor , \\lnot , ( , ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "be the alphabet. A Boolean formula is a string defined on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 281, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 143, + 687 + ], + "score": 1.0, + "content": "alphabet", + "type": "text" + }, + { + "bbox": [ + 143, + 673, + 151, + 682 + ], + "score": 0.81, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 671, + 281, + 687 + ], + "score": 1.0, + "content": "by the following recursive way:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 691, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 689, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 118, + 689, + 506, + 704 + ], + "score": 1.0, + "content": "4While the authors only proved that predicting a binary label using Transformer encoder can be implemented", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 118, + 713 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 118, + 701, + 135, + 711 + ], + "score": 0.86, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "circuits, it is straightforward to extend the result to autoregressive Transformers predicting a label in a", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 710, + 171, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 171, + 724 + ], + "score": 1.0, + "content": "finite vocabulary.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 523, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 496, + 525, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 496, + 525, + 504, + 534 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 753 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 753 + ], + "score": 1.0, + "content": "28", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 72, + 222, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 224, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 224, + 86 + ], + "score": 1.0, + "content": "D.2 Proof of Theorem 3.1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 92, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 92, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 506, + 105 + ], + "score": 1.0, + "content": "We now prove that evaluating arithmetic expressions without CoT is extremely difficult for bounded-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 455, + 117 + ], + "score": 1.0, + "content": "depth autoregressive Transformers. We will make the widely-believed assumption that", + "type": "text" + }, + { + "bbox": [ + 455, + 103, + 504, + 116 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "(see Appendix A.2 for definitions of these complexity classes). We further need the notion of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "uniformity: informally, this condition says that there exists an efficient algorithm to construct the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 398, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 398, + 150 + ], + "score": 1.0, + "content": "circuits. For a rigorous definition, we refer readers to Arora & Barak [2].", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 92, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 504, + 177 + ], + "lines": [ + { + "bbox": [ + 104, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 353, + 168 + ], + "score": 1.0, + "content": "We first present some lemmas on the expressive power of the", + "type": "text" + }, + { + "bbox": [ + 353, + 154, + 372, + 165 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "circuits, based on which we will", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 166, + 265, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 265, + 179 + ], + "score": 1.0, + "content": "give a reduction proof of Theorem 3.1.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 153, + 505, + 179 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 104, + 178, + 504, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 302, + 196 + ], + "score": 1.0, + "content": "Lemma D.2. For any n n-bits binary integers", + "type": "text" + }, + { + "bbox": [ + 302, + 184, + 362, + 193 + ], + "score": 0.83, + "content": "a _ { 1 } , a _ { 2 } , \\cdots , a _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 178, + 380, + 196 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 380, + 180, + 483, + 194 + ], + "score": 0.92, + "content": "f : \\{ 0 , 1 \\} ^ { n ^ { 2 } } \\{ 0 , 1 \\} ^ { 2 n }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 178, + 497, + 196 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 497, + 183, + 504, + 192 + ], + "score": 0.39, + "content": "a", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 190, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 190, + 220, + 209 + ], + "score": 1.0, + "content": "boolean function such that", + "type": "text" + }, + { + "bbox": [ + 221, + 193, + 348, + 207 + ], + "score": 0.9, + "content": "\\begin{array} { r } { f ( a _ { 1 } , a _ { 2 } , \\cdot \\cdot \\cdot , a _ { n } ) = \\sum _ { i \\in [ n ] } a _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 190, + 381, + 209 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 382, + 194, + 389, + 205 + ], + "score": 0.8, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 190, + 506, + 209 + ], + "score": 1.0, + "content": "can be implemented by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 196, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 140, + 221 + ], + "score": 1.0, + "content": "uniform", + "type": "text" + }, + { + "bbox": [ + 141, + 206, + 159, + 218 + ], + "score": 0.78, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 205, + 196, + 221 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 178, + 506, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 318, + 239 + ], + "score": 1.0, + "content": "This lemma demonstrates that computing the sum of", + "type": "text" + }, + { + "bbox": [ + 318, + 229, + 334, + 237 + ], + "score": 0.45, + "content": "_ { n n }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "-bit integers (known as iterated addition or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 237, + 252 + ], + "score": 1.0, + "content": "simply summation) is in uniform", + "type": "text" + }, + { + "bbox": [ + 238, + 238, + 256, + 250 + ], + "score": 0.84, + "content": "{ \\mathsf { T } } { \\bar { \\mathsf { C } } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 239, + 505, + 252 + ], + "score": 1.0, + "content": ". The detailed proof of this lemma can be found in the previous", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 169, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 169, + 263 + ], + "score": 1.0, + "content": "works [29, 15].", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 227, + 505, + 263 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 506, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 244, + 277 + ], + "score": 1.0, + "content": "Lemma D.3. Consider any string", + "type": "text" + }, + { + "bbox": [ + 244, + 266, + 307, + 276 + ], + "score": 0.89, + "content": "s = s _ { 1 } s _ { 2 } \\cdot \\cdot \\cdot s _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 263, + 436, + 277 + ], + "score": 1.0, + "content": "of length n containing brackets", + "type": "text" + }, + { + "bbox": [ + 436, + 264, + 460, + 276 + ], + "score": 0.85, + "content": "^ { \\ast } ( ^ { \\prime } , \\ ^ { \\prime } ) ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 263, + 506, + 277 + ], + "score": 1.0, + "content": ", and other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 296, + 288 + ], + "score": 1.0, + "content": "characters, and all brackets in s are paired. Let", + "type": "text" + }, + { + "bbox": [ + 297, + 276, + 304, + 287 + ], + "score": 0.76, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "be a boolean function taking s as input and output", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 259, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 259, + 298 + ], + "score": 1.0, + "content": "n pairs of integers defined as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 263, + 506, + 298 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 302, + 486, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 486, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 486, + 344 + ], + "score": 0.34, + "content": "f _ { i } ( s ) = \\left\\{ \\begin{array} { l l } { ( - 1 , j ) } & { i f s _ { i } i s a l e f t b r a c k e t a n d s _ { i } , s _ { j } a r e p a i r e d . } \\\\ { ( j , - 1 ) } & { i f s _ { i } i s a r i g h t b r a c k e t a n d s _ { i } , s _ { j } a r e p a i r e d . } \\\\ { ( j , k ) } & { i f s _ { i } i s n o t a b r a c k e t , a n d s _ { j } , s _ { k } i s t h e n e a r e s t p a i r o f m a t c h i n g b r a c k e t s c a l o r b a r a m e t a n d s _ { i } , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "58d9864e319bef12f9de8bbdaaaff5bc8424ad01aa0178ed31de1b4bce90834f.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 106, + 302, + 486, + 316.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 106, + 316.0, + 486, + 330.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 106, + 330.0, + 486, + 344.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 300, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 300, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 128, + 361 + ], + "score": 1.0, + "content": "Then", + "type": "text" + }, + { + "bbox": [ + 128, + 349, + 136, + 360 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 346, + 246, + 361 + ], + "score": 1.0, + "content": "can be implemented by the", + "type": "text" + }, + { + "bbox": [ + 246, + 347, + 265, + 359 + ], + "score": 0.72, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 346, + 300, + 361 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 346, + 300, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 506, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "Note that the input characters and the output integers are all encoded in binary. This lemma demon-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 376, + 309, + 394 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 286, + 394 + ], + "score": 1.0, + "content": "strates that the task of bracket matching is in", + "type": "text" + }, + { + "bbox": [ + 287, + 379, + 305, + 390 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 376, + 309, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 367, + 506, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 223, + 415 + ], + "score": 1.0, + "content": "Proof. Given the input string", + "type": "text" + }, + { + "bbox": [ + 224, + 406, + 230, + 413 + ], + "score": 0.69, + "content": "\\pmb { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 403, + 317, + 415 + ], + "score": 1.0, + "content": ", we first get the index", + "type": "text" + }, + { + "bbox": [ + 317, + 405, + 322, + 413 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 403, + 392, + 415 + ], + "score": 1.0, + "content": "of each character", + "type": "text" + }, + { + "bbox": [ + 392, + 406, + 401, + 414 + ], + "score": 0.85, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 403, + 505, + 415 + ], + "score": 1.0, + "content": ", which can be hard-coded", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 414, + 415, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 271, + 426 + ], + "score": 1.0, + "content": "into the circuits. Then for each character", + "type": "text" + }, + { + "bbox": [ + 271, + 416, + 280, + 425 + ], + "score": 0.85, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 414, + 415, + 426 + ], + "score": 1.0, + "content": ", we calculate the following term:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 403, + 505, + 426 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 429, + 409, + 457 + ], + "lines": [ + { + "bbox": [ + 201, + 429, + 409, + 457 + ], + "spans": [ + { + "bbox": [ + 201, + 429, + 409, + 457 + ], + "score": 0.91, + "content": "r _ { i } = \\sum _ { j < i } \\mathbb { I } [ s _ { j } = { ^ { * } ( \\cdot ( \\cdot ) } - \\sum _ { j < i } \\mathbb { I } [ s _ { j } = { ^ { * } } ) ^ { * } ] + \\mathbb { I } [ s _ { i } = { ^ { * } ( \\cdot ( \\cdot ) } ] .", + "type": "interline_equation", + "image_path": "b8275307ebe52b542c70abf524567630220c93ec9a0a77b373dfd14e59c5ad83.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 201, + 429, + 409, + 457 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 358, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 359, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 185, + 474 + ], + "score": 1.0, + "content": "Then, the output of", + "type": "text" + }, + { + "bbox": [ + 185, + 460, + 207, + 472 + ], + "score": 0.92, + "content": "f _ { i } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 459, + 359, + 474 + ], + "score": 1.0, + "content": "can be written as the following form:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 459, + 359, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 476, + 471, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 471, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 471, + 519 + ], + "score": 0.9, + "content": "f _ { i } ( s ) = \\left\\{ \\begin{array} { l l } { ( - 1 , \\operatorname* { m i n } \\{ j > i : s _ { j } = { ^ { \\circ } } \\} , r _ { j } = r _ { i } \\} ) } & { \\mathrm { i f ~ } s _ { i } = { ^ { \\circ } } , } \\\\ { ( \\operatorname* { m a x } \\{ j < i : s _ { j } = { ^ { \\circ } } ( { ^ { \\circ } } , r _ { j } = r _ { i } \\} , - 1 ) } & { \\mathrm { i f ~ } s _ { i } = { ^ { \\circ } } \\} , } \\\\ { ( \\operatorname* { m a x } \\{ j < i : s _ { j } = { ^ { \\circ } } ( { ^ { \\circ } } , r _ { j } = r _ { i } \\} , \\operatorname* { m i n } \\{ j > i : s _ { j } = { ^ { \\circ } } \\} , r _ { j } = r _ { i } \\} ) } & { \\mathrm { i f ~ } s _ { i } \\mathrm { ~ i s ~ n o t ~ a ~ } \\} , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "13c6d0284903e092d54dd87a057c65acdc60d3835004bbfff136c2d86a4ff019.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 106, + 476, + 471, + 490.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 106, + 490.3333333333333, + 471, + 504.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 106, + 504.66666666666663, + 471, + 519.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 422, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 423, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 368, + 536 + ], + "score": 1.0, + "content": "It is simple to verify that the construction can be implemented by", + "type": "text" + }, + { + "bbox": [ + 369, + 521, + 388, + 534 + ], + "score": 0.79, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 521, + 423, + 536 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 521, + 423, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 201, + 555 + ], + "score": 1.0, + "content": "Theorem D.4. Assume", + "type": "text" + }, + { + "bbox": [ + 202, + 540, + 285, + 554 + ], + "score": 0.73, + "content": "\\mathsf { T C } ^ { 0 } \\neq u n i f o r m \\mathsf { N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 540, + 381, + 555 + ], + "score": 1.0, + "content": ". For any prime number", + "type": "text" + }, + { + "bbox": [ + 381, + 544, + 387, + 553 + ], + "score": 0.45, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 540, + 438, + 555 + ], + "score": 1.0, + "content": ", any integer", + "type": "text" + }, + { + "bbox": [ + 438, + 542, + 446, + 552 + ], + "score": 0.56, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 540, + 506, + 555 + ], + "score": 1.0, + "content": ", and any poly-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 136, + 565 + ], + "score": 1.0, + "content": "nomial", + "type": "text" + }, + { + "bbox": [ + 136, + 553, + 145, + 564 + ], + "score": 0.79, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 553, + 253, + 565 + ], + "score": 1.0, + "content": ", there exists a problem size", + "type": "text" + }, + { + "bbox": [ + 253, + 555, + 260, + 563 + ], + "score": 0.41, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "such that no log-precision autoregressive Transformer defined", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 563, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 200, + 577 + ], + "score": 1.0, + "content": "in Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 200, + 564, + 208, + 574 + ], + "score": 0.62, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 563, + 300, + 577 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 300, + 564, + 341, + 576 + ], + "score": 0.93, + "content": "d \\leq Q ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 563, + 477, + 577 + ], + "score": 1.0, + "content": "can solve the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 477, + 564, + 500, + 576 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 563, + 504, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 540, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 295, + 600 + ], + "score": 1.0, + "content": "Proof. Our proof is based on leveraging the", + "type": "text" + }, + { + "bbox": [ + 295, + 587, + 314, + 599 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "-completeness of a classic problem: Boolean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "Formula Evaluation. According to the Buss reduction [12], calculating whether a Boolean formula is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 249, + 624 + ], + "score": 1.0, + "content": "true or false is complete for uniform", + "type": "text" + }, + { + "bbox": [ + 250, + 610, + 268, + 622 + ], + "score": 0.82, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 610, + 505, + 624 + ], + "score": 1.0, + "content": ". Based on this theorem, it suffices to prove that the Boolean", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "Formula Evaluation problem can be reduced to evaluating the arithmetic expression. This will yield", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "the conclusion by using the result that bounded-depth log-precision Transformers with polynomial", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 642, + 401, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 149, + 657 + ], + "score": 1.0, + "content": "size are in", + "type": "text" + }, + { + "bbox": [ + 150, + 644, + 191, + 657 + ], + "score": 0.89, + "content": "{ \\mathsf { T C } } ^ { 0 } { \\mathsf { \\Pi } } [ 4 2 ] ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 642, + 313, + 657 + ], + "score": 1.0, + "content": "as well as the assumption that", + "type": "text" + }, + { + "bbox": [ + 314, + 644, + 398, + 657 + ], + "score": 0.73, + "content": "\\mathsf { T C } ^ { 0 } \\ne \\mathsf { u n i f o r m } \\mathsf { N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 642, + 401, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 587, + 506, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 162, + 675 + ], + "score": 1.0, + "content": "Formally, let", + "type": "text" + }, + { + "bbox": [ + 162, + 662, + 260, + 674 + ], + "score": 0.93, + "content": "\\Sigma = \\{ 0 , 1 , \\land , \\lor , \\lnot , ( , ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "be the alphabet. A Boolean formula is a string defined on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 281, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 143, + 687 + ], + "score": 1.0, + "content": "alphabet", + "type": "text" + }, + { + "bbox": [ + 143, + 673, + 151, + 682 + ], + "score": 0.81, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 671, + 281, + 687 + ], + "score": 1.0, + "content": "by the following recursive way:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 660, + 505, + 687 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 123, + 72, + 505, + 118 + ], + "lines": [ + { + "bbox": [ + 127, + 71, + 260, + 85 + ], + "spans": [ + { + "bbox": [ + 127, + 71, + 260, + 85 + ], + "score": 1.0, + "content": "• 0 and 1 are Boolean formulae;", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 127, + 88, + 375, + 101 + ], + "spans": [ + { + "bbox": [ + 127, + 88, + 145, + 101 + ], + "score": 1.0, + "content": "• If", + "type": "text" + }, + { + "bbox": [ + 146, + 91, + 154, + 101 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 88, + 263, + 101 + ], + "score": 1.0, + "content": "is a Boolean formula, then", + "type": "text" + }, + { + "bbox": [ + 263, + 89, + 284, + 101 + ], + "score": 0.87, + "content": "\\left( \\neg \\varphi \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 88, + 375, + 101 + ], + "score": 1.0, + "content": "is a Boolean formula;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 126, + 104, + 507, + 119 + ], + "spans": [ + { + "bbox": [ + 126, + 104, + 145, + 119 + ], + "score": 1.0, + "content": "• If", + "type": "text" + }, + { + "bbox": [ + 145, + 107, + 172, + 117 + ], + "score": 0.84, + "content": "\\varphi _ { 1 } , \\varphi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 104, + 319, + 119 + ], + "score": 1.0, + "content": "are two Boolean formulae, then both", + "type": "text" + }, + { + "bbox": [ + 319, + 105, + 358, + 118 + ], + "score": 0.9, + "content": "\\left( \\varphi _ { 1 } \\wedge \\varphi _ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 104, + 376, + 119 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 105, + 416, + 118 + ], + "score": 0.9, + "content": "\\left( \\varphi _ { 1 } \\lor \\varphi _ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 104, + 507, + 119 + ], + "score": 1.0, + "content": "are Boolean formulae.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 134 + ], + "score": 1.0, + "content": "The Boolean Formula Evaluation problem aims to compute whether a Boolean formula is true (1) or", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "false (0). We now show that we can translate this problem into the problem of evaluating arithmetic", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 269, + 156 + ], + "score": 1.0, + "content": "expressions. Given a Boolean formula", + "type": "text" + }, + { + "bbox": [ + 269, + 146, + 275, + 154 + ], + "score": 0.65, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 143, + 378, + 156 + ], + "score": 1.0, + "content": ", the translation function", + "type": "text" + }, + { + "bbox": [ + 378, + 144, + 386, + 155 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "generates the corresponding", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 192, + 168 + ], + "score": 1.0, + "content": "arithmetic expression", + "type": "text" + }, + { + "bbox": [ + 192, + 155, + 211, + 167 + ], + "score": 0.92, + "content": "f ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 154, + 314, + 168 + ], + "score": 1.0, + "content": "that has the same result as", + "type": "text" + }, + { + "bbox": [ + 315, + 157, + 320, + 164 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "under evaluation. The translation is recursively", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 184, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 184, + 177 + ], + "score": 1.0, + "content": "defined as follows:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 127, + 181, + 469, + 244 + ], + "lines": [ + { + "bbox": [ + 128, + 180, + 236, + 195 + ], + "spans": [ + { + "bbox": [ + 128, + 180, + 136, + 195 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 181, + 175, + 194 + ], + "score": 0.91, + "content": "f ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 180, + 192, + 195 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 193, + 181, + 231, + 194 + ], + "score": 0.92, + "content": "f ( 1 ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 180, + 236, + 195 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 127, + 197, + 408, + 211 + ], + "spans": [ + { + "bbox": [ + 127, + 197, + 239, + 211 + ], + "score": 1.0, + "content": "• For any Boolean formula", + "type": "text" + }, + { + "bbox": [ + 239, + 200, + 247, + 210 + ], + "score": 0.59, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 197, + 250, + 211 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 251, + 198, + 316, + 210 + ], + "score": 0.89, + "content": "f ( \\lnot \\varphi ) = 1 - \\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 197, + 334, + 211 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 198, + 404, + 211 + ], + "score": 0.92, + "content": "f ( \\left( \\varphi \\right) ) = \\left( f ( \\varphi ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 197, + 408, + 211 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 127, + 213, + 403, + 228 + ], + "spans": [ + { + "bbox": [ + 127, + 213, + 246, + 228 + ], + "score": 1.0, + "content": "• For any Boolean formulae", + "type": "text" + }, + { + "bbox": [ + 246, + 214, + 399, + 227 + ], + "score": 0.85, + "content": "\\rho _ { 1 } , \\varphi _ { 2 } , f ( \\varphi _ { 1 } \\wedge \\varphi _ { 2 } ) = f ( \\varphi _ { 1 } ) \\times f ( \\varphi _ { 2 } ) ;", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 213, + 403, + 228 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 127, + 230, + 467, + 245 + ], + "spans": [ + { + "bbox": [ + 127, + 230, + 243, + 245 + ], + "score": 1.0, + "content": "• For any Boolean formulae", + "type": "text" + }, + { + "bbox": [ + 244, + 230, + 467, + 244 + ], + "score": 0.79, + "content": "\\varphi _ { 1 } , \\varphi _ { 2 } , f \\big ( \\varphi _ { 1 } \\vee \\varphi _ { 2 } \\big ) = 1 - \\big ( 1 - f \\big ( \\varphi _ { 1 } \\big ) \\big ) \\times \\big ( 1 - f \\big ( \\varphi _ { 2 } \\big ) \\big ) .", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 506, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 507, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 284, + 261 + ], + "score": 1.0, + "content": "It is easy to see that for any Boolean formula", + "type": "text" + }, + { + "bbox": [ + 284, + 250, + 290, + 257 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 247, + 345, + 261 + ], + "score": 1.0, + "content": ", the length of", + "type": "text" + }, + { + "bbox": [ + 345, + 247, + 365, + 259 + ], + "score": 0.92, + "content": "f ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 247, + 448, + 261 + ], + "score": 1.0, + "content": "is upper-bounded by", + "type": "text" + }, + { + "bbox": [ + 449, + 247, + 475, + 259 + ], + "score": 0.9, + "content": "O ( | s | )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 247, + 507, + 261 + ], + "score": 1.0, + "content": ". More-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 257, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 257, + 437, + 274 + ], + "score": 1.0, + "content": "over, the translation function can be simply implemented using the circuits within", + "type": "text" + }, + { + "bbox": [ + 437, + 259, + 456, + 270 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 257, + 507, + 274 + ], + "score": 1.0, + "content": "complexity.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 257, + 284 + ], + "score": 1.0, + "content": "To do so, we first replace the symbols", + "type": "text" + }, + { + "bbox": [ + 257, + 273, + 265, + 281 + ], + "score": 0.47, + "content": "\\neg", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 270, + 268, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 268, + 272, + 277, + 281 + ], + "score": 0.67, + "content": "\\wedge .", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 270, + 296, + 284 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 297, + 272, + 305, + 281 + ], + "score": 0.58, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 270, + 325, + 284 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 326, + 271, + 382, + 282 + ], + "score": 0.38, + "content": "1 - , \\times , \\mathrm { a n d } \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 270, + 506, + 284 + ], + "score": 1.0, + "content": ", respectively. Furthermore, for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 164, + 294 + ], + "score": 1.0, + "content": "each operator", + "type": "text" + }, + { + "bbox": [ + 164, + 283, + 172, + 292 + ], + "score": 0.29, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 282, + 239, + 294 + ], + "score": 1.0, + "content": ", we must insert", + "type": "text" + }, + { + "bbox": [ + 239, + 282, + 279, + 293 + ], + "score": 0.68, + "content": "\\mathbf { \\hat { \\theta } } ^ { 1 } - ( 1 - \\mathbf { \\hat { \\theta } } ^ { }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 282, + 494, + 294 + ], + "score": 1.0, + "content": "after the nearest left bracket containing the operator", + "type": "text" + }, + { + "bbox": [ + 495, + 282, + 502, + 292 + ], + "score": 0.43, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 282, + 506, + 294 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 217, + 305 + ], + "score": 1.0, + "content": "insert a right bracket before", + "type": "text" + }, + { + "bbox": [ + 217, + 294, + 226, + 303 + ], + "score": 0.78, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 293, + 261, + 305 + ], + "score": 1.0, + "content": ", insert ‘(", + "type": "text" + }, + { + "bbox": [ + 261, + 293, + 278, + 304 + ], + "score": 0.73, + "content": "1 - \\ '", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 293, + 299, + 305 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 300, + 294, + 309, + 303 + ], + "score": 0.75, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 293, + 505, + 305 + ], + "score": 1.0, + "content": ", and insert a right bracket before the nearest right", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 232, + 316 + ], + "score": 1.0, + "content": "bracket containing the operator", + "type": "text" + }, + { + "bbox": [ + 232, + 305, + 240, + 314 + ], + "score": 0.31, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 304, + 505, + 316 + ], + "score": 1.0, + "content": ". According to Lemmas D.2 and D.3, all of these operations can be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 187, + 329 + ], + "score": 1.0, + "content": "implemented by the", + "type": "text" + }, + { + "bbox": [ + 187, + 314, + 206, + 326 + ], + "score": 0.86, + "content": "{ \\mathsf { T C } } ^ { \\bar { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "circuits. Therefore, this translation function can be simply implemented by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 125, + 338 + ], + "score": 0.73, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 326, + 439, + 341 + ], + "score": 1.0, + "content": "circuits. Also, note that the above construction does not depend on the modulus", + "type": "text" + }, + { + "bbox": [ + 439, + 329, + 446, + 339 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 326, + 506, + 341 + ], + "score": 1.0, + "content": ". Therefore, by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 413, + 353 + ], + "score": 1.0, + "content": "reduction, we obtain that the problem of evaluating arithmetic expressions is", + "type": "text" + }, + { + "bbox": [ + 414, + 338, + 432, + 350 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 339, + 458, + 353 + ], + "score": 1.0, + "content": "-hard.", + "type": "text" + }, + { + "bbox": [ + 495, + 340, + 505, + 350 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 369, + 270, + 383 + ], + "lines": [ + { + "bbox": [ + 104, + 368, + 271, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 271, + 385 + ], + "score": 1.0, + "content": "E System of Linear Equations", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 506, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "In this section, we will prove that the autoregressive Transformer equipped with CoT can solve a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 493, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 493, + 419 + ], + "score": 1.0, + "content": "system of linear equations, whereas the autoregressive Transformer without CoT cannot solve it.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 433, + 223, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 223, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 223, + 446 + ], + "score": 1.0, + "content": "E.1 Proof of Theorem 3.4", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 312, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 313, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 313, + 467 + ], + "score": 1.0, + "content": "For ease of reading, we restate Theorem 3.3 below:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 276, + 483 + ], + "score": 1.0, + "content": "Theorem E.1. For any prime p and integer", + "type": "text" + }, + { + "bbox": [ + 276, + 471, + 304, + 481 + ], + "score": 0.88, + "content": "m > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 470, + 506, + 483 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer defined", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 228, + 494 + ], + "score": 1.0, + "content": "in Section 2 with hidden size", + "type": "text" + }, + { + "bbox": [ + 228, + 482, + 296, + 493 + ], + "score": 0.91, + "content": "d = { \\cal O } ( \\mathrm { p o l y } ( p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 481, + 365, + 494 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 366, + 484, + 375, + 492 + ], + "score": 0.54, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 481, + 408, + 494 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 409, + 482, + 437, + 492 + ], + "score": 0.88, + "content": "L = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 481, + 459, + 494 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 460, + 482, + 467, + 491 + ], + "score": 0.36, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "heads in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 367, + 505 + ], + "score": 1.0, + "content": "each layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 367, + 493, + 375, + 502 + ], + "score": 0.32, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 492, + 477, + 505 + ], + "score": 1.0, + "content": "for all inputs in Equation", + "type": "text" + }, + { + "bbox": [ + 477, + 492, + 502, + 504 + ], + "score": 0.92, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 492, + 506, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 503, + 430, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 375, + 516 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 504, + 426, + 515 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( m ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 503, + 430, + 516 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "Proof. The proof technique is similar to that of Theorem 3.3. We recommend readers to read the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "proof Theorem 3.3 first as we will omit redundant details in the subsequent proof. Below, without", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 556, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 103, + 556, + 209, + 574 + ], + "score": 1.0, + "content": "abuse of notation, we use", + "type": "text" + }, + { + "bbox": [ + 210, + 556, + 226, + 570 + ], + "score": 0.93, + "content": "\\mathbf { \\pmb { x } } _ { i } ^ { ( l ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 556, + 353, + 574 + ], + "score": 1.0, + "content": "to denote the output at position", + "type": "text" + }, + { + "bbox": [ + 354, + 559, + 359, + 568 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 556, + 395, + 574 + ], + "score": 1.0, + "content": "after the", + "type": "text" + }, + { + "bbox": [ + 395, + 559, + 399, + 568 + ], + "score": 0.73, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 556, + 505, + 574 + ], + "score": 1.0, + "content": "-th Transformer layer, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 122, + 582 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 122, + 571, + 132, + 581 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 570, + 187, + 582 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + }, + { + "bbox": [ + 188, + 570, + 192, + 579 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 570, + 389, + 582 + ], + "score": 1.0, + "content": "-th variable in linear equations. We also note that", + "type": "text" + }, + { + "bbox": [ + 389, + 571, + 399, + 579 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "is the upper bound on the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "number of variables, and we will construct Transformer parameters such that the Transformer can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 592, + 394, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 380, + 603 + ], + "score": 1.0, + "content": "solve all linear equations with the number of variables no more than", + "type": "text" + }, + { + "bbox": [ + 380, + 593, + 390, + 601 + ], + "score": 0.59, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 592, + 394, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 108, + 607, + 503, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 382, + 622 + ], + "score": 1.0, + "content": "Token Embeddings. Assume that we have a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 382, + 610, + 436, + 619 + ], + "score": 0.87, + "content": "s _ { 1 } , s _ { 2 } , \\ldots , s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 606, + 506, + 622 + ], + "score": 1.0, + "content": "and we want to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 103, + 614, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 103, + 614, + 201, + 637 + ], + "score": 1.0, + "content": "generate the next token", + "type": "text" + }, + { + "bbox": [ + 201, + 622, + 221, + 633 + ], + "score": 0.89, + "content": "s _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 614, + 325, + 637 + ], + "score": 1.0, + "content": ". We can embed the token", + "type": "text" + }, + { + "bbox": [ + 325, + 623, + 335, + 632 + ], + "score": 0.85, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 614, + 403, + 637 + ], + "score": 1.0, + "content": "using the format", + "type": "text" + }, + { + "bbox": [ + 404, + 619, + 497, + 634 + ], + "score": 0.93, + "content": "\\pmb { x } _ { i } ^ { ( 0 ) } = ( e _ { \\mathrm { i d } ( s _ { i } ) } , l _ { i } , i , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 614, + 504, + 637 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 124, + 638, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 124, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 124, + 639, + 182, + 650 + ], + "score": 1.0, + "content": "1. 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Since we hope the embedding dimension is a constant", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 136, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 136, + 661, + 362, + 673 + ], + "score": 1.0, + "content": "and does not depend on the number of variable tokens", + "type": "text" + }, + { + "bbox": [ + 362, + 663, + 372, + 671 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 661, + 505, + 673 + ], + "score": 1.0, + "content": ", we consider representing them", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 136, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 136, + 671, + 418, + 683 + ], + "score": 1.0, + "content": "using a unified (single) encoding and distinguishing them via the term", + "type": "text" + }, + { + "bbox": [ + 418, + 672, + 426, + 682 + ], + "score": 0.84, + "content": "\\mathbf { \\xi } _ { l _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 671, + 506, + 683 + ], + "score": 1.0, + "content": ". This means that if", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 136, + 682, + 434, + 696 + ], + "spans": [ + { + "bbox": [ + 136, + 684, + 145, + 693 + ], + "score": 0.83, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 682, + 163, + 696 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 164, + 684, + 174, + 695 + ], + "score": 0.86, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 682, + 319, + 696 + ], + "score": 1.0, + "content": "are two different variables, we have", + "type": "text" + }, + { + "bbox": [ + 320, + 682, + 383, + 695 + ], + "score": 0.93, + "content": "\\mathrm { i d } ( s _ { i } ) = \\mathrm { i d } ( s _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 682, + 401, + 696 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 683, + 430, + 695 + ], + "score": 0.91, + "content": "\\boldsymbol { l } _ { i } \\neq \\boldsymbol { l } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 682, + 434, + 696 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 123, + 699, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 123, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 123, + 699, + 136, + 712 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 136, + 699, + 168, + 711 + ], + "score": 0.9, + "content": "\\boldsymbol { l } _ { i } \\in \\mathbb { R } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "is a vector used to distinguish between different variables. Its first element, denoted", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 136, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 136, + 711, + 148, + 723 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 148, + 711, + 178, + 723 + ], + "score": 0.82, + "content": "\\operatorname { v a r } ( s _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 711, + 333, + 723 + ], + "score": 1.0, + "content": ", represents the index of the variable", + "type": "text" + }, + { + "bbox": [ + 333, + 713, + 342, + 722 + ], + "score": 0.85, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 711, + 400, + 723 + ], + "score": 1.0, + "content": ". 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We now show that we can translate this problem into the problem of evaluating arithmetic", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 269, + 156 + ], + "score": 1.0, + "content": "expressions. 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More-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 257, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 257, + 437, + 274 + ], + "score": 1.0, + "content": "over, the translation function can be simply implemented using the circuits within", + "type": "text" + }, + { + "bbox": [ + 437, + 259, + 456, + 270 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 257, + 507, + 274 + ], + "score": 1.0, + "content": "complexity.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 257, + 284 + ], + "score": 1.0, + "content": "To do so, we first replace the symbols", + "type": "text" + }, + { + "bbox": [ + 257, + 273, + 265, + 281 + ], + "score": 0.47, + "content": "\\neg", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 270, + 268, + 284 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 268, + 272, + 277, + 281 + ], + "score": 0.67, + "content": "\\wedge .", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 270, + 296, + 284 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 297, + 272, + 305, + 281 + ], + "score": 0.58, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 270, + 325, + 284 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 326, + 271, + 382, + 282 + ], + "score": 0.38, + "content": "1 - , \\times , \\mathrm { a n d } \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 270, + 506, + 284 + ], + "score": 1.0, + "content": ", respectively. Furthermore, for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 164, + 294 + ], + "score": 1.0, + "content": "each operator", + "type": "text" + }, + { + "bbox": [ + 164, + 283, + 172, + 292 + ], + "score": 0.29, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 282, + 239, + 294 + ], + "score": 1.0, + "content": ", we must insert", + "type": "text" + }, + { + "bbox": [ + 239, + 282, + 279, + 293 + ], + "score": 0.68, + "content": "\\mathbf { \\hat { \\theta } } ^ { 1 } - ( 1 - \\mathbf { \\hat { \\theta } } ^ { }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 282, + 494, + 294 + ], + "score": 1.0, + "content": "after the nearest left bracket containing the operator", + "type": "text" + }, + { + "bbox": [ + 495, + 282, + 502, + 292 + ], + "score": 0.43, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 282, + 506, + 294 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 217, + 305 + ], + "score": 1.0, + "content": "insert a right bracket before", + "type": "text" + }, + { + "bbox": [ + 217, + 294, + 226, + 303 + ], + "score": 0.78, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 293, + 261, + 305 + ], + "score": 1.0, + "content": ", insert ‘(", + "type": "text" + }, + { + "bbox": [ + 261, + 293, + 278, + 304 + ], + "score": 0.73, + "content": "1 - \\ '", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 293, + 299, + 305 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 300, + 294, + 309, + 303 + ], + "score": 0.75, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 293, + 505, + 305 + ], + "score": 1.0, + "content": ", and insert a right bracket before the nearest right", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 232, + 316 + ], + "score": 1.0, + "content": "bracket containing the operator", + "type": "text" + }, + { + "bbox": [ + 232, + 305, + 240, + 314 + ], + "score": 0.31, + "content": "\\vee", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 304, + 505, + 316 + ], + "score": 1.0, + "content": ". According to Lemmas D.2 and D.3, all of these operations can be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 187, + 329 + ], + "score": 1.0, + "content": "implemented by the", + "type": "text" + }, + { + "bbox": [ + 187, + 314, + 206, + 326 + ], + "score": 0.86, + "content": "{ \\mathsf { T C } } ^ { \\bar { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "circuits. Therefore, this translation function can be simply implemented by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 125, + 338 + ], + "score": 0.73, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 326, + 439, + 341 + ], + "score": 1.0, + "content": "circuits. Also, note that the above construction does not depend on the modulus", + "type": "text" + }, + { + "bbox": [ + 439, + 329, + 446, + 339 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 326, + 506, + 341 + ], + "score": 1.0, + "content": ". Therefore, by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 413, + 353 + ], + "score": 1.0, + "content": "reduction, we obtain that the problem of evaluating arithmetic expressions is", + "type": "text" + }, + { + "bbox": [ + 414, + 338, + 432, + 350 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 339, + 458, + 353 + ], + "score": 1.0, + "content": "-hard.", + "type": "text" + }, + { + "bbox": [ + 495, + 340, + 505, + 350 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 247, + 507, + 353 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 369, + 270, + 383 + ], + "lines": [ + { + "bbox": [ + 104, + 368, + 271, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 271, + 385 + ], + "score": 1.0, + "content": "E System of Linear Equations", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 506, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "In this section, we will prove that the autoregressive Transformer equipped with CoT can solve a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 493, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 493, + 419 + ], + "score": 1.0, + "content": "system of linear equations, whereas the autoregressive Transformer without CoT cannot solve it.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 395, + 506, + 419 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 433, + 223, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 223, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 223, + 446 + ], + "score": 1.0, + "content": "E.1 Proof of Theorem 3.4", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 312, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 313, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 313, + 467 + ], + "score": 1.0, + "content": "For ease of reading, we restate Theorem 3.3 below:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 454, + 313, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 276, + 483 + ], + "score": 1.0, + "content": "Theorem E.1. For any prime p and integer", + "type": "text" + }, + { + "bbox": [ + 276, + 471, + 304, + 481 + ], + "score": 0.88, + "content": "m > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 470, + 506, + 483 + ], + "score": 1.0, + "content": ", there exists an autoregressive Transformer defined", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 228, + 494 + ], + "score": 1.0, + "content": "in Section 2 with hidden size", + "type": "text" + }, + { + "bbox": [ + 228, + 482, + 296, + 493 + ], + "score": 0.91, + "content": "d = { \\cal O } ( \\mathrm { p o l y } ( p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 481, + 365, + 494 + ], + "score": 1.0, + "content": "(independent of", + "type": "text" + }, + { + "bbox": [ + 366, + 484, + 375, + 492 + ], + "score": 0.54, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 481, + 408, + 494 + ], + "score": 1.0, + "content": "), depth", + "type": "text" + }, + { + "bbox": [ + 409, + 482, + 437, + 492 + ], + "score": 0.88, + "content": "L = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 481, + 459, + 494 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 460, + 482, + 467, + 491 + ], + "score": 0.36, + "content": "5", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "heads in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 367, + 505 + ], + "score": 1.0, + "content": "each layer that can generate the CoT solution defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 367, + 493, + 375, + 502 + ], + "score": 0.32, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 492, + 477, + 505 + ], + "score": 1.0, + "content": "for all inputs in Equation", + "type": "text" + }, + { + "bbox": [ + 477, + 492, + 502, + 504 + ], + "score": 0.92, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 492, + 506, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 503, + 430, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 375, + 516 + ], + "score": 1.0, + "content": "Moreover, all parameter values in the Transformer are bounded by", + "type": "text" + }, + { + "bbox": [ + 375, + 504, + 426, + 515 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( m ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 503, + 430, + 516 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 470, + 506, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "Proof. The proof technique is similar to that of Theorem 3.3. We recommend readers to read the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "proof Theorem 3.3 first as we will omit redundant details in the subsequent proof. Below, without", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 556, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 103, + 556, + 209, + 574 + ], + "score": 1.0, + "content": "abuse of notation, we use", + "type": "text" + }, + { + "bbox": [ + 210, + 556, + 226, + 570 + ], + "score": 0.93, + "content": "\\mathbf { \\pmb { x } } _ { i } ^ { ( l ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 556, + 353, + 574 + ], + "score": 1.0, + "content": "to denote the output at position", + "type": "text" + }, + { + "bbox": [ + 354, + 559, + 359, + 568 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 556, + 395, + 574 + ], + "score": 1.0, + "content": "after the", + "type": "text" + }, + { + "bbox": [ + 395, + 559, + 399, + 568 + ], + "score": 0.73, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 556, + 505, + 574 + ], + "score": 1.0, + "content": "-th Transformer layer, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 122, + 582 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 122, + 571, + 132, + 581 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 570, + 187, + 582 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + }, + { + "bbox": [ + 188, + 570, + 192, + 579 + ], + "score": 0.68, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 570, + 389, + 582 + ], + "score": 1.0, + "content": "-th variable in linear equations. 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We also use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "additional attention heads to perform some auxiliary calculations that will be used in subsequent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 477, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 477, + 462 + ], + "score": 1.0, + "content": "layers. Concretely, the second layer uses four attention heads to perform the following tasks:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 123, + 467, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 122, + 465, + 508, + 481 + ], + "spans": [ + { + "bbox": [ + 122, + 465, + 202, + 481 + ], + "score": 1.0, + "content": "1. Copy the value", + "type": "text" + }, + { + "bbox": [ + 203, + 466, + 217, + 481 + ], + "score": 0.91, + "content": "n _ { j } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 465, + 276, + 481 + ], + "score": 1.0, + "content": "with position", + "type": "text" + }, + { + "bbox": [ + 277, + 468, + 283, + 479 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 465, + 406, + 481 + ], + "score": 1.0, + "content": "corresponding to the nearest", + "type": "text" + }, + { + "bbox": [ + 406, + 468, + 430, + 478 + ], + "score": 0.86, + "content": "\\xrightarrow { \\cdot } \\implies", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 465, + 457, + 481 + ], + "score": 1.0, + "content": "token", + "type": "text" + }, + { + "bbox": [ + 458, + 468, + 468, + 480 + ], + "score": 0.68, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 468, + 502, + 479 + ], + "score": 0.7, + "content": "( j \\ \\leq \\ i )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 465, + 508, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 135, + 478, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 135, + 478, + 294, + 493 + ], + "score": 1.0, + "content": "Clearly, the value is well-defined when", + "type": "text" + }, + { + "bbox": [ + 294, + 480, + 328, + 492 + ], + "score": 0.92, + "content": "n _ { i } ^ { \\mathrm { c o t } } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 478, + 466, + 493 + ], + "score": 1.0, + "content": ", and we define the value to be 0 if", + "type": "text" + }, + { + "bbox": [ + 467, + 480, + 501, + 492 + ], + "score": 0.91, + "content": "n _ { i } ^ { \\mathrm { c o t } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 478, + 507, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 120, + 493, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 120, + 493, + 175, + 510 + ], + "score": 1.0, + "content": "2. Compute", + "type": "text" + }, + { + "bbox": [ + 176, + 495, + 257, + 510 + ], + "score": 0.93, + "content": "d _ { \\underline { { i } } } ^ { \\mathrm { e q } } = n _ { i } ^ { \\mathrm { e q } } - n _ { j } ^ { \\mathrm { e q } } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 493, + 507, + 510 + ], + "score": 1.0, + "content": ", which corresponds to the index of the current equation in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 135, + 506, + 209, + 520 + ], + "spans": [ + { + "bbox": [ + 135, + 506, + 209, + 520 + ], + "score": 1.0, + "content": "current CoT step.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 120, + 518, + 507, + 540 + ], + "spans": [ + { + "bbox": [ + 120, + 518, + 219, + 540 + ], + "score": 1.0, + "content": "3. 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Note that while", + "type": "text" + }, + { + "bbox": [ + 373, + 290, + 411, + 301 + ], + "score": 0.88, + "content": "r _ { j _ { 1 } } = r _ { j _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 287, + 506, + 304 + ], + "score": 1.0, + "content": "may hold for different", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 145, + 313 + ], + "score": 1.0, + "content": "positions", + "type": "text" + }, + { + "bbox": [ + 146, + 300, + 168, + 311 + ], + "score": 0.89, + "content": "j _ { 1 } , j _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 299, + 297, + 313 + ], + "score": 1.0, + "content": ", their values are the same (i.e.,", + "type": "text" + }, + { + "bbox": [ + 297, + 301, + 338, + 312 + ], + "score": 0.88, + "content": "v _ { j _ { 1 } } = v _ { j _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "), so the COPY operation still works and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 307, + 225, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 137, + 326 + ], + "score": 1.0, + "content": "obtains", + "type": "text" + }, + { + "bbox": [ + 138, + 312, + 154, + 324 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { v a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 307, + 183, + 326 + ], + "score": 1.0, + "content": "(when", + "type": "text" + }, + { + "bbox": [ + 183, + 311, + 216, + 324 + ], + "score": 0.91, + "content": "n _ { i } ^ { \\mathrm { e q } } \\geq 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 307, + 225, + 326 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 255, + 506, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 102, + 325, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 102, + 325, + 292, + 344 + ], + "score": 1.0, + "content": "Then, we use MLPs in parallel to calculate", + "type": "text" + }, + { + "bbox": [ + 292, + 328, + 365, + 341 + ], + "score": 0.93, + "content": "n _ { i } ^ { \\mathrm { e q } } = ( n _ { i } ^ { \\mathrm { e q } } / i ) \\cdot i", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 325, + 386, + 344 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 329, + 463, + 341 + ], + "score": 0.92, + "content": "\\begin{array} { r } { n _ { i } ^ { \\mathrm { c o t } } ~ = ~ ( n _ { i } ^ { \\mathrm { c o t } } / i ) \\cdot i } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 325, + 507, + 344 + ], + "score": 1.0, + "content": "based on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 381, + 352 + ], + "score": 1.0, + "content": "Lemma C.1. Besides, we use an MLP to compute the auxiliary term", + "type": "text" + }, + { + "bbox": [ + 382, + 340, + 390, + 350 + ], + "score": 0.85, + "content": "i ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "that will be used in the next", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 351, + 294, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 294, + 363 + ], + "score": 1.0, + "content": "layer. 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As described in Appendix B, each CoT step eliminates one variable, and thus at the current", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 257, + 409 + ], + "score": 1.0, + "content": "position we are eliminating variable", + "type": "text" + }, + { + "bbox": [ + 257, + 398, + 276, + 410 + ], + "score": 0.89, + "content": "\\mathbf { \\Pi } _ { x } ^ { x _ { n _ { i } ^ { \\mathrm { c o t } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 397, + 506, + 409 + ], + "score": 1.0, + "content": ". By the uniqueness of the solution, there must exist an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 307, + 419 + ], + "score": 1.0, + "content": "equation with a nonzero coefficient for variable", + "type": "text" + }, + { + "bbox": [ + 308, + 408, + 326, + 421 + ], + "score": 0.87, + "content": "x _ { \\underline { { n } } _ { i } ^ { \\mathrm { c o t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 407, + 506, + 419 + ], + "score": 1.0, + "content": ". In the second Transformer layer, we can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "determine which equation satisfies this condition. More precisely, we record whether the current", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 298, + 443 + ], + "score": 1.0, + "content": "equation will be used to eliminate the variable", + "type": "text" + }, + { + "bbox": [ + 298, + 430, + 327, + 443 + ], + "score": 0.89, + "content": "\\boldsymbol { x } _ { n _ { i } ^ { \\mathrm { c o t } } + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 428, + 415, + 443 + ], + "score": 1.0, + "content": "in the next CoT step", + "type": "text" + }, + { + "bbox": [ + 415, + 429, + 449, + 441 + ], + "score": 0.92, + "content": "n _ { i } ^ { \\mathrm { c o t } } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 428, + 506, + 443 + ], + "score": 1.0, + "content": ". We also use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "additional attention heads to perform some auxiliary calculations that will be used in subsequent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 477, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 477, + 462 + ], + "score": 1.0, + "content": "layers. Concretely, the second layer uses four attention heads to perform the following tasks:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 385, + 506, + 462 + ] + }, + { + "type": "list", + "bbox": [ + 123, + 467, + 506, + 626 + ], + "lines": [ + { + "bbox": [ + 122, + 465, + 508, + 481 + ], + "spans": [ + { + "bbox": [ + 122, + 465, + 202, + 481 + ], + "score": 1.0, + "content": "1. Copy the value", + "type": "text" + }, + { + "bbox": [ + 203, + 466, + 217, + 481 + ], + "score": 0.91, + "content": "n _ { j } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 465, + 276, + 481 + ], + "score": 1.0, + "content": "with position", + "type": "text" + }, + { + "bbox": [ + 277, + 468, + 283, + 479 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 465, + 406, + 481 + ], + "score": 1.0, + "content": "corresponding to the nearest", + "type": "text" + }, + { + "bbox": [ + 406, + 468, + 430, + 478 + ], + "score": 0.86, + "content": "\\xrightarrow { \\cdot } \\implies", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 465, + 457, + 481 + ], + "score": 1.0, + "content": "token", + "type": "text" + }, + { + "bbox": [ + 458, + 468, + 468, + 480 + ], + "score": 0.68, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 468, + 502, + 479 + ], + "score": 0.7, + "content": "( j \\ \\leq \\ i )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 465, + 508, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 478, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 135, + 478, + 294, + 493 + ], + "score": 1.0, + "content": "Clearly, the value is well-defined when", + "type": "text" + }, + { + "bbox": [ + 294, + 480, + 328, + 492 + ], + "score": 0.92, + "content": "n _ { i } ^ { \\mathrm { c o t } } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 478, + 466, + 493 + ], + "score": 1.0, + "content": ", and we define the value to be 0 if", + "type": "text" + }, + { + "bbox": [ + 467, + 480, + 501, + 492 + ], + "score": 0.91, + "content": "n _ { i } ^ { \\mathrm { c o t } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 478, + 507, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 120, + 493, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 120, + 493, + 175, + 510 + ], + "score": 1.0, + "content": "2. Compute", + "type": "text" + }, + { + "bbox": [ + 176, + 495, + 257, + 510 + ], + "score": 0.93, + "content": "d _ { \\underline { { i } } } ^ { \\mathrm { e q } } = n _ { i } ^ { \\mathrm { e q } } - n _ { j } ^ { \\mathrm { e q } } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 493, + 507, + 510 + ], + "score": 1.0, + "content": ", which corresponds to the index of the current equation in the", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 506, + 209, + 520 + ], + "spans": [ + { + "bbox": [ + 135, + 506, + 209, + 520 + ], + "score": 1.0, + "content": "current CoT step.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + }, + { + "bbox": [ + 120, + 518, + 507, + 540 + ], + "spans": [ + { + "bbox": [ + 120, + 518, + 219, + 540 + ], + "score": 1.0, + "content": "3. Copy the embedding", + "type": "text" + }, + { + "bbox": [ + 220, + 525, + 247, + 537 + ], + "score": 0.91, + "content": "e _ { \\operatorname { i d } ( s _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 518, + 317, + 540 + ], + "score": 1.0, + "content": "with the smallest", + "type": "text" + }, + { + "bbox": [ + 317, + 524, + 323, + 535 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 518, + 363, + 540 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 364, + 523, + 405, + 537 + ], + "score": 0.93, + "content": "n _ { j } ^ { \\mathrm { e q } } = n _ { i } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 518, + 422, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 422, + 525, + 432, + 536 + ], + "score": 0.85, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 518, + 507, + 540 + ], + "score": 1.0, + "content": "is a number. Note", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 533, + 308, + 549 + ], + "spans": [ + { + "bbox": [ + 135, + 533, + 154, + 549 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 154, + 536, + 181, + 548 + ], + "score": 0.91, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 533, + 268, + 549 + ], + "score": 1.0, + "content": "is well-defined when", + "type": "text" + }, + { + "bbox": [ + 268, + 535, + 302, + 546 + ], + "score": 0.9, + "content": "s _ { i } = \\cdot = \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 533, + 308, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35, + "is_list_end_line": true + }, + { + "bbox": [ + 120, + 550, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 120, + 550, + 280, + 568 + ], + "score": 1.0, + "content": "4. Compute a Boolean flag (denoted as", + "type": "text" + }, + { + "bbox": [ + 281, + 554, + 291, + 565 + ], + "score": 0.85, + "content": "f _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 550, + 393, + 568 + ], + "score": 1.0, + "content": "), which is true only when", + "type": "text" + }, + { + "bbox": [ + 394, + 553, + 456, + 566 + ], + "score": 0.86, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) } \\neq e _ { \\mathrm { i d } ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 550, + 461, + 568 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 461, + 552, + 502, + 565 + ], + "score": 0.85, + "content": "d _ { i } ^ { \\mathrm { e q } } > n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 550, + 507, + 568 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 136, + 565, + 153, + 577 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 154, + 565, + 188, + 576 + ], + "score": 0.91, + "content": "s _ { i } = \\cdot = \\overrightarrow { { \\bf { \\Phi } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 565, + 264, + 577 + ], + "score": 1.0, + "content": ". 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The output of the MLP is", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 245, + 464, + 280 + ], + "lines": [ + { + "bbox": [ + 146, + 245, + 464, + 280 + ], + "spans": [ + { + "bbox": [ + 146, + 245, + 464, + 280 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\boldsymbol { x } _ { i } ^ { ( 2 ) } = ( e _ { \\mathrm { i d } ( s _ { i } ) } , l _ { i } , i , i ^ { 2 } , 1 , n _ { i } ^ { \\mathrm { v a r } } , n _ { i } ^ { \\mathrm { c o t } } , ( n _ { i } ^ { \\mathrm { c o t } } ) ^ { 2 } , d _ { i } ^ { \\mathrm { c q } } , f _ { i } , } \\\\ & { \\qquad e _ { \\mathrm { i d } ( s _ { i - 1 } ) } , l _ { i - 1 } , e _ { \\mathrm { i d } ( s _ { i - 2 } ) } , l _ { i - 2 } , ( \\mathrm { v a r } ( s _ { i - 1 } ) ) ^ { 2 } , ( \\mathrm { v a r } ( s _ { i } ) ) ^ { 2 } , \\mathbb { I } [ n _ { i } ^ { \\mathrm { c o t } } = d _ { i } ^ { \\mathrm { e q } } ] ) . } \\end{array}", + "type": "interline_equation", + "image_path": "87f24b16aa34ca77cba28bd2560a23f782ccbf3484978bf6fae00026bd5db080.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 146, + 245, + 464, + 256.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 146, + 256.6666666666667, + 464, + 268.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 146, + 268.33333333333337, + 464, + 280.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 287, + 504, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 506, + 301 + ], + "score": 1.0, + "content": "Layer 3. The third layer of the Transformer uses two attention heads to perform the following tasks:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 120, + 303, + 506, + 467 + ], + "lines": [ + { + "bbox": [ + 123, + 302, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 123, + 302, + 223, + 317 + ], + "score": 1.0, + "content": "1. Copy the embedding", + "type": "text" + }, + { + "bbox": [ + 224, + 303, + 237, + 318 + ], + "score": 0.9, + "content": "d _ { j } ^ { \\mathsf { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 302, + 309, + 317 + ], + "score": 1.0, + "content": "with the smallest", + "type": "text" + }, + { + "bbox": [ + 310, + 305, + 316, + 316 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 302, + 359, + 317 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 360, + 304, + 388, + 317 + ], + "score": 0.92, + "content": "f _ { j } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 302, + 406, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 304, + 469, + 318 + ], + "score": 0.93, + "content": "n _ { j } ^ { \\mathrm { c o t } } = n _ { i } ^ { \\mathrm { c o t } } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 302, + 505, + 317 + ], + "score": 1.0, + "content": ". Denote", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 135, + 316, + 209, + 331 + ], + "spans": [ + { + "bbox": [ + 135, + 316, + 192, + 331 + ], + "score": 1.0, + "content": "the answer as", + "type": "text" + }, + { + "bbox": [ + 193, + 317, + 206, + 331 + ], + "score": 0.9, + "content": "\\hat { d } _ { i } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 316, + 209, + 331 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 122, + 334, + 450, + 348 + ], + "spans": [ + { + "bbox": [ + 122, + 334, + 274, + 347 + ], + "score": 1.0, + "content": "2. Determine whether the next token", + "type": "text" + }, + { + "bbox": [ + 275, + 338, + 293, + 347 + ], + "score": 0.88, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 334, + 428, + 347 + ], + "score": 1.0, + "content": "is a number. Denote the result as", + "type": "text" + }, + { + "bbox": [ + 428, + 335, + 447, + 348 + ], + "score": 0.9, + "content": "f _ { i } ^ { \\mathrm { n u m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 334, + 450, + 347 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 122, + 350, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 122, + 350, + 289, + 365 + ], + "score": 1.0, + "content": "3. Determine the output of the next token", + "type": "text" + }, + { + "bbox": [ + 290, + 354, + 309, + 363 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 350, + 318, + 365 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 318, + 354, + 338, + 363 + ], + "score": 0.89, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 350, + 506, + 365 + ], + "score": 1.0, + "content": "is not a number. We denote its embedding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 133, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 133, + 361, + 147, + 375 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 147, + 363, + 166, + 375 + ], + "score": 0.91, + "content": "e _ { i } ^ { \\mathrm { n e x t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 361, + 352, + 375 + ], + "score": 1.0, + "content": ". Also, we need to determine the variable index", + "type": "text" + }, + { + "bbox": [ + 353, + 363, + 393, + 375 + ], + "score": 0.92, + "content": "\\mathrm { v a r } ( s _ { i + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "of the next token if the next", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 136, + 374, + 213, + 385 + ], + "spans": [ + { + "bbox": [ + 136, + 374, + 213, + 385 + ], + "score": 1.0, + "content": "token is a variable.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 122, + 388, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 122, + 388, + 222, + 403 + ], + "score": 1.0, + "content": "4. Determine the token", + "type": "text" + }, + { + "bbox": [ + 223, + 392, + 242, + 402 + ], + "score": 0.88, + "content": "s _ { i + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 388, + 312, + 403 + ], + "score": 1.0, + "content": "if the next token", + "type": "text" + }, + { + "bbox": [ + 313, + 392, + 332, + 402 + ], + "score": 0.88, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 388, + 474, + 403 + ], + "score": 1.0, + "content": "is a number. There are two cases:", + "type": "text" + }, + { + "bbox": [ + 475, + 392, + 494, + 402 + ], + "score": 0.89, + "content": "s _ { i + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 388, + 505, + 403 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 133, + 397, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 133, + 397, + 198, + 414 + ], + "score": 1.0, + "content": "a variable, and", + "type": "text" + }, + { + "bbox": [ + 199, + 402, + 218, + 412 + ], + "score": 0.89, + "content": "s _ { i + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 397, + 367, + 414 + ], + "score": 1.0, + "content": "is the token ‘;’. 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If the current token", + "type": "text" + }, + { + "bbox": [ + 218, + 430, + 227, + 439 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 425, + 368, + 443 + ], + "score": 1.0, + "content": "is a variable, copy the embedding", + "type": "text" + }, + { + "bbox": [ + 368, + 430, + 404, + 441 + ], + "score": 0.89, + "content": "e _ { \\mathrm { i d } ( s _ { j - 1 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 425, + 506, + 443 + ], + "score": 1.0, + "content": "(which is a number) for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 134, + 439, + 507, + 455 + ], + "spans": [ + { + "bbox": [ + 134, + 439, + 162, + 455 + ], + "score": 1.0, + "content": "index", + "type": "text" + }, + { + "bbox": [ + 162, + 442, + 168, + 453 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 439, + 212, + 455 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 213, + 441, + 259, + 455 + ], + "score": 0.78, + "content": "n _ { j } ^ { \\mathrm { c o t } } = n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 439, + 264, + 455 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 264, + 440, + 308, + 455 + ], + "score": 0.8, + "content": "n _ { j } ^ { \\mathrm { c o t } } = d _ { j } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 439, + 331, + 455 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 331, + 441, + 408, + 453 + ], + "score": 0.9, + "content": "\\operatorname { v a r } ( s _ { j } ) = \\operatorname { v a r } ( s _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 439, + 507, + 455 + ], + "score": 1.0, + "content": ". Denote the answer as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 136, + 447, + 417, + 470 + ], + "spans": [ + { + "bbox": [ + 136, + 453, + 168, + 467 + ], + "score": 0.88, + "content": "e _ { i } ^ { \\mathrm { { c o t \\_ n u m } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 449, + 198, + 470 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 199, + 456, + 208, + 465 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 449, + 285, + 470 + ], + "score": 1.0, + "content": "is not a variable or", + "type": "text" + }, + { + "bbox": [ + 286, + 454, + 327, + 467 + ], + "score": 0.85, + "content": "d _ { i } ^ { \\mathrm { e q } } \\leq n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 447, + 372, + 468 + ], + "score": 1.0, + "content": "ot, ecot_num is", + "type": "text" + }, + { + "bbox": [ + 363, + 453, + 417, + 466 + ], + "score": 1.0, + "content": "undefined.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "We can use an attention head to perform the COPY operation that completes the first task. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "construction is similar to the fourth layer in arithmetic expression and we omit it for clarity. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 492, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 507, + 505 + ], + "score": 1.0, + "content": "second attention head performs the fifth task, which can also be done via the COPY operation.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 102, + 498, + 502, + 522 + ], + "spans": [ + { + "bbox": [ + 102, + 498, + 489, + 522 + ], + "score": 1.0, + "content": "Regarding the second task, whether the next token is a number can be purely determined by", + "type": "text" + }, + { + "bbox": [ + 489, + 503, + 502, + 515 + ], + "score": 0.88, + "content": "d _ { i } ^ { \\mathrm { { e q } } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 509, + 510, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 123, + 527 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 509, + 216, + 532 + ], + "score": 1.0, + "content": ", and the current token eq", + "type": "text" + }, + { + "bbox": [ + 226, + 509, + 284, + 532 + ], + "score": 1.0, + "content": ". Specifically,", + "type": "text" + }, + { + "bbox": [ + 284, + 516, + 304, + 526 + ], + "score": 0.9, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 509, + 365, + 532 + ], + "score": 1.0, + "content": "is a number if", + "type": "text" + }, + { + "bbox": [ + 401, + 509, + 417, + 532 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 453, + 509, + 471, + 532 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 471, + 516, + 499, + 525 + ], + "score": 0.88, + "content": "( s _ { i } = \\mathsf { \\Omega } ^ { \\bullet }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 509, + 510, + 532 + ], + "score": 1.0, + "content": ";’", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 522, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 124, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 525, + 168, + 537 + ], + "score": 0.92, + "content": "d _ { i } ^ { \\mathrm { e q } } > n _ { i } ^ { \\mathrm { c o t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 522, + 507, + 539 + ], + "score": 1.0, + "content": "). Whether the output of the next token is a variable can also be purely determined", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 534, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 201, + 550 + ], + "score": 1.0, + "content": "by the previous tokens", + "type": "text" + }, + { + "bbox": [ + 201, + 539, + 233, + 549 + ], + "score": 0.88, + "content": "s _ { i - 1 } , s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 534, + 271, + 550 + ], + "score": 1.0, + "content": "and also", + "type": "text" + }, + { + "bbox": [ + 271, + 536, + 285, + 550 + ], + "score": 0.91, + "content": "d _ { i } ^ { \\mathrm { { e q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 534, + 303, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 537, + 320, + 549 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 534, + 377, + 550 + ], + "score": 1.0, + "content": ". Specifically,", + "type": "text" + }, + { + "bbox": [ + 378, + 539, + 397, + 549 + ], + "score": 0.87, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 534, + 459, + 550 + ], + "score": 1.0, + "content": "is a variable if", + "type": "text" + }, + { + "bbox": [ + 460, + 537, + 504, + 548 + ], + "score": 0.87, + "content": "s _ { i - 1 } = \\cdot { } + { } ^ { , }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 547, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 123, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 550, + 133, + 560 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 547, + 199, + 563 + ], + "score": 1.0, + "content": "is a number, or", + "type": "text" + }, + { + "bbox": [ + 199, + 550, + 235, + 561 + ], + "score": 0.85, + "content": "s _ { i - 1 } = \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 547, + 261, + 563 + ], + "score": 1.0, + "content": ";’ and", + "type": "text" + }, + { + "bbox": [ + 261, + 551, + 270, + 560 + ], + "score": 0.86, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 547, + 339, + 563 + ], + "score": 1.0, + "content": "is a number, or (", + "type": "text" + }, + { + "bbox": [ + 339, + 550, + 368, + 560 + ], + "score": 0.64, + "content": "s _ { i } = \\mathsf { \\Omega } ^ { \\bullet } ;", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 547, + 385, + 563 + ], + "score": 1.0, + "content": "’ or", + "type": "text" + }, + { + "bbox": [ + 385, + 549, + 439, + 560 + ], + "score": 0.89, + "content": "s _ { i } = \\cdot \\Longrightarrow \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 547, + 460, + 563 + ], + "score": 1.0, + "content": ") and", + "type": "text" + }, + { + "bbox": [ + 460, + 549, + 503, + 561 + ], + "score": 0.92, + "content": "d _ { i } ^ { \\mathrm { e q } } \\leq n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 547, + 507, + 563 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 559, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 300, + 574 + ], + "score": 1.0, + "content": "The variable index can be determined by either", + "type": "text" + }, + { + "bbox": [ + 300, + 561, + 340, + 573 + ], + "score": 0.89, + "content": "\\mathrm { v a r } ( s _ { i - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 559, + 353, + 574 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 353, + 560, + 366, + 573 + ], + "score": 0.89, + "content": "d _ { i } ^ { \\mathrm { { e q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 559, + 507, + 574 + ], + "score": 1.0, + "content": ". When the next token is neither a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 569, + 507, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 268, + 586 + ], + "score": 1.0, + "content": "variable nor a number (i.e., the symbols", + "type": "text" + }, + { + "bbox": [ + 268, + 572, + 280, + 582 + ], + "score": 0.57, + "content": "\\cdot _ { + } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 569, + 327, + 586 + ], + "score": 1.0, + "content": ", ‘=’, ‘;’, or", + "type": "text" + }, + { + "bbox": [ + 327, + 573, + 350, + 582 + ], + "score": 0.7, + "content": "\\xrightarrow { \\bullet } \\implies", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 569, + 507, + 586 + ], + "score": 1.0, + "content": ", we can similarly determine the token", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 580, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 159, + 597 + ], + "score": 1.0, + "content": "by checking", + "type": "text" + }, + { + "bbox": [ + 159, + 582, + 209, + 595 + ], + "score": 0.5, + "content": "s _ { i - 1 } , s _ { i } , d _ { i } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 580, + 231, + 597 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 231, + 583, + 248, + 595 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { v a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 580, + 394, + 597 + ], + "score": 1.0, + "content": ". When the next token is a number,", + "type": "text" + }, + { + "bbox": [ + 394, + 585, + 413, + 594 + ], + "score": 0.88, + "content": "s _ { i + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 580, + 507, + 597 + ], + "score": 1.0, + "content": "can be determined by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 590, + 504, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 195, + 609 + ], + "score": 1.0, + "content": "checking the variable", + "type": "text" + }, + { + "bbox": [ + 195, + 596, + 214, + 605 + ], + "score": 0.88, + "content": "s _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 590, + 302, + 609 + ], + "score": 1.0, + "content": "via three cases: (i) if", + "type": "text" + }, + { + "bbox": [ + 303, + 595, + 322, + 605 + ], + "score": 0.89, + "content": "s _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 590, + 392, + 609 + ], + "score": 1.0, + "content": "is a variable and", + "type": "text" + }, + { + "bbox": [ + 392, + 594, + 461, + 605 + ], + "score": 0.89, + "content": "\\mathrm { v a r } ( s _ { i - 1 } ) < n _ { i } ^ { \\mathrm { v a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 590, + 484, + 609 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 485, + 595, + 504, + 605 + ], + "score": 0.87, + "content": "s _ { i + 2 }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 102, + 601, + 509, + 622 + ], + "spans": [ + { + "bbox": [ + 102, + 601, + 177, + 622 + ], + "score": 1.0, + "content": "is a variable and", + "type": "text" + }, + { + "bbox": [ + 177, + 605, + 289, + 617 + ], + "score": 0.9, + "content": "\\mathrm { v a r } ( s _ { i + 2 } ) = \\mathrm { v a r } ( s _ { i - 1 } ) + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 601, + 319, + 622 + ], + "score": 1.0, + "content": "; (ii) if", + "type": "text" + }, + { + "bbox": [ + 319, + 606, + 338, + 616 + ], + "score": 0.89, + "content": "s _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 601, + 410, + 622 + ], + "score": 1.0, + "content": "is a variable and", + "type": "text" + }, + { + "bbox": [ + 411, + 605, + 481, + 616 + ], + "score": 0.89, + "content": "\\mathrm { v a r } ( s _ { i - 1 } ) = n _ { i } ^ { \\mathrm { v a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 601, + 509, + 622 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 614, + 486, + 629 + ], + "spans": [ + { + "bbox": [ + 107, + 616, + 149, + 628 + ], + "score": 0.84, + "content": "s _ { i + 2 } = \\cdot ;", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 614, + 214, + 629 + ], + "score": 1.0, + "content": "; (iii) otherwise,", + "type": "text" + }, + { + "bbox": [ + 215, + 618, + 234, + 627 + ], + "score": 0.88, + "content": "s _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 614, + 306, + 629 + ], + "score": 1.0, + "content": "is a number, then", + "type": "text" + }, + { + "bbox": [ + 306, + 618, + 325, + 627 + ], + "score": 0.9, + "content": "s _ { i + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 614, + 394, + 629 + ], + "score": 1.0, + "content": "is a variable and", + "type": "text" + }, + { + "bbox": [ + 394, + 616, + 480, + 628 + ], + "score": 0.89, + "content": "\\mathrm { v a r } ( s _ { i + 2 } ) = n _ { i } ^ { \\mathrm { c o t } } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 614, + 486, + 629 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "All these tasks can implemented by MLPs that performs the conditional selection or look-up table", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "based on Lemmas C.4 and C.5. Moreover, the composition of conditional selection and look-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "up table can be merged into a single two-layer MLP (as shown in the construction of the third", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 663, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 103, + 663, + 477, + 679 + ], + "score": 1.0, + "content": "layer in arithmetic expression). We next use multiplication to compute the auxiliary terms", + "type": "text" + }, + { + "bbox": [ + 477, + 664, + 502, + 677 + ], + "score": 0.92, + "content": "( d _ { i } ^ { \\mathrm { { e q } } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 663, + 506, + 679 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 193, + 690 + ], + "score": 0.91, + "content": "( d _ { i } ^ { \\mathrm { e q } } + \\mathbb { I } [ s _ { i + 2 } = \\cdot ; ] ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 676, + 214, + 691 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 215, + 676, + 239, + 690 + ], + "score": 0.91, + "content": "( \\hat { d } _ { i } ^ { \\mathrm { e q } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 676, + 333, + 691 + ], + "score": 1.0, + "content": ". However, to compute", + "type": "text" + }, + { + "bbox": [ + 333, + 677, + 384, + 690 + ], + "score": 0.91, + "content": "( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 676, + 506, + 691 + ], + "score": 1.0, + "content": ", we cannot use multiplication", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "directly as the composition of multiplication and conditional selection will require a deeper MLP.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 178, + 713 + ], + "score": 1.0, + "content": "Instead, note that", + "type": "text" + }, + { + "bbox": [ + 178, + 699, + 229, + 712 + ], + "score": 0.92, + "content": "( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 699, + 312, + 713 + ], + "score": 1.0, + "content": "linearly depends on", + "type": "text" + }, + { + "bbox": [ + 312, + 700, + 363, + 712 + ], + "score": 0.91, + "content": "( \\mathrm { v a r } ( s _ { i - 1 } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 699, + 382, + 713 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 383, + 700, + 423, + 712 + ], + "score": 0.92, + "content": "\\mathrm { v a r } ( s _ { i - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ", or linearly depends", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 708, + 504, + 725 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 120, + 725 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 710, + 148, + 723 + ], + "score": 0.92, + "content": "( n _ { i } ^ { \\mathrm { c o t } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 708, + 168, + 725 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 711, + 185, + 722 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 708, + 452, + 725 + ], + "score": 1.0, + "content": ", all of which is already computed. Therefore, we can compute", + "type": "text" + }, + { + "bbox": [ + 452, + 710, + 504, + 723 + ], + "score": 0.91, + "content": "( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 50 + } + ], + "index": 46.5 + } + ], + "page_idx": 30, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 310, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 312, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 312, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 506, + 118 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 121, + 484, + 139 + ], + "lines": [ + { + "bbox": [ + 125, + 121, + 484, + 139 + ], + "spans": [ + { + "bbox": [ + 125, + 121, + 484, + 139 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\pmb { q } _ { i } = \\pmb { Q x } _ { i } ^ { ( 1 ) } = ( ( i - 1 ) ^ { 2 } , i - 1 , - 1 ) , \\quad \\pmb { k } _ { j } = \\pmb { K } \\pmb { x } _ { j } ^ { ( 1 ) } = ( - 1 , 2 j , j ^ { 2 } ) , \\quad \\pmb { v } _ { j } = ( e _ { \\mathrm { i d } ( s _ { j } ) } , l _ { j } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "14d64f79a3e37db0203234307448335ad95b58feea83caf6c2440ec5cd896763.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 125, + 121, + 484, + 139 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 224, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 222, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 123, + 156 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 144, + 169, + 156 + ], + "score": 0.92, + "content": "\\pmb q _ { i } \\cdot \\pmb k _ { j } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 143, + 182, + 156 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 182, + 144, + 222, + 155 + ], + "score": 0.9, + "content": "j = i - 1", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 143, + 222, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 277, + 174 + ], + "score": 1.0, + "content": "We next use an MLP to correct the value of", + "type": "text" + }, + { + "bbox": [ + 278, + 160, + 292, + 175 + ], + "score": 0.91, + "content": "n _ { j } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 159, + 317, + 174 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 317, + 161, + 352, + 173 + ], + "score": 0.92, + "content": "n _ { i } ^ { \\mathrm { c o t } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 159, + 506, + 174 + ], + "score": 1.0, + "content": "and compute the second task, which is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 171, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 104, + 171, + 318, + 188 + ], + "score": 1.0, + "content": "a linear operation. We also compute an auxiliary flag", + "type": "text" + }, + { + "bbox": [ + 318, + 173, + 369, + 187 + ], + "score": 0.92, + "content": "\\mathbb { I } [ n _ { i } ^ { \\mathrm { c o t } } = d _ { i } ^ { \\mathrm { e q } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 171, + 506, + 188 + ], + "score": 1.0, + "content": "via an MLP. Regarding the fourth", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 184, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 184, + 506, + 198 + ], + "score": 1.0, + "content": "task, it is a multivariate conditional selection operation and can be similarly implemented by an MLP", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 208 + ], + "score": 1.0, + "content": "by extending Lemma C.4. Note that we can compute the second task and the fourth task in parallel", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "using a two-layer MLP because both tasks correspond to (multivariate) conditional selection and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 412, + 231 + ], + "score": 1.0, + "content": "can be merged. 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The output of the MLP is", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 159, + 506, + 243 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 245, + 464, + 280 + ], + "lines": [ + { + "bbox": [ + 146, + 245, + 464, + 280 + ], + "spans": [ + { + "bbox": [ + 146, + 245, + 464, + 280 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\boldsymbol { x } _ { i } ^ { ( 2 ) } = ( e _ { \\mathrm { i d } ( s _ { i } ) } , l _ { i } , i , i ^ { 2 } , 1 , n _ { i } ^ { \\mathrm { v a r } } , n _ { i } ^ { \\mathrm { c o t } } , ( n _ { i } ^ { \\mathrm { c o t } } ) ^ { 2 } , d _ { i } ^ { \\mathrm { c q } } , f _ { i } , } \\\\ & { \\qquad e _ { \\mathrm { i d } ( s _ { i - 1 } ) } , l _ { i - 1 } , e _ { \\mathrm { i d } ( s _ { i - 2 } ) } , l _ { i - 2 } , ( \\mathrm { v a r } ( s _ { i - 1 } ) ) ^ { 2 } , ( \\mathrm { v a r } ( s _ { i } ) ) ^ { 2 } , \\mathbb { I } [ n _ { i } ^ { \\mathrm { c o t } } = d _ { i } ^ { \\mathrm { e q } } ] ) . } \\end{array}", + "type": "interline_equation", + "image_path": "87f24b16aa34ca77cba28bd2560a23f782ccbf3484978bf6fae00026bd5db080.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 146, + 245, + 464, + 256.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 146, + 256.6666666666667, + 464, + 268.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 146, + 268.33333333333337, + 464, + 280.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 287, + 504, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 506, + 301 + ], + "score": 1.0, + "content": "Layer 3. The third layer of the Transformer uses two attention heads to perform the following tasks:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 287, + 506, + 301 + ] + }, + { + "type": "list", + "bbox": [ + 120, + 303, + 506, + 467 + ], + "lines": [ + { + "bbox": [ + 123, + 302, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 123, + 302, + 223, + 317 + ], + "score": 1.0, + "content": "1. Copy the embedding", + "type": "text" + }, + { + "bbox": [ + 224, + 303, + 237, + 318 + ], + "score": 0.9, + "content": "d _ { j } ^ { \\mathsf { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 302, + 309, + 317 + ], + "score": 1.0, + "content": "with the smallest", + "type": "text" + }, + { + "bbox": [ + 310, + 305, + 316, + 316 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 302, + 359, + 317 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 360, + 304, + 388, + 317 + ], + "score": 0.92, + "content": "f _ { j } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 302, + 406, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 304, + 469, + 318 + ], + "score": 0.93, + "content": "n _ { j } ^ { \\mathrm { c o t } } = n _ { i } ^ { \\mathrm { c o t } } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 302, + 505, + 317 + ], + "score": 1.0, + "content": ". Denote", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 316, + 209, + 331 + ], + "spans": [ + { + "bbox": [ + 135, + 316, + 192, + 331 + ], + "score": 1.0, + "content": "the answer as", + "type": "text" + }, + { + "bbox": [ + 193, + 317, + 206, + 331 + ], + "score": 0.9, + "content": "\\hat { d } _ { i } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 316, + 209, + 331 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 122, + 334, + 450, + 348 + ], + "spans": [ + { + "bbox": [ + 122, + 334, + 274, + 347 + ], + "score": 1.0, + "content": "2. Determine whether the next token", + "type": "text" + }, + { + "bbox": [ + 275, + 338, + 293, + 347 + ], + "score": 0.88, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 334, + 428, + 347 + ], + "score": 1.0, + "content": "is a number. Denote the result as", + "type": "text" + }, + { + "bbox": [ + 428, + 335, + 447, + 348 + ], + "score": 0.9, + "content": "f _ { i } ^ { \\mathrm { n u m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 334, + 450, + 347 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 122, + 350, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 122, + 350, + 289, + 365 + ], + "score": 1.0, + "content": "3. 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If the current token", + "type": "text" + }, + { + "bbox": [ + 218, + 430, + 227, + 439 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 425, + 368, + 443 + ], + "score": 1.0, + "content": "is a variable, copy the embedding", + "type": "text" + }, + { + "bbox": [ + 368, + 430, + 404, + 441 + ], + "score": 0.89, + "content": "e _ { \\mathrm { i d } ( s _ { j - 1 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 425, + 506, + 443 + ], + "score": 1.0, + "content": "(which is a number) for", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 134, + 439, + 507, + 455 + ], + "spans": [ + { + "bbox": [ + 134, + 439, + 162, + 455 + ], + "score": 1.0, + "content": "index", + "type": "text" + }, + { + "bbox": [ + 162, + 442, + 168, + 453 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 439, + 212, + 455 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 213, + 441, + 259, + 455 + ], + "score": 0.78, + "content": "n _ { j } ^ { \\mathrm { c o t } } = n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 439, + 264, + 455 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 264, + 440, + 308, + 455 + ], + "score": 0.8, + "content": "n _ { j } ^ { \\mathrm { c o t } } = d _ { j } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 439, + 331, + 455 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 331, + 441, + 408, + 453 + ], + "score": 0.9, + "content": "\\operatorname { v a r } ( s _ { j } ) = \\operatorname { v a r } ( s _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 439, + 507, + 455 + ], + "score": 1.0, + "content": ". Denote the answer as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 136, + 447, + 417, + 470 + ], + "spans": [ + { + "bbox": [ + 136, + 453, + 168, + 467 + ], + "score": 0.88, + "content": "e _ { i } ^ { \\mathrm { { c o t \\_ n u m } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 449, + 198, + 470 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 199, + 456, + 208, + 465 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 449, + 285, + 470 + ], + "score": 1.0, + "content": "is not a variable or", + "type": "text" + }, + { + "bbox": [ + 286, + 454, + 327, + 467 + ], + "score": 0.85, + "content": "d _ { i } ^ { \\mathrm { e q } } \\leq n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 447, + 372, + 468 + ], + "score": 1.0, + "content": "ot, ecot_num is", + "type": "text" + }, + { + "bbox": [ + 363, + 453, + 417, + 466 + ], + "score": 1.0, + "content": "undefined.", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + } + ], + "index": 22.5, + "bbox_fs": [ + 121, + 302, + 507, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "We can use an attention head to perform the COPY operation that completes the first task. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "construction is similar to the fourth layer in arithmetic expression and we omit it for clarity. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 492, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 507, + 505 + ], + "score": 1.0, + "content": "second attention head performs the fifth task, which can also be done via the COPY operation.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 102, + 498, + 502, + 522 + ], + "spans": [ + { + "bbox": [ + 102, + 498, + 489, + 522 + ], + "score": 1.0, + "content": "Regarding the second task, whether the next token is a number can be purely determined by", + "type": "text" + }, + { + "bbox": [ + 489, + 503, + 502, + 515 + ], + "score": 0.88, + "content": "d _ { i } ^ { \\mathrm { { e q } } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 509, + 510, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 123, + 527 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 509, + 216, + 532 + ], + "score": 1.0, + "content": ", and the current token eq", + "type": "text" + }, + { + "bbox": [ + 226, + 509, + 284, + 532 + ], + "score": 1.0, + "content": ". Specifically,", + "type": "text" + }, + { + "bbox": [ + 284, + 516, + 304, + 526 + ], + "score": 0.9, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 509, + 365, + 532 + ], + "score": 1.0, + "content": "is a number if", + "type": "text" + }, + { + "bbox": [ + 401, + 509, + 417, + 532 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 453, + 509, + 471, + 532 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 471, + 516, + 499, + 525 + ], + "score": 0.88, + "content": "( s _ { i } = \\mathsf { \\Omega } ^ { \\bullet }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 509, + 510, + 532 + ], + "score": 1.0, + "content": ";’", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 522, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 124, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 525, + 168, + 537 + ], + "score": 0.92, + "content": "d _ { i } ^ { \\mathrm { e q } } > n _ { i } ^ { \\mathrm { c o t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 522, + 507, + 539 + ], + "score": 1.0, + "content": "). Whether the output of the next token is a variable can also be purely determined", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 534, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 201, + 550 + ], + "score": 1.0, + "content": "by the previous tokens", + "type": "text" + }, + { + "bbox": [ + 201, + 539, + 233, + 549 + ], + "score": 0.88, + "content": "s _ { i - 1 } , s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 534, + 271, + 550 + ], + "score": 1.0, + "content": "and also", + "type": "text" + }, + { + "bbox": [ + 271, + 536, + 285, + 550 + ], + "score": 0.91, + "content": "d _ { i } ^ { \\mathrm { { e q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 534, + 303, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 537, + 320, + 549 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 534, + 377, + 550 + ], + "score": 1.0, + "content": ". Specifically,", + "type": "text" + }, + { + "bbox": [ + 378, + 539, + 397, + 549 + ], + "score": 0.87, + "content": "s _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 534, + 459, + 550 + ], + "score": 1.0, + "content": "is a variable if", + "type": "text" + }, + { + "bbox": [ + 460, + 537, + 504, + 548 + ], + "score": 0.87, + "content": "s _ { i - 1 } = \\cdot { } + { } ^ { , }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 547, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 123, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 550, + 133, + 560 + ], + "score": 0.84, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 547, + 199, + 563 + ], + "score": 1.0, + "content": "is a number, or", + "type": "text" + }, + { + "bbox": [ + 199, + 550, + 235, + 561 + ], + "score": 0.85, + "content": "s _ { i - 1 } = \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 547, + 261, + 563 + ], + "score": 1.0, + "content": ";’ and", + "type": "text" + }, + { + "bbox": [ + 261, + 551, + 270, + 560 + ], + "score": 0.86, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 547, + 339, + 563 + ], + "score": 1.0, + "content": "is a number, or (", + "type": "text" + }, + { + "bbox": [ + 339, + 550, + 368, + 560 + ], + "score": 0.64, + "content": "s _ { i } = \\mathsf { \\Omega } ^ { \\bullet } ;", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 547, + 385, + 563 + ], + "score": 1.0, + "content": "’ or", + "type": "text" + }, + { + "bbox": [ + 385, + 549, + 439, + 560 + ], + "score": 0.89, + "content": "s _ { i } = \\cdot \\Longrightarrow \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 547, + 460, + 563 + ], + "score": 1.0, + "content": ") and", + "type": "text" + }, + { + "bbox": [ + 460, + 549, + 503, + 561 + ], + "score": 0.92, + "content": "d _ { i } ^ { \\mathrm { e q } } \\leq n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 547, + 507, + 563 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 559, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 300, + 574 + ], + "score": 1.0, + "content": "The variable index can be determined by either", + "type": "text" + }, + { + "bbox": [ + 300, + 561, + 340, + 573 + ], + "score": 0.89, + "content": "\\mathrm { v a r } ( s _ { i - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 559, + 353, + 574 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 353, + 560, + 366, + 573 + ], + "score": 0.89, + "content": "d _ { i } ^ { \\mathrm { { e q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 559, + 507, + 574 + ], + "score": 1.0, + "content": ". When the next token is neither a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 569, + 507, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 268, + 586 + ], + "score": 1.0, + "content": "variable nor a number (i.e., the symbols", + "type": "text" + }, + { + "bbox": [ + 268, + 572, + 280, + 582 + ], + "score": 0.57, + "content": "\\cdot _ { + } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 569, + 327, + 586 + ], + "score": 1.0, + "content": ", ‘=’, ‘;’, or", + "type": "text" + }, + { + "bbox": [ + 327, + 573, + 350, + 582 + ], + "score": 0.7, + "content": "\\xrightarrow { \\bullet } \\implies", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 569, + 507, + 586 + ], + "score": 1.0, + "content": ", we can similarly determine the token", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 580, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 159, + 597 + ], + "score": 1.0, + "content": "by checking", + "type": "text" + }, + { + "bbox": [ + 159, + 582, + 209, + 595 + ], + "score": 0.5, + "content": "s _ { i - 1 } , s _ { i } , d _ { i } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 580, + 231, + 597 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 231, + 583, + 248, + 595 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { v a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 580, + 394, + 597 + ], + "score": 1.0, + "content": ". 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Moreover, the composition of conditional selection and look-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "up table can be merged into a single two-layer MLP (as shown in the construction of the third", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 663, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 103, + 663, + 477, + 679 + ], + "score": 1.0, + "content": "layer in arithmetic expression). We next use multiplication to compute the auxiliary terms", + "type": "text" + }, + { + "bbox": [ + 477, + 664, + 502, + 677 + ], + "score": 0.92, + "content": "( d _ { i } ^ { \\mathrm { { e q } } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 663, + 506, + 679 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 193, + 690 + ], + "score": 0.91, + "content": "( d _ { i } ^ { \\mathrm { e q } } + \\mathbb { I } [ s _ { i + 2 } = \\cdot ; ] ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 676, + 214, + 691 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 215, + 676, + 239, + 690 + ], + "score": 0.91, + "content": "( \\hat { d } _ { i } ^ { \\mathrm { e q } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 676, + 333, + 691 + ], + "score": 1.0, + "content": ". However, to compute", + "type": "text" + }, + { + "bbox": [ + 333, + 677, + 384, + 690 + ], + "score": 0.91, + "content": "( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 676, + 506, + 691 + ], + "score": 1.0, + "content": ", we cannot use multiplication", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "directly as the composition of multiplication and conditional selection will require a deeper MLP.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 178, + 713 + ], + "score": 1.0, + "content": "Instead, note that", + "type": "text" + }, + { + "bbox": [ + 178, + 699, + 229, + 712 + ], + "score": 0.92, + "content": "( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 699, + 312, + 713 + ], + "score": 1.0, + "content": "linearly depends on", + "type": "text" + }, + { + "bbox": [ + 312, + 700, + 363, + 712 + ], + "score": 0.91, + "content": "( \\mathrm { v a r } ( s _ { i - 1 } ) ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 699, + 382, + 713 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 383, + 700, + 423, + 712 + ], + "score": 0.92, + "content": "\\mathrm { v a r } ( s _ { i - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ", or linearly depends", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 708, + 504, + 725 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 120, + 725 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 120, + 710, + 148, + 723 + ], + "score": 0.92, + "content": "( n _ { i } ^ { \\mathrm { c o t } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 708, + 168, + 725 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 711, + 185, + 722 + ], + "score": 0.9, + "content": "n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 708, + 452, + 725 + ], + "score": 1.0, + "content": ", all of which is already computed. Therefore, we can compute", + "type": "text" + }, + { + "bbox": [ + 452, + 710, + 504, + 723 + ], + "score": 0.91, + "content": "( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 72, + 349, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 349, + 87 + ], + "score": 1.0, + "content": "without multiplication. The output of this layer has the form", + "type": "text", + "cross_page": true + } + ], + "index": 0 + } + ], + "index": 46.5, + "bbox_fs": [ + 103, + 631, + 506, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 349, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 349, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 349, + 87 + ], + "score": 1.0, + "content": "without multiplication. The output of this layer has the form", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 90, + 499, + 126 + ], + "lines": [ + { + "bbox": [ + 112, + 90, + 499, + 126 + ], + "spans": [ + { + "bbox": [ + 112, + 90, + 499, + 126 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { x _ { i } ^ { ( 3 ) } = ( e _ { \\mathrm { i d } ( s _ { i } ) } , l _ { i } , i , 1 , n _ { i } ^ { \\mathrm { v a r } } , n _ { i } ^ { \\mathrm { c o t } } , ( n _ { i } ^ { \\mathrm { c o t } } ) ^ { 2 } , d _ { i } ^ { \\mathrm { e q } } , ( d _ { i } ^ { \\mathrm { e q } } ) ^ { 2 } , ( d _ { i } ^ { \\mathrm { e q } } + \\mathbb { I } [ s _ { i + 2 } = \\cdot , \\cdot ] ) ^ { 2 } , \\hat { d } _ { i } ^ { \\mathrm { e q } } , ( \\hat { d } _ { i } ^ { \\mathrm { e q } } ) ^ { 2 } , f _ { i } ^ { \\mathrm { n u m } } , } \\\\ & { \\quad \\quad \\quad e _ { i } ^ { \\mathrm { n e x t } } , e _ { i } ^ { \\mathrm { n e x t 2 } } , \\mathrm { v a r } ( s _ { i } ) , \\mathrm { v a r } ( s _ { i + 1 } ) , \\mathrm { v a r } ( s _ { i + 2 } ) , ( \\mathrm { v a r } ( s _ { i } ) ) ^ { 2 } , ( \\mathrm { v a r } ( s _ { i + 2 } ) ) ^ { 2 } , e _ { \\mathrm { i d } ( s _ { i - 1 } ) } , e _ { i } ^ { \\mathrm { c o t . } \\mathrm { n u m } } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e9d370dee6dd5494c408b8178d191b0998fa576180f0c8bc62dfadd8f3a1cbdb.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 112, + 90, + 499, + 102.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 102.0, + 499, + 114.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 112, + 114.0, + 499, + 126.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "Layer 4. The fourth layer of the Transformer performs the the core calculation of equation coefficients", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 443, + 158 + ], + "score": 1.0, + "content": "when the next token is a number. 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There are also two variables related", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 168, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 168, + 232, + 185 + ], + "score": 1.0, + "content": "to the calculation: the variable", + "type": "text" + }, + { + "bbox": [ + 233, + 172, + 272, + 183 + ], + "score": 0.89, + "content": "x _ { \\mathrm { v a r } ( s _ { i + 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 168, + 291, + 185 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 171, + 310, + 183 + ], + "score": 0.88, + "content": "\\underset { \\bf { \\hat { \\phi } } ^ { 2 } } { x } { } _ { n _ { i } ^ { \\mathrm { { c o t } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 168, + 506, + 185 + ], + "score": 1.0, + "content": ". Specifically, we need to copy four coefficients", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 181, + 360, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 281, + 196 + ], + "score": 0.48, + "content": "a _ { \\hat { d } _ { i } ^ { \\mathrm { e q } } , n _ { i } ^ { \\mathrm { c o t } } } , a _ { \\hat { d } _ { i } ^ { \\mathrm { e q } } , \\mathrm { v a r } \\left( s _ { i + 2 } \\right) } , a _ { d _ { i } ^ { \\mathrm { e q } } , n _ { i } ^ { \\mathrm { c o t } } } , a _ { d _ { i } ^ { \\mathrm { e q } } , \\mathrm { v a r } \\left( s _ { i + 2 } \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 181, + 360, + 198 + ], + "score": 1.0, + "content": "defined as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 205, + 476, + 234 + ], + "lines": [], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 226, + 205, + 476, + 234 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 506, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 169, + 252 + ], + "score": 1.0, + "content": "For the case of", + "type": "text" + }, + { + "bbox": [ + 170, + 239, + 212, + 251 + ], + "score": 0.88, + "content": "s _ { i + 2 } = \\cdot ;", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 238, + 335, + 252 + ], + "score": 1.0, + "content": ", we need to copy coefficients", + "type": "text" + }, + { + "bbox": [ + 336, + 240, + 351, + 254 + ], + "score": 0.9, + "content": "b _ { \\hat { d } _ { i } ^ { \\mathrm { s q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 238, + 370, + 252 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 240, + 385, + 253 + ], + "score": 0.9, + "content": "b _ { d _ { i } ^ { \\mathrm { e q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 238, + 505, + 252 + ], + "score": 1.0, + "content": ". To unify the two cases, this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "Transformer layer uses four attention heads to perform the following tasks (note that we define", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 263, + 264, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 156, + 276 + ], + "score": 0.92, + "content": "\\mathrm { v a r } ( s _ { j } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 263, + 181, + 276 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 181, + 264, + 191, + 276 + ], + "score": 0.86, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 263, + 264, + 276 + ], + "score": 1.0, + "content": "is not a variable):", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 123, + 281, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 122, + 279, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 122, + 279, + 224, + 296 + ], + "score": 1.0, + "content": "1. Copy the embedding", + "type": "text" + }, + { + "bbox": [ + 225, + 284, + 261, + 295 + ], + "score": 0.9, + "content": "e _ { \\mathrm { i d } ( s _ { j - 1 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 279, + 313, + 296 + ], + "score": 1.0, + "content": "for position", + "type": "text" + }, + { + "bbox": [ + 313, + 283, + 319, + 294 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 279, + 363, + 296 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 363, + 281, + 427, + 296 + ], + "score": 0.87, + "content": "n _ { j } ^ { \\mathrm { c o t } } = n _ { i } ^ { \\mathrm { c o t } } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 279, + 432, + 296 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 432, + 280, + 472, + 296 + ], + "score": 0.86, + "content": "d _ { j } ^ { \\mathrm { e q } } = \\hat { d } _ { i } ^ { \\mathrm { e q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 279, + 476, + 296 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 476, + 282, + 487, + 295 + ], + "score": 0.6, + "content": "s _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 279, + 506, + 296 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 133, + 290, + 255, + 310 + ], + "spans": [ + { + "bbox": [ + 133, + 290, + 190, + 310 + ], + "score": 1.0, + "content": "variable, and", + "type": "text" + }, + { + "bbox": [ + 190, + 295, + 249, + 307 + ], + "score": 0.92, + "content": "\\mathrm { v a r } ( s _ { j } ) = n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 290, + 255, + 310 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 121, + 310, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 121, + 310, + 220, + 329 + ], + "score": 1.0, + "content": "2. 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Copy the embedding", + "type": "text" + }, + { + "bbox": [ + 224, + 378, + 260, + 389 + ], + "score": 0.9, + "content": "e _ { \\mathrm { i d } ( s _ { j - 1 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 372, + 279, + 392 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 375, + 311, + 389 + ], + "score": 0.91, + "content": "e _ { j } ^ { \\mathrm { { c o t \\_ n u m } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 372, + 362, + 392 + ], + "score": 1.0, + "content": "for position", + "type": "text" + }, + { + "bbox": [ + 362, + 377, + 369, + 388 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 372, + 412, + 392 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 412, + 376, + 475, + 389 + ], + "score": 0.7, + "content": "n _ { j } ^ { \\mathrm { c o t } } = n _ { i } ^ { \\mathrm { c o t } } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 372, + 479, + 392 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 479, + 375, + 506, + 389 + ], + "score": 0.51, + "content": "d _ { j } ^ { \\mathrm { { e q } } } =", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 137, + 384, + 390, + 405 + ], + "spans": [ + { + "bbox": [ + 137, + 388, + 213, + 402 + ], + "score": 0.82, + "content": "d _ { i } ^ { \\mathrm { e q } } + \\mathbb { I } [ s _ { i + 2 } = \\stackrel { \\cdot , \\cdot } { ; } ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 384, + 216, + 405 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 216, + 389, + 278, + 403 + ], + "score": 0.88, + "content": "e _ { \\mathrm { i d } ( s _ { j } ) } = e _ { i } ^ { \\mathrm { n e x t } 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 384, + 299, + 405 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 299, + 389, + 383, + 402 + ], + "score": 0.92, + "content": "\\operatorname { v a r } ( s _ { j } ) = \\operatorname { v a r } ( s _ { i + 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 384, + 390, + 405 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 313, + 419 + ], + "score": 1.0, + "content": "Note that for each task, there is exactly one index", + "type": "text" + }, + { + "bbox": [ + 313, + 407, + 320, + 417 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "satisfying the condition, and thus the copied", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "embeddings contain the four coefficients defined above. 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However, there are several special", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 452, + 214, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 214, + 464 + ], + "score": 1.0, + "content": "cases we have to consider:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 127, + 467, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 126, + 465, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 126, + 465, + 136, + 482 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 137, + 467, + 178, + 480 + ], + "score": 0.9, + "content": "d _ { i } ^ { \\mathrm { e q } } = n _ { i } ^ { \\mathrm { c o t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 465, + 442, + 482 + ], + "score": 1.0, + "content": ". 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\\mathbb { I } [ \\mathrm { i d } ( t ) = \\mathrm { i d } ( \\tilde { t } ) ] - \\mathrm { v a r } ( t ) \\mathrm { v a r } ( \\tilde { t } ) } \\\\ & { \\qquad - m ^ { 4 } \\left( \\sin \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) \\sin \\left( \\frac { 2 \\mathrm { v a r } ( \\tilde { t } ) \\pi } { m } \\right) + \\cos \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) \\cos \\left( \\frac { 2 \\mathrm { v a r } ( \\tilde { t } ) \\pi } { m } \\right) \\right) } \\\\ & { \\qquad = ( 1 - \\mathbb { I } [ \\mathrm { i d } ( t ) = \\mathrm { i d } ( \\tilde { t } ) ] ) + \\mathrm { v a r } ( t ) ( \\mathrm { v a r } ( t ) - \\mathrm { v a r } ( \\tilde { t } ) ) + m ^ { 4 } \\left( 1 - \\cos \\left( \\frac { 2 ( \\mathrm { v a r } ( t ) - \\mathrm { v a r } ( \\tilde { t } ) ) \\pi } { m } \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "64101a73eb45a9d4a071ef373b1c70afc62cb407959ae77ee4a337920a3ce5b8.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 111, + 192, + 505, + 217.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 217.33333333333334, + 505, + 242.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 111, + 242.66666666666669, + 505, + 268.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 414, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 416, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 132, + 288 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 272, + 196, + 285 + ], + "score": 0.93, + "content": "\\mathrm { v a r } ( t ) = \\mathrm { v a r } ( \\tilde { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 270, + 236, + 288 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 236, + 272, + 289, + 285 + ], + "score": 0.92, + "content": "\\operatorname { i d } ( t ) \\neq \\operatorname { i d } ( \\tilde { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 270, + 326, + 288 + ], + "score": 1.0, + "content": "and thus", + "type": "text" + }, + { + "bbox": [ + 327, + 273, + 364, + 285 + ], + "score": 0.86, + "content": "\\mathrm { G a p } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 270, + 416, + 288 + ], + "score": 1.0, + "content": ". Otherwise,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 289, + 386, + 321 + ], + "lines": [ + { + "bbox": [ + 223, + 289, + 386, + 321 + ], + "spans": [ + { + "bbox": [ + 223, + 289, + 386, + 321 + ], + "score": 0.9, + "content": "\\begin{array} { r l r } & { } & { \\mathrm { G a p } \\geq 1 - m ^ { 2 } + m ^ { 4 } \\left( 1 - \\cos ( 2 \\pi / m ) \\right) } \\\\ & { } & { = 1 - m ^ { 2 } + m ^ { 4 } \\sin ^ { 2 } ( \\pi / m ) \\geq 1 , } \\end{array}", + "type": "interline_equation", + "image_path": "f31abfa9fdde18c20e6ce6a1d51aef4643aaf0b1085c90d0de197aeb4dcf760c.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 289, + 386, + 305.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 223, + 305.0, + 386, + 321.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 369, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 369, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 212, + 338 + ], + "score": 1.0, + "content": "where we use the fact that", + "type": "text" + }, + { + "bbox": [ + 212, + 324, + 269, + 336 + ], + "score": 0.93, + "content": "\\sin ( x ) \\geq x / \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 323, + 311, + 338 + ], + "score": 1.0, + "content": "whenever", + "type": "text" + }, + { + "bbox": [ + 312, + 324, + 366, + 336 + ], + "score": 0.9, + "content": "0 < x \\le \\pi / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 323, + 369, + 338 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Now it remains to conduct an error analysis and determine the scale of parameters. Similar to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "the proof of arithemetic expression, we can prove that all parameter values in the Transformer are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 156, + 376 + ], + "score": 1.0, + "content": "bounded by", + "type": "text" + }, + { + "bbox": [ + 156, + 362, + 205, + 375 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 361, + 210, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 363, + 505, + 373 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 386, + 223, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 223, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 223, + 400 + ], + "score": 1.0, + "content": "E.2 Proof of Theorem 3.2", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 504, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "We will now prove that solving a system of linear equations without CoT is extremely difficult for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 286, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 286, + 430 + ], + "score": 1.0, + "content": "bounded-depth autoregressive Transformers.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 506, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 203, + 447 + ], + "score": 1.0, + "content": "Theorem E.2. Assume", + "type": "text" + }, + { + "bbox": [ + 204, + 432, + 253, + 445 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 431, + 410, + 447 + ], + "score": 1.0, + "content": ". For any prime number p, any integer", + "type": "text" + }, + { + "bbox": [ + 410, + 434, + 418, + 444 + ], + "score": 0.54, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 431, + 506, + 447 + ], + "score": 1.0, + "content": ", and any polynomial", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 115, + 456 + ], + "score": 0.75, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 444, + 505, + 457 + ], + "score": 1.0, + "content": ", there exists a problem size m such that no log-precision autoregressive Transformer defined in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 455, + 492, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 190, + 469 + ], + "score": 1.0, + "content": "Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 190, + 456, + 198, + 465 + ], + "score": 0.36, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 455, + 289, + 469 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 290, + 456, + 333, + 468 + ], + "score": 0.93, + "content": "d \\leq Q ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 455, + 462, + 469 + ], + "score": 1.0, + "content": "can solve the problem Equation", + "type": "text" + }, + { + "bbox": [ + 462, + 456, + 488, + 468 + ], + "score": 0.91, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 455, + 492, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 506, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 288, + 493 + ], + "score": 1.0, + "content": "Proof. Our proof is based on leveraging the", + "type": "text" + }, + { + "bbox": [ + 289, + 479, + 307, + 491 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "-completeness of a classic problem: Unsolvable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "Automaton Membership Testing. According to Barrington’s theorem [4, 5], given a fixed unsolvable", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 503, + 415, + 516 + ], + "score": 1.0, + "content": "automaton, judging whether the automaton accepts an input is complete in", + "type": "text" + }, + { + "bbox": [ + 416, + 502, + 434, + 514 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 503, + 505, + 516 + ], + "score": 1.0, + "content": ". Below, we will", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 311, + 528 + ], + "score": 1.0, + "content": "prove that solving the system of linear equations is", + "type": "text" + }, + { + "bbox": [ + 312, + 514, + 330, + 525 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "-hard by demonstrating that the Unsolvable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 287, + 540 + ], + "score": 1.0, + "content": "Automaton Membership Testing problem is", + "type": "text" + }, + { + "bbox": [ + 288, + 526, + 307, + 538 + ], + "score": 0.87, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "reducible to the problem of solving a system of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "linear equations. This will yield the conclusion since bounded-depth log-precision Transformers with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 549, + 240, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 197, + 563 + ], + "score": 1.0, + "content": "polynomial size are in", + "type": "text" + }, + { + "bbox": [ + 198, + 549, + 216, + 561 + ], + "score": 0.86, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 549, + 240, + 563 + ], + "score": 1.0, + "content": "[42].", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 123, + 579 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 124, + 566, + 213, + 578 + ], + "score": 0.92, + "content": "\\boldsymbol { D } = ( \\mathcal { Q } , \\Sigma , \\delta , \\mathcal { F } , q _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 566, + 325, + 579 + ], + "score": 1.0, + "content": "be any automaton, where", + "type": "text" + }, + { + "bbox": [ + 325, + 567, + 335, + 577 + ], + "score": 0.82, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 566, + 412, + 579 + ], + "score": 1.0, + "content": "is a set of states,", + "type": "text" + }, + { + "bbox": [ + 412, + 567, + 421, + 577 + ], + "score": 0.79, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "is a set of symbols", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 576, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 153, + 592 + ], + "score": 1.0, + "content": "(alphabet),", + "type": "text" + }, + { + "bbox": [ + 154, + 578, + 227, + 588 + ], + "score": 0.89, + "content": "\\delta : \\mathcal { Q } \\times \\Sigma \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 576, + 336, + 592 + ], + "score": 1.0, + "content": "is the transition function,", + "type": "text" + }, + { + "bbox": [ + 337, + 578, + 371, + 588 + ], + "score": 0.91, + "content": "\\mathcal { F } \\subset \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 576, + 493, + 592 + ], + "score": 1.0, + "content": "is a set of accept states, and", + "type": "text" + }, + { + "bbox": [ + 494, + 579, + 504, + 589 + ], + "score": 0.79, + "content": "q _ { 0 }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 587, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 267, + 602 + ], + "score": 1.0, + "content": "is the initial state. For any input string", + "type": "text" + }, + { + "bbox": [ + 267, + 590, + 316, + 600 + ], + "score": 0.89, + "content": "\\omega _ { 1 } \\omega _ { 2 } \\cdots \\omega _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 587, + 356, + 602 + ], + "score": 1.0, + "content": ", whether", + "type": "text" + }, + { + "bbox": [ + 356, + 589, + 366, + 599 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 587, + 506, + 602 + ], + "score": 1.0, + "content": "accepts the string can be reduced", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "into solving a system of linear equations defined as follows. The system of linear equations has", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 168, + 623 + ], + "score": 0.92, + "content": "( n + 1 ) | \\mathcal { Q } | + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 610, + 294, + 623 + ], + "score": 1.0, + "content": "variables, which we denote as", + "type": "text" + }, + { + "bbox": [ + 294, + 611, + 306, + 621 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 610, + 324, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 611, + 340, + 623 + ], + "score": 0.67, + "content": "x _ { i , q }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 610, + 440, + 623 + ], + "score": 0.87, + "content": "( i \\in \\{ 0 , \\cdot \\cdot \\cdot , n \\} , q \\in \\mathcal { Q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 610, + 505, + 623 + ], + "score": 1.0, + "content": ". The equations", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 198, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 198, + 633 + ], + "score": 1.0, + "content": "are defined as follows:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 635, + 410, + 686 + ], + "lines": [ + { + "bbox": [ + 196, + 635, + 410, + 686 + ], + "spans": [ + { + "bbox": [ + 196, + 635, + 410, + 686 + ], + "score": 0.93, + "content": "\\left\\{ \\begin{array} { l l } { x ^ { * } = \\sum _ { q \\in \\mathcal { F } } x _ { n , q } } \\\\ { x _ { 0 , q _ { 0 } } = 1 } \\\\ { x _ { 0 , q } = 0 } & { \\mathrm { f o r } q \\in \\mathcal { Q } \\backslash \\{ q _ { 0 } \\} } \\\\ { x _ { i , q } = \\sum _ { \\delta ( r , \\omega _ { i } ) = q } x _ { i - 1 , r } } & { \\mathrm { f o r } 0 < i \\leq n , q \\in \\mathcal { Q } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "4feac5358277813e3dbc67e1a715a2213c169fd99fec392b76d141409ad23ec3.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 196, + 635, + 410, + 652.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 196, + 652.0, + 410, + 669.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 196, + 669.0, + 410, + 686.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 104, + 687, + 504, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 184, + 703 + ], + "score": 1.0, + "content": "It is easy to see that", + "type": "text" + }, + { + "bbox": [ + 185, + 689, + 219, + 700 + ], + "score": 0.9, + "content": "x _ { i , q } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 687, + 347, + 703 + ], + "score": 1.0, + "content": "iff the automaton arrives at state", + "type": "text" + }, + { + "bbox": [ + 347, + 691, + 353, + 700 + ], + "score": 0.82, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 687, + 456, + 703 + ], + "score": 1.0, + "content": "when taking the substring", + "type": "text" + }, + { + "bbox": [ + 457, + 691, + 504, + 700 + ], + "score": 0.87, + "content": "\\omega _ { 1 } \\omega _ { 2 } \\cdot \\cdot \\cdot \\omega _ { i }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 188, + 712 + ], + "score": 1.0, + "content": "as input. Therefore,", + "type": "text" + }, + { + "bbox": [ + 189, + 701, + 218, + 710 + ], + "score": 0.89, + "content": "x ^ { * } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "iff the automaton accepts the input string. Note that the above solution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 501, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 237, + 723 + ], + "score": 1.0, + "content": "does not depend on the modulus", + "type": "text" + }, + { + "bbox": [ + 238, + 713, + 244, + 722 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 711, + 501, + 723 + ], + "score": 1.0, + "content": ", and the solution of these equations always exists and is unique.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "page_idx": 32, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 312, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 312, + 754 + ], + "score": 1.0, + "content": "33", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 505, + 96 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 506, + 96 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 100, + 466, + 158 + ], + "lines": [ + { + "bbox": [ + 144, + 100, + 466, + 158 + ], + "spans": [ + { + "bbox": [ + 144, + 100, + 466, + 158 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathrm { G a p } : = z _ { t } \\left( e _ { \\mathrm { i d } ( t ) } , \\mathrm { v a r } ( t ) , m ^ { 2 } \\sin \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) , m ^ { 2 } \\cos \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) \\right) - } \\\\ & { \\qquad z _ { t } \\left( e _ { \\mathrm { i d } ( \\tilde { t } ) } , \\mathrm { v a r } ( \\tilde { t } ) , m ^ { 2 } \\sin \\left( \\frac { 2 \\mathrm { v a r } ( \\tilde { t } ) \\pi } { m } \\right) , m ^ { 2 } \\cos \\left( \\frac { 2 \\mathrm { v a r } ( \\tilde { t } ) \\pi } { m } \\right) \\right) \\geq \\Theta ( 1 ) . } \\end{array}", + "type": "interline_equation", + "image_path": "356a2d35e05961c704cafc8a21f153b8494f468b854fa0790fcf8832ef7f985f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 144, + 100, + 466, + 119.33333333333333 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 144, + 119.33333333333333, + 466, + 138.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 144, + 138.66666666666666, + 466, + 158.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 504, + 190 + ], + "lines": [ + { + "bbox": [ + 100, + 155, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 100, + 155, + 257, + 193 + ], + "score": 1.0, + "content": "To prove the above result, simply set We have", + "type": "text" + }, + { + "bbox": [ + 258, + 161, + 506, + 182 + ], + "score": 0.79, + "content": "\\begin{array} { r } { { \\pmb w } _ { t } = \\left( e _ { \\mathrm { i d } ( t ) } , \\mathrm { v a r } ( t ) , m ^ { 2 } \\sin \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) , m ^ { 2 } \\cos \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) \\right) . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 100, + 155, + 506, + 193 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 192, + 505, + 268 + ], + "lines": [ + { + "bbox": [ + 111, + 192, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 111, + 192, + 505, + 268 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathrm { \\tilde { \\lambda } a p } = 1 + ( \\mathrm { v a r } ( t ) ) ^ { 2 } + m ^ { 4 } - \\mathbb { I } [ \\mathrm { i d } ( t ) = \\mathrm { i d } ( \\tilde { t } ) ] - \\mathrm { v a r } ( t ) \\mathrm { v a r } ( \\tilde { t } ) } \\\\ & { \\qquad - m ^ { 4 } \\left( \\sin \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) \\sin \\left( \\frac { 2 \\mathrm { v a r } ( \\tilde { t } ) \\pi } { m } \\right) + \\cos \\left( \\frac { 2 \\mathrm { v a r } ( t ) \\pi } { m } \\right) \\cos \\left( \\frac { 2 \\mathrm { v a r } ( \\tilde { t } ) \\pi } { m } \\right) \\right) } \\\\ & { \\qquad = ( 1 - \\mathbb { I } [ \\mathrm { i d } ( t ) = \\mathrm { i d } ( \\tilde { t } ) ] ) + \\mathrm { v a r } ( t ) ( \\mathrm { v a r } ( t ) - \\mathrm { v a r } ( \\tilde { t } ) ) + m ^ { 4 } \\left( 1 - \\cos \\left( \\frac { 2 ( \\mathrm { v a r } ( t ) - \\mathrm { v a r } ( \\tilde { t } ) ) \\pi } { m } \\right) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "64101a73eb45a9d4a071ef373b1c70afc62cb407959ae77ee4a337920a3ce5b8.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 111, + 192, + 505, + 217.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 217.33333333333334, + 505, + 242.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 111, + 242.66666666666669, + 505, + 268.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 414, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 416, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 132, + 288 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 272, + 196, + 285 + ], + "score": 0.93, + "content": "\\mathrm { v a r } ( t ) = \\mathrm { v a r } ( \\tilde { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 270, + 236, + 288 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 236, + 272, + 289, + 285 + ], + "score": 0.92, + "content": "\\operatorname { i d } ( t ) \\neq \\operatorname { i d } ( \\tilde { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 270, + 326, + 288 + ], + "score": 1.0, + "content": "and thus", + "type": "text" + }, + { + "bbox": [ + 327, + 273, + 364, + 285 + ], + "score": 0.86, + "content": "\\mathrm { G a p } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 270, + 416, + 288 + ], + "score": 1.0, + "content": ". Otherwise,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 270, + 416, + 288 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 289, + 386, + 321 + ], + "lines": [ + { + "bbox": [ + 223, + 289, + 386, + 321 + ], + "spans": [ + { + "bbox": [ + 223, + 289, + 386, + 321 + ], + "score": 0.9, + "content": "\\begin{array} { r l r } & { } & { \\mathrm { G a p } \\geq 1 - m ^ { 2 } + m ^ { 4 } \\left( 1 - \\cos ( 2 \\pi / m ) \\right) } \\\\ & { } & { = 1 - m ^ { 2 } + m ^ { 4 } \\sin ^ { 2 } ( \\pi / m ) \\geq 1 , } \\end{array}", + "type": "interline_equation", + "image_path": "f31abfa9fdde18c20e6ce6a1d51aef4643aaf0b1085c90d0de197aeb4dcf760c.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 289, + 386, + 305.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 223, + 305.0, + 386, + 321.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 369, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 369, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 212, + 338 + ], + "score": 1.0, + "content": "where we use the fact that", + "type": "text" + }, + { + "bbox": [ + 212, + 324, + 269, + 336 + ], + "score": 0.93, + "content": "\\sin ( x ) \\geq x / \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 323, + 311, + 338 + ], + "score": 1.0, + "content": "whenever", + "type": "text" + }, + { + "bbox": [ + 312, + 324, + 366, + 336 + ], + "score": 0.9, + "content": "0 < x \\le \\pi / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 323, + 369, + 338 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 323, + 369, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Now it remains to conduct an error analysis and determine the scale of parameters. Similar to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "the proof of arithemetic expression, we can prove that all parameter values in the Transformer are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 361, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 156, + 376 + ], + "score": 1.0, + "content": "bounded by", + "type": "text" + }, + { + "bbox": [ + 156, + 362, + 205, + 375 + ], + "score": 0.91, + "content": "O ( \\mathrm { p o l y } ( n ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 361, + 210, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 363, + 505, + 373 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 340, + 505, + 376 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 386, + 223, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 223, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 223, + 400 + ], + "score": 1.0, + "content": "E.2 Proof of Theorem 3.2", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 504, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "We will now prove that solving a system of linear equations without CoT is extremely difficult for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 286, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 286, + 430 + ], + "score": 1.0, + "content": "bounded-depth autoregressive Transformers.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 407, + 505, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 506, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 203, + 447 + ], + "score": 1.0, + "content": "Theorem E.2. Assume", + "type": "text" + }, + { + "bbox": [ + 204, + 432, + 253, + 445 + ], + "score": 0.92, + "content": "{ \\mathsf { T } } { \\mathsf { C } } ^ { 0 } \\neq { \\mathsf { N } } { \\mathsf { C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 431, + 410, + 447 + ], + "score": 1.0, + "content": ". For any prime number p, any integer", + "type": "text" + }, + { + "bbox": [ + 410, + 434, + 418, + 444 + ], + "score": 0.54, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 431, + 506, + 447 + ], + "score": 1.0, + "content": ", and any polynomial", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 115, + 456 + ], + "score": 0.75, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 444, + 505, + 457 + ], + "score": 1.0, + "content": ", there exists a problem size m such that no log-precision autoregressive Transformer defined in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 455, + 492, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 190, + 469 + ], + "score": 1.0, + "content": "Section 2 with depth", + "type": "text" + }, + { + "bbox": [ + 190, + 456, + 198, + 465 + ], + "score": 0.36, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 455, + 289, + 469 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 290, + 456, + 333, + 468 + ], + "score": 0.93, + "content": "d \\leq Q ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 455, + 462, + 469 + ], + "score": 1.0, + "content": "can solve the problem Equation", + "type": "text" + }, + { + "bbox": [ + 462, + 456, + 488, + 468 + ], + "score": 0.91, + "content": "( m , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 455, + 492, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 431, + 506, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 506, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 288, + 493 + ], + "score": 1.0, + "content": "Proof. Our proof is based on leveraging the", + "type": "text" + }, + { + "bbox": [ + 289, + 479, + 307, + 491 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "-completeness of a classic problem: Unsolvable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "Automaton Membership Testing. According to Barrington’s theorem [4, 5], given a fixed unsolvable", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 503, + 415, + 516 + ], + "score": 1.0, + "content": "automaton, judging whether the automaton accepts an input is complete in", + "type": "text" + }, + { + "bbox": [ + 416, + 502, + 434, + 514 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 503, + 505, + 516 + ], + "score": 1.0, + "content": ". Below, we will", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 311, + 528 + ], + "score": 1.0, + "content": "prove that solving the system of linear equations is", + "type": "text" + }, + { + "bbox": [ + 312, + 514, + 330, + 525 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "-hard by demonstrating that the Unsolvable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 287, + 540 + ], + "score": 1.0, + "content": "Automaton Membership Testing problem is", + "type": "text" + }, + { + "bbox": [ + 288, + 526, + 307, + 538 + ], + "score": 0.87, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "reducible to the problem of solving a system of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "linear equations. This will yield the conclusion since bounded-depth log-precision Transformers with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 549, + 240, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 197, + 563 + ], + "score": 1.0, + "content": "polynomial size are in", + "type": "text" + }, + { + "bbox": [ + 198, + 549, + 216, + 561 + ], + "score": 0.86, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 549, + 240, + 563 + ], + "score": 1.0, + "content": "[42].", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 479, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 123, + 579 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 124, + 566, + 213, + 578 + ], + "score": 0.92, + "content": "\\boldsymbol { D } = ( \\mathcal { Q } , \\Sigma , \\delta , \\mathcal { F } , q _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 566, + 325, + 579 + ], + "score": 1.0, + "content": "be any automaton, where", + "type": "text" + }, + { + "bbox": [ + 325, + 567, + 335, + 577 + ], + "score": 0.82, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 566, + 412, + 579 + ], + "score": 1.0, + "content": "is a set of states,", + "type": "text" + }, + { + "bbox": [ + 412, + 567, + 421, + 577 + ], + "score": 0.79, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "is a set of symbols", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 576, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 153, + 592 + ], + "score": 1.0, + "content": "(alphabet),", + "type": "text" + }, + { + "bbox": [ + 154, + 578, + 227, + 588 + ], + "score": 0.89, + "content": "\\delta : \\mathcal { Q } \\times \\Sigma \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 576, + 336, + 592 + ], + "score": 1.0, + "content": "is the transition function,", + "type": "text" + }, + { + "bbox": [ + 337, + 578, + 371, + 588 + ], + "score": 0.91, + "content": "\\mathcal { F } \\subset \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 576, + 493, + 592 + ], + "score": 1.0, + "content": "is a set of accept states, and", + "type": "text" + }, + { + "bbox": [ + 494, + 579, + 504, + 589 + ], + "score": 0.79, + "content": "q _ { 0 }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 587, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 267, + 602 + ], + "score": 1.0, + "content": "is the initial state. For any input string", + "type": "text" + }, + { + "bbox": [ + 267, + 590, + 316, + 600 + ], + "score": 0.89, + "content": "\\omega _ { 1 } \\omega _ { 2 } \\cdots \\omega _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 587, + 356, + 602 + ], + "score": 1.0, + "content": ", whether", + "type": "text" + }, + { + "bbox": [ + 356, + 589, + 366, + 599 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 587, + 506, + 602 + ], + "score": 1.0, + "content": "accepts the string can be reduced", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "into solving a system of linear equations defined as follows. The system of linear equations has", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 168, + 623 + ], + "score": 0.92, + "content": "( n + 1 ) | \\mathcal { Q } | + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 610, + 294, + 623 + ], + "score": 1.0, + "content": "variables, which we denote as", + "type": "text" + }, + { + "bbox": [ + 294, + 611, + 306, + 621 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 610, + 324, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 611, + 340, + 623 + ], + "score": 0.67, + "content": "x _ { i , q }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 610, + 440, + 623 + ], + "score": 0.87, + "content": "( i \\in \\{ 0 , \\cdot \\cdot \\cdot , n \\} , q \\in \\mathcal { Q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 610, + 505, + 623 + ], + "score": 1.0, + "content": ". The equations", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 198, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 198, + 633 + ], + "score": 1.0, + "content": "are defined as follows:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 566, + 506, + 633 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 635, + 410, + 686 + ], + "lines": [ + { + "bbox": [ + 196, + 635, + 410, + 686 + ], + "spans": [ + { + "bbox": [ + 196, + 635, + 410, + 686 + ], + "score": 0.93, + "content": "\\left\\{ \\begin{array} { l l } { x ^ { * } = \\sum _ { q \\in \\mathcal { F } } x _ { n , q } } \\\\ { x _ { 0 , q _ { 0 } } = 1 } \\\\ { x _ { 0 , q } = 0 } & { \\mathrm { f o r } q \\in \\mathcal { Q } \\backslash \\{ q _ { 0 } \\} } \\\\ { x _ { i , q } = \\sum _ { \\delta ( r , \\omega _ { i } ) = q } x _ { i - 1 , r } } & { \\mathrm { f o r } 0 < i \\leq n , q \\in \\mathcal { Q } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "4feac5358277813e3dbc67e1a715a2213c169fd99fec392b76d141409ad23ec3.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 196, + 635, + 410, + 652.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 196, + 652.0, + 410, + 669.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 196, + 669.0, + 410, + 686.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 104, + 687, + 504, + 703 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 184, + 703 + ], + "score": 1.0, + "content": "It is easy to see that", + "type": "text" + }, + { + "bbox": [ + 185, + 689, + 219, + 700 + ], + "score": 0.9, + "content": "x _ { i , q } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 687, + 347, + 703 + ], + "score": 1.0, + "content": "iff the automaton arrives at state", + "type": "text" + }, + { + "bbox": [ + 347, + 691, + 353, + 700 + ], + "score": 0.82, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 687, + 456, + 703 + ], + "score": 1.0, + "content": "when taking the substring", + "type": "text" + }, + { + "bbox": [ + 457, + 691, + 504, + 700 + ], + "score": 0.87, + "content": "\\omega _ { 1 } \\omega _ { 2 } \\cdot \\cdot \\cdot \\omega _ { i }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 188, + 712 + ], + "score": 1.0, + "content": "as input. Therefore,", + "type": "text" + }, + { + "bbox": [ + 189, + 701, + 218, + 710 + ], + "score": 0.89, + "content": "x ^ { * } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "iff the automaton accepts the input string. Note that the above solution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 501, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 237, + 723 + ], + "score": 1.0, + "content": "does not depend on the modulus", + "type": "text" + }, + { + "bbox": [ + 238, + 713, + 244, + 722 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 711, + 501, + 723 + ], + "score": 1.0, + "content": ", and the solution of these equations always exists and is unique.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 687, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "score": 1.0, + "content": "Furthermore, the coefficient of each equation only depends on at most one input symbol. This implies", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "that these equations can be efficiently constructed using a highly parallelizable algorithm within a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 165, + 109 + ], + "score": 1.0, + "content": "complexity of", + "type": "text" + }, + { + "bbox": [ + 166, + 94, + 185, + 106 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 94, + 505, + 109 + ], + "score": 1.0, + "content": ". Therefore, by reduction, we obtain that the problem of judging whether there", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 295, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 211, + 119 + ], + "score": 1.0, + "content": "exists a solution such that", + "type": "text" + }, + { + "bbox": [ + 211, + 108, + 241, + 118 + ], + "score": 0.9, + "content": "x ^ { * } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 106, + 251, + 119 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 251, + 106, + 270, + 118 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 106, + 295, + 119 + ], + "score": 1.0, + "content": "-hard.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "score": 1.0, + "content": "Now consider solving linear equations using a Transformer without CoT. While the output of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 134, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 506, + 147 + ], + "score": 1.0, + "content": "Transformer contains multiple tokens, we can arrange the order of variables such that the Transformer", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 211, + 158 + ], + "score": 1.0, + "content": "has to output the value of", + "type": "text" + }, + { + "bbox": [ + 212, + 146, + 223, + 156 + ], + "score": 0.86, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "first. The parallel complexity of outputting the first token is bounded", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 120, + 171 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 157, + 139, + 168 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "according to [42]. Therefore, it cannot judge whether there exists a solution satisfying", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 168, + 504, + 181 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 136, + 179 + ], + "score": 0.89, + "content": "x ^ { * } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 168, + 141, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 171, + 504, + 179 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 106, + 197, + 303, + 211 + ], + "lines": [ + { + "bbox": [ + 104, + 196, + 304, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 304, + 213 + ], + "score": 1.0, + "content": "F Discussion on Other Architectures", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 223, + 261, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 263, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 263, + 236 + ], + "score": 1.0, + "content": "F.1 Encoder-Decoder Transformer", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "In this paper, we choose the autoregressive Transformer as it is simple and the de facto standard for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "LLMs (e.g., GPT). However, all of the theoretical results of this paper can be easily transferred to an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 405, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 405, + 279 + ], + "score": 1.0, + "content": "encoder-decoder Transformer (such as T5) using the following arguments.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 127, + 282, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 127, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 127, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "• Given a bounded-depth polynomial-size log-precision encoder-decoder Transformer, its", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 134, + 293, + 448, + 307 + ], + "spans": [ + { + "bbox": [ + 134, + 293, + 311, + 307 + ], + "score": 1.0, + "content": "computation complexity is still bounded by", + "type": "text" + }, + { + "bbox": [ + 312, + 293, + 330, + 306 + ], + "score": 0.81, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 293, + 448, + 307 + ], + "score": 1.0, + "content": ", so our negative results hold.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 127, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 127, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "• Any finite-depth autoregressive Transformer can be mimicked by a finite-depth encoder-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 135, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 135, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "decoder Transformer of roughly the same size, since causal masking for the input sequence", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 135, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 135, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "can be mimicked through the use of positional encoding and joint attention can be mimicked", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 135, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 135, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "by the integration of cross-attention and self-attention. Thus, all positive results in this paper", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 136, + 353, + 333, + 364 + ], + "spans": [ + { + "bbox": [ + 136, + 353, + 333, + 364 + ], + "score": 1.0, + "content": "are not exclusive to autoregressive Transformers.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 273, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 275, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 275, + 391 + ], + "score": 1.0, + "content": "F.2 Recurrent Neural Network(RNN)", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "Although the theorems in this paper can be naturally extended to other popular Transformer ar-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "chitectures, such as the encoder-decoder Transformer (e.g., T5), RNNs do not have this property.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "Theorem F.1 shows that RNNs cannot generate the CoT sequence using the same format proposed in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "our paper for the arithmetic formula task and the linear equation task unless the hidden dimension of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 103, + 438, + 402, + 460 + ], + "spans": [ + { + "bbox": [ + 103, + 438, + 184, + 460 + ], + "score": 1.0, + "content": "the RNN is at least", + "type": "text" + }, + { + "bbox": [ + 185, + 442, + 219, + 457 + ], + "score": 0.93, + "content": "\\Omega ( { \\frac { n } { \\log n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 438, + 250, + 460 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 250, + 444, + 257, + 452 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 438, + 402, + 460 + ], + "score": 1.0, + "content": "is the length of the input sequence.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 231, + 473 + ], + "score": 1.0, + "content": "Theorem F.1. For any integer", + "type": "text" + }, + { + "bbox": [ + 231, + 462, + 238, + 471 + ], + "score": 0.48, + "content": "p _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 459, + 505, + 473 + ], + "score": 1.0, + "content": ", any log-precision RNN with constant layers can neither generate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 466, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 453, + 488 + ], + "score": 1.0, + "content": "the CoT solution for the problem Arithmetic(n, p) with hidden dimension of o( nlog n )", + "type": "text" + }, + { + "bbox": [ + 449, + 472, + 505, + 483 + ], + "score": 1.0, + "content": "nor generate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 103, + 483, + 454, + 505 + ], + "spans": [ + { + "bbox": [ + 103, + 483, + 272, + 505 + ], + "score": 1.0, + "content": "the CoT solution for the problem Eqution", + "type": "text" + }, + { + "bbox": [ + 273, + 487, + 296, + 499 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 483, + 415, + 505 + ], + "score": 1.0, + "content": "with the hidden dimension of", + "type": "text" + }, + { + "bbox": [ + 416, + 484, + 447, + 501 + ], + "score": 0.94, + "content": "O \\big ( \\frac { n ^ { \\overline { { 2 } } } } { \\log n } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 483, + 454, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "Proof. Given an RNN model, if the hidden embeddings in each layer of the last symbols of two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "different input sequences are the same, the two output sequences will also be exactly the same. When", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "the RNN finishes processing the input sequence, the input sequence has been compressed into a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 168, + 563 + ], + "score": 1.0, + "content": "hidden state of", + "type": "text" + }, + { + "bbox": [ + 168, + 550, + 215, + 562 + ], + "score": 0.92, + "content": "O ( D \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 549, + 263, + 563 + ], + "score": 1.0, + "content": "bits, where", + "type": "text" + }, + { + "bbox": [ + 263, + 550, + 273, + 560 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "is the hidden dimension and each scalar is represented by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 107, + 561, + 144, + 573 + ], + "score": 0.91, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 560, + 478, + 573 + ], + "score": 1.0, + "content": "bits (by definition of log-precision). Therefore, an RNN with a hidden dimension of", + "type": "text" + }, + { + "bbox": [ + 478, + 561, + 488, + 571 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 195, + 586 + ], + "score": 1.0, + "content": "only generate at most", + "type": "text" + }, + { + "bbox": [ + 195, + 571, + 215, + 582 + ], + "score": 0.89, + "content": "n 2 ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 570, + 479, + 586 + ], + "score": 1.0, + "content": "different output sequences. However, for the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 572, + 502, + 584 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 570, + 506, + 586 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 581, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 353, + 596 + ], + "score": 1.0, + "content": "the first step of the CoT needs to output a sequence of length", + "type": "text" + }, + { + "bbox": [ + 353, + 583, + 376, + 595 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 581, + 443, + 596 + ], + "score": 1.0, + "content": ", which contains", + "type": "text" + }, + { + "bbox": [ + 443, + 583, + 466, + 595 + ], + "score": 0.91, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 581, + 507, + 596 + ], + "score": 1.0, + "content": "numbers.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 592, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 217, + 609 + ], + "score": 1.0, + "content": "Therefore, there are at least", + "type": "text" + }, + { + "bbox": [ + 217, + 594, + 256, + 607 + ], + "score": 0.92, + "content": "O ( 2 ^ { O ( n ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 592, + 479, + 609 + ], + "score": 1.0, + "content": "different solution sequences for the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 595, + 503, + 607 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 592, + 507, + 609 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 605, + 507, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 218, + 623 + ], + "score": 1.0, + "content": "Similarly, there are at least", + "type": "text" + }, + { + "bbox": [ + 218, + 607, + 262, + 621 + ], + "score": 0.92, + "content": "O ( 2 ^ { O ( n ^ { 2 } ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 605, + 479, + 623 + ], + "score": 1.0, + "content": "different solution sequences for the problem Eqution", + "type": "text" + }, + { + "bbox": [ + 479, + 608, + 503, + 621 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 605, + 507, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 618, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 618, + 354, + 635 + ], + "score": 1.0, + "content": "Thus, by the Pigeon Hole Principle, to be able to generate all", + "type": "text" + }, + { + "bbox": [ + 354, + 619, + 378, + 631 + ], + "score": 0.9, + "content": "2 ^ { O ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 618, + 507, + 635 + ], + "score": 1.0, + "content": "different output sequences, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 629, + 508, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 629, + 248, + 649 + ], + "score": 1.0, + "content": "must have a hidden dimension of", + "type": "text" + }, + { + "bbox": [ + 248, + 631, + 308, + 646 + ], + "score": 0.94, + "content": "\\begin{array} { r } { D = \\Omega \\big ( \\frac { n } { \\log n } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 629, + 423, + 649 + ], + "score": 1.0, + "content": "for the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 423, + 632, + 447, + 644 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 629, + 508, + 649 + ], + "score": 1.0, + "content": "and a hidden", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 644, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 644, + 161, + 667 + ], + "score": 1.0, + "content": "dimension of", + "type": "text" + }, + { + "bbox": [ + 162, + 644, + 218, + 662 + ], + "score": 0.94, + "content": "\\begin{array} { r } { D = \\Omega ( \\frac { n ^ { 2 } } { \\log n } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 644, + 317, + 667 + ], + "score": 1.0, + "content": "for the problem Eqution", + "type": "text" + }, + { + "bbox": [ + 317, + 648, + 340, + 660 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 644, + 346, + 667 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 648, + 505, + 659 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "Due to the great difference in the model architecture between the RNN and the transformer, most of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "the theorems in this paper cannot be simply generalized to the RNN. Moreover, To the best of our", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "knowledge, there is very little work to investigate the chain of thought in RNN model, either from the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 711, + 246, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 246, + 723 + ], + "score": 1.0, + "content": "theoretical or the empirical aspect.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 505, + 86 + ], + "score": 1.0, + "content": "Furthermore, the coefficient of each equation only depends on at most one input symbol. This implies", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "that these equations can be efficiently constructed using a highly parallelizable algorithm within a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 165, + 109 + ], + "score": 1.0, + "content": "complexity of", + "type": "text" + }, + { + "bbox": [ + 166, + 94, + 185, + 106 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 94, + 505, + 109 + ], + "score": 1.0, + "content": ". Therefore, by reduction, we obtain that the problem of judging whether there", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 295, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 211, + 119 + ], + "score": 1.0, + "content": "exists a solution such that", + "type": "text" + }, + { + "bbox": [ + 211, + 108, + 241, + 118 + ], + "score": 0.9, + "content": "x ^ { * } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 106, + 251, + 119 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 251, + 106, + 270, + 118 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 106, + 295, + 119 + ], + "score": 1.0, + "content": "-hard.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 71, + 506, + 119 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "score": 1.0, + "content": "Now consider solving linear equations using a Transformer without CoT. While the output of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 134, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 506, + 147 + ], + "score": 1.0, + "content": "Transformer contains multiple tokens, we can arrange the order of variables such that the Transformer", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 211, + 158 + ], + "score": 1.0, + "content": "has to output the value of", + "type": "text" + }, + { + "bbox": [ + 212, + 146, + 223, + 156 + ], + "score": 0.86, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "first. The parallel complexity of outputting the first token is bounded", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 120, + 171 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 157, + 139, + 168 + ], + "score": 0.85, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "according to [42]. Therefore, it cannot judge whether there exists a solution satisfying", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 168, + 504, + 181 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 136, + 179 + ], + "score": 0.89, + "content": "x ^ { * } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 168, + 141, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 171, + 504, + 179 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 124, + 506, + 181 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 197, + 303, + 211 + ], + "lines": [ + { + "bbox": [ + 104, + 196, + 304, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 304, + 213 + ], + "score": 1.0, + "content": "F Discussion on Other Architectures", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 223, + 261, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 263, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 263, + 236 + ], + "score": 1.0, + "content": "F.1 Encoder-Decoder Transformer", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "In this paper, we choose the autoregressive Transformer as it is simple and the de facto standard for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "LLMs (e.g., GPT). However, all of the theoretical results of this paper can be easily transferred to an", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 405, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 405, + 279 + ], + "score": 1.0, + "content": "encoder-decoder Transformer (such as T5) using the following arguments.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 244, + 505, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 127, + 282, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 127, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 127, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "• Given a bounded-depth polynomial-size log-precision encoder-decoder Transformer, its", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 134, + 293, + 448, + 307 + ], + "spans": [ + { + "bbox": [ + 134, + 293, + 311, + 307 + ], + "score": 1.0, + "content": "computation complexity is still bounded by", + "type": "text" + }, + { + "bbox": [ + 312, + 293, + 330, + 306 + ], + "score": 0.81, + "content": "{ \\mathsf { T C } } ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 293, + 448, + 307 + ], + "score": 1.0, + "content": ", so our negative results hold.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 127, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 127, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "• Any finite-depth autoregressive Transformer can be mimicked by a finite-depth encoder-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 135, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 135, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "decoder Transformer of roughly the same size, since causal masking for the input sequence", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 135, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 135, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "can be mimicked through the use of positional encoding and joint attention can be mimicked", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 135, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 135, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "by the integration of cross-attention and self-attention. Thus, all positive results in this paper", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 136, + 353, + 333, + 364 + ], + "spans": [ + { + "bbox": [ + 136, + 353, + 333, + 364 + ], + "score": 1.0, + "content": "are not exclusive to autoregressive Transformers.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 127, + 281, + 506, + 364 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 273, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 275, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 275, + 391 + ], + "score": 1.0, + "content": "F.2 Recurrent Neural Network(RNN)", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "Although the theorems in this paper can be naturally extended to other popular Transformer ar-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "chitectures, such as the encoder-decoder Transformer (e.g., T5), RNNs do not have this property.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 420, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 506, + 434 + ], + "score": 1.0, + "content": "Theorem F.1 shows that RNNs cannot generate the CoT sequence using the same format proposed in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "our paper for the arithmetic formula task and the linear equation task unless the hidden dimension of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 103, + 438, + 402, + 460 + ], + "spans": [ + { + "bbox": [ + 103, + 438, + 184, + 460 + ], + "score": 1.0, + "content": "the RNN is at least", + "type": "text" + }, + { + "bbox": [ + 185, + 442, + 219, + 457 + ], + "score": 0.93, + "content": "\\Omega ( { \\frac { n } { \\log n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 438, + 250, + 460 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 250, + 444, + 257, + 452 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 438, + 402, + 460 + ], + "score": 1.0, + "content": "is the length of the input sequence.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 103, + 398, + 506, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 231, + 473 + ], + "score": 1.0, + "content": "Theorem F.1. For any integer", + "type": "text" + }, + { + "bbox": [ + 231, + 462, + 238, + 471 + ], + "score": 0.48, + "content": "p _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 459, + 505, + 473 + ], + "score": 1.0, + "content": ", any log-precision RNN with constant layers can neither generate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 466, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 453, + 488 + ], + "score": 1.0, + "content": "the CoT solution for the problem Arithmetic(n, p) with hidden dimension of o( nlog n )", + "type": "text" + }, + { + "bbox": [ + 449, + 472, + 505, + 483 + ], + "score": 1.0, + "content": "nor generate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 103, + 483, + 454, + 505 + ], + "spans": [ + { + "bbox": [ + 103, + 483, + 272, + 505 + ], + "score": 1.0, + "content": "the CoT solution for the problem Eqution", + "type": "text" + }, + { + "bbox": [ + 273, + 487, + 296, + 499 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 483, + 415, + 505 + ], + "score": 1.0, + "content": "with the hidden dimension of", + "type": "text" + }, + { + "bbox": [ + 416, + 484, + 447, + 501 + ], + "score": 0.94, + "content": "O \\big ( \\frac { n ^ { \\overline { { 2 } } } } { \\log n } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 483, + 454, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 103, + 459, + 505, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "Proof. Given an RNN model, if the hidden embeddings in each layer of the last symbols of two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "different input sequences are the same, the two output sequences will also be exactly the same. When", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "the RNN finishes processing the input sequence, the input sequence has been compressed into a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 168, + 563 + ], + "score": 1.0, + "content": "hidden state of", + "type": "text" + }, + { + "bbox": [ + 168, + 550, + 215, + 562 + ], + "score": 0.92, + "content": "O ( D \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 549, + 263, + 563 + ], + "score": 1.0, + "content": "bits, where", + "type": "text" + }, + { + "bbox": [ + 263, + 550, + 273, + 560 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "is the hidden dimension and each scalar is represented by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 107, + 561, + 144, + 573 + ], + "score": 0.91, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 560, + 478, + 573 + ], + "score": 1.0, + "content": "bits (by definition of log-precision). Therefore, an RNN with a hidden dimension of", + "type": "text" + }, + { + "bbox": [ + 478, + 561, + 488, + 571 + ], + "score": 0.83, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 195, + 586 + ], + "score": 1.0, + "content": "only generate at most", + "type": "text" + }, + { + "bbox": [ + 195, + 571, + 215, + 582 + ], + "score": 0.89, + "content": "n 2 ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 570, + 479, + 586 + ], + "score": 1.0, + "content": "different output sequences. However, for the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 572, + 502, + 584 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 570, + 506, + 586 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 581, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 353, + 596 + ], + "score": 1.0, + "content": "the first step of the CoT needs to output a sequence of length", + "type": "text" + }, + { + "bbox": [ + 353, + 583, + 376, + 595 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 581, + 443, + 596 + ], + "score": 1.0, + "content": ", which contains", + "type": "text" + }, + { + "bbox": [ + 443, + 583, + 466, + 595 + ], + "score": 0.91, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 581, + 507, + 596 + ], + "score": 1.0, + "content": "numbers.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 592, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 217, + 609 + ], + "score": 1.0, + "content": "Therefore, there are at least", + "type": "text" + }, + { + "bbox": [ + 217, + 594, + 256, + 607 + ], + "score": 0.92, + "content": "O ( 2 ^ { O ( n ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 592, + 479, + 609 + ], + "score": 1.0, + "content": "different solution sequences for the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 479, + 595, + 503, + 607 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 592, + 507, + 609 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 605, + 507, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 218, + 623 + ], + "score": 1.0, + "content": "Similarly, there are at least", + "type": "text" + }, + { + "bbox": [ + 218, + 607, + 262, + 621 + ], + "score": 0.92, + "content": "O ( 2 ^ { O ( n ^ { 2 } ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 605, + 479, + 623 + ], + "score": 1.0, + "content": "different solution sequences for the problem Eqution", + "type": "text" + }, + { + "bbox": [ + 479, + 608, + 503, + 621 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 605, + 507, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 618, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 618, + 354, + 635 + ], + "score": 1.0, + "content": "Thus, by the Pigeon Hole Principle, to be able to generate all", + "type": "text" + }, + { + "bbox": [ + 354, + 619, + 378, + 631 + ], + "score": 0.9, + "content": "2 ^ { O ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 618, + 507, + 635 + ], + "score": 1.0, + "content": "different output sequences, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 629, + 508, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 629, + 248, + 649 + ], + "score": 1.0, + "content": "must have a hidden dimension of", + "type": "text" + }, + { + "bbox": [ + 248, + 631, + 308, + 646 + ], + "score": 0.94, + "content": "\\begin{array} { r } { D = \\Omega \\big ( \\frac { n } { \\log n } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 629, + 423, + 649 + ], + "score": 1.0, + "content": "for the problem Arithmetic", + "type": "text" + }, + { + "bbox": [ + 423, + 632, + 447, + 644 + ], + "score": 0.91, + "content": "( n , p )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 629, + 508, + 649 + ], + "score": 1.0, + "content": "and a hidden", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 644, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 644, + 161, + 667 + ], + "score": 1.0, + "content": "dimension of", + "type": "text" + }, + { + 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the model architecture between the RNN and the transformer, most of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "the theorems in this paper cannot be simply generalized to the RNN. Moreover, To the best of our", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "knowledge, there is very little work to investigate the chain of thought in RNN model, either from the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 711, + 246, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 246, + 723 + ], + "score": 1.0, + "content": "theoretical or the empirical aspect.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 250, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 68, + 252, + 90 + ], + "spans": [ + { + "bbox": [ + 104, + 68, + 252, + 90 + ], + "score": 1.0, + "content": "G Dynamic Programming", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 96, + 176, + 108 + ], + "lines": [ + { + "bbox": [ + 105, + 95, + 176, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 176, + 110 + ], + "score": 1.0, + "content": "G.1 Examples", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 115, + 506, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "Longest Increasing Subsequence (LIS). The LIS problem aims to compute the length of the longest", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 302, + 140 + ], + "score": 1.0, + "content": "increasing subsequence given an input sequence", + "type": "text" + }, + { + "bbox": [ + 302, + 128, + 333, + 138 + ], + "score": 0.9, + "content": "s \\in \\mathbb { N } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 127, + 378, + 140 + ], + "score": 1.0, + "content": ". Formally,", + "type": "text" + }, + { + "bbox": [ + 379, + 128, + 385, + 137 + ], + "score": 0.62, + "content": "\\tilde { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 127, + 466, + 140 + ], + "score": 1.0, + "content": "is a subsequence of", + "type": "text" + }, + { + "bbox": [ + 466, + 129, + 473, + 137 + ], + "score": 0.74, + "content": "\\pmb { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "if there", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 138, + 504, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 138, + 162, + 153 + ], + "score": 1.0, + "content": "exists indices", + "type": "text" + }, + { + "bbox": [ + 162, + 139, + 281, + 150 + ], + "score": 0.89, + "content": "1 \\leq i _ { 1 } \\leq i _ { 2 } \\leq \\cdot \\cdot \\cdot \\leq i _ { | \\tilde { s } | } \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 138, + 320, + 153 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 321, + 139, + 356, + 150 + ], + "score": 0.91, + "content": "\\tilde { s } _ { k } = s _ { i _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 138, + 409, + 153 + ], + "score": 1.0, + "content": "holds for all", + "type": "text" + }, + { + "bbox": [ + 409, + 138, + 444, + 150 + ], + "score": 0.92, + "content": "k \\in [ | \\tilde { s } | ]", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 138, + 497, + 153 + ], + "score": 1.0, + "content": ". A sequence", + "type": "text" + }, + { + "bbox": [ + 497, + 139, + 504, + 148 + ], + "score": 0.64, + "content": "\\tilde { s }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 148, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 192, + 163 + ], + "score": 1.0, + "content": "is called increasing if", + "type": "text" + }, + { + "bbox": [ + 192, + 150, + 277, + 161 + ], + "score": 0.85, + "content": "\\tilde { s } _ { 1 } < \\tilde { s } _ { 2 } < \\cdots < s _ { | \\tilde { s } | }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 148, + 506, + 163 + ], + "score": 1.0, + "content": ". The LIS problem aims to find an increasing subsequence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 117, + 174 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 162, + 124, + 170 + ], + "score": 0.63, + "content": "\\pmb { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 158, + 506, + 174 + ], + "score": 1.0, + "content": "with maximal length. A standard DP solution is to compute the length of the longest increasing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 262, + 183 + ], + "score": 1.0, + "content": "subsequence that ends at each position", + "type": "text" + }, + { + "bbox": [ + 263, + 172, + 267, + 181 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 171, + 352, + 183 + ], + "score": 1.0, + "content": ", which we denote as", + "type": "text" + }, + { + "bbox": [ + 352, + 171, + 374, + 183 + ], + "score": 0.88, + "content": "\\mathsf { d p } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 171, + 505, + 183 + ], + "score": 1.0, + "content": ". It is easy to write the transition", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 187, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 187, + 194 + ], + "score": 1.0, + "content": "function as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 192, + 364, + 212 + ], + "lines": [ + { + "bbox": [ + 246, + 192, + 364, + 212 + ], + "spans": [ + { + "bbox": [ + 246, + 192, + 364, + 212 + ], + "score": 0.94, + "content": "{ \\mathsf { d p } } ( i ) = 1 + \\operatorname* { m a x } _ { j < i , s _ { j } < s _ { i } } { \\mathsf { d p } } ( j ) .", + "type": "interline_equation", + "image_path": "a3b6bc8b8de26694c5b68b483750d4a669c7c2a683f1dbf538377c613fa5f418.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 246, + 192, + 364, + 212 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 268, + 229 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 266, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 205, + 231 + ], + "score": 1.0, + "content": "The final answer will be", + "type": "text" + }, + { + "bbox": [ + 205, + 217, + 266, + 230 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } _ { i \\in [ n ] } { \\mathsf { d p } } ( i )", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 419, + 246 + ], + "score": 1.0, + "content": "However, the above DP transition function does not match the form of (5), since", + "type": "text" + }, + { + "bbox": [ + 420, + 234, + 442, + 246 + ], + "score": 0.88, + "content": "\\mathsf { d p } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "may depend on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 260, + 257 + ], + "score": 1.0, + "content": "(an unbounded number of) all previous", + "type": "text" + }, + { + "bbox": [ + 261, + 245, + 314, + 257 + ], + "score": 0.71, + "content": "{ \\mathsf { d p } } ( j ) \\left( j < i \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 245, + 505, + 257 + ], + "score": 1.0, + "content": ". Nevertheless, this issue can be easily addressed", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 269, + 268 + ], + "score": 1.0, + "content": "by using a different DP formulation. Let", + "type": "text" + }, + { + "bbox": [ + 270, + 257, + 302, + 268 + ], + "score": 0.92, + "content": "\\mathsf { d p } ( j , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "be the longest increasing subsequence that ends at", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 141, + 280 + ], + "score": 1.0, + "content": "position", + "type": "text" + }, + { + "bbox": [ + 141, + 268, + 147, + 279 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 266, + 325, + 280 + ], + "score": 1.0, + "content": "and the second last position is no more than", + "type": "text" + }, + { + "bbox": [ + 326, + 267, + 333, + 277 + ], + "score": 0.61, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 267, + 364, + 279 + ], + "score": 0.65, + "content": "( k < j )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 266, + 505, + 280 + ], + "score": 1.0, + "content": ". In this case, it is easy to write the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 228, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 228, + 290 + ], + "score": 1.0, + "content": "transition function as follows:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 293, + 454, + 321 + ], + "lines": [ + { + "bbox": [ + 153, + 293, + 454, + 321 + ], + "spans": [ + { + "bbox": [ + 153, + 293, + 454, + 321 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { d } \\mathsf { p } ( j , k ) = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { i f ~ } k = 0 } \\\\ { \\operatorname* { m a x } ( \\mathrm { d } \\mathsf { p } ( j , k - 1 ) , \\mathrm { d } \\mathsf { p } ( k , k - 1 ) \\cdot \\mathbb { I } [ s _ { j } > s _ { k } ] + 1 ) } & { \\mathrm { i f ~ } k > 0 } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "3e1c896acfa57b0ef472db55413f4cbd85a0c31c33b2447215a012b110f4e36f.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 153, + 293, + 454, + 321 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 326, + 471, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 474, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 205, + 340 + ], + "score": 1.0, + "content": "The final answer will be", + "type": "text" + }, + { + "bbox": [ + 205, + 326, + 291, + 340 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { i \\in [ n ] } { \\mathsf { d } } { \\mathsf { p } } ( i , i - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 325, + 474, + 340 + ], + "score": 1.0, + "content": ". This DP formulation fits our framework (5).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "Edit Distance (ED). The ED problem aims to find the minimum operation cost required to convert", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 353, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 353, + 154, + 368 + ], + "score": 1.0, + "content": "a sequence", + "type": "text" + }, + { + "bbox": [ + 154, + 354, + 190, + 365 + ], + "score": 0.91, + "content": "\\pmb { u } \\in \\Sigma ^ { n _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 353, + 276, + 368 + ], + "score": 1.0, + "content": "to another sequence", + "type": "text" + }, + { + "bbox": [ + 276, + 354, + 312, + 364 + ], + "score": 0.91, + "content": "\\pmb { v } \\in \\Sigma ^ { n _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 353, + 505, + 368 + ], + "score": 1.0, + "content": ". There are three types of operations: inserting", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "a letter into any position, deleting a letter from any position, and replacing a letter at any position", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 335, + 388 + ], + "score": 1.0, + "content": "by a new one. The costs of insert, delete, and replace are", + "type": "text" + }, + { + "bbox": [ + 335, + 376, + 352, + 386 + ], + "score": 0.39, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 375, + 372, + 388 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 372, + 378, + 378, + 385 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 375, + 505, + 388 + ], + "score": 1.0, + "content": ", respectively. These operations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "are sequentially executed and the total operation cost is the summation of all costs of individual", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 154, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 154, + 410 + ], + "score": 1.0, + "content": "operations.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 411, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 411, + 457, + 429 + ], + "score": 1.0, + "content": "A standard DP solution is to compute the minimum operation cost to convert the substring", + "type": "text" + }, + { + "bbox": [ + 457, + 416, + 504, + 426 + ], + "score": 0.88, + "content": "u _ { 1 } u _ { 2 } \\cdot \\cdot \\cdot u _ { j }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 169, + 438 + ], + "score": 1.0, + "content": "to the substring", + "type": "text" + }, + { + "bbox": [ + 169, + 426, + 214, + 436 + ], + "score": 0.89, + "content": "v _ { 1 } v _ { 2 } \\cdots v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 424, + 297, + 438 + ], + "score": 1.0, + "content": ", which we denote as", + "type": "text" + }, + { + "bbox": [ + 297, + 425, + 330, + 437 + ], + "score": 0.92, + "content": "\\mathsf { d p } ( j , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 424, + 506, + 438 + ], + "score": 1.0, + "content": ". It is easy to write the transition function as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 142, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 142, + 448 + ], + "score": 1.0, + "content": "follows:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 450, + 458, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 458, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 458, + 492 + ], + "score": 0.91, + "content": "\\mathbf { d } \\mathsf { p } ( j , k ) = \\left\\{ \\begin{array} { l l } { a k } & { \\mathsf { d } \\mathsf { p } ( j , k ) = \\mathsf { d } , } \\\\ { b j } & { \\mathsf { d } \\mathsf { p } ( j , k - 1 ) + a , \\mathsf { d } \\mathsf { p } ( j - 1 , k ) + b , \\mathsf { d } \\mathsf { p } ( j - 1 , k - 1 ) + c \\mathbb { I } [ s _ { j } ^ { ( 1 ) } \\neq s _ { k } ^ { ( 2 ) } ] ) } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "ea1ea44162596c7c1266fd721828c4974405ee219b5df3ddfac7f2781847f340.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 105, + 450, + 458, + 464.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 105, + 464.0, + 458, + 478.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 105, + 478.0, + 458, + 492.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 432, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 432, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 205, + 513 + ], + "score": 1.0, + "content": "The final answer will be", + "type": "text" + }, + { + "bbox": [ + 205, + 499, + 249, + 511 + ], + "score": 0.91, + "content": "\\mathsf { d p } ( n _ { 1 } , n _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 497, + 432, + 513 + ], + "score": 1.0, + "content": ". 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A context-free grammar (CFG) is a 4-tuple, denoted as", + "type": "text" + }, + { + "bbox": [ + 480, + 516, + 505, + 527 + ], + "score": 0.85, + "content": "G \\ =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 187, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 156, + 539 + ], + "score": 0.91, + "content": "( \\mathcal { V } , \\Sigma , R , S )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 525, + 187, + 539 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 127, + 543, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 126, + 542, + 301, + 555 + ], + "spans": [ + { + "bbox": [ + 126, + 542, + 136, + 555 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 543, + 145, + 553 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 542, + 301, + 555 + ], + "score": 1.0, + "content": "is a finite set of non-terminal symbols.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 126, + 555, + 349, + 568 + ], + "spans": [ + { + "bbox": [ + 126, + 555, + 136, + 568 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 556, + 145, + 566 + ], + "score": 0.77, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 555, + 338, + 568 + ], + "score": 1.0, + "content": "is a finite set of terminal symbols, disjoint from", + "type": "text" + }, + { + "bbox": [ + 338, + 557, + 346, + 567 + ], + "score": 0.63, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 555, + 349, + 568 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 126, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 126, + 567, + 136, + 582 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 570, + 145, + 579 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 567, + 403, + 582 + ], + "score": 1.0, + "content": "is a finite set of production rules, where each rule has the form", + "type": "text" + }, + { + "bbox": [ + 403, + 570, + 434, + 581 + ], + "score": 0.91, + "content": "A \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 567, + 458, + 582 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 459, + 570, + 487, + 580 + ], + "score": 0.9, + "content": "A \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 136, + 579, + 398, + 593 + ], + "spans": [ + { + "bbox": [ + 136, + 580, + 194, + 592 + ], + "score": 0.91, + "content": "\\beta \\in ( V \\cup \\Sigma ) ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 579, + 398, + 593 + ], + "score": 1.0, + "content": "(the asterisk represents the Kleene star operation).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 127, + 593, + 242, + 605 + ], + "spans": [ + { + "bbox": [ + 127, + 593, + 136, + 605 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 594, + 164, + 604 + ], + "score": 0.87, + "content": "S \\in \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 593, + 242, + 605 + ], + "score": 1.0, + "content": "is the start symbol.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 223, + 623 + ], + "score": 1.0, + "content": "The non-terminal symbols in", + "type": "text" + }, + { + "bbox": [ + 224, + 611, + 232, + 621 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "represent (abstract) syntactic categories, while the terminal symbols", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 117, + 633 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 621, + 125, + 631 + ], + "score": 0.81, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 620, + 419, + 633 + ], + "score": 1.0, + "content": "represent the actual words/tokens of the language. 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Let", + "type": "text" + }, + { + "bbox": [ + 270, + 257, + 302, + 268 + ], + "score": 0.92, + "content": "\\mathsf { d p } ( j , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "be the longest increasing subsequence that ends at", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 141, + 280 + ], + "score": 1.0, + "content": "position", + "type": "text" + }, + { + "bbox": [ + 141, + 268, + 147, + 279 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 266, + 325, + 280 + ], + "score": 1.0, + "content": "and the second last position is no more than", + "type": "text" + }, + { + "bbox": [ + 326, + 267, + 333, + 277 + ], + "score": 0.61, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 267, + 364, + 279 + ], + "score": 0.65, + "content": "( k < j )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 266, + 505, + 280 + ], + "score": 1.0, + "content": ". In this case, it is easy to write the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 228, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 228, + 290 + ], + "score": 1.0, + "content": "transition function as follows:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 234, + 505, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 293, + 454, + 321 + ], + "lines": [ + { + "bbox": [ + 153, + 293, + 454, + 321 + ], + "spans": [ + { + "bbox": [ + 153, + 293, + 454, + 321 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { d } \\mathsf { p } ( j , k ) = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathrm { i f ~ } k = 0 } \\\\ { \\operatorname* { m a x } ( \\mathrm { d } \\mathsf { p } ( j , k - 1 ) , \\mathrm { d } \\mathsf { p } ( k , k - 1 ) \\cdot \\mathbb { I } [ s _ { j } > s _ { k } ] + 1 ) } & { \\mathrm { i f ~ } k > 0 } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "3e1c896acfa57b0ef472db55413f4cbd85a0c31c33b2447215a012b110f4e36f.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 153, + 293, + 454, + 321 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 326, + 471, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 474, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 205, + 340 + ], + "score": 1.0, + "content": "The final answer will be", + "type": "text" + }, + { + "bbox": [ + 205, + 326, + 291, + 340 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { i \\in [ n ] } { \\mathsf { d } } { \\mathsf { p } } ( i , i - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 325, + 474, + 340 + ], + "score": 1.0, + "content": ". This DP formulation fits our framework (5).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 325, + 474, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "Edit Distance (ED). The ED problem aims to find the minimum operation cost required to convert", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 353, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 353, + 154, + 368 + ], + "score": 1.0, + "content": "a sequence", + "type": "text" + }, + { + "bbox": [ + 154, + 354, + 190, + 365 + ], + "score": 0.91, + "content": "\\pmb { u } \\in \\Sigma ^ { n _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 353, + 276, + 368 + ], + "score": 1.0, + "content": "to another sequence", + "type": "text" + }, + { + "bbox": [ + 276, + 354, + 312, + 364 + ], + "score": 0.91, + "content": "\\pmb { v } \\in \\Sigma ^ { n _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 353, + 505, + 368 + ], + "score": 1.0, + "content": ". There are three types of operations: inserting", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "a letter into any position, deleting a letter from any position, and replacing a letter at any position", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 335, + 388 + ], + "score": 1.0, + "content": "by a new one. The costs of insert, delete, and replace are", + "type": "text" + }, + { + "bbox": [ + 335, + 376, + 352, + 386 + ], + "score": 0.39, + "content": "a , b", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 375, + 372, + 388 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 372, + 378, + 378, + 385 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 375, + 505, + 388 + ], + "score": 1.0, + "content": ", respectively. These operations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "are sequentially executed and the total operation cost is the summation of all costs of individual", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 154, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 154, + 410 + ], + "score": 1.0, + "content": "operations.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 343, + 506, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 411, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 411, + 457, + 429 + ], + "score": 1.0, + "content": "A standard DP solution is to compute the minimum operation cost to convert the substring", + "type": "text" + }, + { + "bbox": [ + 457, + 416, + 504, + 426 + ], + "score": 0.88, + "content": "u _ { 1 } u _ { 2 } \\cdot \\cdot \\cdot u _ { j }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 169, + 438 + ], + "score": 1.0, + "content": "to the substring", + "type": "text" + }, + { + "bbox": [ + 169, + 426, + 214, + 436 + ], + "score": 0.89, + "content": "v _ { 1 } v _ { 2 } \\cdots v _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 424, + 297, + 438 + ], + "score": 1.0, + "content": ", which we denote as", + "type": "text" + }, + { + "bbox": [ + 297, + 425, + 330, + 437 + ], + "score": 0.92, + "content": "\\mathsf { d p } ( j , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 424, + 506, + 438 + ], + "score": 1.0, + "content": ". It is easy to write the transition function as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 434, + 142, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 142, + 448 + ], + "score": 1.0, + "content": "follows:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 411, + 506, + 448 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 450, + 458, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 458, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 458, + 492 + ], + "score": 0.91, + "content": "\\mathbf { d } \\mathsf { p } ( j , k ) = \\left\\{ \\begin{array} { l l } { a k } & { \\mathsf { d } \\mathsf { p } ( j , k ) = \\mathsf { d } , } \\\\ { b j } & { \\mathsf { d } \\mathsf { p } ( j , k - 1 ) + a , \\mathsf { d } \\mathsf { p } ( j - 1 , k ) + b , \\mathsf { d } \\mathsf { p } ( j - 1 , k - 1 ) + c \\mathbb { I } [ s _ { j } ^ { ( 1 ) } \\neq s _ { k } ^ { ( 2 ) } ] ) } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "ea1ea44162596c7c1266fd721828c4974405ee219b5df3ddfac7f2781847f340.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 105, + 450, + 458, + 464.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 105, + 464.0, + 458, + 478.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 105, + 478.0, + 458, + 492.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 432, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 432, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 205, + 513 + ], + "score": 1.0, + "content": "The final answer will be", + "type": "text" + }, + { + "bbox": [ + 205, + 499, + 249, + 511 + ], + "score": 0.91, + "content": "\\mathsf { d p } ( n _ { 1 } , n _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 497, + 432, + 513 + ], + "score": 1.0, + "content": ". This DP formulation fits our framework (5).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 497, + 432, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 515, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 480, + 529 + ], + "score": 1.0, + "content": "CFG Membership Testing. A context-free grammar (CFG) is a 4-tuple, denoted as", + "type": "text" + }, + { + "bbox": [ + 480, + 516, + 505, + 527 + ], + "score": 0.85, + "content": "G \\ =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 187, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 156, + 539 + ], + "score": 0.91, + "content": "( \\mathcal { V } , \\Sigma , R , S )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 525, + 187, + 539 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 514, + 505, + 539 + ] + }, + { + "type": "list", + "bbox": [ + 127, + 543, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 126, + 542, + 301, + 555 + ], + "spans": [ + { + "bbox": [ + 126, + 542, + 136, + 555 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 543, + 145, + 553 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 542, + 301, + 555 + ], + "score": 1.0, + "content": "is a finite set of non-terminal symbols.", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 555, + 349, + 568 + ], + "spans": [ + { + "bbox": [ + 126, + 555, + 136, + 568 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 556, + 145, + 566 + ], + "score": 0.77, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 555, + 338, + 568 + ], + "score": 1.0, + "content": "is a finite set of terminal symbols, disjoint from", + "type": "text" + }, + { + "bbox": [ + 338, + 557, + 346, + 567 + ], + "score": 0.63, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 555, + 349, + 568 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 126, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 126, + 567, + 136, + 582 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 570, + 145, + 579 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 567, + 403, + 582 + ], + "score": 1.0, + "content": "is a finite set of production rules, where each rule has the form", + "type": "text" + }, + { + "bbox": [ + 403, + 570, + 434, + 581 + ], + "score": 0.91, + "content": "A \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 567, + 458, + 582 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 459, + 570, + 487, + 580 + ], + "score": 0.9, + "content": "A \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 579, + 398, + 593 + ], + "spans": [ + { + "bbox": [ + 136, + 580, + 194, + 592 + ], + "score": 0.91, + "content": "\\beta \\in ( V \\cup \\Sigma ) ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 579, + 398, + 593 + ], + "score": 1.0, + "content": "(the asterisk represents the Kleene star operation).", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 127, + 593, + 242, + 605 + ], + "spans": [ + { + "bbox": [ + 127, + 593, + 136, + 605 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 136, + 594, + 164, + 604 + ], + "score": 0.87, + "content": "S \\in \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 593, + 242, + 605 + ], + "score": 1.0, + "content": "is the start symbol.", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 35, + "bbox_fs": [ + 126, + 542, + 506, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 223, + 623 + ], + "score": 1.0, + "content": "The non-terminal symbols in", + "type": "text" + }, + { + "bbox": [ + 224, + 611, + 232, + 621 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "represent (abstract) syntactic categories, while the terminal symbols", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 117, + 633 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 621, + 125, + 631 + ], + "score": 0.81, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 620, + 419, + 633 + ], + "score": 1.0, + "content": "represent the actual words/tokens of the language. The production rules in", + "type": "text" + }, + { + "bbox": [ + 420, + 622, + 429, + 631 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "specify how a non-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "terminal symbol can be replaced by sequences of terminal and non-terminal symbols concatenated", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 437, + 656 + ], + "score": 1.0, + "content": "together. 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This problem is known to be", + "type": "text" + }, + { + "bbox": [ + 456, + 84, + 464, + 94 + ], + "score": 0.36, + "content": "\\mathsf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "-complete", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "[33]. The CYK algorithm [53], which is based on DP, is a classic algorithm to solve the CFG", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "Membership Testing problem. 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Denote", + "type": "text" + }, + { + "bbox": [ + 244, + 116, + 328, + 128 + ], + "score": 0.94, + "content": "R = \\{ R _ { 1 } , \\cdots , R _ { m } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 116, + 360, + 129 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 360, + 117, + 393, + 128 + ], + "score": 0.93, + "content": "n = | \\pmb { v } |", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 116, + 506, + 129 + ], + "score": 1.0, + "content": ". 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This DP formulation fits our framework 5.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 504, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "Regarding Remark 4.6. It can be easily verified that the state spaces of the three problems mentioned", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "above are of polynomial size, satisfying Assumption 4.2. 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According to Lemma C.2, these functions can be efficiently", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "approximated by a perceptron of constant size with GeLU activation. 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Consider any", + "type": "text" + }, + { + "bbox": [ + 225, + 261, + 240, + 271 + ], + "score": 0.25, + "content": "D P", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 259, + 476, + 273 + ], + "score": 1.0, + "content": "problem satisfying Assumptions 4.2 to 4.5. 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Assume that we have a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 366, + 318, + 407, + 327 + ], + "score": 0.89, + "content": "s _ { 1 } , \\cdots , s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "and we want to generate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 326, + 297, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 165, + 340 + ], + "score": 1.0, + "content": "the next token", + "type": "text" + }, + { + "bbox": [ + 165, + 328, + 185, + 339 + ], + "score": 0.88, + "content": "s _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 326, + 272, + 340 + ], + "score": 1.0, + "content": ". We embed the token", + "type": "text" + }, + { + "bbox": [ + 273, + 329, + 283, + 338 + ], + "score": 0.86, + "content": "s _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 326, + 297, + 340 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 343, + 392, + 360 + ], + "lines": [ + { + "bbox": [ + 218, + 343, + 392, + 360 + ], + "spans": [ + { + "bbox": [ + 218, + 343, + 392, + 360 + ], + "score": 0.92, + "content": "\\pmb { x } _ { k } ^ { ( 0 ) } = ( \\mathbf { e } _ { k } ^ { \\mathrm { i n p u t } } , \\mathbf { e } _ { k } ^ { \\mathrm { s t a t e } } , \\mathbf { e } _ { k } ^ { \\mathrm { d p } } , \\mathbf { e } _ { k } ^ { \\mathrm { a n s w e r } } , \\mathbf { e } _ { k } ^ { \\mathrm { s e p } } , k , 1 ) ,", + "type": "interline_equation", + "image_path": "c380f14adceff3d99848b002eef9693c7214ca5a178b852126510486e653f1ad.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 218, + 343, + 392, + 360 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 330, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 364, + 330, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 330, + 377 + ], + "score": 1.0, + "content": "where each part of the embedding is defined as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 123, + 382, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 123, + 379, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 123, + 382, + 136, + 394 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 136, + 381, + 158, + 396 + ], + "score": 0.89, + "content": "e _ { k } ^ { \\mathrm { i n p u t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 379, + 304, + 396 + ], + "score": 1.0, + "content": "is the embedding of the input token", + "type": "text" + }, + { + "bbox": [ + 304, + 383, + 335, + 394 + ], + "score": 0.91, + "content": "s _ { k } \\in \\mathcal X", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 379, + 505, + 396 + ], + "score": 1.0, + "content": ". If the current position does not represent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 134, + 393, + 263, + 412 + ], + "spans": [ + { + "bbox": [ + 134, + 393, + 218, + 412 + ], + "score": 1.0, + "content": "an input token, then", + "type": "text" + }, + { + "bbox": [ + 218, + 394, + 258, + 409 + ], + "score": 0.92, + "content": "e _ { k } ^ { \\mathrm { i n p u t } } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 393, + 263, + 412 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 123, + 407, + 509, + 431 + ], + "spans": [ + { + "bbox": [ + 123, + 407, + 136, + 431 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 136, + 412, + 156, + 425 + ], + "score": 0.88, + "content": "e _ { k } ^ { \\mathrm { s t a t e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 407, + 300, + 431 + ], + "score": 1.0, + "content": "is the embedding of the DP state in", + "type": "text" + }, + { + "bbox": [ + 300, + 413, + 308, + 423 + ], + "score": 0.82, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 407, + 353, + 431 + ], + "score": 1.0, + "content": "at position", + "type": "text" + }, + { + "bbox": [ + 354, + 413, + 360, + 423 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 407, + 509, + 431 + ], + "score": 1.0, + "content": ". If the current position corresponds", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 132, + 419, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 132, + 419, + 307, + 442 + ], + "score": 1.0, + "content": "kto an input token or the final answer, then", + "type": "text" + }, + { + "bbox": [ + 308, + 424, + 347, + 436 + ], + "score": 0.92, + "content": "e _ { k } ^ { \\mathrm { s t a t e } } = { \\bf 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 419, + 463, + 442 + ], + "score": 1.0, + "content": ". We also assume that for all", + "type": "text" + }, + { + "bbox": [ + 487, + 419, + 505, + 442 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 136, + 434, + 270, + 447 + ], + "spans": [ + { + "bbox": [ + 136, + 434, + 214, + 447 + ], + "score": 1.0, + "content": "embedding of state", + "type": "text" + }, + { + "bbox": [ + 219, + 434, + 270, + 447 + ], + "score": 1.0, + "content": "is non-zero.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 123, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 123, + 452, + 136, + 465 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 136, + 451, + 151, + 465 + ], + "score": 0.9, + "content": "e _ { k } ^ { \\mathrm { d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 451, + 294, + 465 + ], + "score": 1.0, + "content": "is the embedding of the DP value in", + "type": "text" + }, + { + "bbox": [ + 294, + 453, + 303, + 464 + ], + "score": 0.85, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 451, + 347, + 465 + ], + "score": 1.0, + "content": "at position", + "type": "text" + }, + { + "bbox": [ + 347, + 453, + 354, + 463 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 451, + 505, + 465 + ], + "score": 1.0, + "content": ". If the current position corresponds to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 131, + 459, + 331, + 486 + ], + "spans": [ + { + "bbox": [ + 131, + 459, + 294, + 486 + ], + "score": 1.0, + "content": "an input token or the final answer, then", + "type": "text" + }, + { + "bbox": [ + 294, + 465, + 327, + 479 + ], + "score": 0.94, + "content": "e _ { k } ^ { \\mathrm { d p } } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 459, + 331, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 123, + 475, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 123, + 483, + 135, + 494 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 131, + 475, + 398, + 500 + ], + "score": 1.0, + "content": ". eanswerk is the embedding of the answer token in Z, and eanswerk", + "type": "text" + }, + { + "bbox": [ + 412, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "if the current position", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 135, + 494, + 384, + 506 + ], + "spans": [ + { + "bbox": [ + 135, + 494, + 384, + 506 + ], + "score": 1.0, + "content": "corresponds to an input token or an intermediate DP position.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 123, + 505, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 123, + 509, + 135, + 522 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 130, + 508, + 160, + 523 + ], + "score": 1.0, + "content": ". esep", + "type": "text" + }, + { + "bbox": [ + 153, + 505, + 464, + 527 + ], + "score": 1.0, + "content": "is the embedding of the separator | separating different input sequences. We set", + "type": "text" + }, + { + "bbox": [ + 464, + 509, + 504, + 522 + ], + "score": 0.92, + "content": "e _ { k } ^ { \\mathrm { s e p } } = e _ { j }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 135, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 135, + 520, + 219, + 534 + ], + "score": 1.0, + "content": "if the current token", + "type": "text" + }, + { + "bbox": [ + 219, + 523, + 230, + 532 + ], + "score": 0.86, + "content": "s _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 520, + 258, + 534 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 258, + 522, + 264, + 532 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 520, + 348, + 534 + ], + "score": 1.0, + "content": "-th separator, where", + "type": "text" + }, + { + "bbox": [ + 348, + 523, + 358, + 533 + ], + "score": 0.86, + "content": "e _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 520, + 486, + 534 + ], + "score": 1.0, + "content": "is the one-hot vector with the", + "type": "text" + }, + { + "bbox": [ + 487, + 523, + 492, + 533 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "-th", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 136, + 533, + 204, + 544 + ], + "spans": [ + { + "bbox": [ + 136, + 533, + 204, + 544 + ], + "score": 1.0, + "content": "element begin 1.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 123, + 548, + 437, + 561 + ], + "spans": [ + { + "bbox": [ + 123, + 548, + 235, + 561 + ], + "score": 1.0, + "content": "6. The position embedding", + "type": "text" + }, + { + "bbox": [ + 235, + 549, + 242, + 558 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 548, + 437, + 561 + ], + "score": 1.0, + "content": "indicates the index of the token in the sequence.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 493, + 577 + ], + "score": 1.0, + "content": "Block 1. The first block of the autoregressive Transformer contains several layers. It first uses", + "type": "text" + }, + { + "bbox": [ + 494, + 565, + 504, + 575 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 576, + 290, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 290, + 588 + ], + "score": 1.0, + "content": "attention heads to perform the following task:", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 126, + 590, + 438, + 609 + ], + "spans": [ + { + "bbox": [ + 126, + 591, + 289, + 609 + ], + "score": 1.0, + "content": "• Copy the positional embedding of the", + "type": "text" + }, + { + "bbox": [ + 289, + 594, + 299, + 604 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 591, + 343, + 609 + ], + "score": 1.0, + "content": "separators", + "type": "text" + }, + { + "bbox": [ + 344, + 592, + 434, + 607 + ], + "score": 0.9, + "content": "p _ { k } ^ { \\mathrm { s e p } , 1 } , \\cdot \\cdot \\cdot , p _ { k } ^ { \\mathrm { s e p } , N } \\in \\mathbb { N } .", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 590, + 438, + 607 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 503, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 442, + 622 + ], + "score": 1.0, + "content": "Similar to the previous proofs, this can be achieved via the COPY operation with", + "type": "text" + }, + { + "bbox": [ + 443, + 610, + 506, + 622 + ], + "score": 0.91, + "content": "S _ { k } = \\{ j \\leq k :", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 619, + 503, + 635 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 150, + 635 + ], + "score": 0.92, + "content": "e _ { j } ^ { \\mathrm { s e p } } = e _ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 619, + 165, + 635 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 166, + 621, + 197, + 633 + ], + "score": 0.92, + "content": "t \\in [ N ]", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 619, + 215, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 622, + 244, + 634 + ], + "score": 0.92, + "content": "v _ { j } = j", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 619, + 503, + 635 + ], + "score": 1.0, + "content": ". Then, several MLPs follow, which perform the following tasks:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 128, + 638, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 127, + 638, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 127, + 640, + 242, + 654 + ], + "score": 1.0, + "content": "• Calculate the problem size", + "type": "text" + }, + { + "bbox": [ + 243, + 638, + 505, + 655 + ], + "score": 0.82, + "content": "\\pmb { n } _ { k } = ( p _ { k } ^ { \\mathrm { s e p } , 1 } - 1 , p _ { k } ^ { \\mathrm { s e p } , 2 } - p _ { k } ^ { \\mathrm { s e p } , 1 } - 1 , \\cdots , p _ { k } ^ { \\mathrm { s e p } , N } - p _ { k } ^ { \\mathrm { s e p } , N - 1 } - 1 ) .", + "type": "inline_equation", + "image_path": "2ef07f968619649028fe26ed40212208c7a1fd8fb295626a2cc578bf7701012a.jpg" + } + ], + "index": 36 + }, + { + "bbox": [ + 122, + 651, + 504, + 680 + ], + "spans": [ + { + "bbox": [ + 122, + 651, + 221, + 680 + ], + "score": 1.0, + "content": "• Obtain the next state", + "type": "text" + }, + { + "bbox": [ + 221, + 659, + 256, + 672 + ], + "score": 0.9, + "content": "e _ { k } ^ { \\mathrm { n e x t \\_ s t a t e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 651, + 444, + 680 + ], + "score": 1.0, + "content": ". 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Consider any", + "type": "text" + }, + { + "bbox": [ + 225, + 261, + 240, + 271 + ], + "score": 0.25, + "content": "D P", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 259, + 476, + 273 + ], + "score": 1.0, + "content": "problem satisfying Assumptions 4.2 to 4.5. 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Assume that we have a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 366, + 318, + 407, + 327 + ], + "score": 0.89, + "content": "s _ { 1 } , \\cdots , s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "and we want to generate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 326, + 297, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 165, + 340 + ], + "score": 1.0, + "content": "the next token", + "type": "text" + }, + { + "bbox": [ + 165, + 328, + 185, + 339 + ], + "score": 0.88, + "content": "s _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 326, + 272, + 340 + ], + "score": 1.0, + "content": ". 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If the current position does not represent", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 134, + 393, + 263, + 412 + ], + "spans": [ + { + "bbox": [ + 134, + 393, + 218, + 412 + ], + "score": 1.0, + "content": "an input token, then", + "type": "text" + }, + { + "bbox": [ + 218, + 394, + 258, + 409 + ], + "score": 0.92, + "content": "e _ { k } ^ { \\mathrm { i n p u t } } = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 393, + 263, + 412 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 123, + 407, + 509, + 431 + ], + "spans": [ + { + "bbox": [ + 123, + 407, + 136, + 431 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 136, + 412, + 156, + 425 + ], + "score": 0.88, + "content": "e _ { k } ^ { \\mathrm { s t a t e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 407, + 300, + 431 + ], + "score": 1.0, + "content": "is the embedding of the DP state in", + "type": "text" + }, + { + "bbox": [ + 300, + 413, + 308, + 423 + ], + "score": 0.82, + "content": "\\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 407, + 353, + 431 + ], + "score": 1.0, + "content": "at position", + "type": "text" + }, + { + "bbox": [ + 354, + 413, + 360, + 423 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 407, + 509, + 431 + ], + "score": 1.0, + "content": ". 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This block performs the following tasks:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 127, + 539, + 443, + 552 + ], + "lines": [ + { + "bbox": [ + 127, + 539, + 444, + 553 + ], + "spans": [ + { + "bbox": [ + 127, + 539, + 444, + 553 + ], + "score": 1.0, + "content": "• Aggregate the DP values according to the aggregation function Equation (6).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "For the first task, if the aggregation function is max or min, we use one attention head to simply", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 581, + 508, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 188, + 601 + ], + "score": 1.0, + "content": "copy the embedding", + "type": "text" + }, + { + "bbox": [ + 188, + 583, + 202, + 599 + ], + "score": 0.92, + "content": "e _ { j } ^ { \\mathrm { d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 581, + 241, + 601 + ], + "score": 1.0, + "content": "for index", + "type": "text" + }, + { + "bbox": [ + 241, + 586, + 248, + 597 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 581, + 286, + 601 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 286, + 585, + 332, + 599 + ], + "score": 0.92, + "content": "f _ { j } ^ { \\mathrm { a n s w e r } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 581, + 349, + 601 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 583, + 364, + 599 + ], + "score": 0.94, + "content": "e _ { j } ^ { \\mathrm { d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 581, + 508, + 601 + ], + "score": 1.0, + "content": "is the largest/smallest, according to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 276, + 610 + ], + "score": 1.0, + "content": "Lemma C.7. 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Finally, the second task is a conditional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 633, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 410, + 647 + ], + "score": 1.0, + "content": "selection operation and thus can be implemented by an MLP (Lemma C.4).", + "type": "text" + }, + { + "bbox": [ + 495, + 635, + 504, + 644 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 107, + 657, + 240, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 241, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 241, + 670 + ], + "score": 1.0, + "content": "G.3 Proof of the Theorem 4.8", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 203, + 691 + ], + "score": 1.0, + "content": "Theorem G.2. Assume", + "type": "text" + }, + { + "bbox": [ + 203, + 677, + 252, + 690 + ], + "score": 0.91, + "content": "{ \\sf T C } ^ { 0 } \\neq { \\sf N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 675, + 496, + 691 + ], + "score": 1.0, + "content": ". 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If the aggregation function is", + "type": "text" + }, + { + "bbox": [ + 276, + 598, + 288, + 610 + ], + "score": 0.86, + "content": "\\displaystyle \\sum", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 598, + 505, + 610 + ], + "score": 1.0, + "content": ", we use two attention heads, where one attention head", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 608, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 198, + 624 + ], + "score": 1.0, + "content": "computes the mean of", + "type": "text" + }, + { + "bbox": [ + 198, + 609, + 212, + 625 + ], + "score": 0.91, + "content": "e _ { j } ^ { \\mathrm { d p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 608, + 252, + 624 + ], + "score": 1.0, + "content": "for index", + "type": "text" + }, + { + "bbox": [ + 253, + 612, + 259, + 623 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 608, + 299, + 624 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 299, + 611, + 344, + 624 + ], + "score": 0.88, + "content": "f _ { j } ^ { \\mathrm { a n s w e r } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 608, + 506, + 624 + ], + "score": 1.0, + "content": ", and the other attention head calculates", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 300, + 636 + ], + "score": 1.0, + "content": "the fraction of elements in the sequence such that", + "type": "text" + }, + { + "bbox": [ + 300, + 624, + 345, + 636 + ], + "score": 0.82, + "content": "f _ { j } ^ { \\mathrm { a n s w e r } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 620, + 506, + 636 + ], + "score": 1.0, + "content": ". Finally, the second task is a conditional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 633, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 410, + 647 + ], + "score": 1.0, + "content": "selection operation and thus can be implemented by an MLP (Lemma C.4).", + "type": "text" + }, + { + "bbox": [ + 495, + 635, + 504, + 644 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 572, + 508, + 647 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 657, + 240, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 241, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 241, + 670 + ], + "score": 1.0, + "content": "G.3 Proof of the Theorem 4.8", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 675, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 203, + 691 + ], + "score": 1.0, + "content": "Theorem G.2. Assume", + "type": "text" + }, + { + "bbox": [ + 203, + 677, + 252, + 690 + ], + "score": 0.91, + "content": "{ \\sf T C } ^ { 0 } \\neq { \\sf N C } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 675, + 496, + 691 + ], + "score": 1.0, + "content": ". There exists a context-free language such that for any depth", + "type": "text" + }, + { + "bbox": [ + 496, + 678, + 504, + 688 + ], + "score": 0.5, + "content": "L", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 188, + 702 + ], + "score": 1.0, + "content": "and any polynomial", + "type": "text" + }, + { + "bbox": [ + 189, + 689, + 197, + 700 + ], + "score": 0.79, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 688, + 322, + 702 + ], + "score": 1.0, + "content": ", there exists a sequence length", + "type": "text" + }, + { + "bbox": [ + 323, + 689, + 349, + 699 + ], + "score": 0.89, + "content": "n \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "where no log-precision autoregressive", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 201, + 713 + ], + "score": 1.0, + "content": "transformer with depth", + "type": "text" + }, + { + "bbox": [ + 201, + 700, + 209, + 710 + ], + "score": 0.57, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 700, + 302, + 713 + ], + "score": 1.0, + "content": "and hidden dimension", + "type": "text" + }, + { + "bbox": [ + 302, + 700, + 343, + 712 + ], + "score": 0.93, + "content": "d \\leq Q ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "can generate the correct answer for the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 375, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 364, + 724 + ], + "score": 1.0, + "content": "CFG Membership Testing problem for all input strings of length", + "type": "text" + }, + { + "bbox": [ + 365, + 713, + 371, + 721 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 710, + 375, + 724 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 675, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "Proof. According to Barrington’s theorem [4, 5], given a fixed unsolvable automaton, judging whether", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 288, + 98 + ], + "score": 1.0, + "content": "the automaton accepts an input is complete in", + "type": "text" + }, + { + "bbox": [ + 288, + 83, + 307, + 95 + ], + "score": 0.85, + "content": "{ \\mathsf { N C } } ^ { \\bar { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 83, + 506, + 98 + ], + "score": 1.0, + "content": ", which is a special case for the CFG Membership", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 96, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 378, + 109 + ], + "score": 1.0, + "content": "Testing problem. Therefore, the CFG Membership Testing problem is", + "type": "text" + }, + { + "bbox": [ + 379, + 96, + 398, + 108 + ], + "score": 0.88, + "content": "{ \\mathsf { N C } } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 96, + 505, + 109 + ], + "score": 1.0, + "content": "-hard. 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The constructions of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 265, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 265, + 272 + ], + "score": 1.0, + "content": "different datasets are introduced below.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 216, + 507, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "Arithmetic Expression. All arithmetic expression problems are generated according to Algorithm", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "1. In Algorithm 1, we first create a number that serves as the answer to the problem. We then", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "decompose the number using sampled operators sequentially, serving as the problem, until the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "maximum number of operators is met. The CoT procedure is precisely defined by reversing this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 328, + 437, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 437, + 340 + ], + "score": 1.0, + "content": "problem generation process. 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The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "CoT solution to the problem is the DP array plus the final answer, which is defined in (20). Here,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "we consider the DP formulation (20) because the CoT output length is much shorter than the one", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "corresponding to formulation (21). 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The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "CoT solution to the problem is the DP array plus the final answer, which is defined in (20). Here,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "we consider the DP formulation (20) because the CoT output length is much shorter than the one", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "corresponding to formulation (21). 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Algorithm4:EDData Generation
Input :Length of the First String n Input :Alphabet V = {a,b...z}
Output :Sequence S1, S2
Sample t uniformly from {3,4...10} ;
2 T ← Sample t letters from V ;
3 S1 ← Sample n letters uniformly from T ;
4 Sample p uniformly from [0,1] ;
5 if p<0.4 then
6Sample l uniformly from {n- 3,n- 2,,n + 2};
7S2 ← Sample l letters uniformly from T ;
8 else
9do
10 11S2↑S1；
12fori←1 to n do
13Sample p uniformly from {0,1...len(s2) -1};
14Sample l uniformly from T; Randomly conduct one of the followings: pop s2[p],substitute s2[p] with l, insert l
into s2[p];
15 16end
17while len(s2) not in [n -3,n + 2];
end
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Algorithm4:EDData Generation
Input :Length of the First String n Input :Alphabet V = {a,b...z}
Output :Sequence S1, S2
Sample t uniformly from {3,4...10} ;
2 T ← Sample t letters from V ;
3 S1 ← Sample n letters uniformly from T ;
4 Sample p uniformly from [0,1] ;
5 if p<0.4 then
6Sample l uniformly from {n- 3,n- 2,,n + 2};
7S2 ← Sample l letters uniformly from T ;
8 else
9do
10 11S2↑S1；
12fori←1 to n do
13Sample p uniformly from {0,1...len(s2) -1};
14Sample l uniformly from T; Randomly conduct one of the followings: pop s2[p],substitute s2[p] with l, insert l
into s2[p];
15 16end
17while len(s2) not in [n -3,n + 2];
end
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The total number of training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 190, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 190, + 352 + ], + "score": 1.0, + "content": "epochs is set to 100.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 264, + 506, + 352 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 367, + 311, + 381 + ], + "lines": [ + { + "bbox": [ + 104, + 365, + 312, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 312, + 384 + ], + "score": 1.0, + "content": "I Robustness in Training Data Quality", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "In the training of language models, the quality of data, especially intermediate steps, is crucial", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "for optimal performance. However, real-world training datasets may contain corrupted or missing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "intermediate steps. This highlights the importance of evaluating model resilience under imperfect", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "conditions. To investigate this, we conducted a series of experiments focused on arithmetic tasks,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 507, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 441, + 449 + ], + "score": 1.0, + "content": "introducing varying rates of corruption and omission, encapsulated by the parameter", + "type": "text" + }, + { + "bbox": [ + 442, + 438, + 449, + 447 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 435, + 507, + 449 + ], + "score": 1.0, + "content": ". For instance,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 130, + 460 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 131, + 447, + 163, + 459 + ], + "score": 0.91, + "content": "\\gamma = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 446, + 227, + 460 + ], + "score": 1.0, + "content": ", it denotes that", + "type": "text" + }, + { + "bbox": [ + 227, + 447, + 246, + 457 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 446, + 415, + 460 + ], + "score": 1.0, + "content": "of the intermediate steps are omitted and", + "type": "text" + }, + { + "bbox": [ + 415, + 447, + 434, + 457 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "of the remaining", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "steps are corrupted through a single-token random replacement. Our motivation for introducing this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 480 + ], + "score": 1.0, + "content": "metric is to simulate potential imperfections in real-world datasets, thus assessing the robustness of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 456, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 456, + 491 + ], + "score": 1.0, + "content": "our model more thoroughly. The number of operators in the arithmetic task is set to 10.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 392, + 507, + 491 + ] + }, + { + "type": "table", + "bbox": [ + 199, + 519, + 413, + 544 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 216, + 508, + 394, + 519 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 216, + 507, + 394, + 521 + ], + "spans": [ + { + "bbox": [ + 216, + 507, + 394, + 521 + ], + "score": 1.0, + "content": "Table 1: Experimental results of robustness.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "table_body", + "bbox": [ + 199, + 519, + 413, + 544 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 199, + 519, + 413, + 544 + ], + "spans": [ + { + "bbox": [ + 199, + 519, + 413, + 544 + ], + "score": 0.966, + "html": "
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As the rate of corruption and omission increases,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "there is a slight decrease in accuracy. These results elucidate the inherent robustness of training CoT", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 601, + 504, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 504, + 613 + ], + "score": 1.0, + "content": "demonstrations. It is noteworthy that the accuracy of the model remains commendably high despite", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "significant distortions present in the training data, highlighting its potential utility in real-world", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 623, + 313, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 313, + 635 + ], + "score": 1.0, + "content": "scenarios where data quality may be compromised.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 578, + 506, + 635 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/dev/qHrADgAdYu/qHrADgAdYu_model.json b/parse/dev/qHrADgAdYu/qHrADgAdYu_model.json new file mode 100644 index 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ProblemLongest increasing subsequenceEdit distance
InputA string s of length nTwo strings g(1), g(2) of length n1 = |s(1)|and n2 = |s(2)|, concatenated together
State space{(j,k):j∈[n],k ∈{0,..,j-1}}{0,..·,n1} × {0,..,n2}
Transitionfunction1 ifk=0dp(j,k)= max(dp(j,k-1),ifk>0dp(k,k-1)×I[sj>Sk]+1)dp(j,k)=ak if j=0bj ifk=0min(dp(j,k-1) +a,dp(j-1,k)+b,dp(j-1,k-1) otherwise+[1)≠s2)]）
Aggregationfunctionmaxi∈[n] dp(i,i-1)dp(n1,n2)
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Input :Number of Operators n Input :Vocabulary of numbers V = {0,1...1O}// number field p = 11 Output:Arithmetic expression s
1 Sample the first number t uniformly from V; 2 s=[;
Append t to s; 4 fori←1tondo
5 Sample p uniformly from {0,1,..,len(s)-1},satisfying s[p] is a number;
6 Sampleouniformly from{+,-,×,÷};
7 Sample numbers t1,t2, satisfying the result of o(t1,t2) equals s[pl; 8 if s[p-1]=÷or(o∈{+,-}and s[p-1]∈{-,x})or(o∈{+,-}and
s[p+1]∈{×,÷})then
9 pop s[p]; insert[(],[t1],[o], [t2], D] sequentially into s[p]; 10 11 else
12 pop s[p]；
13 insert [ti],[o], [t2] sequentially into s[p]; 14 end
15 end
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Input:Number ofVariable n Output :Linear Equation sInput :Vocabulary of numbers V = {0,1...1O}// number field p = 11
Sample b uniformly from Vn×1;
2 do
3Sample A uniformly from Vn ×n; 一
4whileAis not invertible;
5 S ← "A11x1 +...+ AinXn = b1,..,An1x1 + ... + AnnXn = bn"
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Algorithm4:EDData Generation
Input :Length of the First String n Input :Alphabet V = {a,b...z}
Output :Sequence S1, S2
Sample t uniformly from {3,4...10} ;
2 T ← Sample t letters from V ;
3 S1 ← Sample n letters uniformly from T ;
4 Sample p uniformly from [0,1] ;
5 if p<0.4 then
6Sample l uniformly from {n- 3,n- 2,,n + 2};
7S2 ← Sample l letters uniformly from T ;
8 else
9do
10 11S2↑S1；
12fori←1 to n do
13Sample p uniformly from {0,1...len(s2) -1};
14Sample l uniformly from T; Randomly conduct one of the followings: pop s2[p],substitute s2[p] with l, insert l
into s2[p];
15 16end
17while len(s2) not in [n -3,n + 2];
end
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Input :Sequence Length n Input:Vocabulary of numbers V = {101,101...250}
Output :Sequence s
Sample l uniformly from {3,4...n} ;
2 Sample t uniformly from {1,2,3} ;
3a=；
4 push O to a ; 5 ift=2 then
6Sample j uniformly from {1,2...[l/2」+1} ;
7push j to a ;
8 else if t= 3 then
9Sample j uniformly from {1,2...[𝑙/3」+1} ;
10Sample k uniformly from {1,2..[(𝑙- j)/2]+1} ;
11push j to a ;
12push j+ k to a ;
13push l to a;
14 s ← Sample l numbers from V;
15 fori←1 to t do
16Sort s[ali -1] :a[i]];// This process makes sure the LIS of the generated
sequence s is at least [l/t].
17 end
18 r ← Sample n - l numbers from V;
19 Randomly insert r into s;
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WMTEN-FR39 (10 -82)
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classifier4.51×1.15×2.042.97×1.22×2.001.37×5.09×1.11
Prrsrorssrr 000softmax3.75×0.47×1.963.19×0.67×2.101.65×3.15×1.63
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state5.22×1.11×1.642.72×2.01×1.582.00×3.65×1.68
classifier2.30×1.25×2.093.08×1.21×1.982.59×2.37×1.10
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testSouthAfrica-bGntotiatig84tiilackCapscdfrstCufiiElit'sutiatei Eliot,6,adotpadinteaialcketfor14montsenedinisindicaiofrteacigrandplaedunderdon McCullum.New Zealand play the winner of the semi-final between Australia or India.
Yfull：GrantElititZaadrougoteordCupfal.6yedaiSouthAutaturaldKltwil surely never playanotherinningslike his 84.New Zealandwilltake oneither Australiaor India in the finalon Sunday.
Y)： early_Grant_Eliott_hit_a_six_to_put_New_Zealand_through__to__the_World_Cup_final_._Elliott tt_was__born_in_South__Africa_but_ a_naturalised_Kiwi_ The_36-year-old_will_surely_never__play_another_innings_like_his_unbeaten_84_ New_Zealand_will_now__take_on_either_Australia_or__India_in_the_final_on_Sunday_.<EOS>
Y(2) early ‘Grant_Ellott_hit_84_in_the_Black_Caps_chase__New_Zealand_reached_the_World_Cup_final._lliott tt_was__born__in_So uthAfrica_butanaturalised_KiwiEliowillsurelynevepayanotheinninglikehisunbeate84>
y(1) YearlyD(Yearly,Yfull)
0.02Rearty-RfullAverage layersSpeedup
Y(2） early0.01 -0.32.1 1.9X 2.9 X3.6Exit layer-colormapping:12345678 D and R are computed with ROUGE-L
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