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sha256:9457dbfb44616dc98950ebb073fa38d5f7675175b20a7721cf8164b45d6c0701 +size 257518 diff --git a/parse/dev/6NT1a56mNim/6NT1a56mNim.md b/parse/dev/6NT1a56mNim/6NT1a56mNim.md new file mode 100644 index 0000000000000000000000000000000000000000..a4b568e21ce1a9d4dc13fb78ad48651c4a56ea80 --- /dev/null +++ b/parse/dev/6NT1a56mNim/6NT1a56mNim.md @@ -0,0 +1,286 @@ +# LANGUAGE MODELS AS ZERO-SHOT PLANNERS: EXTRACTING ACTIONABLE KNOWLEDGE FOR EMBODIED AGENTS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Can world knowledge learned by large language models (LLMs) be used to act in interactive environments? In this paper, we investigate the possibility of grounding high-level tasks, expressed in natural language (i.e. “make breakfast”), to a chosen set of actionable steps (i.e. “open fridge”). While prior work focused on learning from explicit step-by-step examples of how to act, we surprisingly find that if pre-trained LMs are large enough and prompted appropriately, they can effectively decompose high-level tasks into low-level plans without any further training. However, the plans produced naively by LLMs often cannot map precisely to admissible actions. We propose a procedure that conditions on existing demonstrations and semantically translates the plans to admissible actions. Our evaluation in the recent VirtualHome environment shows that the resulting method substantially improves executability over the LLM baseline. The conducted human evaluation reveals a trade-off between executability and correctness but shows a promising sign towards extracting actionable knowledge from language models1. + +# 1 INTRODUCTION + +Large language models (LLMs) have made impressive advances in language generation and understanding in recent years (Devlin et al., 2018; Radford et al., 2019; Raffel et al., 2019; Brown et al., 2020). See Bommasani et al. (2021) for a recent summary of their capabilities and impacts. Being trained on large corpora of human-produced language, these models are thought to contain a lot of information about the world (Roberts et al., 2020; Li et al., 2021; BIG-bench collaboration, 2021) - albeit in linguistic form. + +We ask whether we can use such knowledge contained in LLMs not just for linguistic tasks, but to make goal-driven decisions to that can be enacted in interactive, embodied environments. But we are not simply interested in whether we can train models on a dataset of demonstrations collected for some specific environment – we are instead interested in whether LLMs already contain information necessary to accomplish goals without any additional training. + +More specifically, we ask whether world knowledge about how to perform high-level tasks (such as “make breakfast”) can be expanded to a series of groundable actions (such as “open fridge”, “grab milk”, “close fridge”, etc) that can be executed in the environment. For our investigation, we use recently proposed VirtualHome environment (Puig et al., 2018). It can simulate a large variety of realistic human activities in a household environment and supports ability to perform them via embodied actions defined with a verb-object syntax. However, due to open-ended nature of the tasks, it is difficult to autonomously evaluate their success. We rely on human evaluation (conducted on Mechanical Turk) to decide whether sequences of actions meaningfully accomplish posed tasks. + +We find that large GPT-3 (Brown et al., 2020) and Codex (Chen et al., 2021) models, when prompted with a single fixed example of a task description and its associated sequence of actions, can produce very plausible action plans for the task we’re interested in. Such completions reflect the information already stored in the model – no model fine-tuning is involved. Additionally, we only observe this effect in the larger models. Unfortunately, despite their semantic correctness, the produced action plans are often not executable in the environment. Produced actions may not map precisely to admissible actions, or may contain various linguistic ambiguities. + +![](images/a24d1ce4158585dea15fe7aba14314c25e020e09e7ef5ac52d4f0a0daf0d9eca.jpg) +Figure 1: Executability v.s. semantic correctness of generated action plans (left) and sample action plans generated by different models (right). Large models can produce action plans indistinguishable from plans created by humans, but frequently are not executable in the environment. Using our techniques, we can significantly improve executability, albeit at the cost of correctness. + +We propose several tools to improve executability of the model’s outputs. First, we enumerate all admissible action phrases and map the model’s output action to the most semantically-similar admissible action (we use similarity measure between sentence embeddings produced by a RoBERTa model Liu et al. (2019) in this work, but other choices are possible). Second, we use the model to autoregressively generate actions in a plan by conditioning past actions that have been made admissible via the technique above. Such on the fly correction can keep generation anchored to admissible actions. Third, we provide weak supervision to the model by prompting the model with a known task example similar to the query task. This is somewhat reminiscent of prompt tuning approaches, but does not require access to gradients or internals of the model. + +Using above tools to bias model generation, we find that we improve executability of instructions from $18 \%$ to $79 \%$ (see Figure 1) without any invasive modifications to model parameters or any extra gradient or internal information beyond what is returned from the model’s forward pass. This is advantageous because it does not require any modifications to model training procedure and can fit within existing model serving pipelines. However, we do find there to be a significant drop in correctness of the instruction sequences generated with above tools (as judged by humans), indicating a promising step, but requiring more research on the topic. + +To summarize, our paper’s contributions are as follows: + +• We show that without any training, large language models can be prompted to generate plausible goal-driven action plans, but such plans are frequently not executable in interactive environments. +• We propose several tools to improve executability of the model generation without invasive probing or modifications to the model. +• We conduct a human evaluation of multiple techniques and models and report on the tradeoffs between executabiltiy and semantic correctness. + +# 2 EVALUATION FRAMEWORK + +Simulating open-ended tasks that resemble naturalistic human activities requires an environment to support a rich set of diverse interactions, rendering most existing embodied environments unsuitable for our investigation. One exception is VirtualHome (Puig et al., 2018), which models human activities in a typical household. Therefore, we only provide evaluation in this environment. To further measure correctness given open-ended tasks, we conduct a human evaluation. We note that since no further training is involved throughout our investigations, the observations and findings presented in this paper should also translate to similar embodied environments. + +# 2.1 EVALUATED ENVIRONMENT: VIRTUALHOME + +Preliminaries In VirtualHome, activities are expressed as programs. Each program consists of a sequence of steps, where each step is written as: [action] $\langle a r g _ { 1 } \rangle ( i d _ { 1 } )$ ... $\langle a r g _ { n } \rangle ( i d _ { n } )$ . Each action refers to atomic actions such as “walk”, “open”, and “put”. A total of 45 atomic actions are supported by VirtualHome. Different actions take in different numbers of arg necessary for specifying an interaction. Associated with each arg is a unique $i d$ specifying the corresponding node in the environment graph, in case of multiple instances of the same object class are present in the graph. For the sake of simplicity, we omit the $i d$ in the remaining discussions of this paper and allow automatic assignment by the environment. An example program is shown in Appendix 4. + +Evaluated Tasks We use the knowledge base collected by VirtualHome for evaluation. The knowledge base contains household activities crowd-sourced from Amazon Mechanical Turk (MTurk). The MTurk workers were asked to provide natural language descriptions of daily household activities and all actionable steps necessary for completing the activities. The descriptions are both given as high-level task descriptions and step-by-step instructions. We omit the use of stepby-step instructions in this work as we desire direct extraction of executable programs from only task descriptions. For evaluations, we randomly sample a subset of 88 high-level tasks, each having one or more annotated ground-truth programs. The remaining 204 tasks are used as demonstration set, from which we are allowed to select as example(s) for prompting language models. Note that no training or fine-tuning is performed using these tasks and their annotations. More details of the evaluated tasks can be found in Appendix 8.6. + +# 2.2 METRICS + +A program that commands the agent to wander around in a household environment is highly executable but may not complete the desired task. On the other hand, a program composed of step instructions from knowledge bases can likely complete the task but cannot be executed. The reason is that free-form instructions can be ambiguous and may lack necessary common-sense actions. To this end, we consider two axes for evaluation: executability and correctness. + +Executability Executability measures whether an action plan can be correctly parsed and satisfies the common-sense constraints of the environment. To be correctly parsed, an action plan must be syntatically correct and contain only allowed actions and recognizable objects. To satisfy the common-sense constraints, each action step must not violate the set of its pre-conditions (e.g. the agent cannot grab milk from the fridge before opening it) and post-conditions (e.g. the state of the fridge changes from “closed” to “open” after the agent opens it). We report the average executability across all 88 tasks and across all 7 VirtualHome scenes. + +Correctness Unlike most embodied environments where the completion of a task can be easily judged, the ambiguous and multimodal nature of natural language task specification makes it impractical to obtain a gold-standard measurement of correctness. One approach could be measuring similarity of the final environment state produced by executing predicted and ground-truth programs, but VirtualHome initializes an environment differently based on the to-be-executed program, making comparisons difficult if measured in such way. Therefore, we conduct human evaluation for the highlighted methods. More details of the human evaluations can be found in Appendix 8.5. For the remaining methods and ablation studies, we rely on a match-based metric that measures how similar a generated program is to human annotations. Specifically, we follow Puig et al. (2018) and calculate the longest common subsequence (LCS) between two programs, normalized by the maximum length of the two. In the presence of multiple ground-truth programs for a single task, we take the maximum LCS across the ground-truth programs. However, we note that the majority of the tasks only have one ground-truth annotation, but there are often many plausible ways to complete a certain task, making this metric imperfect at evaluation program correctness2. Although correlation between the two is shown by Puig et al. (2018), we consider it only as a proxy metric in replacement of unscalable human evaluation. + +![](images/3fc27a6d1f4ab6c9de1366e50d5832a5f1d55a9a48f4da6dea52f4f5857ed62d.jpg) +Zero-Shot Planning via Causal LLM + +![](images/c42e3b664e14a2fdb2efd720431d14e61a741bc129346c810bb24bdb37bdaa56.jpg) +Translation to Admissible Action + +![](images/e09c408546dd11c5b347cced2e90ba8fc498ebc4998ea5901b86071d56b1750a.jpg) +Figure 2: We investigate the possibility of extracting actionable knowledge from pre-trained language models without any additional training. We first show surprising finding that large language models (LLMs) can decompose high-level tasks into sensible low-level action plans (left). To make the action plans executable, we propose to translate each step into admissible action via another LLM (middle). The translated action is appended to the original prompt used for generating the remaining steps (right). + +# 3 METHOD + +In this section, we investigate the possibility of extracting actionable knowledge from pre-trained language models without further training. We first give an overview of the common approach to query large language models (LLMs) and how it may be used for embodied agents. Then we describe an inference-time procedure that addresses several deficiencies of the LLM baseline and offers better executability in embodied environments. We break down the proposed procedure into three individual components, each discussed in Sec 3.2, 3.3, and 3.4. + +Since LMs excel at dealing with natural language text instead of the specific format required by VirtualHome as described in Section 2.1, we only expose natural language text to LMs. To do this, we define a mapping for each atomic action that parses a natural language phrase to the required format. For instance, “Walk to living room” is converted to “[Walk] hliving roomi(1)”. When an LM output cannot be parsed to any of the allowed action, the entire program is considered syntactically incorrect and thus not executable. + +# 3.1 PROMPTING + +Previous works have shown that large language models pre-trained on a colossal amount of data contain useful world knowledge that can be probed to perform various down-stream tasks (Radford et al., 2019; Brown et al., 2020). Notably, autoregressive LLMs can even perform in-context learning, an approach to solve tasks using only contextual information without gradient updates (Brown et al., 2020). Contextual information is given as part of the input prompt and LMs are asked to complete the remaining text. It often consists of natural language instructions and/or a number of examples containing the desired input/output pairs. + +We adopt the same approach to query LLMs to generate action plans for high-level tasks. Specifically, we prepend one example task description sentence and its annotated action plan from the demonstration set to the query task description, as shown in Fig 2. To obtain text completion results, we sample from autoregressive LLM using temperature sampling and nucleus sampling (Holtzman et al., 2019). We refer to this LM as Planning LM and the approach using this LM for plan generation as Vanilla $I L M ]$ , where $I L M J$ is replaced by specific language model such as GPT-3 or Codex. + +To further improve the quality of the generated output, we follow Chen et al. (2021) that uses LMs for program synthesis to sample multiple output for each task. However, unlike prior works in program synthesis that choose the sample with highest unit test pass rate, we only consider the setting where one sample is allowed to be evaluated for each task. This is because repetitive trialand-errors can be dangerous in the real world, and executing many action plans is equivalent to probing the environment for privileged information, which is often considered not viable. + +# 3.2 ROBUST PARSING BY SEMANTIC TRANSLATION + +One issue arises when naively following the above approach to generate action plans for high-level tasks: the action plan is often not executable because LMs are allowed to generate free-form text. + +Therefore, most of the time the output cannot be mapped to one unambiguous actionable step. And many reasons can cause such failures: 1) the output does not follow pre-defined mappings of any atomic action (i.e. “I first walk to the bedroom” does not follow “Walk to $\langle \mathrm { P L A C E } \rangle ^ { \prime \prime } )$ , 2) the output may refer to atomic action and objects using words unrecognizable by the environment (i.e. “Clean the dirty dishes in the sink” where “clean” and “dirty dishes in the sink” cannot be mapped to precise action and object), 3) the output contains lexical ambiguous words (i.e. “Open TV” should instead be “Switch on TV”), or 4) the output may use disallowed action (i.e. “Microwave the cup”). + +Instead of developing a set of rules to transform the free-form text into admissible action steps, we propose to again leverage world knowledge learned by large language models to semantically translate the action. For each step in the action plan $\hat { a }$ , we aim to find the most similar admissible environment action $a _ { e }$ as measured by cosine similarity: + +$\operatorname { a r g m a x } \frac { f ( \hat { a } ) \cdot f ( a _ { e } ) } { \| f ( \hat { a } ) \| \| f ( a _ { e } ) \| }$ where $f$ is an embedding function. + +To embed the output text and environment actions, we use a BERT-style LM (Devlin et al., 2018; Liu et al., 2019) trained with Sentence-BERT (Reimers & Gurevych, 2019) objective because of its suitability for sentence modeling. The sentence embedding is obtained by mean-pooling the last layer hidden states across all tokens. We refer to this LM as Translation LM. Note that this is a different LM than the GPT-style Planning LM discussed in the text so far. Using a single LM for both purposes could as well be possible and likely more efficient, but we leave such investigation to future works. While the set of actions in our environment is discrete and possible to exhaustively enumerate, sampling or projection can be employed in larger discrete or continuous action spaces. + +Since Translation LM can guarantee the parsed action is allowed by the environment, we can tradeoff semantic soundness of an LM step by how likely it can be mapped to an admissible action in the environment. This can be achieved by a simple modification to the scheme that we use to choose the best sample from the LM output. Instead of only using mean token log probability as a ranking metric, we choose the sample with the highest score calculated as $s = C + \beta \cdot l o g p r o b$ , where $C$ is the cosine similarity to the closest allowed action and $\beta$ is a weighting parameter. + +# 3.3 AUTOREGRESSIVE TRAJECTORY CORRECTION + +Translating each step of the program after the entire program has been synthesized is analogous to open-loop planning and is subject to compounding errors. In practice, LLMs might output compounded instructions for a single step, even though it cannot be completed using one admissible action in the environment. To this end, we can instead interleave plan generation and action translation to allow for automatic trajectory correction. At each step, we first query Planning LM to generate $k$ samples for a single action. Then we calculate score $s$ for each sample using Translation LM and append the translated action to the unfinished text completion. This way all subsequent steps will be conditioned on admissible actions instead of free-form text output generated by Planning LM. Furthermore, we can use Translation LM to detect out-of-distribution actions, those outside the capabilities of a robot, and terminate a program early instead of mapping to a faulty action. This can be easily implemented by setting a threshold $\epsilon$ such that if $C _ { \mathrm { m a x } } ^ { t } < \epsilon$ at step $t$ , the program is terminated early. We empirically show this leads to better executability while maintaining similar correctness of the generated action plans. + +# 3.4 DYNAMIC EXAMPLE SELECTION FOR IMPROVED KNOWLEDGE EXTRACTION + +So far in the text, we always give the same example in the prompt for all evaluated high-level tasks. However, consider the task of “ordering pizza”. Prompting LLMs with this task may give the assumption that the agent is initialized in front of a computer, and the LLMs may guide the agent to search for a pizza store and click “checkout my cart”. Although these are reasonable and feasible in the real world, such assumption cannot always be made as these interactions may not be supported in simulated environments like VirtualHome. In fact, the closest series of actions that human experts give may be “walking to a computer”, “switching on the computer”, and “typing the keyboard”. Without being finetuned on these data, LLMs would often fail at these tasks. To provide weak supervision at inference time, we propose to use Translation LM to select the most similar task from the demonstration set to be used as the example in the prompt. Specifically, we choose the task whose high-level description matches closely the query task as measured by cosine similarity. This allows Planning LM to reason how to perform a similar task given a human-annotated example. In our evaluations, we make sure the demonstration set is not overlapping with our test queries. + +![](images/8bcdc34cf82e9ec412ef334ee9cfec2fa6c4106d1a23c2e2af1e86633dc3553c.jpg) +Figure 3: Visualization of VirtualHome programs generated by our approach. The top row shows the execution of the task “Complete Amazon Turk Surveys”, and the bottom row shows the task “Get Glass of Milk”. We show LLMs not only can generate sensible action plans given only high-level tasks but also contains the actionable knowledge that can be extracted for grounding in embodied environments. + +Combining the various improvement discussed above, we refer to the final approach as Translated $I L M ]$ , where $I L M ]$ is replaced by specific language model used such as GPT-3 and Codex. + +# 4 RESULTS + +In this section, we first show that language models can generate sensible action plans for many highlevel tasks, even without any additional training. Then we highlight its inadequacy when naively applied to embodied environments and demonstrate how this can be improved by again leveraging world knowledge learned by LLMs. Visualization of generated programs are shown in Fig 3. + +Sampling from LMs Pre-trained LMs are sensitive to sampling parameters and the specific example given in the prompt. For all evaluated methods, we perform hyper-parameter search over various sampling parameters, and for methods using fixed prompt example, we report metrics averaged across three randomly chosen examples. To select the best run for each method, we rank the runs by LCS + executability, each normalized by human-expert scores3. Further details can be found in Appendix 8.3. + +Model Choices We find empirically the combination of Codex-12B and Sentence-RoBERTa355M work well in our setting. Codex and GPT-3 are accessed using OpenAI API. The remaining models are accessed through open-source packages, Hugging Face Transformers (Wolf et al., 2019) and SentenceTransformers (Reimers & Gurevych, 2019), without additional modifications. + +# 4.1 DO LLMS CONTAIN ACTIONABLE KNOWLEDGE FOR HIGH-LEVEL TASKS? + +We first investigate whether LLMs can generate sensible action plans expressed in free-form language. We use the approach described in Section 3.1 to query pre-trained LLMs. To evaluate the correctness of generated action plans, we conduct human evaluations. For each model, we ask 10 human annotators to determine – by answering “Yes” or “No” – whether each task can be completed using provided action steps. To provide a reference of how humans might rate the action plans provided by other humans, we also ask annotators to rate the ground-truth action plans provided in the VirtualHome dataset for the same set of tasks. In contrast to the free-form text output by LLMs, the ground-truth action plans from VirtualHome are generated via a graphical programming interface that enforces strict syntax, although annotators were allowed to compose necessary actions. + +We show the human evaluation results in Fig 1, where y-axis shows correctness averaged across all tasks and all annotators. Surprisingly, when LLMs are large enough and without imposed syntactic constraints, they can generate highly realistic action plans whose correctness – as deemed by human annotators – even surpasses human-labeled ground-truth. Yet another interesting finding is that Codex outperforms GPT-3 significantly under the same number of model parameters. One hypothesis could be that by fine-tuning on structured data (docstrings and code), Codex specializes at decomposing a high-level objective into a number of basic operations, even with those operations described in natural language. We also observe some level of correctness for smaller models such as GPT-2. However, inspection of its produced output indicates that it often generates significantly shorter plans by ignoring common-sense actions or by simply rephrasing the given task (e.g. the task “Go to sleep” produces only a single step “Go to bed”). These failure modes sometimes mislead human annotators to mark them correct as the annotators may ignore common-sense actions in their judgment as well, resulting in a higher correctness rate than the quality of the output shows. + +Table 1: Human-evaluated correctness and evaluation results in VirtualHome. Although action plans generated by GPT-3 and Codex can even surpass the annotated GT in correctness measure, they are rarely executable. By translating the naive action plans, we show an important step towards grounding LLMs in embodied environments, but we observe room to achieve this without trading executability for correctness. We also observe a failure mode among smaller models that lead to high executability. + +
Language ModelExecutabilityLCSCorrectness
Vanilla GPT-2117M18.66%3.19%14.27%
Vanilla GPT-21.5B39.40%7.78%19.51%
Vanilla Codex 2.5B17.62%15.57%53.44%
Vanilla GPT-Neo 2.7B29.92%11.52%50.74%
Vanilla Codex 12B18.07%16.97%56.06%
Vanilla GPT-312B Vanilla GPT-3175B25.87% 7.79%13.40%30.98%
Annotated GT100.00%17.82% N/A65.19% 54.80%
Fine-tuned GPT-3 12B66.07%34.08%50.56%
OurFinal Methods
Translated Codex 12B78.57%24.72%34.97%
Translated GPT-3175B73.05%24.09%51.73%
+ +# 4.2 HOW EXECUTABLE ARE THE LLM ACTION PLANS? + +We analyze the executability of LLM plans by evaluating them in all 7 household scenes in VirtualHome. As shown in Table 1, we find action plans generated naively by LLMs are generally not very executable. Although smaller models seem to have higher executability, we find that the majority of these executable plans are produced by ignoring the queried task and repeating the given GT example of a different task. This is validated by the fact that smaller models have lower LCS than larger models despite having higher executability, showing that this failure mode is prevalent among smaller models. In contrast, larger models do not suffer severely from this failure mode. Yet as a result of being more expressive, their generated programs are substantially less executable. + +# 4.3 CAN LLM ACTION PLANS BE MADE EXECUTABLE BY ACTION TRANSLATION? + +In this section, we evaluate the effectiveness of our proposed procedure of action translation. We first create a bank of all allowed 47522 action steps in the environment, including all possible combinations of atomic actions and allowed arguments/objects. Then we use an off-the-shelf SentenceRoBERTa (Liu et al., 2019; Reimers & Gurevych, 2019) as Translation LM to create embeddings for actions and output text. For better computational efficiency, we pre-compute the embeddings for all allowed actions, leaving minor computation overhead for our procedure over the baseline methods at inference time. As shown in Table 1, executability of generated programs is significantly improved. Furthermore, we also observe improved LCS because the translated action steps precisely follow the program syntax and thus are more similar to ground-truth annotations. One sample output is shown in Fig 1 and a larger random subset of generated samples can be found in Appendix 8.7. + +To validate their correctness, we again perform human studies using the same procedure in Sec 4.1. Results are shown in Table 1. We find that despite being more similar to GT, the programs are deemed less correct by humans. By examining the generated output, we observe two main sources of errors. First, we find Translation LM is poor at mapping compounded instructions to a succinct admissible action. This is partly due to that Translation LM is trained on a much smaller dataset and contains much a smaller number of parameters, so we expect further improvement by using a larger pre-trained model for translation. The second source of error comes from imperfect expressivity of the environment; we find that for many tasks we evaluate, certain necessary actions or objects are not implemented. This is also reflected by out human evaluation results of the GT programs, as only half of the programs are considered complete by the human annotators. + +# 5 ANALYSIS AND DISCUSSIONS + +# 5.1 ABLATION OF DESIGN DECISIONS + +We perform ablation studies to show the effectiveness and necessity of three components of our proposed procedure, each described Sec 3.2, 3.3, and 3.4. As shown in Table 2, leaving out any of the three components would all + +lead to decreased performance in both executability and LCS. Notably, not doing action translation leads to the most significant executability drop, showing the importance of action translation in extracting executable action plans from LLMs. + +Table 2: Ablation of three proposed techniques. + +
MethodsExecutabilityLCS
Translated Codex 12B78.57%24.72%
- w/o Action Translation31.49%22.53%
- w/o Dynamic Example50.86%22.84%
- w/o Iterative55.19%24.43%
+ +# 5.2 CAN LLMS GENERATE ACTIONABLE PROGRAMS BY FOLLOWING DETAILED INSTRUCTIONS? + +Prior works often focus on translating step-by-step instructions into executable programs. We evaluate LLMs under this setting using a prompt format shown in Appendix 8.2. Although this setting is easier as it does not require rich actionable knowledge, detailed instructions can help resolve much ambiguity of exactly how to perform a high-level task when multiple solutions are possible. Therefore, Translated Codex 12B achieves executability of $7 8 . 5 7 \%$ and LCS of $3 2 . 8 7 \%$ , where LCS sees a considerable bump from the setting without detailed instructions. Surprisingly, the LCS result is very close to that of a supervised LSTM (Hochreiter & Schmidhuber, 1997) baseline from VirtualHome trained on human-annotated data, which is at $3 4 . 0 0 \%$ . Note that since code to train the baseline and the specific train/test split is not publicly released, we only show results reported in Puig et al. (2018) as a reference. We also cannot compare executability as it is not reported. + +The investigations of this paper are two-fold: 1) Is actionable knowledge present in LLMs? 2) Can we ground this actionable knowledge in interactive environment? In this section, we focus our attention on the second question by conditioning on the assumption that first question is true. To do this, since successful execution of correct action plans directly measures grounding, we select only the correct plans generated by LLMs and measure how executable they are. We deem an action plan to be correct if $70 \%$ or more human annotators decide it is correct. + +5.3 IS ACTION TRANSLATION NECESSARY FOR ACTIONABLE KNOWLEDGE GROUNDING? +Table 3: Count of correct/executable programs and percentage of executable among correct. C indicates correct and E indicates executable. + +
Methods# ofC#of C and EE/C
GPT-21.5B800.00%
GPT-312B2428.33%
GPT-3175B6857.35%
Codex 12B45817.78%
Translated Codex 12B181583.33%
+ +As shown by Table 3, when an LM is not large enough (e.g. GPT-2), not only it contains little actionable knowledge, but this knowledge cannot be grounded at all. GPT-3 and Codex, on the other hand, can generate highly correct action plans in free-form language. However, they do not have the capability to ground their actionable knowledge in interactive environments. What’s more interesting, by comparing GPT-3 of both 12B parameters and 175B parameters, ratio of executable plans does not improve with the parameter count. This shows that simply training larger models does not necessarily lead to better knowledge grounding. In the meantime, action translation offers a promising way towards grounding actionable knowledge by producing highly executable plans. However, we again note that it comes at a trade-off of producing less correct plans as compared to its vanilla counterpart, and we hope to see future endeavors for bridging the gap. + +# 6 RELATED WORKS + +Large-scale natural language modeling has witnessed rapid advances since the inception of the Transformer architecture (Vaswani et al., 2017). It has been shown by recent works that large language models (LLMs) pre-trained on large unstructured text corpus not only can perform strongly on various down-stream NLP tasks (Devlin et al., 2018; Radford et al., 2019; Raffel et al., 2019; Brown et al., 2020) but also can internalize an implicit knowledge base containing rich information about the world (Petroni et al., 2019; Jiang et al., 2020; Davison et al., 2019; Talmor et al., 2020; Roberts et al., 2020). Furthermore, the learned representations can be used to model relations of entities (Li et al., 2021), retrieve matching visual features (Ilharco et al., 2020), and even as valuable priors when applied to diverse tasks from different modalities (Lu et al., 2021; Tsimpoukelli et al., 2021). Compared to prior works in knowledge extraction that extract single-step factual answers memorized by the models (e.g. “Dante was born in [PLACE]”), we aim to extract sequential action plans to complete an open-ended human activity (e.g. “make breakfast”). We further require these plans to only contain allowed actions and satisfy the pre/post-conditions of actions in order to be executed by an embodied agent. + +At the same time, there has also been growing interest and development in grounding language in embodied environment. A series of prior works have investigated the possibility of parsing language instructions into formal logic to resolve various linguistic ambiguities for embodied agents (Artzi & Zettlemoyer, 2013; Misra et al., 2015; Tenorth et al., 2010). However, they often scale poorly to complex tasks and environments. Recently, more research efforts have been put into creating better and more realistic environments with the goal to further advances in this area (Puig et al., 2018; Shridhar et al., 2020a;b; Kolve et al., 2017; Savva et al., 2019). At the same time, by leveraging the better representation power of neural architectures, a number of works have looked into creating instruction-following agents that can perform manipulation (Lynch & Sermanet, 2020), navigation (Majumdar et al., 2020), or both (Suglia et al., 2021; Hill et al., 2020). + +Notably, most of these prior works do not leverage full-blown pre-trained LLMs (Suglia et al., 2021) or do not scale to complex human activities (Hill et al., 2020; Lynch & Sermanet, 2020). Perhaps more importantly, few works have evaluated LLMs in an embodiment setting that realizes the full potential of the world knowledge these models contain: the tasks evaluated are often “pick”, “grab”, “open”, and etc, which do not resemble the highly diverse activities that humans perform in daily lives. The development of VirtualHome environment Puig et al. (2018) enables such possibility. However, relevant works (Puig et al., 2020; Liao et al., 2019) rely on human-annotated data and perform supervised training from scratch. Due to the lack of rich world knowledge, these models can only generate action plans given step-by-step instructions of how to act or video demonstrations. In this work, we take a step further by conditioning only on the high-level descriptions and by extracting executable action plans from LLMs without any additional training. + +# 7 LIMITATIONS AND CONCLUSION + +There are several notable limitations of this work. First, although our approach presents a viable way to ground world knowledge in embodied environments, it is still a trade-off rather than one best solution since we observe considerable drop in correctness. Second, we focus on high-level to midlevel grounding, assuming there is a controller that can execute mid-level tasks (such as “grab cup”). Our work does not investigate usefulness of LLMs for low-level sensorimotor behavior grounding. The third limitation is that we do not incorporate observation context or feedback into our models. To some extent, we approach LLMs in the same way as how VirtualHome asks human annotators to give action plans for a given huamn activity by imagination, in which case the human-generated action plans also do not incorporate observation context. However, we do see incorporating observation context for complex activities as an exciting future direction. + +# REFERENCES + +Yoav Artzi and Luke Zettlemoyer. Weakly supervised learning of semantic parsers for mapping instructions to actions. Transactions of the Association for Computational Linguistics, 1:49–62, 2013. + +BIG-bench collaboration. Beyond the imitation game: Measuring and extrapolating the capabilities of language models. In preparation, 2021. URL https://github.com/google/ BIG-bench/. + +Rishi Bommasani, Drew A. 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Huggingface’s transformers: ´ State-of-the-art natural language processing. arXiv preprint arXiv:1910.03771, 2019. + +# 8 APPENDIX + +# 8.1 EXAMPLE PROGRAM IN VIRTUALHOME + +[Walk] h living room i (1) [Walk] h television i (1) [Find] h television i (1) [SwitchOn] h television i (1) [Find] h sofa i (1) [Sit] h sofa i (1) [TurnTo] h television i (1) [Watch] h television i (1) + +![](images/cdd10b5b1764575509d058912492130d6873463de86c074dab324e7ca787b63c.jpg) +Figure 4: An example prompt containing step-by-step instructions. + +# 8.3 HYPERPARAMETER SEARCH + +For each evaluated method, we perform grid search over the following hyperparameters: + +
NameDescriptionSearch Values
epsilon (e)OOD cutoff threshold used in iterative action{0,O.4,0.8} translation
temperaturesampling parameter adjusting relative probabili-{0.1, O.3, 0.6} ties across tokens
knumber of samples generated when querying LMs{1, 10} each time
frequence_penaltyOpenAI API specific; penalize new tokens based on their existing frequency in the text so far{0.1, 0.3, 0.6, 0.9}
presence_penaltyOpenAI API specific; penalize new tokens based on whether they appear in the text so far{0.3,0.5, 0.8}
repetition_penaltyHugging Face Transformers specific; penalize new tokens based on whether they are repeating existing text{1.0, 1.2, 1.5, 1.8}
+ +For methods that use fixed example across evaluated tasks, we search over the following three randomly chosen examples: + +
Example 1Example 2Example 3
Task: Use computerStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9: Find keyboardStep 10: Type on keyboardTask: Relax on sofaStep 1: Walk to home officeStep 2: Walk to couchStep 3: Find couchStep 4: Sit on couchStep 5: Find pillowStep 6: Lie on couchTask:Read bookStep 1: Walk to home officeStep 2: Walk to novelStep 3: Find novelStep 4: Grab novelStep 5: Find chairStep 6: Sit on chairStep 7: Read novel
+ +# 8.4 ANALYSIS OF PROGRAM LENGTH + +Shorter programs have a natural advantage of being more executable. Consider a task “wash hands” and a corresponding program that only commands an agent to “go to bathroom” without additional steps. The program is obviously incorrect yet trivially executable. To validate our approach does not simply generate very short program, we calculate the average program length across the 88 evaluated tasks. Results are shown in Table 6. In addition to the failure mode discussed in Section 4.2 that leads to incorrect yet executable programs, smaller LMs such as GPT-2 also generate programs significantly shorter than larger models, making them more executable. In contrast, larger models like Codex-12B generate more expressive program of high correctness, but they often suffer from executability. We show action translation can lead to benefits of both worlds, generating programs that are highly executable while maintaining similar expressiveness in terms of program length. + +
MethodsExecutability Average Length
Vanilla GPT-2 1.5B39.40%4.24
Vanilla Codex 12B18.07%7.22
Translated Codex 12B78.57%7.13
Ground-Truth100.00%9.66
+ +Table 6: Average executability & program length of different methods and human annotations. + +# 8.5 DETAILS OF HUMAN EVALUATIONS + +Human evaluations are conducted on Amazon Mechanical Turk. For each method, we generate action plans for all 88 high-level tasks. To account for expressivity of the VirtualHome environment (Puig et al., 2018), we further include ground-truth action plans from the VirtualHome dataset in our human evaluations. The human evaluations are conducted in the form of questionnaires containing all action plans with unknown corresponding methods. The questionnaire contains the following instructions at the top: + +For every question below, determine whether the task can be completed in any reasonable scenario using the provided steps. In other words, can the task be decomposed into these steps? Note that simply re-stating the task does not mean completing it. + +Human annotators are required to answer all the questions in the questionnaire, where each question is an action plan generated by a method unknown to the annotator. The order of the questions is randomly permuted before presented to each annotator. For each question, the annotators need to answer either “Yes” or “No” indicating if they believe the action plan completes the task. For each method, we report correctness percentage averaged across 10 participated human annotators. + +# 8.6 DETAILS OF VIRTUALHOME TASKS + +VirtualHome ActivityPrograms is a knowledge base collected on Mechanical Turk by Puig et al. (2018). The knowledge base features real household activities of humans and correspondingly sequences of basic actions for robots to perform these activities. Each entry contains a high-level task name (i.e. “Watch TV”), a mid-level detailed description (i.e. “Sit on my couch directly opposite my TV, switch on my TV with the remote control and watch”), and a sequence of allowed actions (a.k.a. a program) to perform the task in VirtualHome. There are 292 distinct high-level tasks and a total of 2821 different entries in the knowledge base, where each high-level task may correspond one or several programs. The knowledge base covers a diverse set of real human activities and a wide range of different actions and objects. Puig et al. (2018) also conduct additional human evaluation on 100 randomly sampled programs and show that $6 4 \%$ of them are complete (i.e. contain all necessary steps). Readers are encouraged to refer to Puig et al. (2018) for more comprehensive analysis of the dataset. + +Train/Test Split From all 292 distinct high-level tasks from the ActivityPrograms knowledge base, we randomly sample 88 high-level tasks for evaluation. The remaining 204 tasks are used as demonstration set from which we are allowed to select as example(s) for prompting language models, or in the case of supervised fine-tuning, they are used to fine-tune pre-trained language models. + +8.7 RANDOM SAMPLES OF GT, VANILLA CODEX, AND TRANSLATED CODEX + +
Annotated Ground-TruthCodexTranslated Codex (Ours)
Task: Go to sleep Step 1: Walk to bedroom Step 2: Walk to bed Step 3: Find bed Step 4: Turn to bedTask: Go to sleep Step 1: Walk to bedroom Step 2: Walk to bed Step 3: Find bed Step 4: Sit on bedTask: Go to sleep Step 1: Walk to bedroom Step 2: Walk to bed Step 3: Lie on bed Step 4: Sleep
Task:Browse internet Step 1: Walk to home office Step 2: Walk to chair Step 3: Find chair Step 4: Sit on chair Step 5: Find computer Step 6: Switch on computer Step 7: Turn to computer Step 8: Look at computer Step 9: Find keyboard Step 10: Type on keyboard Task:Listen to musicTask:Browse internet Step 1: Walk to home office Step 2: Walk to chair Step 3: Find chair Step 4: Sit on chair Step 5: Find computer Step 6: Switch on computer Step 7: Turn to computer Step 8: Look at computer Step 9: Find keyboardTask:Browse internet Step 1: Walk to home office Step 2: Walk to desk Step 3: Find chair Step 4: Sit on chair Step 5: Find computer Step 6: Switch on computer Step 7: Find mouse Step 8: Grab mouse Step 9: Turn to computer Step 10: Point at computer
Step 1: Walk to bedroom Step 2: Walk to computer Step 3: Find computer Step 4: Turn to computer Step 5: Look at computer Step 6: Walk to computer Step 7: Find chair Step 8: Pull chair Step 9: Sit on chair Step 10: Find headset Step 11: Grab headset Step 12: Put on headset Step 13: Switch on computer Step 14: Find keyboard Step 15: Type on keyboard Task:Brush teeth Step 1: Walk to bathroom Step 2: Walk to tooth paste Step 3: Find tooth paste Step 4: Grab tooth paste Step 5: Find toothbrush Step 6: Grab toothbrush Step 6:Move brush around mouth for two minutes Step 7: Spit out toothpaste and brush into sinkStep 10: Type on keyboard Task:Listen to music Step 1: Walk to home office Step 2: Walk to chair Step 3: Find chair Step 4: Sit on chair Step 5: Find computer Step 6: Switch on computer Step 7: Turn to computer Step 8: Look at computer Step 9: Find keyboard Step 10: Type on keyboard Task:Brush teeth Step 1: Walk to bathroom Step 2: Walk to sink Step 3: Find toothbrush Step 4: Pick up toothbrush Step 5: Put toothbrush in mouthTask:Listen to music Step 1: Walk to home office Step 2: Walk to computer Step 3: Sit on chair Step 4: Switch on computer Step 5: Find music stand Task:Brush teeth Step 1: Walk to bathroom Step 2: Open door Step 3: Walk to sink Step 4: Put pot on sink Step 5: Put brush on toothbrush
Task: Dry soap bottles Step 1: Find towel Step 2: Find soap Step 3: Grab soap Step 4: Wipe soap Step 5: Put back soap Step 6: Find shampoo Step 7: Grab shampoo Step 8: Wipe shampoo Step 9: Put back shampoo Step 10: Find shampoo Step 11: Grab shampoo Step 12: Wipe shampoo Step 13: Put back shampooStep 8: Turn on water in sink and rinse brush for one minute Step 9: Turn off water in sink and return brush to cupboard Task:Dry soap bottles Step 1: Walk to kitchen Step 2: Walk to sink Step 3: Find sink Step 4: Turn on water tap Step 5: Wait for water to be hot enough Step 6: Fill container with hot water Step 7: Find soap bottles Step 8: Pour soap into container with hot water Step 9: Wait for soap to dissolve in waterTask: Dry soap bottles Step 1: Walk to bathroom Step 2: Walk to shower Step 3: Find soap
+ +
Task: Vacuum Task: Vacuum Task:VacuumStep 1: Walk to home office Step 1: Walk to hallway Step 1: Walk to home officeStep 2: Walk to computer Step 2: Walk to closet Step 2: Walk to dresserStep 3: Find vacuum cleaner Step 3:Find closet Step 3: Open dresserStep 4:Switch on vacuum Step 4: Open closet Step 4: Find vacuum cleanercleaner Step 5:Find vacuum cleaner Step 5: Grab vacuum cleaner Step 5:Switch off vacuum Step 6: Turn on vacuum cleaner Step 6: Pull vacuum cleanercleaner Step 7: Walk to living room Step 7: Plug in vacuum cleanerStep 8: Vacuum floor in living Step 8: Switch on vacuumroom cleanerStep 9: Pull vacuum cleanerStep 10: Push vacuum cleaner
Task: Vacuum Step 1: Walk to hallwayStep 2: Walk to closetStep 3:Find closetStep 4: Open closetStep 5:Find vacuum cleanerStep 6: Turn on vacuum cleaner Step 7: Walk to living roomStep 8: Vacuum floor in livingroom
Task: Find dictionaryStep 1: Walk to home officeStep 2: Walk to bookshelfTask: Find dictionaryStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find dictionary Step 6: Open dictionaryStep 7: Look at dictionaryTask: Find dictionaryStep 1: Walk to home officeStep 2: Walk to bookStep 3: Find textbook
f
Step 3: Turn to bookshelfStep 4: Look at bookshelfStep 5: Turn to bookshelfStep 6: Look at bookshelfStep 7: Turn to bookshelfStep 8: Look at bookshelfStep 9: Turn to bookshelfStep 10: Look at bookshelfStep 11: Turn to bookshelfStep 12: Look at bookshelfStep 13: Turn to bookshelfStep 14: Look at bookshelfStep 15: Find novelStep 16: Grab novel
Task:Wash monitorStep 1: Walk to home officeStep 2: Walk to computerStep 3: Find computerStep 4: Turn to computerStep 5: Look at computerStep 6: Switch off computerStep 7: Find brushStep 8: Turn to brushStep 9: Look at brushStep 10: Grab brush Step 11: Wipe computerStep 12: Drop brushStep 13:Find napkinStep 14: Grab napkinStep 15: Wash napkinStep 16: Wipe computerTask:Wash monitorStep 1: Walk to home officeStep 2: Walk to chairStep 3:Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9: Find keyboardStep 10: Type on keyboardTask:Wash monitorStep 1:Walk to checkStep 2: Walk to desk
Task: Read newspaperStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find newspaperStep 6: Grab newspaperStep 7: Open newspaperStep 8: Read newspaperTask: Read newspaperStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find newspaperStep 6: Open newspaperStep 7: Read newspaperTask: Read newspaperStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find newspaperStep 6: Grab newspaperStep 7: Read newspaper
Task:ApplylotionStep 1: Find bedStep 2: Turn to bedStep 3: Point at bedStep 4: Wash bedStep 5: Turn to bedStep 6: Look at bedStep 7: Push bedStep 8: Sit on bedTask:ApplylotionStep 1: Walk to bathroomStep 2: Walk to cabinetStep 3:Open cabinetStep 4: Find lotionStep 5: Apply lotion on handsStep 6: Rub hands together for10 secondsStep 7: Apply lotion on faceTask:ApplylotionStep 1: Walk to bathroomStep 2: Turn to lotionbottleStep 3:Grab lotionbottle
Task: Draw pictureStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find deskStep 6: Open deskStep 7: Find checkStep 8: Grab checkStep 9: Put check on deskStep 10: Find pencilStep 11: Grab pencilStep 12: Close deskTask: Draw pictureStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9: Find keyboardStep 10: Type on keyboardTask: Draw pictureStep 1: Walk to home officeStep 2: Walk to wallStep 3: Walk to drawingStep 4: Find drawingStep 5: Turn to drawingStep 6: Look at drawingStep 7: Touch drawingStep 8: Pull drawingStep 9: Push drawing
Task: Prepare pot of boiling wa-terStep 1: Walk to dining roomStep 2: Walk to potStep 3: Find potStep 4: Grab potStep 5: Walk to sinkStep 6: Find faucetStep 7: Switch on faucetStep 8: Put pot on sinkStep 9: Grab potStep 10: Switch off faucetStep 11: Grab potStep 12: Walk to oven Step 13: Put pot on ovenStep 14: Switch on ovenTask: Prepare pot of boiling wa-terStep1: Walk to kitchenStep 2: Walk to sinkStep 3:Find sink Step 4: Turn on tapStep 5: Fill kettle with waterStep 6: Put kettle on stoveStep 7: Turn on stoveStep 8: Wait for water to boilTask: Prepare pot of boiling wa-terStep 1: Walk to kitchenStep 2: Walk to cabinetStep 3: Find pot Step 4: Grab pot Step 5: Walk to stoveStep 6: Find stoveStep 7: Switch on stove
Task: Look at mirrorStep 1: Walk to bathroomStep 2: Walk to mirrorStep 3:Find mirrorStep 4: Turn to mirrorStep 5: Look at mirrorTask: Look at mirrorStep 1: Walk to bathroomStep 2: Walk to mirrorStep 3: Look at mirrorTask:Look at mirrorStep 1: Walk to bedroomStep 2: Walk to mirrorStep 3:Find mirrorStep 4: Turn to mirrorStep 5: Look at mirror
Task:Wipe down sinkStep 1: Walk to dining roomStep 2: Walk to ragStep 3: Find ragStep 4: Grab ragStep 5: Walk to sinkStep 6: Wipe sinkStep 7: Put back ragTask:Wipe down sinkStep1: Walk to kitchenStep 2: Walk to sinkStep 3: Find sinkStep 4: Wipe down sinkStep 5: Return to kitchenTask:Wipe down sinkStep1: Walk to bathroomStep 2: Walk to cleaning solutionStep 3: Find cleaning solutionStep 4: Grab cleaning solutionStep 5: Find sinkStep 6: Pour cleaning solutioninto glassesStep 7: Find faucetStep 8: Switch on faucetStep 9: Scrub sinkStep 10:Wash sink
+ +
Task: Complete surveys on ama- Task: Complete surveys on ama- Task: Complete surveys on ama-zon turk zon turk zon turkStep 1: Walk to chair Step 1: Walk to home office Step 1: Walk to home officeStep 2: Sit on chair Step 2: Walk to chair Step 2: Walk to desk Step 3: Find computer Step 3: Find chair Step 3: Find chairStep 4: Switch on computer Step 4: Sit on chair Step 4: Sit on chairStep 5: Turn to computer Step 5: Find computer Step 5: Find computerStep 6: Look at computer Step 6: Switch on computer Step 6: Switch on computerStep 7: Find mouse Step 7: Turn to computerStep 8: Grab mouse Step 8: Look at computerStep 9: Turn to computer Step 9:Find keyboard Step 10: Point at computer Step 10: Type on keyboardStep 11: Find checkStep 12: Grab checkStep 13: Read checkStep 14: Put back mouseStep 15: Find keyboardStep 16: Type on keyboard
Task: Complete surveys on ama-zon turkStep 1: Walk to chairTask: Complete surveys on ama- zon turkStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9:Find keyboardStep 10: Type on keyboard
Step 2: Sit on chair Step 3: Find computerStep 4: Switch on computerStep 5: Turn to computer
Step 6: Look at computerStep 7: Find mouseStep 8: Grab mouseStep 9: Turn to computer Step 10: Point at computerStep 11: Find checkStep 12: Grab checkStep 13: Read checkStep 14: Put back mouseStep 15: Find keyboardStep 16: Type on keyboard
Task:Write book Step 1: Walk to home office Step 2: Walk to laptopStep 3: Find laptopStep 4: Grab laptopStep 5: Find electrical outletStep 6: Plug in laptopStep 7: Walk to love seatStep 8: Sit on love seatTask: Write bookStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9: Find keyboardStep 10: Type on keyboardTask:Write bookStep 1: Walk to home officeStep 2: Walk to lightStep 3: Find lightStep 4: Switch on lightStep 5: Find penStep 6: Grab penStep 7: Find chairStep 8: Sit on chairStep 9: Turn to novel
Step 7: Walk to love seatStep 8: Sit on love seatStep 9: Switch on laptopStep 10: Find keyboardStep 11: Type on keyboard
Task: Paint ceiling Step 1: Find drawing Step 2: Turn to drawingStep 3: Point at drawing Step 4: Watch drawingStep 5: Touch drawingStep 6: Walk to bedroomTask: Paint ceilingStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find paint canStep 6: Open paint can Step 7: Pour paint into paint canStep 8: Close paint canStep 9: Find roller Step 10: Roll paint onto ceilingTask: Paint ceilingStep 1: Walk to home office Step 2: Walk to ceilingStep 3: Walk to paintingStep 4: Find paintingStep 5: Turn to paintingStep 6: Look at paintingStep 7: Touch paintingStep 8: Pull paintingStep 9: Push painting
Task: Change clothesStep 1: Walk to bedroomStep 2: Walk to dresserStep 3: Open dresserStep 4: Find dressStep 5: Grab dressStep 6: Put on dressStep 7: Close dresserTask: Change clothesStep 1: Walk to closetStep 2: Open closetStep 3: Find clothesStep 4: Put on clothesStep 5: Close closetTask: Change clothesStep 1: Walk to bedroomStep 2: Walk to dresserStep 3: Find dresserStep 4: Open dresserStep 5: Find pantsStep 6: Grab pantsStep 7: Put on pantsStep 8: Find shirtStep 9: Grab shirtStep 10: Put on shirtStep 11: Close dresser
Task: Make bed Task: Make bed Task:Make bedStep 1: Walk to bedroom Step 1: Walk to bedroom Step 1: Stand upStep 2: Walk to bed Step 2: Walk to bed Step 2: Walk to bedroomStep 3: Find sheets Step 3: Find bed Step 3: Walk to bedStep 4: Grab sheets Step 4: Sit on bed Step 4: Turn to bedStep 5: Pull sheets Step 5: Find sheet Step 5: Put sheets on bedStep 6: Pick up sheetStep 7: Find pillowStep 8: Pick up pillowStep 9: Find blanketStep 10: Pick up blanket
Step 2: Walk to bedStep 3: Find sheetsStep 4: Grab sheetsStep 5: Pull sheets
Task: Read yourself to sleepStep 1: Walk to bedroomStep 2: Walk to novelStep 3: Find novelStep 4: Grab novelStep 5: Find bedStep 6: Sit on bedStep 7: Find bookmarkStep 8: Grab bookmarkStep 9: Read novelStep 10: Put bookmark on novelStep 11: SleepTask: Read yourself to sleep Step 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9: Find keyboardStep 10: Type on keyboardTask: Read yourself to sleepStep 1: Walk to bedroomStep 2: Walk to bookStep 3: Sit on chairStep 4: Read checkStep 5: Sleep
Task: Hang keysStep 1: Walk to bedroomStep 2: Walk to dresserStep 3: Find purseStep 4: Grab purseStep 5: Open purseStep 6: Find keysStep 7: Grab keysStep 8: Find bowlStep 9: Put keys on bowlStep 10: Close purseStep 11: Put back purseTask: Hang keysStep 1: Walk to home officeStep 2: Walk to chairStep 3: Find chairStep 4: Sit on chairStep 5: Find computerStep 6: Switch on computerStep 7: Turn to computerStep 8: Look at computerStep 9: Find keyboardStep 10: Type on keyboardTask: Hang keysStep 1: Walk to home officeStep 2: Walk to tableStep 3: Find keysStep 4: Grab keysStep 5: Walk to wallStep 6: Find hangerStep 7: Put keys on hanger
Task:Eat cheeseStep 1: Walk to dining roomStep 2: Walk to freezerStep 3: Open freezerStep 4: Find cheeseStep 5: Grab cheeseStep 6: Eat cheeseTask:Eat cheeseStep 1: Walk to fridge Step 2: Open fridgeStep 3: Find cheeseStep 4: Pick up cheeseStep 5: Walk to tableStep 6: Place cheese on tableStep 7: Close fridgeStep 8: Walk to chairStep 9: Sit on chairStep 10: Eat cheeseTask: Eat cheeseStep 1: Walk to dining roomStep 2: Walk to tableStep 3: Find tableStep 4: Turn to tableStep 5: Find chairStep 6: Sit on chairStep 7: Find foodStep 8: Grab foodStep 9: Find plateStep 10: Put food on plate
\ No newline at end of file diff --git a/parse/dev/716PvHoDct/716PvHoDct.md b/parse/dev/716PvHoDct/716PvHoDct.md new file mode 100644 index 0000000000000000000000000000000000000000..109d9d74d245f28e1cc659d2d08e65295df16c81 --- /dev/null +++ b/parse/dev/716PvHoDct/716PvHoDct.md @@ -0,0 +1,367 @@ +# VPGTrans: Transfer Visual Prompt Generator across LLMs + +Ao Zhang 1 Hao Fei 1∗ Yuan $\mathbf { Y a o } ^ { 2 * }$ Wei Ji 1 Li Li 1 Zhiyuan Liu 2 Tat-Seng Chua 1 1 $\mathbf { N E x T + + }$ Lab, School of Computing, National University of Singapore 2Department of Computer Science and Technology, Tsinghua University zhanga6@outlook.com haofei37@nus.edu.sg yaoyuanthu@163.com + +Project: https://vpgtrans.github.io + +# Abstract + +Since developing a new multimodal LLM (MLLM) by pre-training on a tremendous amount of image-text pairs from scratch is exceedingly resource-consuming, connecting an existing LLM with a comparatively lightweight visual prompt generator (VPG) becomes a feasible paradigm. However, further tuning the VPG component of the MLLM still incurs significant computational costs, such as thousands of GPU hours and millions of training data points. An alternative solution is to transfer an existing VPG from one MLLM to the target MLLM. In this work, we investigate VPG transferability across LLMs for the first time, aiming to reduce the cost of VPG transfer. Specifically, we explore VPG transfer across different LLM sizes (e.g., small-to-large) and types. We identify key factors to maximize the transfer efficiency, based on which we develop a simple yet highly effective two-stage transfer framework, called VPGTrans. Notably, it enables VPG transfer from BLIP-2 $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ to BLIP-2 $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ with less than $10 \%$ of GPU hours using only $1 0 . 7 \%$ of the training data compared to training a VPG for $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ from scratch. Furthermore, we provide a series of intriguing findings and discuss potential explanations behind them. Finally, we showcase the practical value of our VPGTrans approach, by customizing two novel MLLMs, including VL-LLaMA and VL-Vicuna, with the recently released LLaMA and Vicuna LLMs. + +![](images/47f10b1dd10c7582861b3a81730cf2754b8fef72df8b0732479b81981c6d570f.jpg) +Figure 1: (a) The general architecture of MLLMs, e.g., BLIP-2 [30] and PaLM-E [16], including a visual prompt generator (VPG), a linear projector and a backbone LLM. Typically, to tune the MLLM, only the VPG and the projector are updated, while the LLM is kept frozen. (b) This work investigates the VPG transferability across LLMs, including different LLM sizes and LLM types. + +# 1 Introduction + +Background. Recent years have witnessed a great rise in large-scale language models (LLMs) in ushering the human-like artificial intelligence. Text-based LLMs [42, 7, 44] are further enhanced by associating with other modalities such as vision, leading to the multimodal LLMs (MLLMs), such as BLIP-2 [30], Flamingo [2], GPT-4 [8] for multimodal dialog system, and PaLM-E [16] for embodied AI system. To construct a MLLM, a visual prompt generator (VPG) module (cf. Fig. 1(a)) that produces soft prompts for the input images/videos is added2 for bridging the gap between vision and language modalities. Currently, such architecture has been frequently adopted by many popular MLLMs [30, 27]. For example, BLIP-2 pre-trains a CLIP-ViT [43] combined with a Q-Former as VPG. To obtain the final MLLM, the VPG needs to be tuned. Ideally, the vast LLM backbone can remain untouched, leaving only the relatively lightweight VPG module to be fully or partially updated.3 + +Motivation. However, building a MLLM is inevitably computation-expensive, due to the huge overhead brought by the LLM. For example, training a BLIP-2 Flan ${ \mathrm { 7 } } 5 _ { \mathrm { X X L } }$ needs over 600 A100-GPU hours on over 100 million image-text pairs. Hopefully, transferring a pre-trained VPG (which is the main body of trainable parts) from an existing MLLM to a novel LLM instead of training from scratch,4 offers a promising solution. Intuitively, all the MLLMs literally can share the same VPG infrastructure and utility,5 which makes the VPG transfer theoretically feasible. In this work, we thus investigate the potential of transferring VPG across LLMs. + +Proposal. Specifically, this paper examines the transferability of VPG across LLMs: 1) with different sizes (the same type), i.e. , transfer across LLM sizes, and 2) across different LLM types, i.e. , transfer across LLM type, as illustrated in Fig. 1(b). + +• [Transfer across LLM Sizes (TaS)]. It has been a typical practice for LLM-related research [8] to validate the training strategy and the hyperparameter on smaller models (e.g., $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ ) and then scale up to larger ones (e.g., $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ ). It is thus worth exploring whether a VPG trained on a smaller LLM can be transferred to a larger LLM, resulting in reduced computational costs & data, and maintaining comparable performance. • [Transfer across LLM Types (TaT)]. With a well-tuned VPG for a type of LLM, it is interesting to see if VPG can be transferred to other types of LLMs even with different architectures (e.g., decoder v.s. encoder-decoder). If the transfer can be achieved, how to make it more efficient? + +We conduct a series of exploratory analyses (cf. $\ S 3 . 1 \ r ,$ ) to identify the key factors for transfer efficiency. Based on our empirical study, we design a two-stage transfer learning framework $( c f . \ \ S 3 . 2 )$ , namely VPGTrans, that includes a projector warm-up (stage-1) and vanilla fine-tuning (stage-2). For stage-1, we find that warming up the projector before VPG tuning can effectively reduce the training step for adapting a pre-trained VPG to a new LLM, and avoid the potential performance drop in the adaptation. To achieve an efficient warm-up, the projector will be well-initialized and then trained with an extremely large learning rate $( 5 \times l r )$ . For stage-2, there is a vanilla fine-tuning of both the VPG and projector. Despite its simplicity, VPGTrans is able to significantly speed up the VPG-transfer process without harming the performance. + +Results and Findings. Via extensive experiments on the transfer across LLM sizes and types (cf. $\ S 4$ & §5), we gain the following key observations: + +• VPGTrans helps to avoid the performance drop caused by directly inheriting the VPG and achieves at most 10 times acceleration for the small-to-large transfer across LLMs in the same type. • VPGTrans can also achieve comparable or better performance than training from scratch and achieve at most 5 times acceleration for the transfers between different model types. • Notably, our VPGTrans helps to achieve a BLIP-2 ViT-G $\mathbf { O P T _ { 2 . 7 B 6 . 7 B } }$ transfer with less than $10 \%$ of GPU hours and $1 0 . 7 \%$ of training data required for the original model training. • Furthermore, our framework can even outperform the original BLIP-2 $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ on most of the evaluated datasets, with a $\mathbf { + 2 . 9 }$ improvement on VQAv2 and a $\mathbf { + 3 . 4 }$ improvement on OKVQA. + +Our investigation further reveals some intriguing findings, for which we provide possible explanations: + +![](images/fe9874330a5d4f954361378d30db421b9ee5eb4c7b892fe951e704e0517ee88f.jpg) +Figure 2: Comparing the cost between training VPG from scratch vs. transferring VPG via our VPGTrans strategy. Note the LLM via VPGTrans is FlanT5 $\bf { \Phi } _ { \mathrm { X L \to X X L } }$ and $\mathrm { O P T } _ { 2 . 7 \mathrm { B } 6 . 7 \mathrm { B } }$ , respectively. + +• When conducting TaS from $\mathbf { L L M } _ { \mathrm { s r c } }$ to $\mathrm { L L M _ { \mathrm { t g t } } }$ , the size of $\mathbf { L L M } _ { \mathrm { s r c } }$ is not the larger the better. The transfer sometimes even follows a counterintuitive principle of “the smaller the $L L M _ { s r c }$ size, the more speed-up and better performance” $( c f . \ \ S 4 . 2 )$ . • When conducting TaT, efficient VPG transfer can not be achieved between two small LLMs with our VPGTrans, due to the large gap between small LLMs’ embedding space (cf. $\ S 5 . 2 )$ . + +Contributions. In this study, we show for the first time that effective VPG transfer across LLMs can be achieved under most conditions, suggesting that it is possible to build a new MLLM with considerably lower computational cost, as seen in Fig. 2. To summarize, we make the following key contributions: + +• Effective approach. We investigate the key factors for VPG-transfer efficiency and propose a two-stage transfer framework VPGTrans. The approach helps to achieve a highly-efficient VPG transfer across LLMs with less training data and even task improvements. • Intriguing findings. By exploring the VPG transfer across LLMs, we reveal several intriguing findings and provide potential explanations that will shed light on further research. • Open source. We showcase how to customize a novel GPT-4-like MLLM with our VPGTrans (cf. $\ S 6$ , and release two multimodal-version MLLMs: VL-LLaMA and VL-Vicuna. All codes and models is released at https://github.com/VPGTrans/VPGTrans. + +# 2 Preliminary + +This section will outlines the existing prevailing MLLMs, and elaborates on the settings of the exploratory analyses of these MLLMs. + +# 2.1 MLLM + +Architecture. As illustrated in Fig. 1(a), current MLLMs mostly adopt a common architecture, including a visual prompt generator (VPG), a projector, and a backbone LLM. Typically, VPG takes images/videos as inputs, and encodes the visual input into a fixed length of soft prompts. Then, a linear projector is employed to align the soft prompt’s dimension to LLM’s word embedding dimension. Finally, the LLM will generate sentences based on the information from the soft prompt. We list some of the recent representative MLLMs in Table 1. + +Table 1: MLLMs architectures and pre-training paradigm. †: it is a GPT-2-like LLM with relative position embeddings. + +
MLLMsVPGVPG TrainableLLMLLMTrainable
KOSMOS-1[19]CLIP [43]AllRand.Init.LMAll
Frozen [54]NF-ResNet-50 [6]NF-ResNet-50GPT-2-like† [42]No
Flamingo [2]NFNet-F6 [6]+Resampler [21]ResamplerChinchilla [18]Xattn-Dense
PaLM-E[16]ViT[15]/OSRT[49]AllPaLM[11]No
BLIP-2 [30]EVA-CLIP [52] + Q-Former [30]Q-FormerOPT[60]/Flan-T5[12]No
+ +Training Paradigm. Given a MLLM, typically the VPG and linear projector will be trained, fully or partially. For example, PaLM-E updates all of the parameters of VPG in the pre-training stage, while BLIP-2 and Flamingo freeze the ViTs and tune their Q-Former and Resampler, respectively. As the main part of the whole architecture, the LLM is usually frozen during the training or tuned only a small portion (e.g., 10B for Flamingo-80B). KOSMOS-1 is an exception, which does not use a pre-trained LLM but trains the LLM from scratch. Such a training paradigm typically results in much longer training time and data (both multimodal and pure text corpus). Recent works [30, 16] show that adopting an existing LLM and freezing all of its parameters can also achieve excellent performance with significantly reduced computational cost, which leads to the trend of adapting frozen pre-trained LLM. For example, BLIP-2 FlanT $\mathfrak { j } _ { \mathrm { X X L } }$ (12.1B) can achieve better zero-shot VQAv2 performance $( 6 5 . 0 \%$ in Acc.) compared with KOSMOS-1 $5 1 . 0 \%$ in Acc.) and Flamingo-80B $5 6 . 3 \%$ in Acc.). Thus, in this paper, we mainly focus on VPG transfer across frozen LLMs. + +# 2.2 Experiment Settings + +Architecture. We adopt BLIP-2’s architecture and training paradigm. In our exploration experiments, we consider using the VPG that consists of a CLIP ViT-L/14 [43], and a Q-Former that has already undergone a BLIP-like pre-training (the 1st stage pre-training in BLIP-2’s paper [30]). + +Training Data. For all of the exploration experiments, we adopt human-annotated COCO caption dataset [34] and web image-text pairs SBU dataset [40], which results in 1.4 million image-text pairs. + +Transfer Direction. For the small-to-large model transfer among the same type of LLMs, we investigate: 1) OPT [60] (decoder-only) series including 125M, 350M, 1.3B, and 2.7B, and 2) FlanT5 [12] (encoder-decoder) ranging base, large, and XL. For the transfer across different types of LLMs, we consider the ones of OPT and FlanT5 with similar sizes. + +Evaluation. To evaluate the performance of MLLMs, we choose two caption datasets: (1) COCO caption [34] (2) NoCaps [1], and three VQA datasets: (3) VQAv2 [4] (4) GQA [20] (5) OKVQA [37]. We make evaluations after the pre-training without task-specific fine-tuning and report the CIDEr [55] for all caption tasks and accuracy for all VQA tasks. + +Implementation Details. We follow the same implementation details of BLIP-2, via the open code.6 Concretely, we use FP16 and BFloat16 for OPT and FlanT5 respectively in the model training. For the learning rate, we first conduct a linear warm-up from 1e-6 to 1e-4, and then use a cosine learning rate schedule with the minimal $l r { = } 1 \mathrm { e } { - } 5$ for 10 epochs. Due to the limited data amount, we slightly decrease the batch size, which we find beneficial for the final performance. Specifically, we set the batch size of 1,728 and 1,152 for OPT and FlanT5-based models, respectively. + +# 3 Maximizing the Transfer Efficiency with a Two-stage Transfer Strategy + +In this section, we first identify the key factors for maximizing transfer efficiency, based on which we then motivate our solution for better transfer. + +# 3.1 Exploratory Analysis: Identifying Key Factors for VPG Transfer + +Via selected experiments of small-to-large transfer among OPT models, we can obtain the following key observations. More systematical comparisons are conducted in the later section (cf. §4). + +• Inheriting the trained VPG can accelerate training. To demonstrate this, we compare the convergence rates of VPG training on $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ from scratch, and inheriting VPG trained on $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ The patterns are shown in Fig. 3. Overall, we find that inheriting VPG trained on $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ accelerates convergence, particularly for two caption tasks. However, for datasets that require fine-grained visual perception such as VQAv2 and GQA, directly conduct continue training with an inherited VPG will harm the performance. We hypothesize that tuning VPG with a randomly initialized projector will compromise the existing fine-grained visual perception ability of VPG. The possible reason can be that, the VPG is typically a pre-trained model with powerful visual perception ability, and thus updating based on the gradient passed through a random projector will mislead the VPG at the initial steps [2, 33, 23]. + +• Warming up the linear projector can prevent performance drop and expedite VPG training. To verify this, we first conduct a warm-up training of the linear projector for 3 epochs, during which both VPG and LLM are frozen. Subsequently, we jointly train VPG and the projector and plot the performance curve in Fig. 4 (the warm-up process is not included in this figure). The results show that the performance drop observed in Fig.3 can be avoided in Fig. 4. Additionally, we observe that the warm-up training leads to fewer training steps required for VPG and projector joint training. However, we must emphasize that warming up is a costly step. In the case of a large LLM, such as 6.7B, the trainable parameters of BLIP-2’s VPG will account for less than $10 \%$ of the total parameters, where freezing VPG can only lead to a reduction of $5 . 4 \%$ of A100 hours (36.9 out of 684.0 A100 hours). We will elaborate on how to accelerate the linear projector warm-up in our later discussion (cf. 3.1). + +![](images/77bfd9db8e4a48b0aed1c3c4bbc1af6ae6f361ece62d3323ca16cbef7ebc2455.jpg) +Figure 3: Comparisons between i) inheriting VPG from $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ and training it with randomly initialized projector for $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ and ii) training VPG and randomly initialized projector for OPT350M from scratch. + +• Initializing $\mathbf { L L M _ { \mathrm { t g t } } } ^ { \prime }$ s projector with the help of the word converter can accelerate the linear projector warm-up. In fact, the VPG and projector trained on $\mathbf { L L M } _ { \mathrm { s r c } }$ have already learned how to map the visual content to $\mathbf { L L M } _ { \mathrm { s r c } }$ ’s understandable soft prompt [39]. If we can convert the $\mathbf { L L M } _ { \mathrm { s r c } }$ ’s soft prompt to $\mathrm { L L M _ { \mathrm { t g t } } }$ ’s soft prompt, we can directly get a VPG suitable for $\mathrm { L L M } _ { t g t }$ . One natural idea is to leverage the word embeddings of both models as a proxy for the soft prompt [25]. The intuition behind the scene is that, the soft prompt works in the same format as normal words. + +![](images/7cd279fb90695a457a34a0af9cf6c126e10153fd3c1f56570ca729680105284c.jpg) +Figure 4: First warming-up then transferring can avoid performance drop on VQAv2 and accelerate convergence for COCO caption. + +To validate our hypothesis, we conduct an experiment on the transfer from $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ to $\mathrm { O P T } _ { 1 . 3 \mathrm { B } }$ . After training a linear word embedding converter (cf. $\ S 3 . 2 ( \mathsf { b } ) )$ , we initialize the projector for $\mathrm { O P T } _ { 1 . 3 \mathrm { B } }$ with the merged linear operation of the projector for $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ and converter. As shown in Table 2, we observe that the initialization can reduce the 3 epochs’ warm-up to 2 epochs. + +• Linear projector warm-up enables faster convergence with an extremely large learning rate. To determine the most efficient transfer practice, we experiment with training the projector using different learning rates. Surprisingly, we find that the linear projector enables fast and stable convergence with an extremely large learning rate. Specifically, by setting the learning rate to 5 times of the original value, the COCO caption’s CIDEr score can reach 133.1 with 1 epoch training, which is higher than the 3 epochs results of w/o init. as shown in Table 2. + +Table 2: Comparison between linear projector warm-up with/without word embedding initialization. The metric is COCO caption’s CIDEr. + +
Epochw/ init.w/o init.
1130.2126.1
2132.7131.6
3133.4132.8
+ +# 3.2 A Two-stage VPG Transfer Framework + +By connecting all the dots as discussed above in $\ S 3 . 1$ , we now design our two-stage VPGTrans framework for more efficient VPG transfer. As shown in Fig. 5, the stage-1 of VPGTrans performs projector warm-up and the stage-2 carries out a vanilla fine-tuning. Our results demonstrate that the VPGTrans is simple yet effective that can significantly speed up the transfer without compromising performance. Detailed results are given in the later sections (cf. §4 & 5). + +# ▶ Stage-1: Projector Warm-up. + +(a) Inherit VPG. We first initialize the VPG for $\mathrm { L L M _ { \mathrm { t g t } } }$ with the VPG trained on $\mathbf { L L M } _ { \mathrm { s r c } }$ + +$( b )$ Projector Initialization. Then, we initialize the projector for $\mathrm { L L M _ { \mathrm { t g t } } }$ merged from the projector of $\mathbf { L L M } _ { \mathrm { s r c } }$ and a linear word converter. Formally, we define the linear projector of $\mathbf { L L M } _ { \mathrm { s r c } }$ as ${ \bar { f } } _ { s } ( x ) = W _ { s } x + b _ { s }$ , the linear projector for $\mathrm { L L M _ { \mathrm { t g t } } }$ as ${ \dot { \boldsymbol { f } } } _ { t } ( { \boldsymbol { x } } ) = W _ { t } { \boldsymbol { x } } + b _ { t }$ , and the word converter as $g _ { c } ( x ) = W _ { c } x + b _ { c }$ . + +The word converter is a linear layer trained with text-only caption data to convert the $\mathbf { L L M } _ { \mathrm { s r c } }$ ’s word embeddings to $\mathrm { L L M _ { \mathrm { t g t } } }$ ’s word embeddings. We experiment with optimizing losses based on cosine similarity or Euclidean distance, and observe no significant difference between the two losses. Thus we simply use cosine similarity in our experiments. In cases where $\mathbf { L L M } _ { \mathrm { s r c } }$ and $\mathrm { L L M _ { \mathrm { t g t } } }$ use different tokenization methods, we optimize based on the overlapped tokens. Formally, for every given token $k$ , we denote its word embeddings of $\mathbf { L L M } _ { \mathrm { s r c } }$ and $\mathrm { L L M _ { \mathrm { t g t } } }$ as $x _ { s }$ and $x _ { t }$ . Then, we minimize the loss: + +![](images/9f6e9e59a458377f952659afdc462b5701c4d509e76a8037345325d7cb356240.jpg) +Figure 5: Our two-stage VPGTrans framework. Stage-1 is to first (a) inherit the VPG of $\mathbf { L L M } _ { \mathrm { s r c } }$ and (b) initialize the projector by merging the projector of $\mathbf { L L M } _ { \mathrm { s r c } }$ and word converter. (c) Then the projector will be warmed up for 1 epoch with a large learning rate. Stage-2 is to (d) conduct a vanilla fine-tuning for the VPG and projector for $n$ epochs. + +$$ +\mathcal { L } = 1 - s i m ( g _ { c } ( x _ { s } ) , x _ { t } ) . +$$ + +Once we obtain the word converter $g _ { c } ( \cdot )$ , we can easily merge it with the projector of $\mathbf { L L M } _ { \mathrm { s r c } }$ as: + +$$ +f _ { t } ( x ) = f _ { s } ( g _ { c } ( x ) ) = W _ { s } ( W _ { c } x + b _ { c } ) + b _ { s } , +$$ + +resulting in $f _ { t }$ ’s weight and bias as $W _ { t } = W _ { s } W _ { c }$ and $b _ { t } = W _ { s } b _ { c } + b _ { s }$ . + +(c) Warm-up Training. Then, we only train the projector in this stage with a frozen VPG and LLM. Specifically, we train the projector for 1 epoch with 5 times of the normal learning rate. + +$\blacktriangleright$ Stage-2: Vanilla Fine-tuning. $( d )$ Vanilla Fine-tuning. In the final step, we conduct a joint training of VPG and projector for $n$ +epochs with a normal learning rate. + +# 4 Exp-I: Transfer across Different Model Sizes + +In this section, we conduct experiments to systematically illustrate the effectiveness of our VPGTrans and analyze the relationship between transfer efficiency and model size. For simplicity, we use TaS to represent the transfer across different model sizes. + +# 4.1 Experimental Settings + +In this part, we introduce baselines and transfer variants. For details about training data and implementation details, please refer to the experiment settings in the Preliminary (cf. 2.2). + +Baselines. We mainly compare our VPGTrans with training from scratch (TFS) and VPG inheritance (VPG Inherit), where we report their performance on the aforementioned 5 tasks without further task-specific fine-tuning. For our VPGTrans, the word converter training only requires updating a linear layer on tokenized text data and typically takes less than 10 minutes on 1 A100 GPU with less than 15G GPU memory. Meanwhile, freezing the VPG can lead to at least 14 A100 minutes speed-up per epoch. Therefore, we consider the whole stage-1 training as the 1st epoch for simplicity. + +Transfer Variants. We conducted experiments on transfer learning using 1) the OPT model across four different sizes: 125M, 350M, 1.3B, and 2.7B, and 2) the FlanT5 model across three sizes: base, large, and XL. However, we encountered significant instability during training with FlanT5large. As a result, we mainly present the transfer results between FlanT $\bar { \mathsf { \Sigma } } _ { \mathrm { b a s e } }$ and FlanT ${ \mathrm { 5 } } _ { \mathrm { X L } }$ . + +# 4.2 VPGTrans Enabling Faster Convergence without Performance Drop under TaS + +First of all, as shown in Fig. 6, our VPGTrans can consistently accelerate the model convergence. For COCO caption and NoCaps that require more training steps to converge, our VPGTrans (green line) can be higher than the other two lines (blue and orange lines). To give a quantitative evaluation of the speed-up rate, we show the speed-up rate in Table 3. The speed-up rate is calculated by considering the number of epochs reduced to achieve the best TFS performance on a particular dataset. Formally, given a dataset $D$ , TFS obtains the best performance $p$ on $D$ at epoch $e _ { \mathrm { { t f s } } }$ , whereas VPGTrans first achieves a better performance than $p$ at epoch $e _ { \mathrm { v t } }$ . The speed-up rate on $D$ is given by $\frac { e _ { \mathrm { t f s } } } { e _ { \mathrm { v t } } }$ . According to Table 3, our VPGTrans can achieve at least 4 times speed-up on $40 \%$ of Transfer-Task variants. Furthermore, for the two caption datasets, which take a long time to converge, our VPGTrans $\mathrm { O P T } _ { 1 2 5 \mathrm { M } 2 . 7 \mathrm { B } }$ delivers a 10 times speed-up. + +![](images/fa99f2b6f62c2ce0b26adb7222ba147e3060579929d0401acfa7be373dfb66d8.jpg) +Figure 6: Comparison between different methods across 3 TaS variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. Please refer to Appendix§C for other transfer variants. + +Table 3: The speed-up rate of our VPGTrans compared with training from scratch (TFS). The symbol "-" means VPGTrans can not achieve better performance than TFS. + +
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.73.01.05.05.0
OPT125M→1.3B9.010.09.0=2.0
OPT350M→1.3B4.55.09.02.02.0
OPT125M→2.7B10.010.02.02.03.0
OPT350M→2.7B10.010.02.013.0
OPT1.3B-→2.7B3.33.32.0-1.5
FlanT5base→XL1.01.13.04.02.0
FlanT5xL →OPT2.7B5.05.02.02.03.0
OPT2.7B→FlanT5xL1.72.012.01
+ +Moreover, when compared with the VPG inherit in Fig. 6, our VPGTrans can achieve a higher speed-up rate on all of the variants on caption tasks, and achieve better performance on most variants except for $\mathrm { O P T } _ { 1 . 3 \mathrm { B } 2 . 7 \mathrm { B } }$ . We refer the readers to Appendix $\mathrm { \ : \ : \xi { 8 C . 3 } }$ for more comparisons. + +We provide interesting findings with respect to the efficiency transfer by VPGTrans in the following. + +• The smaller size of $\mathbf { L L M _ { s r c } } ,$ the easier the transfer. In our OPT based experiments, we notice an interesting phenomenon: when transferring to a given $\mathrm { L L M _ { \mathrm { t g t } } }$ , both the convergence rate and optimal performance are roughly inversely proportional to the size of $\mathbf { L L M } _ { \mathrm { s r c } }$ . For example, as shown in Table 3, the $\mathrm { O P T } _ { 1 2 5 \mathrm { M } 2 . 7 \mathrm { B } }$ and $\mathrm { O P T } _ { 3 5 0 \mathbf { M } 2 . 7 \mathbf { B } }$ have much higher speed-up rate than $\mathrm { O P T } _ { 1 . 3 \mathrm { B } 2 . 7 \mathrm { B } }$ on all of the datasets. Meanwhile, as demonstrated in Fig. 6, the optimal performance of $\mathrm { O P T } _ { 1 2 5 \mathrm { M } 2 . 7 \mathrm { B } }$ is better than $\mathrm { O P T } _ { 1 . 3 \mathrm { B } 2 . 7 \mathrm { B } }$ on 3 VQA tasks. + +Table 4: Comparison between models built with our VPGTrans and the original BLIP-2 ViT-G $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ and BLIP-2 ViT-G FlanT $5 _ { \mathrm { X X L } }$ . + +
ModelsVQAv2 valGQA test-devOKVQA testGPU hourstraining data
BLIP-2 ViT-G OPT6.7B54.336.436.4631.5129M
BLIP-2 ViT-G OPT2.7B-→6.7B (0urs)57.236.239.859.013.8M
VL-LLaMA7B (ours)58.137.537.467.113.8M
BLIP-2 ViT-G FlanT5xXL65.244.745.9684.0121.6M
BLIP-2 ViT-G FlanT5xL-→XXL (ours)65.245.045.032.45.3M
+ +![](images/f11488c7c1f59614c4c51593b02dafbaba22e12307d37ed4aa376c6986e89da4.jpg) +Figure 8: Comparison between different methods across $2 \ \mathrm { T a T }$ variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. Please refer to Appendix§D for other transfer variants. + +We hypothesize that training VPG on larger OPT will have a worse influence on VPG’s existing fine-grained perception ability, which might be caused by the enlarging embedding dimensions. To validate our hypothesis, we fix the VPG weight and only tune linear projectors to test VPGs trained on different $\mathbf { L L M } _ { s r c }$ through cross-size transfer. The SPICE [3] metric on COCO caption is used to evaluate the VPG’s visual perception ability, where SPICE is specifically designed for visual concept perception in captions. As shown in Fig. 7, for each row, given the $\mathbf { L L M } _ { t a r }$ , the performance of VPG trained on smaller $\mathbf { L L M } _ { s r c }$ can outperform the larger ones in most conditions, which indicates a better visual perception ability of VPG trained on smaller $\mathbf { L L M } _ { s r c }$ . Therefore, adapting a VPG from a smaller OPT model which is less affected, is helpful to take fewer steps to reach the TFS’s best performance and achieve even better performance. + +# 4.3 Scale-up Experiments + +To validate the effectiveness of our VPGTrans on the real-world application level, we experiment on transferring from BLIP-2 ViT-G $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ to $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ and from BLIP-2 ViT-G FlanT $5 _ { \mathrm { X L } }$ to FlanT5XXL. Please refer to Appendix $\mathrm { \ : \ S C . 4 }$ for implementation details. + +Speed-up with non-degenerated performances. As shown in Table 4, we can see that (1) $\mathbf { O P T } _ { 2 . 7 \mathbf { B } 6 . 7 \mathbf { B } } ;$ : our VPGTrans achieves a 10.7 times speed-up with only $1 0 . 7 \%$ training data, while the performance on VQAv2 and OKVQA have over 2 points improvement. (2) FlanT $\bf \Pi ^ { \prime } 5 _ { X L X X L }$ : VPGTrans can achieve 21.1 times speed-up with less than $5 \%$ training data while achieving the same performance on VQAv2, higher performance on GQA and slightly lower performance on OKVQA. Note that continuing training the FlanT $\mathsf { \Pi } _ { \mathrm { X L X X L } }$ only shows improvement on VQAv2 and GQA. Thus, we do not show a checkpoint with more training steps. + +![](images/0c22991255013c317bd03c3ea2c511cd0e2a0a970e3a3141779a5f18e0221710.jpg) +Figure 7: The confusion matrix. Only linear layers are trained for VPG evaluation. Models are tested on COCO caption with SPICE metric to compare the VPGs trained on different $\mathrm { L L M } _ { s r c }$ . + +Table 5: Comparison between our VL-Vicuna and SOTA MLLMs for multimodal conversation. The evaluation is done by Multimodality Chatbot Arena platform via user voting. + +
RankModel Elo Rating
1LLaMA-Adapter v2 [17] 1023.0
2 LLaVA [35]1019.9
3VL-Vicuna (ours) 1012.1
4MiniGPT-4 [62] 1011.9
5InstructBLIP[13] 999.5
6mPLUG-Owl[59] 996.3
7 Otter [26]981.5
8 BLIP-2 [30]955.8
+ +# 5 Exp-II: Transfer across Different Model Types + +In this section, we further investigate the transfer across different model types. For simplicity, we mark this type of transfer as TaT. + +# 5.1 Experimental Settings + +In this part, we introduce baselines and transfer variants. For details about training data and implementation details, please refer to the experiment settings in the Preliminary (cf. 2.2). + +Baselines. We mainly compare our VPGTrans with training from scratch (TFS), and report the performance on the aforementioned 5 tasks. Other details (cf. 4.1) are totally the same with TaS experiments. + +Transfer Variants. We conducted experiments on transfer learning between 1) $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ and FlanT $\mathrm { \Delta } 5 _ { \mathrm { b a s e } }$ , and 2) $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ and FlanT $5 _ { \mathrm { X L } }$ . + +# 5.2 VPGTrans Enabling Faster Convergence only on Large LLMs under TaT + +• There is no speed-up of TaT between two small LLMs. A finding is that on TaT our VPGTrans does not show speed-up for small models, and even shows a degeneration of training speed in the initial several epochs. As shown in Fig. 8, when transferring from $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ to FlanT $5 _ { \mathrm { b a s e } }$ , the convergence speed of VPGTrans is even slower than TFS in the initial several epochs. + +• Speed-up of VPGTrans happens in large LLMs. However, when moving to the large LLMs like $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ and FlanT ${ \mathrm { 5 } } _ { \mathrm { X L } }$ , there is an obvious speed-up. As shown in Table 3, we can see at least 2 times speed-up when transferring from $\mathrm { F l a n T 5 } _ { \mathrm { X L } }$ to $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ . We empirically find that the soft prompts for larger LLM are more linear transferrable among different LLM types. As shown in Fig. 8, when transferring between ${ \mathrm { F l a n T } } 5 _ { \mathrm { b a s e } }$ and $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ , the VGPTrans’ 1st epoch results on two caption datasets are limited, where only a linear operation can be trained. The result of $\mathrm { O P T } _ { 3 5 0 \mathrm { M } } { } \mathrm { F l a n T } 5 _ { \mathrm { b a s e } }$ on the COCO caption is even near to zero. By contrast, the result of $\mathrm { F l a n T } 5 _ { \mathrm { X L } } { } \mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ with our VPGTrans are obviously higher than TFS. We hypothesize that larger LLM typically learned more generalizable text embeddings and share more similarity among relative word distances, which enables an easier VPG transfer. + +# 6 Customizing New MLLMs with Any LLMs + +Above, we thoroughly certify the efficacy of our proposed VPGTrans approach for higher efficient transfer of VPG. In this section, we illustrate how to apply the VPGTrans framework for VPG transfer to customize new MLLMs with any LLMs. + +VL-LLaMA. By applying our VPGTrans, we can equip the recently released LLaMA [53] model with a VPG trained on BLIP-2 $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ to perceive the visual information. As shown in Table 4, we can see that our VL-LLaMA can outperform the original BLIP-2 $\mathrm { O P T } _ { 6 . 7 B }$ on all datasets. + +VL-Vicuna. An exciting application of our VPGTrans is to build a GPT-4 [8] style multimodal conversation chatbot. To achieve our goal, we employ Vicuna [10] as our base LLM. Similarly, we transfer the VPG from BLIP-2 $\mathrm { O P T } _ { 6 . 7 \mathrm { B } }$ , and add an extra instruction tuning using MiniGPT-4’s selfinstruct data [62]. We compare our model with MiniGPT-4 in Fig. 9. When compared to MiniGPT-4, our VL-Vicuna shows better visual perception ability. Please refer to Appendix $\mathrm { \Omega ^ { \ S E } }$ for more cases. In addition, we also report the ranking and ELO ratings for our VL-Vicuna compared with the other 7 SOTA multimodal chatbots in Table 5 (The evaluation is done by Multimodality Chatbot Arena platform7). The results show the effectiveness of our VL-Vicuna compared with existing SOTA models. + +![](images/26ccc1e5deaa7a37d5a1df096d13105e4a9097978362507df2699ccb5e3303ae.jpg) +Figure 9: Comparison between MiniGPT-4 and our VL-Vicuna. + +# 7 Conclusion + +In this work, we conduct a comprehensive investigation to the problem of VPG transferability across LLMs. We first explore the key factors for maximizing the transfer efficiency under the VPG transfer across different LLM sizes and types. Based on the key findings, we propose a novel two-stage transfer framework, namely VPGTrans, which can help to achieve comparable or better performance while significantly reducing the training costs. Moreover, a list of important findings and possible reasons behind them are shown and discussed. Finally, we demonstrate the practical value of our VPGTrans, by customizing new MLLMs via VPG transfer from existing MLLMs. + +# Acknowledgments + +This research is supported by ${ \mathrm { N E x T } } + +$ Lab, Singapore Ministry of Education Academic Research Fund Tier 2 under MOE’s official grant number T2EP20221-0023, and CCF-Baidu Open Fund. + +# References + +IEEE/CVF international conference on computer vision, pages 8948–8957, 2019. +[2] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. Advances in Neural Information Processing Systems, 35:23716–23736, 2022. +[3] Peter Anderson, Basura Fernando, Mark Johnson, and Stephen Gould. Spice: Semantic propositional image caption evaluation. 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MiniGPT-4: Enhancing vision-language understanding with advanced large language models. arXiv preprint arXiv:2304.10592, 2023. + +# A Related Work + +# A.1 Vision and Language Models + +Vision and language models (VLMs) [50, 29, 32, 31] aim at understanding visual and textual information with a single model. Previously, the VLM mainly employs a pre-trained object detector as its feature extractor and conducts unsupervised training on a huge amount of image-text pairs. For example, VL-Bert [50], ViL-Bert [36] and Uniter [9] adopt Faster-RCNN [47] to extract image information into object features and take advantage of the masked language model as their pre-training task. Later, due to the prevalence of vision transformer (ViT), the VLM paradigm is turned into end-to-end training with a ViT as the visual encoder. The representative works include ALBEF [28], BLIP [29], and BEIT-v3 [56], which show the state-of-the-arts supervised training performance on a wide range of downstream tasks. + +Recently, LLMs have shown their remarkable capability as zero/few-shot learners [7] and a series of emergent abilities [57] like in-context learning [7], and chain-of-thoughts reasoning [58]. A new paradigm, i.e., MLLMs is created by associating the VLM or pure vision encoders with LLMs. As we illustrated before, the VLM or visual encoders are typically able to convert the input vision signals into LLM-understandable soft prompts, and thus we call them VPG. The MLLMs advance in inheriting the great potentials of the backbone LLMs, and thus are capable of achieving excellent zero/few-shot performances [2, 30] on downstream tasks or be equipped with visual planning ability [16]. However, connecting the VPG to the existing LLMs with further tuning is costly. Even the BLIP-2 [30], targeted at efficient training, will take over 600 A100 GPU hours on over 100M image-text pairs for its largest model. With this regard, our proposed VPGTrans can effectively reduce the cost of building new MLLMs with the help of existing ones. + +# A.2 Prompt Transfer + +In this paper, we investigate the VPG transfer, where the soft prompt is to represent the content of specific inputs like images and videos. In addition to the content prompt, the more explored soft prompt is the task prompt [24, 61, 22], where a sequence of soft prompts are tuned to assist the pre-trained models to achieve better performance on specific tasks. There have already been some works exploring the transferability of task prompts. For example, Su et al. [51] conducts a series of experiments to illustrate the transferability across tasks and models. Specifically, Su et al. [51] find that the transfer between similar tasks is beneficial for training speed-up and better performance. Lester et al. [25] proposes to recycle soft prompts across models with vocab-to-vocab transformations, or linear-combination transformations. Note that our word converter initialization is similar to the idea of vocab-to-vocab transformations. However, we do not observe a zero-shot transfer in our experiments like them, which indicates a potential difference between the content prompts and task prompts. Another way of soft prompt transfer [14] is to conduct prompt tuning on discrete prompts, and thus the discrete prompts can be directly shared across models. Different from these task prompts transfer works, our VPG transfer scenario actually suffers from fewer limitations. For example, the task soft prompts transfer suffers from the dimension change problem, where the main body of the trainable parameters should be processed. However, our VPG (the main trainable parameters) can naturally be shared among LLMs with different embedding dimensions and leave the dimension change problem to a simple projector with ignorable parameters. + +# B Extended Findings in Exploratory Analysis + +In this section, we show extended findings of exploratory analysis (cf. §3.1). + +• 1. Merely tuning the projector can not achieve the best performance. We want to clarify that merely tuning the projector is insufficient for achieving the best performance. Notably, as shown in Fig. 10, significant performance gaps are observed between the “only linear” (green curve) and “train from scratch” (orange curve) approaches for COCO caption and NoCaps. Therefore, if the goal is to build a multimodal conversation robot using carefully collected dialog data, training only the linear projector is insufficient to align with the provided data. + +![](images/33540963e9f2bd811dc970d5f78f00224ddc6d489b9621249332294a34459900.jpg) +Figure 10: Comparisons between i) inheriting VPG from $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ and training it with randomly initialized projector for $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ and ii) training VPG and randomly initialized projector for $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ from scratch. iii) training only the projector. + +![](images/32f1e17321f3f1584e1644e783c867e1bde2b301cc63f5fd1793a651ad638427.jpg) +Figure 11: Interpreting generated soft prompts for $\mathrm { O P T } _ { 1 2 5 \mathrm { M } }$ with nearest words. + +• 2. Word embedding converter can not replace a trained linear projector. As demonstrated in Fig. 11, we observe a common pattern: the last token of soft prompts is closest to EOS, while the middle tokens represent the image content. Such a phenomenon indicates a similarity between soft prompts and word embeddings. However, they are not identical. For instance, the norm of soft prompts is typically around 10 times the average norm of word embeddings. It is important to note that the linear projector initialization cannot replace the warm-up training. Using only the linear projector initialization even yields a random performance. We believe that a better understanding of how prompt works will further benefit the VPG’s transfer learning. + +• 3. The projector warm-up is robust to a larger learning rate, while VPG can not. The first thing we want to clarify is that the 5 times normal learning rate will result in a training crash for VPG. Additionally, we find that although increasing the learning rate to 10 times in the projector warm-up does not yield any additional acceleration, the projector can converge without crashing during training. + +# C Extended TaS Experiments + +In this section, we first illustrate extending findings of TaS experiments (cf. $\ S 4 )$ . Then, we introduce the implementation details of scale-up experiments. We also plot a more complete version of Fig. 6 in Fig. 13. + +# C.1 VPGTrans Enabling Stable Training under TaS + +As we illustrated before, $\mathrm { F l a n T } 5 _ { \mathrm { l a r g e } }$ training is extremely unstable. Even when the learning rate is adjusted to one-tenth of its original value, the model does not converge or shows a very slow convergence rate after 4 epochs. However, we find that by lowering the learning rate for stage-2 training, our VPGTrans can achieve stable training on $\mathrm { F l a n T } 5 _ { \mathrm { l a r g e } }$ . We plot the performance curve of the COCO caption in Fig. 12. + +# C.2 VPGTrans Enabling Training with Less Data under TaS + +We empirically find that TaS can reduce the requirement for the amount of training data. By reducing the training data to only COCO, we find no obvious performance drop. However, we want to stress that the retained data should be of high quality, which means that if the same number of SBU data + +![](images/3e4dd23620bef89b02685655c6d2638d085099edb3dda84cf3da77d5e5a29fd1.jpg) +Figure 12: Comparison of training FlanT5 $\mathrm { l a r g e }$ using different methods. + +Table 6: The speed-up rate of our VPGTrans compared with VPG inherit. The symbol "-" means VPGTrans can not achieve better performance than VPG inherit. + +
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.12.710.06.06.0
OPT125M→1.3B-2.71-1
OPT350M→1.3B11.71.02.01.0
OPT125M→2.7B3.03.03.01.03.0
OPT350M→2.7B3.34.51.01.01.0
OPT1.3B→2.7B2.33.0111
FlanT5base-→XL11.01.89.00.4
+ +is retained, the performance especially for captioning will drop. The conclusion can also be found in Table 4. We refer the readers to the next subsection for more details. + +# C.3 Comparison between VPGTrans and VPG Inherit + +As shown in Fig. 13, the green line (our VPGTrans) can be higher than the orange line (VPG inherit) for the majority of various conditions. Especially when considering the best performance of different tasks, our VPGTrans can achieve better performance than VPG inherit (non “-” in Table 6) for over $74 \%$ Transfer-Task variants. Moreover, among the variants that our VPGTrans can achieve better performance, our VPGTrans can also achieve a speed-up on $6 9 . 2 \%$ conditions. + +# C.4 Scale-up Experiment Implementation Details + +The scale-up experiment refers to the results in Table 4. We try to imitate BLIP-2’s pre-training data composition. First of all, two human-annotated datasets COCO and VG are used. SBU is also used. Then, BLIP-2 uses BLIP to generate captions for the 115M web images and rank them with CLIP ViT-L/14. We also adopt similar synthetic data from Laion-COCO.8 We report the concrete number of data we use in Table 4. For the stage-1 training, we keep the same as the previous validation experiments where COCO and SBU are used for warm-up with a 5 times the learning rate. Then, we use COCO, VG, and Laion-COCO for the stage-2 training. Note that we have tried to include Laion-COCO and VG for the stage-1 training, but found no obvious difference and thus use COCO and SBU for simplicity. For VL-Vicuna, to align with the conversation scenario, we further fine-tune our VL-Vicuna with MiniGPT-4’s self-instruct data, which is 3,439 image-text pairs. + +![](images/4392bb6f62287ea11fe623c93cb50e6fb2a414be6c70d57c54cdd18ec14375cf.jpg) +Figure 13: Comparison between different methods across 7 TaS variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. + +Table 7: Comparison between tuning only the projector (i.e. linear transfer) and the best results the VPGTrans achieve. + +
TransferCOCO CaptionVQAv2
linearbestlinearbest
FlanT5base → OPT350M110.8136.540.444.2
FlanT5xL →OPT2.7B132.1139.350.450.3
OPT350M→FlanT5base1.1122.134.049.9
OPT2.7B→FlanT5xL106.5133.251.353.5
+ +# D Extended TaT Experiments + +In this section, we mainly illustrate extending findings of TaT experiments $( c f . \ \ S 5 )$ . We plot a more complete version of Fig. 8 in Fig. 14. + +# D.1 Linear Transfer Gap between Different LLM’s Visual Prompts + +As illustrated in Section 5.2, it is more difficult to transfer between two small LLMs with our VPGTrans due to the weaker linear transferability between two small LLMs’ visual prompts. To better support our results, we compare the results of tuning only the projector (i.e. linear transfer) and the best results the VPGTrans can achieve. As shown in Table 7, we can see that when conducting transfers between two large models $( \mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ and ${ \mathrm { F l a n T } } 5 _ { \mathrm { b a s e } }$ ), the performance is typically far from the optimal results. However, when we transfer between $\mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ and $\mathrm { F l a n T 5 } _ { \mathrm { X L } }$ , the linear performance is near to the optimal performance. There is a 25.7 points gap between $\mathrm { F l a n T 5 _ { b a s e } }$ $\mathrm { O P T } _ { 3 5 0 \mathrm { M } }$ ’s linear and best on COCO caption datasets but only 7.2 points gap between FlanT $5 _ { \mathrm { X L } }$ $ \mathrm { O P T } _ { 2 . 7 \mathrm { B } }$ ’s linear and best. If considering the transfer between the small to large LLMs under TaT, both the VPG’s visual perception ability and transfer gap should be considered. We leave the systematical exploration for future works. + +# E Extended Results for MLLM Customization + +We show more comparisons between VL-Vicuna and MiniGPT-4 in Fig. 15. We can see that our VL-Vicuna has better visual perception ability. For example, when MiniGPT-4 falsely recognizes the three people in the image as two in the first example, our VL-Vicuna can not only recognize the number of people but also tell their roles. Moreover, our VL-Vicuna can successfully link the vision content with external knowledge. In the third example, our VL-Vicuna can recognize Leonardo and link the content with his films like Titanic. + +# F Potential Impact and Limitations + +Our VPGTrans is designed for building new MLLMs with lower computational cost, i.e., shorter training time and less training data. With an already pre-trained MLLM, VPGTrans enables fast VPG transfer to build either a larger MLLM or a MLLM with a different type of LLM. We hope VPGTrans can facilitate teams in LLM communicty to customize their MLLMs with reduced cost. There are also possible limitations of the current version of VPGTrans. The first one is that our VPGTrans should rely on some already-aligned VPGs. The second potential limitation is that the VPGTrans-built MLLMs still suffer from the common problems of content generation AI systems [45, 46, 48, 7]. For example, the VL-Vicuna may make up some sentences with falsely recognized visual facts, like what is shown in Fig. 16. It is worth exploring associating our VPGTrans with training safer models [5, 41, 38]. + +![](images/6b71e6150cb974ca3f4ff4a2b8513068f693caa29c5ead48ea547b61a244d464.jpg) +Figure 14: Comparison between different methods across 4 TaS variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. + +![](images/782e3db48f255a657cbf6437cf9fcc5416f9d47fc7e9c50b97780d26130971cd.jpg) +Figure 15: Comparison between MiniGPT-4 and our VL-Vicuna. + +![](images/ca663ea28691202a79fd05df32d71dd60e6ff115c315efa2a9179c903804c7cf.jpg) +Figure 16: Failure cases of our VL-Vicuna. \ No newline at end of file diff --git a/parse/dev/716PvHoDct/716PvHoDct_content_list.json b/parse/dev/716PvHoDct/716PvHoDct_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cc7bc81af8d0337289cf3179cf1dac5a2c3e493b --- /dev/null +++ b/parse/dev/716PvHoDct/716PvHoDct_content_list.json @@ -0,0 +1,1734 @@ +[ + { + "type": "text", + "text": "VPGTrans: Transfer Visual Prompt Generator across LLMs ", + "text_level": 1, + "bbox": [ + 215, + 122, + 781, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ao Zhang 1 Hao Fei 1∗ Yuan $\\mathbf { Y a o } ^ { 2 * }$ Wei Ji 1 Li Li 1 Zhiyuan Liu 2 Tat-Seng Chua 1 1 $\\mathbf { N E x T + + }$ Lab, School of Computing, National University of Singapore 2Department of Computer Science and Technology, Tsinghua University zhanga6@outlook.com haofei37@nus.edu.sg yaoyuanthu@163.com ", + "bbox": [ + 184, + 219, + 843, + 280 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Project: https://vpgtrans.github.io ", + "bbox": [ + 331, + 289, + 663, + 306 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 315, + 535, + 332 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since developing a new multimodal LLM (MLLM) by pre-training on a tremendous amount of image-text pairs from scratch is exceedingly resource-consuming, connecting an existing LLM with a comparatively lightweight visual prompt generator (VPG) becomes a feasible paradigm. However, further tuning the VPG component of the MLLM still incurs significant computational costs, such as thousands of GPU hours and millions of training data points. An alternative solution is to transfer an existing VPG from one MLLM to the target MLLM. In this work, we investigate VPG transferability across LLMs for the first time, aiming to reduce the cost of VPG transfer. Specifically, we explore VPG transfer across different LLM sizes (e.g., small-to-large) and types. We identify key factors to maximize the transfer efficiency, based on which we develop a simple yet highly effective two-stage transfer framework, called VPGTrans. Notably, it enables VPG transfer from BLIP-2 $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ to BLIP-2 $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ with less than $10 \\%$ of GPU hours using only $1 0 . 7 \\%$ of the training data compared to training a VPG for $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ from scratch. Furthermore, we provide a series of intriguing findings and discuss potential explanations behind them. Finally, we showcase the practical value of our VPGTrans approach, by customizing two novel MLLMs, including VL-LLaMA and VL-Vicuna, with the recently released LLaMA and Vicuna LLMs. ", + "bbox": [ + 232, + 347, + 767, + 595 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/47f10b1dd10c7582861b3a81730cf2754b8fef72df8b0732479b81981c6d570f.jpg", + "image_caption": [ + "Figure 1: (a) The general architecture of MLLMs, e.g., BLIP-2 [30] and PaLM-E [16], including a visual prompt generator (VPG), a linear projector and a backbone LLM. Typically, to tune the MLLM, only the VPG and the projector are updated, while the LLM is kept frozen. (b) This work investigates the VPG transferability across LLMs, including different LLM sizes and LLM types. " + ], + "image_footnote": [], + "bbox": [ + 178, + 617, + 815, + 744 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 825, + 312, + 842 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Background. Recent years have witnessed a great rise in large-scale language models (LLMs) in ushering the human-like artificial intelligence. Text-based LLMs [42, 7, 44] are further enhanced by associating with other modalities such as vision, leading to the multimodal LLMs (MLLMs), such as BLIP-2 [30], Flamingo [2], GPT-4 [8] for multimodal dialog system, and PaLM-E [16] for embodied AI system. To construct a MLLM, a visual prompt generator (VPG) module (cf. Fig. 1(a)) that produces soft prompts for the input images/videos is added2 for bridging the gap between vision and language modalities. Currently, such architecture has been frequently adopted by many popular MLLMs [30, 27]. For example, BLIP-2 pre-trains a CLIP-ViT [43] combined with a Q-Former as VPG. To obtain the final MLLM, the VPG needs to be tuned. Ideally, the vast LLM backbone can remain untouched, leaving only the relatively lightweight VPG module to be fully or partially updated.3 ", + "bbox": [ + 174, + 849, + 825, + 878 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Motivation. However, building a MLLM is inevitably computation-expensive, due to the huge overhead brought by the LLM. For example, training a BLIP-2 Flan ${ \\mathrm { 7 } } 5 _ { \\mathrm { X X L } }$ needs over 600 A100-GPU hours on over 100 million image-text pairs. Hopefully, transferring a pre-trained VPG (which is the main body of trainable parts) from an existing MLLM to a novel LLM instead of training from scratch,4 offers a promising solution. Intuitively, all the MLLMs literally can share the same VPG infrastructure and utility,5 which makes the VPG transfer theoretically feasible. In this work, we thus investigate the potential of transferring VPG across LLMs. ", + "bbox": [ + 173, + 224, + 825, + 321 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Proposal. Specifically, this paper examines the transferability of VPG across LLMs: 1) with different sizes (the same type), i.e. , transfer across LLM sizes, and 2) across different LLM types, i.e. , transfer across LLM type, as illustrated in Fig. 1(b). ", + "bbox": [ + 178, + 330, + 821, + 372 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• [Transfer across LLM Sizes (TaS)]. It has been a typical practice for LLM-related research [8] to validate the training strategy and the hyperparameter on smaller models (e.g., $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ ) and then scale up to larger ones (e.g., $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ ). It is thus worth exploring whether a VPG trained on a smaller LLM can be transferred to a larger LLM, resulting in reduced computational costs & data, and maintaining comparable performance. • [Transfer across LLM Types (TaT)]. With a well-tuned VPG for a type of LLM, it is interesting to see if VPG can be transferred to other types of LLMs even with different architectures (e.g., decoder v.s. encoder-decoder). If the transfer can be achieved, how to make it more efficient? ", + "bbox": [ + 192, + 378, + 825, + 492 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We conduct a series of exploratory analyses (cf. $\\ S 3 . 1 \\ r ,$ ) to identify the key factors for transfer efficiency. Based on our empirical study, we design a two-stage transfer learning framework $( c f . \\ \\ S 3 . 2 )$ , namely VPGTrans, that includes a projector warm-up (stage-1) and vanilla fine-tuning (stage-2). For stage-1, we find that warming up the projector before VPG tuning can effectively reduce the training step for adapting a pre-trained VPG to a new LLM, and avoid the potential performance drop in the adaptation. To achieve an efficient warm-up, the projector will be well-initialized and then trained with an extremely large learning rate $( 5 \\times l r )$ . For stage-2, there is a vanilla fine-tuning of both the VPG and projector. Despite its simplicity, VPGTrans is able to significantly speed up the VPG-transfer process without harming the performance. ", + "bbox": [ + 173, + 497, + 825, + 623 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Results and Findings. Via extensive experiments on the transfer across LLM sizes and types (cf. $\\ S 4$ & §5), we gain the following key observations: ", + "bbox": [ + 176, + 632, + 825, + 660 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• VPGTrans helps to avoid the performance drop caused by directly inheriting the VPG and achieves at most 10 times acceleration for the small-to-large transfer across LLMs in the same type. • VPGTrans can also achieve comparable or better performance than training from scratch and achieve at most 5 times acceleration for the transfers between different model types. • Notably, our VPGTrans helps to achieve a BLIP-2 ViT-G $\\mathbf { O P T _ { 2 . 7 B 6 . 7 B } }$ transfer with less than $10 \\%$ of GPU hours and $1 0 . 7 \\%$ of training data required for the original model training. • Furthermore, our framework can even outperform the original BLIP-2 $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ on most of the evaluated datasets, with a $\\mathbf { + 2 . 9 }$ improvement on VQAv2 and a $\\mathbf { + 3 . 4 }$ improvement on OKVQA. ", + "bbox": [ + 194, + 666, + 826, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our investigation further reveals some intriguing findings, for which we provide possible explanations: ", + "bbox": [ + 176, + 804, + 825, + 820 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/fe9874330a5d4f954361378d30db421b9ee5eb4c7b892fe951e704e0517ee88f.jpg", + "image_caption": [ + "Figure 2: Comparing the cost between training VPG from scratch vs. transferring VPG via our VPGTrans strategy. Note the LLM via VPGTrans is FlanT5 $\\bf { \\Phi } _ { \\mathrm { X L \\to X X L } }$ and $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } 6 . 7 \\mathrm { B } }$ , respectively. " + ], + "image_footnote": [], + "bbox": [ + 179, + 89, + 820, + 190 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• When conducting TaS from $\\mathbf { L L M } _ { \\mathrm { s r c } }$ to $\\mathrm { L L M _ { \\mathrm { t g t } } }$ , the size of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ is not the larger the better. The transfer sometimes even follows a counterintuitive principle of “the smaller the $L L M _ { s r c }$ size, the more speed-up and better performance” $( c f . \\ \\ S 4 . 2 )$ . • When conducting TaT, efficient VPG transfer can not be achieved between two small LLMs with our VPGTrans, due to the large gap between small LLMs’ embedding space (cf. $\\ S 5 . 2 )$ . ", + "bbox": [ + 192, + 238, + 825, + 311 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Contributions. In this study, we show for the first time that effective VPG transfer across LLMs can be achieved under most conditions, suggesting that it is possible to build a new MLLM with considerably lower computational cost, as seen in Fig. 2. To summarize, we make the following key contributions: ", + "bbox": [ + 176, + 318, + 825, + 373 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Effective approach. We investigate the key factors for VPG-transfer efficiency and propose a two-stage transfer framework VPGTrans. The approach helps to achieve a highly-efficient VPG transfer across LLMs with less training data and even task improvements. • Intriguing findings. By exploring the VPG transfer across LLMs, we reveal several intriguing findings and provide potential explanations that will shed light on further research. • Open source. We showcase how to customize a novel GPT-4-like MLLM with our VPGTrans (cf. $\\ S 6$ , and release two multimodal-version MLLMs: VL-LLaMA and VL-Vicuna. All codes and models is released at https://github.com/VPGTrans/VPGTrans. ", + "bbox": [ + 194, + 380, + 825, + 497 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Preliminary ", + "text_level": 1, + "bbox": [ + 173, + 515, + 305, + 531 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This section will outlines the existing prevailing MLLMs, and elaborates on the settings of the exploratory analyses of these MLLMs. ", + "bbox": [ + 176, + 539, + 823, + 566 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 MLLM ", + "text_level": 1, + "bbox": [ + 173, + 575, + 266, + 589 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Architecture. As illustrated in Fig. 1(a), current MLLMs mostly adopt a common architecture, including a visual prompt generator (VPG), a projector, and a backbone LLM. Typically, VPG takes images/videos as inputs, and encodes the visual input into a fixed length of soft prompts. Then, a linear projector is employed to align the soft prompt’s dimension to LLM’s word embedding dimension. Finally, the LLM will generate sentences based on the information from the soft prompt. We list some of the recent representative MLLMs in Table 1. ", + "bbox": [ + 173, + 594, + 826, + 678 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/94453217e65d59a603b7f1daf46d79a8af881dcfce2be3530856d09bbad2fe11.jpg", + "table_caption": [ + "Table 1: MLLMs architectures and pre-training paradigm. †: it is a GPT-2-like LLM with relative position embeddings. " + ], + "table_footnote": [], + "table_body": "
MLLMsVPGVPG TrainableLLMLLMTrainable
KOSMOS-1[19]CLIP [43]AllRand.Init.LMAll
Frozen [54]NF-ResNet-50 [6]NF-ResNet-50GPT-2-like† [42]No
Flamingo [2]NFNet-F6 [6]+Resampler [21]ResamplerChinchilla [18]Xattn-Dense
PaLM-E[16]ViT[15]/OSRT[49]AllPaLM[11]No
BLIP-2 [30]EVA-CLIP [52] + Q-Former [30]Q-FormerOPT[60]/Flan-T5[12]No
", + "bbox": [ + 176, + 723, + 825, + 804 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Training Paradigm. Given a MLLM, typically the VPG and linear projector will be trained, fully or partially. For example, PaLM-E updates all of the parameters of VPG in the pre-training stage, while BLIP-2 and Flamingo freeze the ViTs and tune their Q-Former and Resampler, respectively. As the main part of the whole architecture, the LLM is usually frozen during the training or tuned only a small portion (e.g., 10B for Flamingo-80B). KOSMOS-1 is an exception, which does not use a pre-trained LLM but trains the LLM from scratch. Such a training paradigm typically results in much longer training time and data (both multimodal and pure text corpus). Recent works [30, 16] show that adopting an existing LLM and freezing all of its parameters can also achieve excellent performance with significantly reduced computational cost, which leads to the trend of adapting frozen pre-trained LLM. For example, BLIP-2 FlanT $\\mathfrak { j } _ { \\mathrm { X X L } }$ (12.1B) can achieve better zero-shot VQAv2 performance $( 6 5 . 0 \\%$ in Acc.) compared with KOSMOS-1 $5 1 . 0 \\%$ in Acc.) and Flamingo-80B $5 6 . 3 \\%$ in Acc.). Thus, in this paper, we mainly focus on VPG transfer across frozen LLMs. ", + "bbox": [ + 173, + 814, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 Experiment Settings ", + "text_level": 1, + "bbox": [ + 174, + 170, + 354, + 184 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Architecture. We adopt BLIP-2’s architecture and training paradigm. In our exploration experiments, we consider using the VPG that consists of a CLIP ViT-L/14 [43], and a Q-Former that has already undergone a BLIP-like pre-training (the 1st stage pre-training in BLIP-2’s paper [30]). ", + "bbox": [ + 174, + 189, + 825, + 231 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Training Data. For all of the exploration experiments, we adopt human-annotated COCO caption dataset [34] and web image-text pairs SBU dataset [40], which results in 1.4 million image-text pairs. ", + "bbox": [ + 171, + 238, + 825, + 266 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Transfer Direction. For the small-to-large model transfer among the same type of LLMs, we investigate: 1) OPT [60] (decoder-only) series including 125M, 350M, 1.3B, and 2.7B, and 2) FlanT5 [12] (encoder-decoder) ranging base, large, and XL. For the transfer across different types of LLMs, we consider the ones of OPT and FlanT5 with similar sizes. ", + "bbox": [ + 174, + 273, + 825, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Evaluation. To evaluate the performance of MLLMs, we choose two caption datasets: (1) COCO caption [34] (2) NoCaps [1], and three VQA datasets: (3) VQAv2 [4] (4) GQA [20] (5) OKVQA [37]. We make evaluations after the pre-training without task-specific fine-tuning and report the CIDEr [55] for all caption tasks and accuracy for all VQA tasks. ", + "bbox": [ + 174, + 338, + 825, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Implementation Details. We follow the same implementation details of BLIP-2, via the open code.6 Concretely, we use FP16 and BFloat16 for OPT and FlanT5 respectively in the model training. For the learning rate, we first conduct a linear warm-up from 1e-6 to 1e-4, and then use a cosine learning rate schedule with the minimal $l r { = } 1 \\mathrm { e } { - } 5$ for 10 epochs. Due to the limited data amount, we slightly decrease the batch size, which we find beneficial for the final performance. Specifically, we set the batch size of 1,728 and 1,152 for OPT and FlanT5-based models, respectively. ", + "bbox": [ + 174, + 401, + 825, + 484 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 Maximizing the Transfer Efficiency with a Two-stage Transfer Strategy ", + "text_level": 1, + "bbox": [ + 174, + 503, + 799, + 521 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we first identify the key factors for maximizing transfer efficiency, based on which we then motivate our solution for better transfer. ", + "bbox": [ + 173, + 529, + 823, + 556 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Exploratory Analysis: Identifying Key Factors for VPG Transfer ", + "text_level": 1, + "bbox": [ + 176, + 565, + 665, + 580 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Via selected experiments of small-to-large transfer among OPT models, we can obtain the following key observations. More systematical comparisons are conducted in the later section (cf. §4). ", + "bbox": [ + 174, + 584, + 825, + 612 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Inheriting the trained VPG can accelerate training. To demonstrate this, we compare the convergence rates of VPG training on $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ from scratch, and inheriting VPG trained on $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ The patterns are shown in Fig. 3. Overall, we find that inheriting VPG trained on $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ accelerates convergence, particularly for two caption tasks. However, for datasets that require fine-grained visual perception such as VQAv2 and GQA, directly conduct continue training with an inherited VPG will harm the performance. We hypothesize that tuning VPG with a randomly initialized projector will compromise the existing fine-grained visual perception ability of VPG. The possible reason can be that, the VPG is typically a pre-trained model with powerful visual perception ability, and thus updating based on the gradient passed through a random projector will mislead the VPG at the initial steps [2, 33, 23]. ", + "bbox": [ + 173, + 618, + 825, + 756 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Warming up the linear projector can prevent performance drop and expedite VPG training. To verify this, we first conduct a warm-up training of the linear projector for 3 epochs, during which both VPG and LLM are frozen. Subsequently, we jointly train VPG and the projector and plot the performance curve in Fig. 4 (the warm-up process is not included in this figure). The results show that the performance drop observed in Fig.3 can be avoided in Fig. 4. Additionally, we observe that the warm-up training leads to fewer training steps required for VPG and projector joint training. However, we must emphasize that warming up is a costly step. In the case of a large LLM, such as 6.7B, the trainable parameters of BLIP-2’s VPG will account for less than $10 \\%$ of the total parameters, where freezing VPG can only lead to a reduction of $5 . 4 \\%$ of A100 hours (36.9 out of 684.0 A100 hours). We will elaborate on how to accelerate the linear projector warm-up in our later discussion (cf. 3.1). ", + "bbox": [ + 173, + 763, + 826, + 887 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/77bfd9db8e4a48b0aed1c3c4bbc1af6ae6f361ece62d3323ca16cbef7ebc2455.jpg", + "image_caption": [ + "Figure 3: Comparisons between i) inheriting VPG from $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ and training it with randomly initialized projector for $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ and ii) training VPG and randomly initialized projector for OPT350M from scratch. " + ], + "image_footnote": [], + "bbox": [ + 174, + 88, + 821, + 185 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 247, + 825, + 275 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Initializing $\\mathbf { L L M _ { \\mathrm { t g t } } } ^ { \\prime }$ s projector with the help of the word converter can accelerate the linear projector warm-up. In fact, the VPG and projector trained on $\\mathbf { L L M } _ { \\mathrm { s r c } }$ have already learned how to map the visual content to $\\mathbf { L L M } _ { \\mathrm { s r c } }$ ’s understandable soft prompt [39]. If we can convert the $\\mathbf { L L M } _ { \\mathrm { s r c } }$ ’s soft prompt to $\\mathrm { L L M _ { \\mathrm { t g t } } }$ ’s soft prompt, we can directly get a VPG suitable for $\\mathrm { L L M } _ { t g t }$ . One natural idea is to leverage the word embeddings of both models as a proxy for the soft prompt [25]. The intuition behind the scene is that, the soft prompt works in the same format as normal words. ", + "bbox": [ + 174, + 285, + 517, + 450 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/7cd279fb90695a457a34a0af9cf6c126e10153fd3c1f56570ca729680105284c.jpg", + "image_caption": [ + "Figure 4: First warming-up then transferring can avoid performance drop on VQAv2 and accelerate convergence for COCO caption. " + ], + "image_footnote": [], + "bbox": [ + 531, + 285, + 818, + 393 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To validate our hypothesis, we conduct an experiment on the transfer from $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ to $\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } }$ . After training a linear word embedding converter (cf. $\\ S 3 . 2 ( \\mathsf { b } ) )$ , we initialize the projector for $\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } }$ with the merged linear operation of the projector for $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ and converter. As shown in Table 2, we observe that the initialization can reduce the 3 epochs’ warm-up to 2 epochs. ", + "bbox": [ + 173, + 457, + 826, + 512 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Linear projector warm-up enables faster convergence with an extremely large learning rate. To determine the most efficient transfer practice, we experiment with training the projector using different learning rates. Surprisingly, we find that the linear projector enables fast and stable convergence with an extremely large learning rate. Specifically, by setting the learning rate to 5 times of the original value, the COCO caption’s CIDEr score can reach 133.1 with 1 epoch training, which is higher than the 3 epochs results of w/o init. as shown in Table 2. ", + "bbox": [ + 173, + 522, + 581, + 660 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/cd33d023b6f6689d7c4442d01a49092578bbc66b247c7fa4e7c6b9ae28ddb257.jpg", + "table_caption": [ + "Table 2: Comparison between linear projector warm-up with/without word embedding initialization. The metric is COCO caption’s CIDEr. " + ], + "table_footnote": [], + "table_body": "
Epochw/ init.w/o init.
1130.2126.1
2132.7131.6
3133.4132.8
", + "bbox": [ + 604, + 595, + 810, + 666 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 A Two-stage VPG Transfer Framework ", + "text_level": 1, + "bbox": [ + 174, + 671, + 488, + 686 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "By connecting all the dots as discussed above in $\\ S 3 . 1$ , we now design our two-stage VPGTrans framework for more efficient VPG transfer. As shown in Fig. 5, the stage-1 of VPGTrans performs projector warm-up and the stage-2 carries out a vanilla fine-tuning. Our results demonstrate that the VPGTrans is simple yet effective that can significantly speed up the transfer without compromising performance. Detailed results are given in the later sections (cf. §4 & 5). ", + "bbox": [ + 173, + 688, + 825, + 757 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "▶ Stage-1: Projector Warm-up. ", + "text_level": 1, + "bbox": [ + 176, + 763, + 397, + 776 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(a) Inherit VPG. We first initialize the VPG for $\\mathrm { L L M _ { \\mathrm { t g t } } }$ with the VPG trained on $\\mathbf { L L M } _ { \\mathrm { s r c } }$ ", + "bbox": [ + 196, + 776, + 784, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$( b )$ Projector Initialization. Then, we initialize the projector for $\\mathrm { L L M _ { \\mathrm { t g t } } }$ merged from the projector of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ and a linear word converter. Formally, we define the linear projector of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ as ${ \\bar { f } } _ { s } ( x ) = W _ { s } x + b _ { s }$ , the linear projector for $\\mathrm { L L M _ { \\mathrm { t g t } } }$ as ${ \\dot { \\boldsymbol { f } } } _ { t } ( { \\boldsymbol { x } } ) = W _ { t } { \\boldsymbol { x } } + b _ { t }$ , and the word converter as $g _ { c } ( x ) = W _ { c } x + b _ { c }$ . ", + "bbox": [ + 174, + 794, + 825, + 849 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The word converter is a linear layer trained with text-only caption data to convert the $\\mathbf { L L M } _ { \\mathrm { s r c } }$ ’s word embeddings to $\\mathrm { L L M _ { \\mathrm { t g t } } }$ ’s word embeddings. We experiment with optimizing losses based on cosine similarity or Euclidean distance, and observe no significant difference between the two losses. Thus we simply use cosine similarity in our experiments. In cases where $\\mathbf { L L M } _ { \\mathrm { s r c } }$ and $\\mathrm { L L M _ { \\mathrm { t g t } } }$ use different tokenization methods, we optimize based on the overlapped tokens. Formally, for every given token $k$ , we denote its word embeddings of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ and $\\mathrm { L L M _ { \\mathrm { t g t } } }$ as $x _ { s }$ and $x _ { t }$ . Then, we minimize the loss: ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/9f6e9e59a458377f952659afdc462b5701c4d509e76a8037345325d7cb356240.jpg", + "image_caption": [ + "Figure 5: Our two-stage VPGTrans framework. Stage-1 is to first (a) inherit the VPG of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ and (b) initialize the projector by merging the projector of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ and word converter. (c) Then the projector will be warmed up for 1 epoch with a large learning rate. Stage-2 is to (d) conduct a vanilla fine-tuning for the VPG and projector for $n$ epochs. " + ], + "image_footnote": [], + "bbox": [ + 178, + 88, + 818, + 263 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 343, + 823, + 372 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/edf5dd7ab5c5cd15ca97dde9e49f32a7095d71c219935d5719170eb0d12aaa3b.jpg", + "text": "$$\n\\mathcal { L } = 1 - s i m ( g _ { c } ( x _ { s } ) , x _ { t } ) .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 378, + 588, + 396 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Once we obtain the word converter $g _ { c } ( \\cdot )$ , we can easily merge it with the projector of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ as: ", + "bbox": [ + 169, + 409, + 805, + 424 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/3e11151cb993ca4517eeedfcb5e4b960020e044f8e94b8a460cbfc290e08d2d4.jpg", + "text": "$$\nf _ { t } ( x ) = f _ { s } ( g _ { c } ( x ) ) = W _ { s } ( W _ { c } x + b _ { c } ) + b _ { s } ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 430, + 642, + 448 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "resulting in $f _ { t }$ ’s weight and bias as $W _ { t } = W _ { s } W _ { c }$ and $b _ { t } = W _ { s } b _ { c } + b _ { s }$ . ", + "bbox": [ + 171, + 454, + 637, + 469 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(c) Warm-up Training. Then, we only train the projector in this stage with a frozen VPG and LLM. Specifically, we train the projector for 1 epoch with 5 times of the normal learning rate. ", + "bbox": [ + 178, + 476, + 825, + 503 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\blacktriangleright$ Stage-2: Vanilla Fine-tuning. $( d )$ Vanilla Fine-tuning. In the final step, we conduct a joint training of VPG and projector for $n$ \nepochs with a normal learning rate. ", + "bbox": [ + 174, + 511, + 825, + 554 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Exp-I: Transfer across Different Model Sizes ", + "text_level": 1, + "bbox": [ + 173, + 565, + 578, + 583 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we conduct experiments to systematically illustrate the effectiveness of our VPGTrans and analyze the relationship between transfer efficiency and model size. For simplicity, we use TaS to represent the transfer across different model sizes. ", + "bbox": [ + 174, + 589, + 825, + 632 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Experimental Settings ", + "text_level": 1, + "bbox": [ + 174, + 640, + 367, + 656 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this part, we introduce baselines and transfer variants. For details about training data and implementation details, please refer to the experiment settings in the Preliminary (cf. 2.2). ", + "bbox": [ + 174, + 659, + 821, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. We mainly compare our VPGTrans with training from scratch (TFS) and VPG inheritance (VPG Inherit), where we report their performance on the aforementioned 5 tasks without further task-specific fine-tuning. For our VPGTrans, the word converter training only requires updating a linear layer on tokenized text data and typically takes less than 10 minutes on 1 A100 GPU with less than 15G GPU memory. Meanwhile, freezing the VPG can lead to at least 14 A100 minutes speed-up per epoch. Therefore, we consider the whole stage-1 training as the 1st epoch for simplicity. ", + "bbox": [ + 174, + 694, + 825, + 779 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Transfer Variants. We conducted experiments on transfer learning using 1) the OPT model across four different sizes: 125M, 350M, 1.3B, and 2.7B, and 2) the FlanT5 model across three sizes: base, large, and XL. However, we encountered significant instability during training with FlanT5large. As a result, we mainly present the transfer results between FlanT $\\bar { \\mathsf { \\Sigma } } _ { \\mathrm { b a s e } }$ and FlanT ${ \\mathrm { 5 } } _ { \\mathrm { X L } }$ . ", + "bbox": [ + 174, + 785, + 825, + 842 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 VPGTrans Enabling Faster Convergence without Performance Drop under TaS ", + "text_level": 1, + "bbox": [ + 171, + 851, + 766, + 866 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "First of all, as shown in Fig. 6, our VPGTrans can consistently accelerate the model convergence. For COCO caption and NoCaps that require more training steps to converge, our VPGTrans (green line) can be higher than the other two lines (blue and orange lines). To give a quantitative evaluation of the speed-up rate, we show the speed-up rate in Table 3. The speed-up rate is calculated by considering the number of epochs reduced to achieve the best TFS performance on a particular dataset. Formally, given a dataset $D$ , TFS obtains the best performance $p$ on $D$ at epoch $e _ { \\mathrm { { t f s } } }$ , whereas VPGTrans first achieves a better performance than $p$ at epoch $e _ { \\mathrm { v t } }$ . The speed-up rate on $D$ is given by $\\frac { e _ { \\mathrm { t f s } } } { e _ { \\mathrm { v t } } }$ . According to Table 3, our VPGTrans can achieve at least 4 times speed-up on $40 \\%$ of Transfer-Task variants. Furthermore, for the two caption datasets, which take a long time to converge, our VPGTrans $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } 2 . 7 \\mathrm { B } }$ delivers a 10 times speed-up. ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/fa99f2b6f62c2ce0b26adb7222ba147e3060579929d0401acfa7be373dfb66d8.jpg", + "image_caption": [ + "Figure 6: Comparison between different methods across 3 TaS variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. Please refer to Appendix§C for other transfer variants. " + ], + "image_footnote": [], + "bbox": [ + 173, + 87, + 818, + 385 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/735cfabd5eaf835fdab0f525c67b4238badd21a0a5333b44ae186f0d0a64bb2e.jpg", + "table_caption": [ + "Table 3: The speed-up rate of our VPGTrans compared with training from scratch (TFS). The symbol \"-\" means VPGTrans can not achieve better performance than TFS. " + ], + "table_footnote": [], + "table_body": "
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.73.01.05.05.0
OPT125M→1.3B9.010.09.0=2.0
OPT350M→1.3B4.55.09.02.02.0
OPT125M→2.7B10.010.02.02.03.0
OPT350M→2.7B10.010.02.013.0
OPT1.3B-→2.7B3.33.32.0-1.5
FlanT5base→XL1.01.13.04.02.0
FlanT5xL →OPT2.7B5.05.02.02.03.0
OPT2.7B→FlanT5xL1.72.012.01
", + "bbox": [ + 238, + 482, + 758, + 632 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 645, + 826, + 744 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Moreover, when compared with the VPG inherit in Fig. 6, our VPGTrans can achieve a higher speed-up rate on all of the variants on caption tasks, and achieve better performance on most variants except for $\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } 2 . 7 \\mathrm { B } }$ . We refer the readers to Appendix $\\mathrm { \\ : \\ : \\xi { 8 C . 3 } }$ for more comparisons. ", + "bbox": [ + 174, + 751, + 825, + 792 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We provide interesting findings with respect to the efficiency transfer by VPGTrans in the following. ", + "bbox": [ + 171, + 797, + 823, + 814 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• The smaller size of $\\mathbf { L L M _ { s r c } } ,$ the easier the transfer. In our OPT based experiments, we notice an interesting phenomenon: when transferring to a given $\\mathrm { L L M _ { \\mathrm { t g t } } }$ , both the convergence rate and optimal performance are roughly inversely proportional to the size of $\\mathbf { L L M } _ { \\mathrm { s r c } }$ . For example, as shown in Table 3, the $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } 2 . 7 \\mathrm { B } }$ and $\\mathrm { O P T } _ { 3 5 0 \\mathbf { M } 2 . 7 \\mathbf { B } }$ have much higher speed-up rate than $\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } 2 . 7 \\mathrm { B } }$ on all of the datasets. Meanwhile, as demonstrated in Fig. 6, the optimal performance of $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } 2 . 7 \\mathrm { B } }$ is better than $\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } 2 . 7 \\mathrm { B } }$ on 3 VQA tasks. ", + "bbox": [ + 174, + 827, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d0874583b76a417e4b0dfbb2d89721189385fb1b3587a24f0151f4dc299270ea.jpg", + "table_caption": [ + "Table 4: Comparison between models built with our VPGTrans and the original BLIP-2 ViT-G $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ and BLIP-2 ViT-G FlanT $5 _ { \\mathrm { X X L } }$ . " + ], + "table_footnote": [], + "table_body": "
ModelsVQAv2 valGQA test-devOKVQA testGPU hourstraining data
BLIP-2 ViT-G OPT6.7B54.336.436.4631.5129M
BLIP-2 ViT-G OPT2.7B-→6.7B (0urs)57.236.239.859.013.8M
VL-LLaMA7B (ours)58.137.537.467.113.8M
BLIP-2 ViT-G FlanT5xXL65.244.745.9684.0121.6M
BLIP-2 ViT-G FlanT5xL-→XXL (ours)65.245.045.032.45.3M
", + "bbox": [ + 176, + 122, + 825, + 237 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/f11488c7c1f59614c4c51593b02dafbaba22e12307d37ed4aa376c6986e89da4.jpg", + "image_caption": [ + "Figure 8: Comparison between different methods across $2 \\ \\mathrm { T a T }$ variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. Please refer to Appendix§D for other transfer variants. " + ], + "image_footnote": [], + "bbox": [ + 173, + 260, + 818, + 469 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We hypothesize that training VPG on larger OPT will have a worse influence on VPG’s existing fine-grained perception ability, which might be caused by the enlarging embedding dimensions. To validate our hypothesis, we fix the VPG weight and only tune linear projectors to test VPGs trained on different $\\mathbf { L L M } _ { s r c }$ through cross-size transfer. The SPICE [3] metric on COCO caption is used to evaluate the VPG’s visual perception ability, where SPICE is specifically designed for visual concept perception in captions. As shown in Fig. 7, for each row, given the $\\mathbf { L L M } _ { t a r }$ , the performance of VPG trained on smaller $\\mathbf { L L M } _ { s r c }$ can outperform the larger ones in most conditions, which indicates a better visual perception ability of VPG trained on smaller $\\mathbf { L L M } _ { s r c }$ . Therefore, adapting a VPG from a smaller OPT model which is less affected, is helpful to take fewer steps to reach the TFS’s best performance and achieve even better performance. ", + "bbox": [ + 173, + 535, + 825, + 674 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 Scale-up Experiments ", + "text_level": 1, + "bbox": [ + 176, + 689, + 364, + 704 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To validate the effectiveness of our VPGTrans on the real-world application level, we experiment on transferring from BLIP-2 ViT-G $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ to $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ and from BLIP-2 ViT-G FlanT $5 _ { \\mathrm { X L } }$ to FlanT5XXL. Please refer to Appendix $\\mathrm { \\ : \\ S C . 4 }$ for implementation details. ", + "bbox": [ + 174, + 712, + 614, + 767 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Speed-up with non-degenerated performances. As shown in Table 4, we can see that (1) $\\mathbf { O P T } _ { 2 . 7 \\mathbf { B } 6 . 7 \\mathbf { B } } ;$ : our VPGTrans achieves a 10.7 times speed-up with only $1 0 . 7 \\%$ training data, while the performance on VQAv2 and OKVQA have over 2 points improvement. (2) FlanT $\\bf \\Pi ^ { \\prime } 5 _ { X L X X L }$ : VPGTrans can achieve 21.1 times speed-up with less than $5 \\%$ training data while achieving the same performance on VQAv2, higher performance on GQA and slightly lower performance on OKVQA. Note that continuing training the FlanT $\\mathsf { \\Pi } _ { \\mathrm { X L X X L } }$ only shows improvement on VQAv2 and GQA. Thus, we do not show a checkpoint with more training steps. ", + "bbox": [ + 174, + 781, + 614, + 920 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0c22991255013c317bd03c3ea2c511cd0e2a0a970e3a3141779a5f18e0221710.jpg", + "image_caption": [ + "Figure 7: The confusion matrix. Only linear layers are trained for VPG evaluation. Models are tested on COCO caption with SPICE metric to compare the VPGs trained on different $\\mathrm { L L M } _ { s r c }$ . " + ], + "image_footnote": [], + "bbox": [ + 627, + 693, + 823, + 819 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/3cdbe82013fe7a893e96c4ba7285c8552f502c8f7f9744360825f63340768d48.jpg", + "table_caption": [ + "Table 5: Comparison between our VL-Vicuna and SOTA MLLMs for multimodal conversation. The evaluation is done by Multimodality Chatbot Arena platform via user voting. " + ], + "table_footnote": [], + "table_body": "
RankModel Elo Rating
1LLaMA-Adapter v2 [17] 1023.0
2 LLaVA [35]1019.9
3VL-Vicuna (ours) 1012.1
4MiniGPT-4 [62] 1011.9
5InstructBLIP[13] 999.5
6mPLUG-Owl[59] 996.3
7 Otter [26]981.5
8 BLIP-2 [30]955.8
", + "bbox": [ + 338, + 123, + 661, + 261 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Exp-II: Transfer across Different Model Types ", + "text_level": 1, + "bbox": [ + 173, + 286, + 594, + 303 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we further investigate the transfer across different model types. For simplicity, we mark this type of transfer as TaT. ", + "bbox": [ + 173, + 308, + 823, + 335 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.1 Experimental Settings ", + "text_level": 1, + "bbox": [ + 174, + 342, + 367, + 357 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this part, we introduce baselines and transfer variants. For details about training data and implementation details, please refer to the experiment settings in the Preliminary (cf. 2.2). ", + "bbox": [ + 173, + 361, + 823, + 388 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Baselines. We mainly compare our VPGTrans with training from scratch (TFS), and report the performance on the aforementioned 5 tasks. Other details (cf. 4.1) are totally the same with TaS experiments. ", + "bbox": [ + 174, + 392, + 825, + 435 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Transfer Variants. We conducted experiments on transfer learning between 1) $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ and FlanT $\\mathrm { \\Delta } 5 _ { \\mathrm { b a s e } }$ , and 2) $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ and FlanT $5 _ { \\mathrm { X L } }$ . ", + "bbox": [ + 174, + 443, + 823, + 473 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.2 VPGTrans Enabling Faster Convergence only on Large LLMs under TaT ", + "text_level": 1, + "bbox": [ + 173, + 482, + 723, + 497 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• There is no speed-up of TaT between two small LLMs. A finding is that on TaT our VPGTrans does not show speed-up for small models, and even shows a degeneration of training speed in the initial several epochs. As shown in Fig. 8, when transferring from $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ to FlanT $5 _ { \\mathrm { b a s e } }$ , the convergence speed of VPGTrans is even slower than TFS in the initial several epochs. ", + "bbox": [ + 173, + 501, + 825, + 556 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• Speed-up of VPGTrans happens in large LLMs. However, when moving to the large LLMs like $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ and FlanT ${ \\mathrm { 5 } } _ { \\mathrm { X L } }$ , there is an obvious speed-up. As shown in Table 3, we can see at least 2 times speed-up when transferring from $\\mathrm { F l a n T 5 } _ { \\mathrm { X L } }$ to $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ . We empirically find that the soft prompts for larger LLM are more linear transferrable among different LLM types. As shown in Fig. 8, when transferring between ${ \\mathrm { F l a n T } } 5 _ { \\mathrm { b a s e } }$ and $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ , the VGPTrans’ 1st epoch results on two caption datasets are limited, where only a linear operation can be trained. The result of $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } } { } \\mathrm { F l a n T } 5 _ { \\mathrm { b a s e } }$ on the COCO caption is even near to zero. By contrast, the result of $\\mathrm { F l a n T } 5 _ { \\mathrm { X L } } { } \\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ with our VPGTrans are obviously higher than TFS. We hypothesize that larger LLM typically learned more generalizable text embeddings and share more similarity among relative word distances, which enables an easier VPG transfer. ", + "bbox": [ + 173, + 560, + 825, + 699 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Customizing New MLLMs with Any LLMs ", + "text_level": 1, + "bbox": [ + 174, + 712, + 566, + 729 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Above, we thoroughly certify the efficacy of our proposed VPGTrans approach for higher efficient transfer of VPG. In this section, we illustrate how to apply the VPGTrans framework for VPG transfer to customize new MLLMs with any LLMs. ", + "bbox": [ + 174, + 741, + 823, + 782 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "VL-LLaMA. By applying our VPGTrans, we can equip the recently released LLaMA [53] model with a VPG trained on BLIP-2 $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ to perceive the visual information. As shown in Table 4, we can see that our VL-LLaMA can outperform the original BLIP-2 $\\mathrm { O P T } _ { 6 . 7 B }$ on all datasets. ", + "bbox": [ + 173, + 791, + 823, + 833 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "VL-Vicuna. An exciting application of our VPGTrans is to build a GPT-4 [8] style multimodal conversation chatbot. To achieve our goal, we employ Vicuna [10] as our base LLM. Similarly, we transfer the VPG from BLIP-2 $\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }$ , and add an extra instruction tuning using MiniGPT-4’s selfinstruct data [62]. We compare our model with MiniGPT-4 in Fig. 9. When compared to MiniGPT-4, our VL-Vicuna shows better visual perception ability. Please refer to Appendix $\\mathrm { \\Omega ^ { \\ S E } }$ for more cases. In addition, we also report the ranking and ELO ratings for our VL-Vicuna compared with the other 7 SOTA multimodal chatbots in Table 5 (The evaluation is done by Multimodality Chatbot Arena platform7). The results show the effectiveness of our VL-Vicuna compared with existing SOTA models. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/26ccc1e5deaa7a37d5a1df096d13105e4a9097978362507df2699ccb5e3303ae.jpg", + "image_caption": [ + "Figure 9: Comparison between MiniGPT-4 and our VL-Vicuna. " + ], + "image_footnote": [], + "bbox": [ + 176, + 88, + 818, + 513 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 568, + 825, + 622 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 628, + 299, + 643 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this work, we conduct a comprehensive investigation to the problem of VPG transferability across LLMs. We first explore the key factors for maximizing the transfer efficiency under the VPG transfer across different LLM sizes and types. Based on the key findings, we propose a novel two-stage transfer framework, namely VPGTrans, which can help to achieve comparable or better performance while significantly reducing the training costs. Moreover, a list of important findings and possible reasons behind them are shown and discussed. Finally, we demonstrate the practical value of our VPGTrans, by customizing new MLLMs via VPG transfer from existing MLLMs. ", + "bbox": [ + 174, + 645, + 825, + 742 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments ", + "text_level": 1, + "bbox": [ + 176, + 761, + 328, + 779 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This research is supported by ${ \\mathrm { N E x T } } + +$ Lab, Singapore Ministry of Education Academic Research Fund Tier 2 under MOE’s official grant number T2EP20221-0023, and CCF-Baidu Open Fund. ", + "bbox": [ + 174, + 792, + 825, + 820 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 176, + 839, + 266, + 854 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "IEEE/CVF international conference on computer vision, pages 8948–8957, 2019. \n[2] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. Advances in Neural Information Processing Systems, 35:23716–23736, 2022. \n[3] Peter Anderson, Basura Fernando, Mark Johnson, and Stephen Gould. Spice: Semantic propositional image caption evaluation. In Computer Vision–ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part V 14, pages 382–398. 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", + "bbox": [ + 171, + 61, + 828, + 914 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 168, + 88, + 828, + 922 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 58, + 828, + 917 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 828, + 407 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A Related Work ", + "text_level": 1, + "bbox": [ + 174, + 89, + 326, + 106 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 Vision and Language Models ", + "text_level": 1, + "bbox": [ + 174, + 123, + 416, + 138 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Vision and language models (VLMs) [50, 29, 32, 31] aim at understanding visual and textual information with a single model. Previously, the VLM mainly employs a pre-trained object detector as its feature extractor and conducts unsupervised training on a huge amount of image-text pairs. For example, VL-Bert [50], ViL-Bert [36] and Uniter [9] adopt Faster-RCNN [47] to extract image information into object features and take advantage of the masked language model as their pre-training task. Later, due to the prevalence of vision transformer (ViT), the VLM paradigm is turned into end-to-end training with a ViT as the visual encoder. The representative works include ALBEF [28], BLIP [29], and BEIT-v3 [56], which show the state-of-the-arts supervised training performance on a wide range of downstream tasks. ", + "bbox": [ + 173, + 151, + 825, + 276 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Recently, LLMs have shown their remarkable capability as zero/few-shot learners [7] and a series of emergent abilities [57] like in-context learning [7], and chain-of-thoughts reasoning [58]. A new paradigm, i.e., MLLMs is created by associating the VLM or pure vision encoders with LLMs. As we illustrated before, the VLM or visual encoders are typically able to convert the input vision signals into LLM-understandable soft prompts, and thus we call them VPG. The MLLMs advance in inheriting the great potentials of the backbone LLMs, and thus are capable of achieving excellent zero/few-shot performances [2, 30] on downstream tasks or be equipped with visual planning ability [16]. However, connecting the VPG to the existing LLMs with further tuning is costly. Even the BLIP-2 [30], targeted at efficient training, will take over 600 A100 GPU hours on over 100M image-text pairs for its largest model. With this regard, our proposed VPGTrans can effectively reduce the cost of building new MLLMs with the help of existing ones. ", + "bbox": [ + 174, + 282, + 825, + 434 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 Prompt Transfer ", + "text_level": 1, + "bbox": [ + 174, + 457, + 333, + 472 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this paper, we investigate the VPG transfer, where the soft prompt is to represent the content of specific inputs like images and videos. In addition to the content prompt, the more explored soft prompt is the task prompt [24, 61, 22], where a sequence of soft prompts are tuned to assist the pre-trained models to achieve better performance on specific tasks. There have already been some works exploring the transferability of task prompts. For example, Su et al. [51] conducts a series of experiments to illustrate the transferability across tasks and models. Specifically, Su et al. [51] find that the transfer between similar tasks is beneficial for training speed-up and better performance. Lester et al. [25] proposes to recycle soft prompts across models with vocab-to-vocab transformations, or linear-combination transformations. Note that our word converter initialization is similar to the idea of vocab-to-vocab transformations. However, we do not observe a zero-shot transfer in our experiments like them, which indicates a potential difference between the content prompts and task prompts. Another way of soft prompt transfer [14] is to conduct prompt tuning on discrete prompts, and thus the discrete prompts can be directly shared across models. Different from these task prompts transfer works, our VPG transfer scenario actually suffers from fewer limitations. For example, the task soft prompts transfer suffers from the dimension change problem, where the main body of the trainable parameters should be processed. However, our VPG (the main trainable parameters) can naturally be shared among LLMs with different embedding dimensions and leave the dimension change problem to a simple projector with ignorable parameters. ", + "bbox": [ + 174, + 484, + 825, + 733 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B Extended Findings in Exploratory Analysis ", + "text_level": 1, + "bbox": [ + 173, + 758, + 570, + 776 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we show extended findings of exploratory analysis (cf. §3.1). ", + "bbox": [ + 174, + 792, + 671, + 808 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• 1. Merely tuning the projector can not achieve the best performance. We want to clarify that merely tuning the projector is insufficient for achieving the best performance. Notably, as shown in Fig. 10, significant performance gaps are observed between the “only linear” (green curve) and “train from scratch” (orange curve) approaches for COCO caption and NoCaps. Therefore, if the goal is to build a multimodal conversation robot using carefully collected dialog data, training only the linear projector is insufficient to align with the provided data. ", + "bbox": [ + 174, + 828, + 823, + 911 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/33540963e9f2bd811dc970d5f78f00224ddc6d489b9621249332294a34459900.jpg", + "image_caption": [ + "Figure 10: Comparisons between i) inheriting VPG from $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ and training it with randomly initialized projector for $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ and ii) training VPG and randomly initialized projector for $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ from scratch. iii) training only the projector. " + ], + "image_footnote": [], + "bbox": [ + 174, + 88, + 821, + 186 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/32f1e17321f3f1584e1644e783c867e1bde2b301cc63f5fd1793a651ad638427.jpg", + "image_caption": [ + "Figure 11: Interpreting generated soft prompts for $\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }$ with nearest words. " + ], + "image_footnote": [], + "bbox": [ + 336, + 246, + 655, + 369 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "• 2. Word embedding converter can not replace a trained linear projector. As demonstrated in Fig. 11, we observe a common pattern: the last token of soft prompts is closest to EOS, while the middle tokens represent the image content. Such a phenomenon indicates a similarity between soft prompts and word embeddings. However, they are not identical. For instance, the norm of soft prompts is typically around 10 times the average norm of word embeddings. It is important to note that the linear projector initialization cannot replace the warm-up training. Using only the linear projector initialization even yields a random performance. We believe that a better understanding of how prompt works will further benefit the VPG’s transfer learning. ", + "bbox": [ + 173, + 424, + 825, + 535 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "• 3. The projector warm-up is robust to a larger learning rate, while VPG can not. The first thing we want to clarify is that the 5 times normal learning rate will result in a training crash for VPG. Additionally, we find that although increasing the learning rate to 10 times in the projector warm-up does not yield any additional acceleration, the projector can converge without crashing during training. ", + "bbox": [ + 173, + 550, + 825, + 619 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C Extended TaS Experiments ", + "text_level": 1, + "bbox": [ + 176, + 638, + 437, + 656 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section, we first illustrate extending findings of TaS experiments (cf. $\\ S 4 )$ . Then, we introduce the implementation details of scale-up experiments. We also plot a more complete version of Fig. 6 in Fig. 13. ", + "bbox": [ + 174, + 671, + 825, + 713 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.1 VPGTrans Enabling Stable Training under TaS ", + "text_level": 1, + "bbox": [ + 174, + 731, + 547, + 746 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "As we illustrated before, $\\mathrm { F l a n T } 5 _ { \\mathrm { l a r g e } }$ training is extremely unstable. Even when the learning rate is adjusted to one-tenth of its original value, the model does not converge or shows a very slow convergence rate after 4 epochs. However, we find that by lowering the learning rate for stage-2 training, our VPGTrans can achieve stable training on $\\mathrm { F l a n T } 5 _ { \\mathrm { l a r g e } }$ . We plot the performance curve of the COCO caption in Fig. 12. ", + "bbox": [ + 174, + 756, + 825, + 827 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.2 VPGTrans Enabling Training with Less Data under TaS ", + "text_level": 1, + "bbox": [ + 173, + 843, + 607, + 858 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We empirically find that TaS can reduce the requirement for the amount of training data. By reducing the training data to only COCO, we find no obvious performance drop. However, we want to stress that the retained data should be of high quality, which means that if the same number of SBU data ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/3e4dd23620bef89b02685655c6d2638d085099edb3dda84cf3da77d5e5a29fd1.jpg", + "image_caption": [ + "Figure 12: Comparison of training FlanT5 $\\mathrm { l a r g e }$ using different methods. " + ], + "image_footnote": [], + "bbox": [ + 372, + 90, + 625, + 281 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/a01109c17f0876641c1a64b088a94ea18d23d988fb0dbcc1a438ae02e0f7ba89.jpg", + "table_caption": [ + "Table 6: The speed-up rate of our VPGTrans compared with VPG inherit. The symbol \"-\" means VPGTrans can not achieve better performance than VPG inherit. " + ], + "table_footnote": [], + "table_body": "
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.12.710.06.06.0
OPT125M→1.3B-2.71-1
OPT350M→1.3B11.71.02.01.0
OPT125M→2.7B3.03.03.01.03.0
OPT350M→2.7B3.34.51.01.01.0
OPT1.3B→2.7B2.33.0111
FlanT5base-→XL11.01.89.00.4
", + "bbox": [ + 238, + 366, + 758, + 494 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "is retained, the performance especially for captioning will drop. The conclusion can also be found in Table 4. We refer the readers to the next subsection for more details. ", + "bbox": [ + 173, + 527, + 823, + 555 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.3 Comparison between VPGTrans and VPG Inherit ", + "text_level": 1, + "bbox": [ + 173, + 580, + 563, + 595 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "As shown in Fig. 13, the green line (our VPGTrans) can be higher than the orange line (VPG inherit) for the majority of various conditions. Especially when considering the best performance of different tasks, our VPGTrans can achieve better performance than VPG inherit (non “-” in Table 6) for over $74 \\%$ Transfer-Task variants. Moreover, among the variants that our VPGTrans can achieve better performance, our VPGTrans can also achieve a speed-up on $6 9 . 2 \\%$ conditions. ", + "bbox": [ + 174, + 609, + 825, + 679 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.4 Scale-up Experiment Implementation Details ", + "text_level": 1, + "bbox": [ + 174, + 704, + 531, + 719 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The scale-up experiment refers to the results in Table 4. We try to imitate BLIP-2’s pre-training data composition. First of all, two human-annotated datasets COCO and VG are used. SBU is also used. Then, BLIP-2 uses BLIP to generate captions for the 115M web images and rank them with CLIP ViT-L/14. We also adopt similar synthetic data from Laion-COCO.8 We report the concrete number of data we use in Table 4. For the stage-1 training, we keep the same as the previous validation experiments where COCO and SBU are used for warm-up with a 5 times the learning rate. Then, we use COCO, VG, and Laion-COCO for the stage-2 training. Note that we have tried to include Laion-COCO and VG for the stage-1 training, but found no obvious difference and thus use COCO and SBU for simplicity. For VL-Vicuna, to align with the conversation scenario, we further fine-tune our VL-Vicuna with MiniGPT-4’s self-instruct data, which is 3,439 image-text pairs. ", + "bbox": [ + 173, + 733, + 825, + 872 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/4392bb6f62287ea11fe623c93cb50e6fb2a414be6c70d57c54cdd18ec14375cf.jpg", + "image_caption": [ + "Figure 13: Comparison between different methods across 7 TaS variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. " + ], + "image_footnote": [], + "bbox": [ + 173, + 132, + 818, + 829 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/2182b9affaae1c808d5c958976a745e860381b23f5b4381e96ddcb0a3ded6391.jpg", + "table_caption": [ + "Table 7: Comparison between tuning only the projector (i.e. linear transfer) and the best results the VPGTrans achieve. " + ], + "table_footnote": [], + "table_body": "
TransferCOCO CaptionVQAv2
linearbestlinearbest
FlanT5base → OPT350M110.8136.540.444.2
FlanT5xL →OPT2.7B132.1139.350.450.3
OPT350M→FlanT5base1.1122.134.049.9
OPT2.7B→FlanT5xL106.5133.251.353.5
", + "bbox": [ + 303, + 119, + 692, + 218 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D Extended TaT Experiments ", + "text_level": 1, + "bbox": [ + 174, + 241, + 441, + 258 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this section, we mainly illustrate extending findings of TaT experiments $( c f . \\ \\ S 5 )$ . We plot a more complete version of Fig. 8 in Fig. 14. ", + "bbox": [ + 174, + 272, + 823, + 301 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D.1 Linear Transfer Gap between Different LLM’s Visual Prompts ", + "text_level": 1, + "bbox": [ + 174, + 315, + 655, + 332 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "As illustrated in Section 5.2, it is more difficult to transfer between two small LLMs with our VPGTrans due to the weaker linear transferability between two small LLMs’ visual prompts. To better support our results, we compare the results of tuning only the projector (i.e. linear transfer) and the best results the VPGTrans can achieve. As shown in Table 7, we can see that when conducting transfers between two large models $( \\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ and ${ \\mathrm { F l a n T } } 5 _ { \\mathrm { b a s e } }$ ), the performance is typically far from the optimal results. However, when we transfer between $\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ and $\\mathrm { F l a n T 5 } _ { \\mathrm { X L } }$ , the linear performance is near to the optimal performance. There is a 25.7 points gap between $\\mathrm { F l a n T 5 _ { b a s e } }$ $\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }$ ’s linear and best on COCO caption datasets but only 7.2 points gap between FlanT $5 _ { \\mathrm { X L } }$ $ \\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }$ ’s linear and best. If considering the transfer between the small to large LLMs under TaT, both the VPG’s visual perception ability and transfer gap should be considered. We leave the systematical exploration for future works. ", + "bbox": [ + 173, + 342, + 825, + 494 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E Extended Results for MLLM Customization ", + "text_level": 1, + "bbox": [ + 173, + 513, + 576, + 531 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We show more comparisons between VL-Vicuna and MiniGPT-4 in Fig. 15. We can see that our VL-Vicuna has better visual perception ability. For example, when MiniGPT-4 falsely recognizes the three people in the image as two in the first example, our VL-Vicuna can not only recognize the number of people but also tell their roles. Moreover, our VL-Vicuna can successfully link the vision content with external knowledge. In the third example, our VL-Vicuna can recognize Leonardo and link the content with his films like Titanic. ", + "bbox": [ + 173, + 545, + 825, + 628 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "F Potential Impact and Limitations ", + "text_level": 1, + "bbox": [ + 176, + 647, + 485, + 665 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Our VPGTrans is designed for building new MLLMs with lower computational cost, i.e., shorter training time and less training data. With an already pre-trained MLLM, VPGTrans enables fast VPG transfer to build either a larger MLLM or a MLLM with a different type of LLM. We hope VPGTrans can facilitate teams in LLM communicty to customize their MLLMs with reduced cost. There are also possible limitations of the current version of VPGTrans. The first one is that our VPGTrans should rely on some already-aligned VPGs. The second potential limitation is that the VPGTrans-built MLLMs still suffer from the common problems of content generation AI systems [45, 46, 48, 7]. For example, the VL-Vicuna may make up some sentences with falsely recognized visual facts, like what is shown in Fig. 16. It is worth exploring associating our VPGTrans with training safer models [5, 41, 38]. ", + "bbox": [ + 173, + 679, + 826, + 818 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/6b71e6150cb974ca3f4ff4a2b8513068f693caa29c5ead48ea547b61a244d464.jpg", + "image_caption": [ + "Figure 14: Comparison between different methods across 4 TaS variants on 5 tasks. Note that the model is directly evaluated after pre-training without further fine-tuning. " + ], + "image_footnote": [], + "bbox": [ + 173, + 285, + 818, + 695 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/782e3db48f255a657cbf6437cf9fcc5416f9d47fc7e9c50b97780d26130971cd.jpg", + "image_caption": [ + "Figure 15: Comparison between MiniGPT-4 and our VL-Vicuna. " + ], + "image_footnote": [], + "bbox": [ + 176, + 114, + 826, + 637 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/ca663ea28691202a79fd05df32d71dd60e6ff115c315efa2a9179c903804c7cf.jpg", + "image_caption": [ + "Figure 16: Failure cases of our VL-Vicuna. 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Ideally, the vast LLM backbone", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "can remain untouched, leaving only the relatively lightweight VPG module to be fully or partially", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 147, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 147, + 172 + ], + "score": 1.0, + "content": "updated.3", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "Motivation. However, building a MLLM is inevitably computation-expensive, due to the huge", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 374, + 202 + ], + "score": 1.0, + "content": "overhead brought by the LLM. For example, training a BLIP-2 Flan", + "type": "text" + }, + { + "bbox": [ + 374, + 189, + 396, + 201 + ], + "score": 0.32, + "content": "{ \\mathrm { 7 } } 5 _ { \\mathrm { X X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "needs over 600 A100-GPU", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "hours on over 100 million image-text pairs. Hopefully, transferring a pre-trained VPG (which is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "the main body of trainable parts) from an existing MLLM to a novel LLM instead of training from", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "scratch,4 offers a promising solution. Intuitively, all the MLLMs literally can share the same VPG", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "infrastructure and utility,5 which makes the VPG transfer theoretically feasible. In this work, we thus", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 244, + 343, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 343, + 256 + ], + "score": 1.0, + "content": "investigate the potential of transferring VPG across LLMs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 109, + 262, + 503, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "Proposal. Specifically, this paper examines the transferability of VPG across LLMs: 1) with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "different sizes (the same type), i.e. , transfer across LLM sizes, and 2) across different LLM types, i.e.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 284, + 321, + 296 + ], + "spans": [ + { + "bbox": [ + 107, + 284, + 321, + 296 + ], + "score": 1.0, + "content": ", transfer across LLM type, as illustrated in Fig. 1(b).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 118, + 300, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 118, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 118, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "• [Transfer across LLM Sizes (TaS)]. It has been a typical practice for LLM-related research [8]", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 127, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 127, + 311, + 449, + 324 + ], + "score": 1.0, + "content": "to validate the training strategy and the hyperparameter on smaller models (e.g.,", + "type": "text" + }, + { + "bbox": [ + 449, + 311, + 484, + 322 + ], + "score": 0.64, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 311, + 506, + 324 + ], + "score": 1.0, + "content": ") and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 127, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 127, + 322, + 258, + 335 + ], + "score": 1.0, + "content": "then scale up to larger ones (e.g.,", + "type": "text" + }, + { + "bbox": [ + 258, + 322, + 293, + 334 + ], + "score": 0.61, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "). It is thus worth exploring whether a VPG trained on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 126, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 126, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "a smaller LLM can be transferred to a larger LLM, resulting in reduced computational costs &", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 126, + 344, + 321, + 357 + ], + "spans": [ + { + "bbox": [ + 126, + 344, + 321, + 357 + ], + "score": 1.0, + "content": "data, and maintaining comparable performance.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 118, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 118, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "• [Transfer across LLM Types (TaT)]. With a well-tuned VPG for a type of LLM, it is interesting", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 126, + 366, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 126, + 366, + 507, + 381 + ], + "score": 1.0, + "content": "to see if VPG can be transferred to other types of LLMs even with different architectures (e.g.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 128, + 379, + 502, + 390 + ], + "spans": [ + { + "bbox": [ + 128, + 379, + 502, + 390 + ], + "score": 1.0, + "content": "decoder v.s. encoder-decoder). If the transfer can be achieved, how to make it more efficient?", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 294, + 408 + ], + "score": 1.0, + "content": "We conduct a series of exploratory analyses (cf.", + "type": "text" + }, + { + "bbox": [ + 294, + 395, + 313, + 407 + ], + "score": 0.4, + "content": "\\ S 3 . 1 \\ r ,", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 395, + 506, + 408 + ], + "score": 1.0, + "content": ") to identify the key factors for transfer efficiency.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 433, + 419 + ], + "score": 1.0, + "content": "Based on our empirical study, we design a two-stage transfer learning framework", + "type": "text" + }, + { + "bbox": [ + 433, + 406, + 469, + 418 + ], + "score": 0.39, + "content": "( c f . \\ \\ S 3 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 406, + 505, + 419 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "VPGTrans, that includes a projector warm-up (stage-1) and vanilla fine-tuning (stage-2). For stage-1,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "we find that warming up the projector before VPG tuning can effectively reduce the training step", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "for adapting a pre-trained VPG to a new LLM, and avoid the potential performance drop in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "adaptation. To achieve an efficient warm-up, the projector will be well-initialized and then trained", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 257, + 474 + ], + "score": 1.0, + "content": "with an extremely large learning rate", + "type": "text" + }, + { + "bbox": [ + 257, + 461, + 289, + 471 + ], + "score": 0.91, + "content": "( 5 \\times l r )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 460, + 505, + 474 + ], + "score": 1.0, + "content": ". For stage-2, there is a vanilla fine-tuning of both the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "VPG and projector. Despite its simplicity, VPGTrans is able to significantly speed up the VPG-transfer", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 483, + 278, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 278, + 496 + ], + "score": 1.0, + "content": "process without harming the performance.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 108, + 501, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "Results and Findings. Via extensive experiments on the transfer across LLM sizes and types (cf.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 511, + 308, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 118, + 523 + ], + "score": 0.46, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 511, + 308, + 524 + ], + "score": 1.0, + "content": "& §5), we gain the following key observations:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 119, + 528, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 118, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 118, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "• VPGTrans helps to avoid the performance drop caused by directly inheriting the VPG and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 127, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 127, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "achieves at most 10 times acceleration for the small-to-large transfer across LLMs in the same", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 126, + 551, + 150, + 563 + ], + "spans": [ + { + "bbox": [ + 126, + 551, + 150, + 563 + ], + "score": 1.0, + "content": "type.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 118, + 561, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 118, + 561, + 505, + 576 + ], + "score": 1.0, + "content": "• VPGTrans can also achieve comparable or better performance than training from scratch and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 126, + 573, + 465, + 587 + ], + "spans": [ + { + "bbox": [ + 126, + 573, + 465, + 587 + ], + "score": 1.0, + "content": "achieve at most 5 times acceleration for the transfers between different model types.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 118, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 118, + 586, + 356, + 600 + ], + "score": 1.0, + "content": "• Notably, our VPGTrans helps to achieve a BLIP-2 ViT-G", + "type": "text" + }, + { + "bbox": [ + 357, + 586, + 413, + 598 + ], + "score": 0.88, + "content": "\\mathbf { O P T _ { 2 . 7 B 6 . 7 B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "transfer with less than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 596, + 487, + 611 + ], + "spans": [ + { + "bbox": [ + 127, + 597, + 147, + 608 + ], + "score": 0.83, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 596, + 227, + 611 + ], + "score": 1.0, + "content": "of GPU hours and", + "type": "text" + }, + { + "bbox": [ + 228, + 597, + 256, + 608 + ], + "score": 0.89, + "content": "1 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 596, + 487, + 611 + ], + "score": 1.0, + "content": "of training data required for the original model training.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 118, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 118, + 609, + 409, + 623 + ], + "score": 1.0, + "content": "• Furthermore, our framework can even outperform the original BLIP-2", + "type": "text" + }, + { + "bbox": [ + 410, + 610, + 444, + 622 + ], + "score": 0.64, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "on most of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 127, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 127, + 621, + 231, + 634 + ], + "score": 1.0, + "content": "evaluated datasets, with a", + "type": "text" + }, + { + "bbox": [ + 231, + 621, + 252, + 632 + ], + "score": 0.84, + "content": "\\mathbf { + 2 . 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 621, + 376, + 634 + ], + "score": 1.0, + "content": "improvement on VQAv2 and a", + "type": "text" + }, + { + "bbox": [ + 377, + 621, + 397, + 632 + ], + "score": 0.82, + "content": "\\mathbf { + 3 . 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "improvement on OKVQA.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 506, + 651 + ], + "score": 1.0, + "content": "Our investigation further reveals some intriguing findings, for which we provide possible explanations:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 659, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 657, + 332, + 671 + ], + "spans": [ + { + "bbox": [ + 118, + 657, + 332, + 671 + ], + "score": 1.0, + "content": "2Also including a linear projector for dimension matching.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 667, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 118, + 667, + 506, + 682 + ], + "score": 1.0, + "content": "3Note that Flamingo also inserts some tunable parameters into the LLM part, but recent works [30, 16] found", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 680, + 255, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 255, + 690 + ], + "score": 1.0, + "content": "that freezing LLM can be more efficient.", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 119, + 690, + 130, + 700 + ], + "score": 0.56, + "content": "^ 4 \\mathrm { I t }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "is not a rigorous expression, because the VPG is typically a pre-trained model, like CLIP [43]. We use it", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 206, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 206, + 713 + ], + "score": 1.0, + "content": "for simplicity in this paper.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 374, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 374, + 723 + ], + "score": 1.0, + "content": "5while the projector can not be shared due to the dimension mismatch.", + "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, + 171 + ], + "lines": [], + "index": 4, + "bbox_fs": [ + 105, + 73, + 506, + 172 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "Motivation. However, building a MLLM is inevitably computation-expensive, due to the huge", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 374, + 202 + ], + "score": 1.0, + "content": "overhead brought by the LLM. For example, training a BLIP-2 Flan", + "type": "text" + }, + { + "bbox": [ + 374, + 189, + 396, + 201 + ], + "score": 0.32, + "content": "{ \\mathrm { 7 } } 5 _ { \\mathrm { X X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "needs over 600 A100-GPU", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "hours on over 100 million image-text pairs. Hopefully, transferring a pre-trained VPG (which is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "the main body of trainable parts) from an existing MLLM to a novel LLM instead of training from", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 235 + ], + "score": 1.0, + "content": "scratch,4 offers a promising solution. Intuitively, all the MLLMs literally can share the same VPG", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "infrastructure and utility,5 which makes the VPG transfer theoretically feasible. In this work, we thus", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 244, + 343, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 343, + 256 + ], + "score": 1.0, + "content": "investigate the potential of transferring VPG across LLMs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 177, + 506, + 256 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 262, + 503, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "Proposal. Specifically, this paper examines the transferability of VPG across LLMs: 1) with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "different sizes (the same type), i.e. , transfer across LLM sizes, and 2) across different LLM types, i.e.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 284, + 321, + 296 + ], + "spans": [ + { + "bbox": [ + 107, + 284, + 321, + 296 + ], + "score": 1.0, + "content": ", transfer across LLM type, as illustrated in Fig. 1(b).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 261, + 505, + 296 + ] + }, + { + "type": "text", + "bbox": [ + 118, + 300, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 118, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 118, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "• [Transfer across LLM Sizes (TaS)]. It has been a typical practice for LLM-related research [8]", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 127, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 127, + 311, + 449, + 324 + ], + "score": 1.0, + "content": "to validate the training strategy and the hyperparameter on smaller models (e.g.,", + "type": "text" + }, + { + "bbox": [ + 449, + 311, + 484, + 322 + ], + "score": 0.64, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 311, + 506, + 324 + ], + "score": 1.0, + "content": ") and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 127, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 127, + 322, + 258, + 335 + ], + "score": 1.0, + "content": "then scale up to larger ones (e.g.,", + "type": "text" + }, + { + "bbox": [ + 258, + 322, + 293, + 334 + ], + "score": 0.61, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "). It is thus worth exploring whether a VPG trained on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 126, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 126, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "a smaller LLM can be transferred to a larger LLM, resulting in reduced computational costs &", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 126, + 344, + 321, + 357 + ], + "spans": [ + { + "bbox": [ + 126, + 344, + 321, + 357 + ], + "score": 1.0, + "content": "data, and maintaining comparable performance.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 118, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 118, + 356, + 506, + 370 + ], + "score": 1.0, + "content": "• [Transfer across LLM Types (TaT)]. With a well-tuned VPG for a type of LLM, it is interesting", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 126, + 366, + 507, + 381 + ], + "spans": [ + { + "bbox": [ + 126, + 366, + 507, + 381 + ], + "score": 1.0, + "content": "to see if VPG can be transferred to other types of LLMs even with different architectures (e.g.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 128, + 379, + 502, + 390 + ], + "spans": [ + { + "bbox": [ + 128, + 379, + 502, + 390 + ], + "score": 1.0, + "content": "decoder v.s. encoder-decoder). If the transfer can be achieved, how to make it more efficient?", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 118, + 300, + 507, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 294, + 408 + ], + "score": 1.0, + "content": "We conduct a series of exploratory analyses (cf.", + "type": "text" + }, + { + "bbox": [ + 294, + 395, + 313, + 407 + ], + "score": 0.4, + "content": "\\ S 3 . 1 \\ r ,", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 395, + 506, + 408 + ], + "score": 1.0, + "content": ") to identify the key factors for transfer efficiency.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 433, + 419 + ], + "score": 1.0, + "content": "Based on our empirical study, we design a two-stage transfer learning framework", + "type": "text" + }, + { + "bbox": [ + 433, + 406, + 469, + 418 + ], + "score": 0.39, + "content": "( c f . \\ \\ S 3 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 406, + 505, + 419 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "VPGTrans, that includes a projector warm-up (stage-1) and vanilla fine-tuning (stage-2). For stage-1,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "we find that warming up the projector before VPG tuning can effectively reduce the training step", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "for adapting a pre-trained VPG to a new LLM, and avoid the potential performance drop in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "adaptation. To achieve an efficient warm-up, the projector will be well-initialized and then trained", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 460, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 257, + 474 + ], + "score": 1.0, + "content": "with an extremely large learning rate", + "type": "text" + }, + { + "bbox": [ + 257, + 461, + 289, + 471 + ], + "score": 0.91, + "content": "( 5 \\times l r )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 460, + 505, + 474 + ], + "score": 1.0, + "content": ". For stage-2, there is a vanilla fine-tuning of both the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "VPG and projector. Despite its simplicity, VPGTrans is able to significantly speed up the VPG-transfer", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 483, + 278, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 278, + 496 + ], + "score": 1.0, + "content": "process without harming the performance.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 395, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 501, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "Results and Findings. Via extensive experiments on the transfer across LLM sizes and types (cf.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 511, + 308, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 118, + 523 + ], + "score": 0.46, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 511, + 308, + 524 + ], + "score": 1.0, + "content": "& §5), we gain the following key observations:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 499, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 119, + 528, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 118, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 118, + 527, + 506, + 541 + ], + "score": 1.0, + "content": "• VPGTrans helps to avoid the performance drop caused by directly inheriting the VPG and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 127, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 127, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "achieves at most 10 times acceleration for the small-to-large transfer across LLMs in the same", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 126, + 551, + 150, + 563 + ], + "spans": [ + { + "bbox": [ + 126, + 551, + 150, + 563 + ], + "score": 1.0, + "content": "type.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 118, + 561, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 118, + 561, + 505, + 576 + ], + "score": 1.0, + "content": "• VPGTrans can also achieve comparable or better performance than training from scratch and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 126, + 573, + 465, + 587 + ], + "spans": [ + { + "bbox": [ + 126, + 573, + 465, + 587 + ], + "score": 1.0, + "content": "achieve at most 5 times acceleration for the transfers between different model types.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 118, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ 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training.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 118, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 118, + 609, + 409, + 623 + ], + "score": 1.0, + "content": "• Furthermore, our framework can even outperform the original BLIP-2", + "type": "text" + }, + { + "bbox": [ + 410, + 610, + 444, + 622 + ], + "score": 0.64, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "on most of the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 127, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 127, + 621, + 231, + 634 + ], + "score": 1.0, + "content": "evaluated datasets, with a", + "type": "text" + }, + { + "bbox": [ + 231, + 621, + 252, + 632 + ], + "score": 0.84, + "content": "\\mathbf { + 2 . 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 621, + 376, + 634 + ], + "score": 1.0, + "content": "improvement on VQAv2 and a", + "type": "text" + }, + { + "bbox": [ + 377, + 621, + 397, + 632 + ], + "score": 0.82, + "content": "\\mathbf { + 3 . 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "improvement on OKVQA.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42, + "bbox_fs": [ + 118, + 527, + 506, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 506, + 651 + ], + "score": 1.0, + "content": "Our investigation further reveals some intriguing findings, for which we provide possible explanations:", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47, + "bbox_fs": [ + 106, + 636, + 506, + 651 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 71, + 502, + 151 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 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"text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 168, + 507, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 344, + 183 + ], + "score": 1.0, + "content": "VPGTrans strategy. Note the LLM via VPGTrans is FlanT5", + "type": "text" + }, + { + "bbox": [ + 344, + 171, + 377, + 181 + ], + "score": 0.72, + "content": "\\bf { \\Phi } _ { \\mathrm { X L \\to X X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 168, + 396, + 183 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 171, + 450, + 182 + ], + "score": 0.86, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 168, + 507, + 183 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 118, + 189, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 118, + 189, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 118, + 189, + 242, + 202 + ], + "score": 1.0, + "content": "• When conducting TaS from", + "type": "text" + }, + { + "bbox": [ + 243, + 190, + 274, + 201 + ], + "score": 0.87, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 189, + 285, + 202 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 285, + 190, + 316, + 202 + ], + "score": 0.88, + "content": "\\mathrm { L L M _ { \\mathrm { t g t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 189, + 365, + 202 + ], + "score": 1.0, + "content": ", the size of", + "type": "text" + }, + { + "bbox": [ + 365, + 190, + 396, + 200 + ], + "score": 0.88, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 189, + 507, + 202 + ], + "score": 1.0, + "content": "is not the larger the better.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 126, + 199, + 507, + 213 + ], + "spans": [ + { + "bbox": [ + 126, + 199, + 456, + 213 + ], + "score": 1.0, + "content": "The transfer sometimes even follows a counterintuitive principle of “the smaller the", + "type": "text" + }, + { + "bbox": [ + 457, + 200, + 486, + 212 + ], + "score": 0.89, + "content": "L L M _ { s r c }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 199, + 507, + 213 + ], + "score": 1.0, + "content": "size,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 127, + 212, + 345, + 224 + ], + "spans": [ + { + "bbox": [ + 127, + 212, + 305, + 224 + ], + "score": 1.0, + "content": "the more speed-up and better performance”", + "type": "text" + }, + { + "bbox": [ + 306, + 212, + 341, + 223 + ], + "score": 0.32, + "content": "( c f . \\ \\ S 4 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 212, + 345, + 224 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 118, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 118, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "• When conducting TaT, efficient VPG transfer can not be achieved between two small LLMs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 126, + 235, + 494, + 248 + ], + "spans": [ + { + "bbox": [ + 126, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "with our VPGTrans, due to the large gap between small LLMs’ embedding space (cf.", + "type": "text" + }, + { + "bbox": [ + 469, + 235, + 490, + 247 + ], + "score": 0.35, + "content": "\\ S 5 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 235, + 494, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 252, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "Contributions. In this study, we show for the first time that effective VPG transfer across LLMs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "score": 1.0, + "content": "can be achieved under most conditions, suggesting that it is possible to build a new MLLM with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "considerably lower computational cost, as seen in Fig. 2. To summarize, we make the following key", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 164, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 164, + 296 + ], + "score": 1.0, + "content": "contributions:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 119, + 301, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 118, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 118, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "• Effective approach. We investigate the key factors for VPG-transfer efficiency and propose a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 127, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 127, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "two-stage transfer framework VPGTrans. The approach helps to achieve a highly-efficient VPG", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 126, + 323, + 423, + 336 + ], + "spans": [ + { + "bbox": [ + 126, + 323, + 423, + 336 + ], + "score": 1.0, + "content": "transfer across LLMs with less training data and even task improvements.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 119, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 119, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "• Intriguing findings. By exploring the VPG transfer across LLMs, we reveal several intriguing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 126, + 346, + 459, + 360 + ], + "spans": [ + { + "bbox": [ + 126, + 346, + 459, + 360 + ], + "score": 1.0, + "content": "findings and provide potential explanations that will shed light on further research.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 120, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "• Open source. We showcase how to customize a novel GPT-4-like MLLM with our VPGTrans", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 127, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 127, + 371, + 143, + 384 + ], + "score": 1.0, + "content": "(cf.", + "type": "text" + }, + { + "bbox": [ + 143, + 371, + 155, + 383 + ], + "score": 0.31, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 371, + 505, + 384 + ], + "score": 1.0, + "content": ", and release two multimodal-version MLLMs: VL-LLaMA and VL-Vicuna. All codes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 127, + 381, + 424, + 394 + ], + "spans": [ + { + "bbox": [ + 127, + 381, + 424, + 394 + ], + "score": 1.0, + "content": "and models is released at https://github.com/VPGTrans/VPGTrans.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 106, + 408, + 187, + 421 + ], + "lines": [ + { + "bbox": [ + 103, + 406, + 189, + 426 + ], + "spans": [ + { + "bbox": [ + 103, + 406, + 189, + 426 + ], + "score": 1.0, + "content": "2 Preliminary", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 427, + 504, + 449 + ], + "lines": [ + { + "bbox": [ + 107, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 107, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "This section will outlines the existing prevailing MLLMs, and elaborates on the settings of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 264, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 264, + 450 + ], + "score": 1.0, + "content": "exploratory analyses of these MLLMs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 106, + 456, + 163, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 164, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 164, + 470 + ], + "score": 1.0, + "content": "2.1 MLLM", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 506, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 484 + ], + "score": 1.0, + "content": "Architecture. As illustrated in Fig. 1(a), current MLLMs mostly adopt a common architecture,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "including a visual prompt generator (VPG), a projector, and a backbone LLM. Typically, VPG takes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 507, + 505 + ], + "score": 1.0, + "content": "images/videos as inputs, and encodes the visual input into a fixed length of soft prompts. Then,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "a linear projector is employed to align the soft prompt’s dimension to LLM’s word embedding", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "dimension. Finally, the LLM will generate sentences based on the information from the soft prompt.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 351, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 351, + 537 + ], + "score": 1.0, + "content": "We list some of the recent representative MLLMs in Table 1.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "table", + "bbox": [ + 108, + 573, + 505, + 637 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 545, + 502, + 568 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "Table 1: MLLMs architectures and pre-training paradigm. †: it is a GPT-2-like LLM with relative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 195, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 195, + 569 + ], + "score": 1.0, + "content": "position embeddings.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 573, + 505, + 637 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 573, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 108, + 573, + 505, + 637 + ], + "score": 0.978, + "html": "
MLLMsVPGVPG TrainableLLMLLMTrainable
KOSMOS-1[19]CLIP [43]AllRand.Init.LMAll
Frozen [54]NF-ResNet-50 [6]NF-ResNet-50GPT-2-like† [42]No
Flamingo [2]NFNet-F6 [6]+Resampler [21]ResamplerChinchilla [18]Xattn-Dense
PaLM-E[16]ViT[15]/OSRT[49]AllPaLM[11]No
BLIP-2 [30]EVA-CLIP [52] + Q-Former [30]Q-FormerOPT[60]/Flan-T5[12]No
", + "type": "table", + "image_path": "94453217e65d59a603b7f1daf46d79a8af881dcfce2be3530856d09bbad2fe11.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 108, + 573, + 505, + 594.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 108, + 594.3333333333334, + 505, + 615.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 108, + 615.6666666666667, + 505, + 637.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + } + ], + "index": 33.75 + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Training Paradigm. Given a MLLM, typically the VPG and linear projector will be trained, fully", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 507, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 670 + ], + "score": 1.0, + "content": "or partially. For example, PaLM-E updates all of the parameters of VPG in the pre-training stage,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "while BLIP-2 and Flamingo freeze the ViTs and tune their Q-Former and Resampler, respectively.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "As the main part of the whole architecture, the LLM is usually frozen during the training or tuned", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "only a small portion (e.g., 10B for Flamingo-80B). KOSMOS-1 is an exception, which does not use a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "pre-trained LLM but trains the LLM from scratch. Such a training paradigm typically results in much", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "longer training time and data (both multimodal and pure text corpus). Recent works [30, 16] show that", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "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": "image", + "bbox": [ + 110, + 71, + 502, + 151 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 71, + 502, + 151 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 71, + 502, + 151 + ], + "spans": [ + { + "bbox": [ + 110, + 71, + 502, + 151 + ], + "score": 0.933, + "type": "image", + "image_path": "fe9874330a5d4f954361378d30db421b9ee5eb4c7b892fe951e704e0517ee88f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 71, + 502, + 97.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 97.66666666666667, + 502, + 124.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 124.33333333333334, + 502, + 151.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 108, + 159, + 506, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "Figure 2: Comparing the cost between training VPG from scratch vs. transferring VPG via our", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 168, + 507, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 344, + 183 + ], + "score": 1.0, + "content": "VPGTrans strategy. Note the LLM via VPGTrans is FlanT5", + "type": "text" + }, + { + "bbox": [ + 344, + 171, + 377, + 181 + ], + "score": 0.72, + "content": "\\bf { \\Phi } _ { \\mathrm { X L \\to X X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 168, + 396, + 183 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 171, + 450, + 182 + ], + "score": 0.86, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 168, + 507, + 183 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 118, + 189, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 118, + 189, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 118, + 189, + 242, + 202 + ], + "score": 1.0, + "content": "• When conducting TaS from", + "type": "text" + }, + { + "bbox": [ + 243, + 190, + 274, + 201 + ], + "score": 0.87, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 189, + 285, + 202 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 285, + 190, + 316, + 202 + ], + "score": 0.88, + "content": "\\mathrm { L L M _ { \\mathrm { t g t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 189, + 365, + 202 + ], + "score": 1.0, + "content": ", the size of", + "type": "text" + }, + { + "bbox": [ + 365, + 190, + 396, + 200 + ], + "score": 0.88, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 189, + 507, + 202 + ], + "score": 1.0, + "content": "is not the larger the better.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 126, + 199, + 507, + 213 + ], + "spans": [ + { + "bbox": [ + 126, + 199, + 456, + 213 + ], + "score": 1.0, + "content": "The transfer sometimes even follows a counterintuitive principle of “the smaller the", + "type": "text" + }, + { + "bbox": [ + 457, + 200, + 486, + 212 + ], + "score": 0.89, + "content": "L L M _ { s r c }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 199, + 507, + 213 + ], + "score": 1.0, + "content": "size,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 127, + 212, + 345, + 224 + ], + "spans": [ + { + "bbox": [ + 127, + 212, + 305, + 224 + ], + "score": 1.0, + "content": "the more speed-up and better performance”", + "type": "text" + }, + { + "bbox": [ + 306, + 212, + 341, + 223 + ], + "score": 0.32, + "content": "( c f . \\ \\ S 4 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 212, + 345, + 224 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 118, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 118, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "• When conducting TaT, efficient VPG transfer can not be achieved between two small LLMs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 126, + 235, + 494, + 248 + ], + "spans": [ + { + "bbox": [ + 126, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "with our VPGTrans, due to the large gap between small LLMs’ embedding space (cf.", + "type": "text" + }, + { + "bbox": [ + 469, + 235, + 490, + 247 + ], + "score": 0.35, + "content": "\\ S 5 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 235, + 494, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 118, + 189, + 507, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 252, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "Contributions. In this study, we show for the first time that effective VPG transfer across LLMs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "score": 1.0, + "content": "can be achieved under most conditions, suggesting that it is possible to build a new MLLM with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "considerably lower computational cost, as seen in Fig. 2. To summarize, we make the following key", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 164, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 164, + 296 + ], + "score": 1.0, + "content": "contributions:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 252, + 506, + 296 + ] + }, + { + "type": "text", + "bbox": [ + 119, + 301, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 118, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 118, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "• Effective approach. We investigate the key factors for VPG-transfer efficiency and propose a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 127, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 127, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "two-stage transfer framework VPGTrans. The approach helps to achieve a highly-efficient VPG", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 126, + 323, + 423, + 336 + ], + "spans": [ + { + "bbox": [ + 126, + 323, + 423, + 336 + ], + "score": 1.0, + "content": "transfer across LLMs with less training data and even task improvements.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 119, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 119, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "• Intriguing findings. By exploring the VPG transfer across LLMs, we reveal several intriguing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 126, + 346, + 459, + 360 + ], + "spans": [ + { + "bbox": [ + 126, + 346, + 459, + 360 + ], + "score": 1.0, + "content": "findings and provide potential explanations that will shed light on further research.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 360, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 120, + 360, + 505, + 371 + ], + "score": 1.0, + "content": "• Open source. We showcase how to customize a novel GPT-4-like MLLM with our VPGTrans", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 127, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 127, + 371, + 143, + 384 + ], + "score": 1.0, + "content": "(cf.", + "type": "text" + }, + { + "bbox": [ + 143, + 371, + 155, + 383 + ], + "score": 0.31, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 371, + 505, + 384 + ], + "score": 1.0, + "content": ", and release two multimodal-version MLLMs: VL-LLaMA and VL-Vicuna. All codes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 127, + 381, + 424, + 394 + ], + "spans": [ + { + "bbox": [ + 127, + 381, + 424, + 394 + ], + "score": 1.0, + "content": "and models is released at https://github.com/VPGTrans/VPGTrans.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 118, + 301, + 506, + 394 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 408, + 187, + 421 + ], + "lines": [ + { + "bbox": [ + 103, + 406, + 189, + 426 + ], + "spans": [ + { + "bbox": [ + 103, + 406, + 189, + 426 + ], + "score": 1.0, + "content": "2 Preliminary", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 427, + 504, + 449 + ], + "lines": [ + { + "bbox": [ + 107, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 107, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "This section will outlines the existing prevailing MLLMs, and elaborates on the settings of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 264, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 264, + 450 + ], + "score": 1.0, + "content": "exploratory analyses of these MLLMs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 426, + 505, + 450 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 456, + 163, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 164, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 164, + 470 + ], + "score": 1.0, + "content": "2.1 MLLM", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 506, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 484 + ], + "score": 1.0, + "content": "Architecture. As illustrated in Fig. 1(a), current MLLMs mostly adopt a common architecture,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "including a visual prompt generator (VPG), a projector, and a backbone LLM. Typically, VPG takes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 493, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 507, + 505 + ], + "score": 1.0, + "content": "images/videos as inputs, and encodes the visual input into a fixed length of soft prompts. Then,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "a linear projector is employed to align the soft prompt’s dimension to LLM’s word embedding", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "dimension. Finally, the LLM will generate sentences based on the information from the soft prompt.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 351, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 351, + 537 + ], + "score": 1.0, + "content": "We list some of the recent representative MLLMs in Table 1.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 469, + 507, + 537 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 573, + 505, + 637 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 545, + 502, + 568 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "Table 1: MLLMs architectures and pre-training paradigm. †: it is a GPT-2-like LLM with relative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 195, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 195, + 569 + ], + "score": 1.0, + "content": "position embeddings.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 573, + 505, + 637 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 573, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 108, + 573, + 505, + 637 + ], + "score": 0.978, + "html": "
MLLMsVPGVPG TrainableLLMLLMTrainable
KOSMOS-1[19]CLIP [43]AllRand.Init.LMAll
Frozen [54]NF-ResNet-50 [6]NF-ResNet-50GPT-2-like† [42]No
Flamingo [2]NFNet-F6 [6]+Resampler [21]ResamplerChinchilla [18]Xattn-Dense
PaLM-E[16]ViT[15]/OSRT[49]AllPaLM[11]No
BLIP-2 [30]EVA-CLIP [52] + Q-Former [30]Q-FormerOPT[60]/Flan-T5[12]No
", + "type": "table", + "image_path": "94453217e65d59a603b7f1daf46d79a8af881dcfce2be3530856d09bbad2fe11.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 108, + 573, + 505, + 594.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 108, + 594.3333333333334, + 505, + 615.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 108, + 615.6666666666667, + 505, + 637.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + } + ], + "index": 33.75 + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Training Paradigm. Given a MLLM, typically the VPG and linear projector will be trained, fully", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 507, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 507, + 670 + ], + "score": 1.0, + "content": "or partially. For example, PaLM-E updates all of the parameters of VPG in the pre-training stage,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "while BLIP-2 and Flamingo freeze the ViTs and tune their Q-Former and Resampler, respectively.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "As the main part of the whole architecture, the LLM is usually frozen during the training or tuned", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "only a small portion (e.g., 10B for Flamingo-80B). KOSMOS-1 is an exception, which does not use a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "pre-trained LLM but trains the LLM from scratch. Such a training paradigm typically results in much", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "longer training time and data (both multimodal and pure text corpus). Recent works [30, 16] show that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "adopting an existing LLM and freezing all of its parameters can also achieve excellent performance", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "with significantly reduced computational cost, which leads to the trend of adapting frozen pre-trained", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 249, + 108 + ], + "score": 1.0, + "content": "LLM. For example, BLIP-2 FlanT", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 250, + 95, + 267, + 106 + ], + "score": 0.44, + "content": "\\mathfrak { j } _ { \\mathrm { X X L } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 267, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "(12.1B) can achieve better zero-shot VQAv2 performance", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 137, + 116 + ], + "score": 0.85, + "content": "( 6 5 . 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 137, + 105, + 293, + 118 + ], + "score": 1.0, + "content": "in Acc.) compared with KOSMOS-1", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 293, + 106, + 321, + 117 + ], + "score": 0.82, + "content": "5 1 . 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 321, + 105, + 440, + 118 + ], + "score": 1.0, + "content": "in Acc.) and Flamingo-80B", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 441, + 106, + 468, + 117 + ], + "score": 0.82, + "content": "5 6 . 3 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 468, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "in Acc.).", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 406, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 406, + 129 + ], + "score": 1.0, + "content": "Thus, in this paper, we mainly focus on VPG transfer across frozen LLMs.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 645, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "adopting an existing LLM and freezing all of its parameters can also achieve excellent performance", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "with significantly reduced computational cost, which leads to the trend of adapting frozen pre-trained", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 249, + 108 + ], + "score": 1.0, + "content": "LLM. For example, BLIP-2 FlanT", + "type": "text" + }, + { + "bbox": [ + 250, + 95, + 267, + 106 + ], + "score": 0.44, + "content": "\\mathfrak { j } _ { \\mathrm { X X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "(12.1B) can achieve better zero-shot VQAv2 performance", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 137, + 116 + ], + "score": 0.85, + "content": "( 6 5 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 105, + 293, + 118 + ], + "score": 1.0, + "content": "in Acc.) compared with KOSMOS-1", + "type": "text" + }, + { + "bbox": [ + 293, + 106, + 321, + 117 + ], + "score": 0.82, + "content": "5 1 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 105, + 440, + 118 + ], + "score": 1.0, + "content": "in Acc.) and Flamingo-80B", + "type": "text" + }, + { + "bbox": [ + 441, + 106, + 468, + 117 + ], + "score": 0.82, + "content": "5 6 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "in Acc.).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 116, + 406, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 406, + 129 + ], + "score": 1.0, + "content": "Thus, in this paper, we mainly focus on VPG transfer across frozen LLMs.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 135, + 217, + 146 + ], + "lines": [ + { + "bbox": [ + 104, + 132, + 218, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 218, + 150 + ], + "score": 1.0, + "content": "2.2 Experiment Settings", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 507, + 162 + ], + "score": 1.0, + "content": "Architecture. We adopt BLIP-2’s architecture and training paradigm. In our exploration experi-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "ments, we consider using the VPG that consists of a CLIP ViT-L/14 [43], and a Q-Former that has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 172, + 484, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 484, + 184 + ], + "score": 1.0, + "content": "already undergone a BLIP-like pre-training (the 1st stage pre-training in BLIP-2’s paper [30]).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 105, + 189, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "score": 1.0, + "content": "Training Data. For all of the exploration experiments, we adopt human-annotated COCO caption", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "dataset [34] and web image-text pairs SBU dataset [40], which results in 1.4 million image-text pairs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "Transfer Direction. For the small-to-large model transfer among the same type of LLMs, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "investigate: 1) OPT [60] (decoder-only) series including 125M, 350M, 1.3B, and 2.7B, and 2)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "FlanT5 [12] (encoder-decoder) ranging base, large, and XL. For the transfer across different types of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 377, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 377, + 262 + ], + "score": 1.0, + "content": "LLMs, we consider the ones of OPT and FlanT5 with similar sizes.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "Evaluation. To evaluate the performance of MLLMs, we choose two caption datasets: (1) COCO", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "caption [34] (2) NoCaps [1], and three VQA datasets: (3) VQAv2 [4] (4) GQA [20] (5) OKVQA [37].", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "score": 1.0, + "content": "We make evaluations after the pre-training without task-specific fine-tuning and report the CIDEr [55]", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 317, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 317, + 313 + ], + "score": 1.0, + "content": "for all caption tasks and accuracy for all VQA tasks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "Implementation Details. We follow the same implementation details of BLIP-2, via the open", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "score": 1.0, + "content": "code.6 Concretely, we use FP16 and BFloat16 for OPT and FlanT5 respectively in the model training.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "For the learning rate, we first conduct a linear warm-up from 1e-6 to 1e-4, and then use a cosine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 267, + 363 + ], + "score": 1.0, + "content": "learning rate schedule with the minimal", + "type": "text" + }, + { + "bbox": [ + 268, + 351, + 300, + 361 + ], + "score": 0.76, + "content": "l r { = } 1 \\mathrm { e } { - } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "for 10 epochs. Due to the limited data amount, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "slightly decrease the batch size, which we find beneficial for the final performance. Specifically, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 450, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 450, + 385 + ], + "score": 1.0, + "content": "set the batch size of 1,728 and 1,152 for OPT and FlanT5-based models, respectively.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 399, + 489, + 413 + ], + "lines": [ + { + "bbox": [ + 102, + 397, + 491, + 417 + ], + "spans": [ + { + "bbox": [ + 102, + 397, + 491, + 417 + ], + "score": 1.0, + "content": "3 Maximizing the Transfer Efficiency with a Two-stage Transfer Strategy", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 504, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "score": 1.0, + "content": "In this section, we first identify the key factors for maximizing transfer efficiency, based on which we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 429, + 288, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 288, + 441 + ], + "score": 1.0, + "content": "then motivate our solution for better transfer.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 407, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 408, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 408, + 464 + ], + "score": 1.0, + "content": "3.1 Exploratory Analysis: Identifying Key Factors for VPG Transfer", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Via selected experiments of small-to-large transfer among OPT models, we can obtain the following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 473, + 474, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 474, + 486 + ], + "score": 1.0, + "content": "key observations. More systematical comparisons are conducted in the later section (cf. §4).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 345, + 502 + ], + "score": 1.0, + "content": "• Inheriting the trained VPG can accelerate training.", + "type": "text" + }, + { + "bbox": [ + 348, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "To demonstrate this, we compare the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 499, + 503, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 499, + 254, + 515 + ], + "score": 1.0, + "content": "convergence rates of VPG training on", + "type": "text" + }, + { + "bbox": [ + 255, + 501, + 292, + 512 + ], + "score": 0.55, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 499, + 466, + 515 + ], + "score": 1.0, + "content": "from scratch, and inheriting VPG trained on", + "type": "text" + }, + { + "bbox": [ + 466, + 501, + 503, + 512 + ], + "score": 0.83, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 421, + 524 + ], + "score": 1.0, + "content": "The patterns are shown in Fig. 3. Overall, we find that inheriting VPG trained on", + "type": "text" + }, + { + "bbox": [ + 422, + 512, + 459, + 523 + ], + "score": 0.84, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "accelerates", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "convergence, particularly for two caption tasks. However, for datasets that require fine-grained visual", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "perception such as VQAv2 and GQA, directly conduct continue training with an inherited VPG will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "harm the performance. We hypothesize that tuning VPG with a randomly initialized projector will", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "compromise the existing fine-grained visual perception ability of VPG. The possible reason can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "be that, the VPG is typically a pre-trained model with powerful visual perception ability, and thus", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "updating based on the gradient passed through a random projector will mislead the VPG at the initial", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 176, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 176, + 601 + ], + "score": 1.0, + "content": "steps [2, 33, 23].", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 506, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 618 + ], + "score": 1.0, + "content": "• Warming up the linear projector can prevent performance drop and expedite VPG training.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "To verify this, we first conduct a warm-up training of the linear projector for 3 epochs, during which", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "both VPG and LLM are frozen. Subsequently, we jointly train VPG and the projector and plot the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "performance curve in Fig. 4 (the warm-up process is not included in this figure). The results show", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "that the performance drop observed in Fig.3 can be avoided in Fig. 4. Additionally, we observe that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 657, + 507, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 507, + 673 + ], + "score": 1.0, + "content": "the warm-up training leads to fewer training steps required for VPG and projector joint training.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "score": 1.0, + "content": "However, we must emphasize that warming up is a costly step. In the case of a large LLM, such as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 681, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 394, + 692 + ], + "score": 1.0, + "content": "6.7B, the trainable parameters of BLIP-2’s VPG will account for less than", + "type": "text" + }, + { + "bbox": [ + 395, + 681, + 414, + 691 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 681, + 506, + 692 + ], + "score": 1.0, + "content": "of the total parameters,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 318, + 703 + ], + "score": 1.0, + "content": "where freezing VPG can only lead to a reduction of", + "type": "text" + }, + { + "bbox": [ + 319, + 692, + 342, + 702 + ], + "score": 0.86, + "content": "5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "of A100 hours (36.9 out of 684.0 A100", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 712, + 288, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 290, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 290, + 723 + ], + "score": 1.0, + "content": "6https://github.com/salesforce/lavis", + "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, + 73, + 505, + 127 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 73, + 506, + 129 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 135, + 217, + 146 + ], + "lines": [ + { + "bbox": [ + 104, + 132, + 218, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 218, + 150 + ], + "score": 1.0, + "content": "2.2 Experiment Settings", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 507, + 162 + ], + "score": 1.0, + "content": "Architecture. We adopt BLIP-2’s architecture and training paradigm. In our exploration experi-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "ments, we consider using the VPG that consists of a CLIP ViT-L/14 [43], and a Q-Former that has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 172, + 484, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 484, + 184 + ], + "score": 1.0, + "content": "already undergone a BLIP-like pre-training (the 1st stage pre-training in BLIP-2’s paper [30]).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 149, + 507, + 184 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 189, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "score": 1.0, + "content": "Training Data. For all of the exploration experiments, we adopt human-annotated COCO caption", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "dataset [34] and web image-text pairs SBU dataset [40], which results in 1.4 million image-text pairs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 188, + 506, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "Transfer Direction. For the small-to-large model transfer among the same type of LLMs, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "investigate: 1) OPT [60] (decoder-only) series including 125M, 350M, 1.3B, and 2.7B, and 2)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "FlanT5 [12] (encoder-decoder) ranging base, large, and XL. For the transfer across different types of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 377, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 377, + 262 + ], + "score": 1.0, + "content": "LLMs, we consider the ones of OPT and FlanT5 with similar sizes.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 217, + 506, + 262 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "Evaluation. To evaluate the performance of MLLMs, we choose two caption datasets: (1) COCO", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "caption [34] (2) NoCaps [1], and three VQA datasets: (3) VQAv2 [4] (4) GQA [20] (5) OKVQA [37].", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 303 + ], + "score": 1.0, + "content": "We make evaluations after the pre-training without task-specific fine-tuning and report the CIDEr [55]", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 317, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 317, + 313 + ], + "score": 1.0, + "content": "for all caption tasks and accuracy for all VQA tasks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 267, + 506, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "Implementation Details. We follow the same implementation details of BLIP-2, via the open", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 507, + 343 + ], + "score": 1.0, + "content": "code.6 Concretely, we use FP16 and BFloat16 for OPT and FlanT5 respectively in the model training.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "For the learning rate, we first conduct a linear warm-up from 1e-6 to 1e-4, and then use a cosine", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 267, + 363 + ], + "score": 1.0, + "content": "learning rate schedule with the minimal", + "type": "text" + }, + { + "bbox": [ + 268, + 351, + 300, + 361 + ], + "score": 0.76, + "content": "l r { = } 1 \\mathrm { e } { - } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "for 10 epochs. Due to the limited data amount, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "slightly decrease the batch size, which we find beneficial for the final performance. Specifically, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 450, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 450, + 385 + ], + "score": 1.0, + "content": "set the batch size of 1,728 and 1,152 for OPT and FlanT5-based models, respectively.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 317, + 507, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 399, + 489, + 413 + ], + "lines": [ + { + "bbox": [ + 102, + 397, + 491, + 417 + ], + "spans": [ + { + "bbox": [ + 102, + 397, + 491, + 417 + ], + "score": 1.0, + "content": "3 Maximizing the Transfer Efficiency with a Two-stage Transfer Strategy", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 504, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 432 + ], + "score": 1.0, + "content": "In this section, we first identify the key factors for maximizing transfer efficiency, based on which we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 429, + 288, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 288, + 441 + ], + "score": 1.0, + "content": "then motivate our solution for better transfer.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 417, + 505, + 441 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 448, + 407, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 408, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 408, + 464 + ], + "score": 1.0, + "content": "3.1 Exploratory Analysis: Identifying Key Factors for VPG Transfer", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Via selected experiments of small-to-large transfer among OPT models, we can obtain the following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 473, + 474, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 474, + 486 + ], + "score": 1.0, + "content": "key observations. More systematical comparisons are conducted in the later section (cf. §4).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 461, + 505, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 345, + 502 + ], + "score": 1.0, + "content": "• Inheriting the trained VPG can accelerate training.", + "type": "text" + }, + { + "bbox": [ + 348, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "To demonstrate this, we compare the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 499, + 503, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 499, + 254, + 515 + ], + "score": 1.0, + "content": "convergence rates of VPG training on", + "type": "text" + }, + { + "bbox": [ + 255, + 501, + 292, + 512 + ], + "score": 0.55, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 499, + 466, + 515 + ], + "score": 1.0, + "content": "from scratch, and inheriting VPG trained on", + "type": "text" + }, + { + "bbox": [ + 466, + 501, + 503, + 512 + ], + "score": 0.83, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 421, + 524 + ], + "score": 1.0, + "content": "The patterns are shown in Fig. 3. Overall, we find that inheriting VPG trained on", + "type": "text" + }, + { + "bbox": [ + 422, + 512, + 459, + 523 + ], + "score": 0.84, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "accelerates", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "convergence, particularly for two caption tasks. However, for datasets that require fine-grained visual", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "perception such as VQAv2 and GQA, directly conduct continue training with an inherited VPG will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "harm the performance. We hypothesize that tuning VPG with a randomly initialized projector will", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "compromise the existing fine-grained visual perception ability of VPG. The possible reason can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "be that, the VPG is typically a pre-trained model with powerful visual perception ability, and thus", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "updating based on the gradient passed through a random projector will mislead the VPG at the initial", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 176, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 176, + 601 + ], + "score": 1.0, + "content": "steps [2, 33, 23].", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 489, + 506, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 506, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 618 + ], + "score": 1.0, + "content": "• Warming up the linear projector can prevent performance drop and expedite VPG training.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "To verify this, we first conduct a warm-up training of the linear projector for 3 epochs, during which", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "both VPG and LLM are frozen. Subsequently, we jointly train VPG and the projector and plot the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 649 + ], + "score": 1.0, + "content": "performance curve in Fig. 4 (the warm-up process is not included in this figure). The results show", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "that the performance drop observed in Fig.3 can be avoided in Fig. 4. Additionally, we observe that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 657, + 507, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 507, + 673 + ], + "score": 1.0, + "content": "the warm-up training leads to fewer training steps required for VPG and projector joint training.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 506, + 683 + ], + "score": 1.0, + "content": "However, we must emphasize that warming up is a costly step. In the case of a large LLM, such as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 681, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 394, + 692 + ], + "score": 1.0, + "content": "6.7B, the trainable parameters of BLIP-2’s VPG will account for less than", + "type": "text" + }, + { + "bbox": [ + 395, + 681, + 414, + 691 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 681, + 506, + 692 + ], + "score": 1.0, + "content": "of the total parameters,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 318, + 703 + ], + "score": 1.0, + "content": "where freezing VPG can only lead to a reduction of", + "type": "text" + }, + { + "bbox": [ + 319, + 692, + 342, + 702 + ], + "score": 0.86, + "content": "5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "of A100 hours (36.9 out of 684.0 A100", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "hours). We will elaborate on how to accelerate the linear projector warm-up in our later discussion", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 205, + 144, + 220 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 144, + 220 + ], + "score": 1.0, + "content": "(cf. 3.1).", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 45, + "bbox_fs": [ + 104, + 602, + 507, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 70, + 503, + 147 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 70, + 503, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 70, + 503, + 147 + ], + "spans": [ + { + "bbox": [ + 107, + 70, + 503, + 147 + ], + "score": 0.966, + "type": "image", + "image_path": "77bfd9db8e4a48b0aed1c3c4bbc1af6ae6f361ece62d3323ca16cbef7ebc2455.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 70, + 503, + 95.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 95.66666666666667, + 503, + 121.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 121.33333333333334, + 503, + 147.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 151, + 504, + 185 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 341, + 164 + ], + "score": 1.0, + "content": "Figure 3: Comparisons between i) inheriting VPG from", + "type": "text" + }, + { + "bbox": [ + 341, + 152, + 379, + 163 + ], + "score": 0.87, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "and training it with randomly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 161, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 199, + 176 + ], + "score": 1.0, + "content": "initialized projector for", + "type": "text" + }, + { + "bbox": [ + 199, + 163, + 236, + 174 + ], + "score": 0.42, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 161, + 506, + 176 + ], + "score": 1.0, + "content": "and ii) training VPG and randomly initialized projector for OPT350M", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 173, + 162, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 162, + 185 + ], + "score": 1.0, + "content": "from scratch.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "hours). We will elaborate on how to accelerate the linear projector warm-up in our later discussion", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 205, + 144, + 220 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 144, + 220 + ], + "score": 1.0, + "content": "(cf. 3.1).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 317, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 317, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 163, + 238 + ], + "score": 1.0, + "content": "• Initializing", + "type": "text" + }, + { + "bbox": [ + 163, + 226, + 199, + 238 + ], + "score": 0.66, + "content": "\\mathbf { L L M _ { \\mathrm { t g t } } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 226, + 317, + 238 + ], + "score": 1.0, + "content": "s projector with the help of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 236, + 317, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 317, + 249 + ], + "score": 1.0, + "content": "the word converter can accelerate the linear pro-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 248, + 317, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 317, + 259 + ], + "score": 1.0, + "content": "jector warm-up. In fact, the VPG and projector", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 258, + 317, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 149, + 271 + ], + "score": 1.0, + "content": "trained on", + "type": "text" + }, + { + "bbox": [ + 149, + 259, + 181, + 270 + ], + "score": 0.9, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 258, + 317, + 271 + ], + "score": 1.0, + "content": "have already learned how to map", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 269, + 318, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 194, + 282 + ], + "score": 1.0, + "content": "the visual content to", + "type": "text" + }, + { + "bbox": [ + 194, + 270, + 226, + 281 + ], + "score": 0.9, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "’s understandable soft", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 318, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 258, + 292 + ], + "score": 1.0, + "content": "prompt [39]. If we can convert the", + "type": "text" + }, + { + "bbox": [ + 258, + 281, + 290, + 291 + ], + "score": 0.88, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 280, + 318, + 292 + ], + "score": 1.0, + "content": "’s soft", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 318, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 148, + 303 + ], + "score": 1.0, + "content": "prompt to", + "type": "text" + }, + { + "bbox": [ + 148, + 291, + 179, + 304 + ], + "score": 0.91, + "content": "\\mathrm { L L M _ { \\mathrm { t g t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 291, + 318, + 303 + ], + "score": 1.0, + "content": "’s soft prompt, we can directly get", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 302, + 317, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 186, + 315 + ], + "score": 1.0, + "content": "a VPG suitable for", + "type": "text" + }, + { + "bbox": [ + 186, + 303, + 220, + 315 + ], + "score": 0.9, + "content": "\\mathrm { L L M } _ { t g t }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 302, + 317, + 315 + ], + "score": 1.0, + "content": ". One natural idea is to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 317, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 317, + 325 + ], + "score": 1.0, + "content": "leverage the word embeddings of both models as a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 324, + 317, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 317, + 335 + ], + "score": 1.0, + "content": "proxy for the soft prompt [25]. The intuition behind", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 336, + 317, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 317, + 346 + ], + "score": 1.0, + "content": "the scene is that, the soft prompt works in the same", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 347, + 206, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 206, + 357 + ], + "score": 1.0, + "content": "format as normal words.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5 + }, + { + "type": "image", + "bbox": [ + 325, + 226, + 501, + 312 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 325, + 226, + 501, + 312 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 325, + 226, + 501, + 312 + ], + "spans": [ + { + "bbox": [ + 325, + 226, + 501, + 312 + ], + "score": 0.961, + "type": "image", + "image_path": "7cd279fb90695a457a34a0af9cf6c126e10153fd3c1f56570ca729680105284c.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 325, + 226, + 501, + 240.33333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 325, + 240.33333333333334, + 501, + 254.66666666666669 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 325, + 254.66666666666669, + 501, + 269.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 325, + 269.0, + 501, + 283.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 325, + 283.3333333333333, + 501, + 297.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 325, + 297.66666666666663, + 501, + 311.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 324, + 317, + 505, + 350 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 323, + 315, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 323, + 315, + 505, + 330 + ], + "score": 1.0, + "content": "Figure 4: First warming-up then transferring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 323, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 323, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "can avoid performance drop on VQAv2 and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 324, + 339, + 497, + 351 + ], + "spans": [ + { + "bbox": [ + 324, + 339, + 497, + 351 + ], + "score": 1.0, + "content": "accelerate convergence for COCO caption.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + } + ], + "index": 24.75 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 406 + ], + "lines": [ + { + "bbox": [ + 104, + 360, + 508, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 418, + 376 + ], + "score": 1.0, + "content": "To validate our hypothesis, we conduct an experiment on the transfer from", + "type": "text" + }, + { + "bbox": [ + 418, + 363, + 456, + 374 + ], + "score": 0.88, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 360, + 468, + 376 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 469, + 363, + 503, + 374 + ], + "score": 0.77, + "content": "\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 360, + 508, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 371, + 503, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 317, + 387 + ], + "score": 1.0, + "content": "After training a linear word embedding converter (cf.", + "type": "text" + }, + { + "bbox": [ + 317, + 374, + 349, + 385 + ], + "score": 0.32, + "content": "\\ S 3 . 2 ( \\mathsf { b } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 371, + 470, + 387 + ], + "score": 1.0, + "content": ", we initialize the projector for", + "type": "text" + }, + { + "bbox": [ + 470, + 374, + 503, + 385 + ], + "score": 0.69, + "content": "\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 384, + 507, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 320, + 397 + ], + "score": 1.0, + "content": "with the merged linear operation of the projector for", + "type": "text" + }, + { + "bbox": [ + 320, + 385, + 358, + 396 + ], + "score": 0.83, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 384, + 507, + 397 + ], + "score": 1.0, + "content": "and converter. As shown in Table 2,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 395, + 428, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 428, + 407 + ], + "score": 1.0, + "content": "we observe that the initialization can reduce the 3 epochs’ warm-up to 2 epochs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 356, + 523 + ], + "lines": [ + { + "bbox": [ + 107, + 414, + 357, + 426 + ], + "spans": [ + { + "bbox": [ + 107, + 414, + 357, + 426 + ], + "score": 1.0, + "content": "• Linear projector warm-up enables faster convergence", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 425, + 356, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 356, + 436 + ], + "score": 1.0, + "content": "with an extremely large learning rate. To determine the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 436, + 356, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 356, + 448 + ], + "score": 1.0, + "content": "most efficient transfer practice, we experiment with training the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 447, + 356, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 356, + 458 + ], + "score": 1.0, + "content": "projector using different learning rates. Surprisingly, we find", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 457, + 357, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 357, + 470 + ], + "score": 1.0, + "content": "that the linear projector enables fast and stable convergence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 468, + 357, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 357, + 481 + ], + "score": 1.0, + "content": "with an extremely large learning rate. Specifically, by setting", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 479, + 356, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 356, + 491 + ], + "score": 1.0, + "content": "the learning rate to 5 times of the original value, the COCO", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 489, + 358, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 358, + 503 + ], + "score": 1.0, + "content": "caption’s CIDEr score can reach 133.1 with 1 epoch training,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 501, + 357, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 357, + 513 + ], + "score": 1.0, + "content": "which is higher than the 3 epochs results of w/o init. as shown", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 511, + 151, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 151, + 524 + ], + "score": 1.0, + "content": "in Table 2.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.0 + }, + { + "type": "table", + "bbox": [ + 370, + 472, + 496, + 528 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 363, + 428, + 505, + 471 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 364, + 428, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 364, + 428, + 506, + 439 + ], + "score": 1.0, + "content": "Table 2: Comparison between lin-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 363, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 363, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "ear projector warm-up with/without", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 363, + 450, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 363, + 450, + 505, + 461 + ], + "score": 1.0, + "content": "word embedding initialization. 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As shown in Fig. 5, the stage-1 of VPGTrans performs", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "projector warm-up and the stage-2 carries out a vanilla fine-tuning. Our results demonstrate that the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "VPGTrans is simple yet effective that can significantly speed up the transfer without compromising", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 588, + 397, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 397, + 601 + ], + "score": 1.0, + "content": "performance. Detailed results are given in the later sections (cf. §4 & 5).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52 + }, + { + "type": "title", + "bbox": [ + 108, + 605, + 243, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 244, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 244, + 620 + ], + "score": 1.0, + "content": "▶ Stage-1: Projector Warm-up.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 55 + }, + { + "type": "text", + "bbox": [ + 120, + 615, + 480, + 627 + ], + "lines": [ + { + "bbox": [ + 122, + 614, + 478, + 629 + ], + "spans": [ + { + "bbox": [ + 122, + 614, + 315, + 629 + ], + "score": 1.0, + "content": "(a) Inherit VPG. 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In fact, the VPG and projector", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 258, + 317, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 149, + 271 + ], + "score": 1.0, + "content": "trained on", + "type": "text" + }, + { + "bbox": [ + 149, + 259, + 181, + 270 + ], + "score": 0.9, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 258, + 317, + 271 + ], + "score": 1.0, + "content": "have already learned how to map", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 269, + 318, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 194, + 282 + ], + "score": 1.0, + "content": "the visual content to", + "type": "text" + }, + { + "bbox": [ + 194, + 270, + 226, + 281 + ], + "score": 0.9, + "content": "\\mathbf { L L M } _ { \\mathrm { s r c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "’s understandable soft", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 318, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 258, + 292 + ], + "score": 1.0, + "content": "prompt [39]. 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The intuition behind", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 336, + 317, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 317, + 346 + ], + "score": 1.0, + "content": "the scene is that, the soft prompt works in the same", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 347, + 206, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 206, + 357 + ], + "score": 1.0, + "content": "format as normal words.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 226, + 318, + 357 + ] + }, + { + "type": "image", + "bbox": [ + 325, + 226, + 501, + 312 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 325, + 226, + 501, + 312 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 325, + 226, + 501, + 312 + ], + "spans": [ + { + "bbox": [ + 325, + 226, + 501, + 312 + ], + "score": 0.961, + "type": "image", + "image_path": "7cd279fb90695a457a34a0af9cf6c126e10153fd3c1f56570ca729680105284c.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 325, + 226, + 501, + 240.33333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 325, + 240.33333333333334, + 501, + 254.66666666666669 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 325, + 254.66666666666669, + 501, + 269.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 325, + 269.0, + 501, + 283.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 325, + 283.3333333333333, + 501, + 297.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 325, + 297.66666666666663, + 501, + 311.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 324, + 317, + 505, + 350 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 323, + 315, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 323, + 315, + 505, + 330 + ], + "score": 1.0, + "content": "Figure 4: First warming-up then transferring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 323, + 328, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 323, + 328, + 505, + 339 + ], + "score": 1.0, + "content": "can avoid performance drop on VQAv2 and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 324, + 339, + 497, + 351 + ], + "spans": [ + { + "bbox": [ + 324, + 339, + 497, + 351 + ], + "score": 1.0, + "content": "accelerate convergence for COCO caption.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + } + ], + "index": 24.75 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 406 + ], + "lines": [ + { + "bbox": [ + 104, + 360, + 508, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 418, + 376 + ], + "score": 1.0, + "content": "To validate our hypothesis, we conduct an experiment on the transfer from", + "type": "text" + }, + { + "bbox": [ + 418, + 363, + 456, + 374 + ], + "score": 0.88, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 360, + 468, + 376 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 469, + 363, + 503, + 374 + ], + "score": 0.77, + "content": "\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 360, + 508, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 371, + 503, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 317, + 387 + ], + "score": 1.0, + "content": "After training a linear word embedding converter (cf.", + "type": "text" + }, + { + "bbox": [ + 317, + 374, + 349, + 385 + ], + "score": 0.32, + "content": "\\ S 3 . 2 ( \\mathsf { b } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 371, + 470, + 387 + ], + "score": 1.0, + "content": ", we initialize the projector for", + "type": "text" + }, + { + "bbox": [ + 470, + 374, + 503, + 385 + ], + "score": 0.69, + "content": "\\mathrm { O P T } _ { 1 . 3 \\mathrm { B } }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 384, + 507, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 320, + 397 + ], + "score": 1.0, + "content": "with the merged linear operation of the projector for", + "type": "text" + }, + { + "bbox": [ + 320, + 385, + 358, + 396 + ], + "score": 0.83, + "content": "\\mathrm { O P T } _ { 1 2 5 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 384, + 507, + 397 + ], + "score": 1.0, + "content": "and converter. As shown in Table 2,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 395, + 428, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 428, + 407 + ], + "score": 1.0, + "content": "we observe that the initialization can reduce the 3 epochs’ warm-up to 2 epochs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 360, + 508, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 356, + 523 + ], + "lines": [ + { + "bbox": [ + 107, + 414, + 357, + 426 + ], + "spans": [ + { + "bbox": [ + 107, + 414, + 357, + 426 + ], + "score": 1.0, + "content": "• Linear projector warm-up enables faster convergence", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 425, + 356, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 356, + 436 + ], + "score": 1.0, + "content": "with an extremely large learning rate. To determine the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 436, + 356, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 356, + 448 + ], + "score": 1.0, + "content": "most efficient transfer practice, we experiment with training the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 447, + 356, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 356, + 458 + ], + "score": 1.0, + "content": "projector using different learning rates. Surprisingly, we find", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 457, + 357, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 357, + 470 + ], + "score": 1.0, + "content": "that the linear projector enables fast and stable convergence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 468, + 357, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 357, + 481 + ], + "score": 1.0, + "content": "with an extremely large learning rate. 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As shown in Fig. 5, the stage-1 of VPGTrans performs", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "projector warm-up and the stage-2 carries out a vanilla fine-tuning. Our results demonstrate that the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "VPGTrans is simple yet effective that can significantly speed up the transfer without compromising", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 588, + 397, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 397, + 601 + ], + "score": 1.0, + "content": "performance. 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For simplicity, we use TaS", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 489, + 320, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 320, + 502 + ], + "score": 1.0, + "content": "to represent the transfer across different model sizes.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 507, + 225, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 226, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 226, + 523 + ], + "score": 1.0, + "content": "4.1 Experimental Settings", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 503, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 504, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 504, + 535 + ], + "score": 1.0, + "content": "In this part, we introduce baselines and transfer variants. For details about training data and imple-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 444, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 444, + 546 + ], + "score": 1.0, + "content": "mentation details, please refer to the experiment settings in the Preliminary (cf. 2.2).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "Baselines. We mainly compare our VPGTrans with training from scratch (TFS) and VPG inheri-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "tance (VPG Inherit), where we report their performance on the aforementioned 5 tasks without further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "task-specific fine-tuning. For our VPGTrans, the word converter training only requires updating a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "linear layer on tokenized text data and typically takes less than 10 minutes on 1 A100 GPU with less", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "than 15G GPU memory. Meanwhile, freezing the VPG can lead to at least 14 A100 minutes speed-up", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 494, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 494, + 619 + ], + "score": 1.0, + "content": "per epoch. Therefore, we consider the whole stage-1 training as the 1st epoch for simplicity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "Transfer Variants. We conducted experiments on transfer learning using 1) the OPT model across", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "four different sizes: 125M, 350M, 1.3B, and 2.7B, and 2) the FlanT5 model across three sizes: base,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "large, and XL. However, we encountered significant instability during training with FlanT5large. 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For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "COCO caption and NoCaps that require more training steps to converge, our VPGTrans (green line)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "can be higher than the other two lines (blue and orange lines). 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Then, we only train the projector in this stage with a frozen VPG and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 387, + 483, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 483, + 401 + ], + "score": 1.0, + "content": "LLM. 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In the final step, we conduct a joint training of VPG and projector for", + "type": "text" + }, + { + "bbox": [ + 497, + 419, + 504, + 426 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 427, + 249, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 249, + 440 + ], + "score": 1.0, + "content": "epochs with a normal learning rate.", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 16, + "bbox_fs": [ + 105, + 405, + 504, + 440 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 448, + 354, + 462 + ], + "lines": [ + { + "bbox": [ + 104, + 447, + 356, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 356, + 464 + ], + "score": 1.0, + "content": "4 Exp-I: Transfer across Different Model Sizes", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "In this section, we conduct experiments to systematically illustrate the effectiveness of our VPGTrans", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "and analyze the relationship between transfer efficiency and model size. For simplicity, we use TaS", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 489, + 320, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 320, + 502 + ], + "score": 1.0, + "content": "to represent the transfer across different model sizes.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 468, + 505, + 502 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 507, + 225, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 226, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 226, + 523 + ], + "score": 1.0, + "content": "4.1 Experimental Settings", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 503, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 504, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 504, + 535 + ], + "score": 1.0, + "content": "In this part, we introduce baselines and transfer variants. For details about training data and imple-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 444, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 444, + 546 + ], + "score": 1.0, + "content": "mentation details, please refer to the experiment settings in the Preliminary (cf. 2.2).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 522, + 504, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "Baselines. We mainly compare our VPGTrans with training from scratch (TFS) and VPG inheri-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "tance (VPG Inherit), where we report their performance on the aforementioned 5 tasks without further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "task-specific fine-tuning. For our VPGTrans, the word converter training only requires updating a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "linear layer on tokenized text data and typically takes less than 10 minutes on 1 A100 GPU with less", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "than 15G GPU memory. Meanwhile, freezing the VPG can lead to at least 14 A100 minutes speed-up", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 494, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 494, + 619 + ], + "score": 1.0, + "content": "per epoch. Therefore, we consider the whole stage-1 training as the 1st epoch for simplicity.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 550, + 506, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "Transfer Variants. We conducted experiments on transfer learning using 1) the OPT model across", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "four different sizes: 125M, 350M, 1.3B, and 2.7B, and 2) the FlanT5 model across three sizes: base,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "large, and XL. However, we encountered significant instability during training with FlanT5large. As a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 426, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 348, + 669 + ], + "score": 1.0, + "content": "result, we mainly present the transfer results between FlanT", + "type": "text" + }, + { + "bbox": [ + 348, + 656, + 364, + 667 + ], + "score": 0.41, + "content": "\\bar { \\mathsf { \\Sigma } } _ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 655, + 405, + 669 + ], + "score": 1.0, + "content": "and FlanT", + "type": "text" + }, + { + "bbox": [ + 406, + 656, + 421, + 667 + ], + "score": 0.37, + "content": "{ \\mathrm { 5 } } _ { \\mathrm { X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 655, + 426, + 669 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 622, + 506, + 669 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 674, + 469, + 686 + ], + "lines": [ + { + "bbox": [ + 104, + 673, + 470, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 673, + 470, + 689 + ], + "score": 1.0, + "content": "4.2 VPGTrans Enabling Faster Convergence without Performance Drop under TaS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "First of all, as shown in Fig. 6, our VPGTrans can consistently accelerate the model convergence. For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "COCO caption and NoCaps that require more training steps to converge, our VPGTrans (green line)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "can be higher than the other two lines (blue and orange lines). To give a quantitative evaluation of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "speed-up rate, we show the speed-up rate in Table 3. 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The speed-up rate on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 429, + 545, + 439, + 554 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 439, + 543, + 490, + 558 + ], + "score": 1.0, + "content": "is given by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 491, + 544, + 502, + 558 + ], + "score": 0.88, + "content": "\\frac { e _ { \\mathrm { t f s } } } { e _ { \\mathrm { v t } } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 503, + 543, + 506, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 416, + 569 + ], + "score": 1.0, + "content": "According to Table 3, our VPGTrans can achieve at least 4 times speed-up on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 416, + 556, + 436, + 567 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 436, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "of Transfer-Task", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "variants. 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Note that the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "model is directly evaluated after pre-training without further fine-tuning. Please refer to Appendix§C", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 330, + 213, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 213, + 342 + ], + "score": 1.0, + "content": "for other transfer variants.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 146, + 382, + 464, + 501 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 354, + 504, + 376 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "Table 3: The speed-up rate of our VPGTrans compared with training from scratch (TFS). The symbol", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 362, + 378, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 378, + 378 + ], + "score": 1.0, + "content": "\"-\" means VPGTrans can not achieve better performance than TFS.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 146, + 382, + 464, + 501 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 382, + 464, + 501 + ], + "spans": [ + { + "bbox": [ + 146, + 382, + 464, + 501 + ], + "score": 0.982, + "html": "
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.73.01.05.05.0
OPT125M→1.3B9.010.09.0=2.0
OPT350M→1.3B4.55.09.02.02.0
OPT125M→2.7B10.010.02.02.03.0
OPT350M→2.7B10.010.02.013.0
OPT1.3B-→2.7B3.33.32.0-1.5
FlanT5base→XL1.01.13.04.02.0
FlanT5xL →OPT2.7B5.05.02.02.03.0
OPT2.7B→FlanT5xL1.72.012.01
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The speed-up rate is calculated by considering", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "the number of epochs reduced to achieve the best TFS performance on a particular dataset. 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TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.73.01.05.05.0
OPT125M→1.3B9.010.09.0=2.0
OPT350M→1.3B4.55.09.02.02.0
OPT125M→2.7B10.010.02.02.03.0
OPT350M→2.7B10.010.02.013.0
OPT1.3B-→2.7B3.33.32.0-1.5
FlanT5base→XL1.01.13.04.02.0
FlanT5xL →OPT2.7B5.05.02.02.03.0
OPT2.7B→FlanT5xL1.72.012.01
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ModelsVQAv2 valGQA test-devOKVQA testGPU hourstraining data
BLIP-2 ViT-G OPT6.7B54.336.436.4631.5129M
BLIP-2 ViT-G OPT2.7B-→6.7B (0urs)57.236.239.859.013.8M
VL-LLaMA7B (ours)58.137.537.467.113.8M
BLIP-2 ViT-G FlanT5xXL65.244.745.9684.0121.6M
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BLIP-2 ViT-G FlanT5xXL65.244.745.9684.0121.6M
BLIP-2 ViT-G FlanT5xL-→XXL (ours)65.245.045.032.45.3M
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RankModel Elo Rating
1LLaMA-Adapter v2 [17] 1023.0
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4MiniGPT-4 [62] 1011.9
5InstructBLIP[13] 999.5
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7 Otter [26]981.5
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As shown in Fig. 8, when transferring from", + "type": "text" + }, + { + "bbox": [ + 390, + 419, + 428, + 430 + ], + "score": 0.87, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 418, + 466, + 431 + ], + "score": 1.0, + "content": "to FlanT", + "type": "text" + }, + { + "bbox": [ + 466, + 419, + 485, + 430 + ], + "score": 0.34, + "content": "5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 418, + 505, + 431 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 429, + 451, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 451, + 442 + ], + "score": 1.0, + "content": "convergence speed of VPGTrans is even slower than TFS in the initial several epochs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "• Speed-up of VPGTrans happens in large LLMs. However, when moving to the large LLMs like", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 140, + 468 + ], + "score": 0.82, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 456, + 181, + 468 + ], + "score": 1.0, + "content": "and FlanT", + "type": "text" + }, + { + "bbox": [ + 181, + 456, + 196, + 467 + ], + "score": 0.48, + "content": "{ \\mathrm { 5 } } _ { \\mathrm { X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 456, + 505, + 468 + ], + "score": 1.0, + "content": ", there is an obvious speed-up. As shown in Table 3, we can see at least 2 times", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 241, + 479 + ], + "score": 1.0, + "content": "speed-up when transferring from", + "type": "text" + }, + { + "bbox": [ + 241, + 467, + 282, + 478 + ], + "score": 0.61, + "content": "\\mathrm { F l a n T 5 } _ { \\mathrm { X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 466, + 293, + 479 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 293, + 467, + 327, + 478 + ], + "score": 0.84, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 466, + 505, + 479 + ], + "score": 1.0, + "content": ". We empirically find that the soft prompts", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "for larger LLM are more linear transferrable among different LLM types. As shown in Fig. 8,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 213, + 501 + ], + "score": 1.0, + "content": "when transferring between", + "type": "text" + }, + { + "bbox": [ + 213, + 489, + 256, + 500 + ], + "score": 0.33, + "content": "{ \\mathrm { F l a n T } } 5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 488, + 273, + 501 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 489, + 311, + 500 + ], + "score": 0.82, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 488, + 506, + 501 + ], + "score": 1.0, + "content": ", the VGPTrans’ 1st epoch results on two caption", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 415, + 514 + ], + "score": 1.0, + "content": "datasets are limited, where only a linear operation can be trained. The result of", + "type": "text" + }, + { + "bbox": [ + 416, + 500, + 504, + 510 + ], + "score": 0.79, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } } { } \\mathrm { F l a n T } 5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 399, + 524 + ], + "score": 1.0, + "content": "on the COCO caption is even near to zero. By contrast, the result of", + "type": "text" + }, + { + "bbox": [ + 399, + 511, + 483, + 522 + ], + "score": 0.85, + "content": "\\mathrm { F l a n T } 5 _ { \\mathrm { X L } } { } \\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "score": 1.0, + "content": "our VPGTrans are obviously higher than TFS. We hypothesize that larger LLM typically learned", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "more generalizable text embeddings and share more similarity among relative word distances, which", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 543, + 234, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 234, + 555 + ], + "score": 1.0, + "content": "enables an easier VPG transfer.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 347, + 578 + ], + "lines": [ + { + "bbox": [ + 104, + 562, + 348, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 348, + 582 + ], + "score": 1.0, + "content": "6 Customizing New MLLMs with Any LLMs", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 504, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "Above, we thoroughly certify the efficacy of our proposed VPGTrans approach for higher efficient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "transfer of VPG. In this section, we illustrate how to apply the VPGTrans framework for VPG transfer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 609, + 281, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 281, + 621 + ], + "score": 1.0, + "content": "to customize new MLLMs with any LLMs.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "VL-LLaMA. By applying our VPGTrans, we can equip the recently released LLaMA [53] model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 230, + 651 + ], + "score": 1.0, + "content": "with a VPG trained on BLIP-2", + "type": "text" + }, + { + "bbox": [ + 230, + 638, + 264, + 649 + ], + "score": 0.71, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "to perceive the visual information. As shown in Table 4, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 467, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 367, + 661 + ], + "score": 1.0, + "content": "can see that our VL-LLaMA can outperform the original BLIP-2", + "type": "text" + }, + { + "bbox": [ + 367, + 649, + 404, + 660 + ], + "score": 0.8, + "content": "\\mathrm { O P T } _ { 6 . 7 B }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 649, + 467, + 661 + ], + "score": 1.0, + "content": "on all datasets.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "VL-Vicuna. An exciting application of our VPGTrans is to build a GPT-4 [8] style multimodal", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "conversation chatbot. To achieve our goal, we employ Vicuna [10] as our base LLM. Similarly, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 230, + 702 + ], + "score": 1.0, + "content": "transfer the VPG from BLIP-2", + "type": "text" + }, + { + "bbox": [ + 230, + 689, + 263, + 700 + ], + "score": 0.8, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 688, + 506, + 702 + ], + "score": 1.0, + "content": ", and add an extra instruction tuning using MiniGPT-4’s self-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "instruct data [62]. We compare our model with MiniGPT-4 in Fig. 9. When compared to MiniGPT-4,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 418, + 724 + ], + "score": 1.0, + "content": "our VL-Vicuna shows better visual perception ability. Please refer to Appendix", + "type": "text" + }, + { + "bbox": [ + 419, + 711, + 430, + 722 + ], + "score": 0.7, + "content": "\\mathrm { \\Omega ^ { \\ S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "for more cases. In", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "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": [ + 207, + 98, + 405, + 207 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 69, + 504, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 70, + 504, + 82 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 504, + 82 + ], + "score": 1.0, + "content": "Table 5: Comparison between our VL-Vicuna and SOTA MLLMs for multimodal conversation. The", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 79, + 415, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 415, + 93 + ], + "score": 1.0, + "content": "evaluation is done by Multimodality Chatbot Arena platform via user voting.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 207, + 98, + 405, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 207, + 98, + 405, + 207 + ], + "spans": [ + { + "bbox": [ + 207, + 98, + 405, + 207 + ], + "score": 0.979, + "html": "
RankModel Elo Rating
1LLaMA-Adapter v2 [17] 1023.0
2 LLaVA [35]1019.9
3VL-Vicuna (ours) 1012.1
4MiniGPT-4 [62] 1011.9
5InstructBLIP[13] 999.5
6mPLUG-Owl[59] 996.3
7 Otter [26]981.5
8 BLIP-2 [30]955.8
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For simplicity, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 255, + 242, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 242, + 266 + ], + "score": 1.0, + "content": "mark this type of transfer as TaT.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 241, + 506, + 266 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 271, + 225, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 226, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 226, + 286 + ], + "score": 1.0, + "content": "5.1 Experimental Settings", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "score": 1.0, + "content": "In this part, we introduce baselines and transfer variants. For details about training data and imple-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 297, + 444, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 444, + 309 + ], + "score": 1.0, + "content": "mentation details, please refer to the experiment settings in the Preliminary (cf. 2.2).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 106, + 286, + 504, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 311, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "Baselines. We mainly compare our VPGTrans with training from scratch (TFS), and report the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "performance on the aforementioned 5 tasks. Other details (cf. 4.1) are totally the same with TaS", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 160, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 160, + 347 + ], + "score": 1.0, + "content": "experiments.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 312, + 505, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 504, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 350, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 447, + 365 + ], + "score": 1.0, + "content": "Transfer Variants. We conducted experiments on transfer learning between 1)", + "type": "text" + }, + { + "bbox": [ + 448, + 352, + 486, + 363 + ], + "score": 0.74, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 350, + 505, + 365 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 362, + 276, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 129, + 376 + ], + "score": 1.0, + "content": "FlanT", + "type": "text" + }, + { + "bbox": [ + 130, + 363, + 149, + 374 + ], + "score": 0.3, + "content": "\\mathrm { \\Delta } 5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 362, + 180, + 376 + ], + "score": 1.0, + "content": ", and 2)", + "type": "text" + }, + { + "bbox": [ + 181, + 363, + 214, + 375 + ], + "score": 0.84, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 362, + 257, + 376 + ], + "score": 1.0, + "content": "and FlanT", + "type": "text" + }, + { + "bbox": [ + 258, + 363, + 272, + 374 + ], + "score": 0.27, + "content": "5 _ { \\mathrm { X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 362, + 276, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 350, + 505, + 376 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 382, + 443, + 394 + ], + "lines": [ + { + "bbox": [ + 104, + 380, + 444, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 444, + 396 + ], + "score": 1.0, + "content": "5.2 VPGTrans Enabling Faster Convergence only on Large LLMs under TaT", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 107, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 107, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "• There is no speed-up of TaT between two small LLMs. A finding is that on TaT our VPGTrans", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "does not show speed-up for small models, and even shows a degeneration of training speed in the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 390, + 431 + ], + "score": 1.0, + "content": "initial several epochs. As shown in Fig. 8, when transferring from", + "type": "text" + }, + { + "bbox": [ + 390, + 419, + 428, + 430 + ], + "score": 0.87, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 418, + 466, + 431 + ], + "score": 1.0, + "content": "to FlanT", + "type": "text" + }, + { + "bbox": [ + 466, + 419, + 485, + 430 + ], + "score": 0.34, + "content": "5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 418, + 505, + 431 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 429, + 451, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 451, + 442 + ], + "score": 1.0, + "content": "convergence speed of VPGTrans is even slower than TFS in the initial several epochs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 397, + 505, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "• Speed-up of VPGTrans happens in large LLMs. However, when moving to the large LLMs like", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 140, + 468 + ], + "score": 0.82, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 456, + 181, + 468 + ], + "score": 1.0, + "content": "and FlanT", + "type": "text" + }, + { + "bbox": [ + 181, + 456, + 196, + 467 + ], + "score": 0.48, + "content": "{ \\mathrm { 5 } } _ { \\mathrm { X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 456, + 505, + 468 + ], + "score": 1.0, + "content": ", there is an obvious speed-up. As shown in Table 3, we can see at least 2 times", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 241, + 479 + ], + "score": 1.0, + "content": "speed-up when transferring from", + "type": "text" + }, + { + "bbox": [ + 241, + 467, + 282, + 478 + ], + "score": 0.61, + "content": "\\mathrm { F l a n T 5 } _ { \\mathrm { X L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 466, + 293, + 479 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 293, + 467, + 327, + 478 + ], + "score": 0.84, + "content": "\\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 466, + 505, + 479 + ], + "score": 1.0, + "content": ". We empirically find that the soft prompts", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "for larger LLM are more linear transferrable among different LLM types. As shown in Fig. 8,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 213, + 501 + ], + "score": 1.0, + "content": "when transferring between", + "type": "text" + }, + { + "bbox": [ + 213, + 489, + 256, + 500 + ], + "score": 0.33, + "content": "{ \\mathrm { F l a n T } } 5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 488, + 273, + 501 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 489, + 311, + 500 + ], + "score": 0.82, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 488, + 506, + 501 + ], + "score": 1.0, + "content": ", the VGPTrans’ 1st epoch results on two caption", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 497, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 415, + 514 + ], + "score": 1.0, + "content": "datasets are limited, where only a linear operation can be trained. The result of", + "type": "text" + }, + { + "bbox": [ + 416, + 500, + 504, + 510 + ], + "score": 0.79, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } } { } \\mathrm { F l a n T } 5 _ { \\mathrm { b a s e } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 399, + 524 + ], + "score": 1.0, + "content": "on the COCO caption is even near to zero. By contrast, the result of", + "type": "text" + }, + { + "bbox": [ + 399, + 511, + 483, + 522 + ], + "score": 0.85, + "content": "\\mathrm { F l a n T } 5 _ { \\mathrm { X L } } { } \\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 504, + 533 + ], + "score": 1.0, + "content": "our VPGTrans are obviously higher than TFS. We hypothesize that larger LLM typically learned", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "more generalizable text embeddings and share more similarity among relative word distances, which", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 543, + 234, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 234, + 555 + ], + "score": 1.0, + "content": "enables an easier VPG transfer.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 445, + 506, + 555 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 564, + 347, + 578 + ], + "lines": [ + { + "bbox": [ + 104, + 562, + 348, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 348, + 582 + ], + "score": 1.0, + "content": "6 Customizing New MLLMs with Any LLMs", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 504, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "Above, we thoroughly certify the efficacy of our proposed VPGTrans approach for higher efficient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "transfer of VPG. In this section, we illustrate how to apply the VPGTrans framework for VPG transfer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 609, + 281, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 281, + 621 + ], + "score": 1.0, + "content": "to customize new MLLMs with any LLMs.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 586, + 506, + 621 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "VL-LLaMA. By applying our VPGTrans, we can equip the recently released LLaMA [53] model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 230, + 651 + ], + "score": 1.0, + "content": "with a VPG trained on BLIP-2", + "type": "text" + }, + { + "bbox": [ + 230, + 638, + 264, + 649 + ], + "score": 0.71, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "to perceive the visual information. As shown in Table 4, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 467, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 367, + 661 + ], + "score": 1.0, + "content": "can see that our VL-LLaMA can outperform the original BLIP-2", + "type": "text" + }, + { + "bbox": [ + 367, + 649, + 404, + 660 + ], + "score": 0.8, + "content": "\\mathrm { O P T } _ { 6 . 7 B }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 649, + 467, + 661 + ], + "score": 1.0, + "content": "on all datasets.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 627, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "VL-Vicuna. An exciting application of our VPGTrans is to build a GPT-4 [8] style multimodal", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "conversation chatbot. To achieve our goal, we employ Vicuna [10] as our base LLM. Similarly, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 230, + 702 + ], + "score": 1.0, + "content": "transfer the VPG from BLIP-2", + "type": "text" + }, + { + "bbox": [ + 230, + 689, + 263, + 700 + ], + "score": 0.8, + "content": "\\mathrm { O P T } _ { 6 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 688, + 506, + 702 + ], + "score": 1.0, + "content": ", and add an extra instruction tuning using MiniGPT-4’s self-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "instruct data [62]. We compare our model with MiniGPT-4 in Fig. 9. When compared to MiniGPT-4,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 418, + 724 + ], + "score": 1.0, + "content": "our VL-Vicuna shows better visual perception ability. Please refer to Appendix", + "type": "text" + }, + { + "bbox": [ + 419, + 711, + 430, + 722 + ], + "score": 0.7, + "content": "\\mathrm { \\Omega ^ { \\ S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "for more cases. In", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "addition, we also report the ranking and ELO ratings for our VL-Vicuna compared with the other", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "7 SOTA multimodal chatbots in Table 5 (The evaluation is done by Multimodality Chatbot Arena", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "platform7). The results show the effectiveness of our VL-Vicuna compared with existing SOTA", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 483, + 140, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 140, + 494 + ], + "score": 1.0, + "content": "models.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 667, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 70, + 501, + 407 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 70, + 501, + 407 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 70, + 501, + 407 + ], + "spans": [ + { + "bbox": [ + 108, + 70, + 501, + 407 + ], + "score": 0.971, + "type": "image", + "image_path": "26ccc1e5deaa7a37d5a1df096d13105e4a9097978362507df2699ccb5e3303ae.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 70, + 501, + 182.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 182.33333333333331, + 501, + 294.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 294.66666666666663, + 501, + 406.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 176, + 418, + 434, + 430 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 176, + 417, + 434, + 431 + ], + "spans": [ + { + "bbox": [ + 176, + 417, + 434, + 431 + ], + "score": 1.0, + "content": "Figure 9: Comparison between MiniGPT-4 and our VL-Vicuna.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "addition, we also report the ranking and ELO ratings for our VL-Vicuna compared with the other", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "7 SOTA multimodal chatbots in Table 5 (The evaluation is done by Multimodality Chatbot Arena", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "platform7). The results show the effectiveness of our VL-Vicuna compared with existing SOTA", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 483, + 140, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 140, + 494 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 498, + 183, + 510 + ], + "lines": [ + { + "bbox": [ + 104, + 495, + 185, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 185, + 514 + ], + "score": 1.0, + "content": "7 Conclusion", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "In this work, we conduct a comprehensive investigation to the problem of VPG transferability across", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "LLMs. We first explore the key factors for maximizing the transfer efficiency under the VPG transfer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "across different LLM sizes and types. 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MiniGPT-4: Enhancing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 124, + 299, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 124, + 299, + 506, + 314 + ], + "score": 1.0, + "content": "vision-language understanding with advanced large language models. arXiv preprint arXiv:2304.10592,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 125, + 310, + 150, + 322 + ], + "spans": [ + { + "bbox": [ + 125, + 310, + 150, + 322 + ], + "score": 1.0, + "content": "2023.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 9.5 + } + ], + "page_idx": 13, + "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": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 105, + 72, + 507, + 323 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 105, + 74, + 507, + 322 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 200, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 201, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 201, + 86 + ], + "score": 1.0, + "content": "A Related Work", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 98, + 255, + 110 + ], + "lines": [ + { + "bbox": [ + 106, + 98, + 257, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 98, + 257, + 112 + ], + "score": 1.0, + "content": "A.1 Vision and Language Models", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 120, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 120, + 504, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 504, + 132 + ], + "score": 1.0, + "content": "Vision and language models (VLMs) [50, 29, 32, 31] aim at understanding visual and textual", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 131, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 144 + ], + "score": 1.0, + "content": "information with a single model. Previously, the VLM mainly employs a pre-trained object detector", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "as its feature extractor and conducts unsupervised training on a huge amount of image-text pairs.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 167 + ], + "score": 1.0, + "content": "For example, VL-Bert [50], ViL-Bert [36] and Uniter [9] adopt Faster-RCNN [47] to extract image", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "information into object features and take advantage of the masked language model as their pre-training", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "task. Later, due to the prevalence of vision transformer (ViT), the VLM paradigm is turned into", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "score": 1.0, + "content": "end-to-end training with a ViT as the visual encoder. The representative works include ALBEF [28],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "score": 1.0, + "content": "BLIP [29], and BEIT-v3 [56], which show the state-of-the-arts supervised training performance on a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 239, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 239, + 220 + ], + "score": 1.0, + "content": "wide range of downstream tasks.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 224, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "Recently, LLMs have shown their remarkable capability as zero/few-shot learners [7] and a series", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "of emergent abilities [57] like in-context learning [7], and chain-of-thoughts reasoning [58]. A new", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "score": 1.0, + "content": "paradigm, i.e., MLLMs is created by associating the VLM or pure vision encoders with LLMs. As we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "illustrated before, the VLM or visual encoders are typically able to convert the input vision signals into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 266, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 506, + 281 + ], + "score": 1.0, + "content": "LLM-understandable soft prompts, and thus we call them VPG. The MLLMs advance in inheriting", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "the great potentials of the backbone LLMs, and thus are capable of achieving excellent zero/few-shot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "performances [2, 30] on downstream tasks or be equipped with visual planning ability [16]. However,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "connecting the VPG to the existing LLMs with further tuning is costly. Even the BLIP-2 [30], targeted", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "at efficient training, will take over 600 A100 GPU hours on over 100M image-text pairs for its largest", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "model. With this regard, our proposed VPGTrans can effectively reduce the cost of building new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 333, + 265, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 265, + 346 + ], + "score": 1.0, + "content": "MLLMs with the help of existing ones.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 204, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 206, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 206, + 376 + ], + "score": 1.0, + "content": "A.2 Prompt Transfer", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 396 + ], + "score": 1.0, + "content": "In this paper, we investigate the VPG transfer, where the soft prompt is to represent the content of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "specific inputs like images and videos. In addition to the content prompt, the more explored soft", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "prompt is the task prompt [24, 61, 22], where a sequence of soft prompts are tuned to assist the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 430 + ], + "score": 1.0, + "content": "pre-trained models to achieve better performance on specific tasks. There have already been some", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "works exploring the transferability of task prompts. For example, Su et al. [51] conducts a series", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "of experiments to illustrate the transferability across tasks and models. Specifically, Su et al. [51]", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "find that the transfer between similar tasks is beneficial for training speed-up and better performance.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "Lester et al. [25] proposes to recycle soft prompts across models with vocab-to-vocab transformations,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "or linear-combination transformations. Note that our word converter initialization is similar to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 483, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 494 + ], + "score": 1.0, + "content": "idea of vocab-to-vocab transformations. However, we do not observe a zero-shot transfer in our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "experiments like them, which indicates a potential difference between the content prompts and task", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 504, + 507, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 507, + 518 + ], + "score": 1.0, + "content": "prompts. Another way of soft prompt transfer [14] is to conduct prompt tuning on discrete prompts,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "and thus the discrete prompts can be directly shared across models. Different from these task prompts", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "transfer works, our VPG transfer scenario actually suffers from fewer limitations. For example, the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "task soft prompts transfer suffers from the dimension change problem, where the main body of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "trainable parameters should be processed. However, our VPG (the main trainable parameters) can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 506, + 571 + ], + "score": 1.0, + "content": "naturally be shared among LLMs with different embedding dimensions and leave the dimension", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 570, + 367, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 367, + 583 + ], + "score": 1.0, + "content": "change problem to a simple projector with ignorable parameters.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 106, + 601, + 349, + 615 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 350, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 350, + 618 + ], + "score": 1.0, + "content": "B Extended Findings in Exploratory Analysis", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 411, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 412, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 412, + 642 + ], + "score": 1.0, + "content": "In this section, we show extended findings of exploratory analysis (cf. §3.1).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 405, + 668 + ], + "score": 1.0, + "content": "• 1. Merely tuning the projector can not achieve the best performance.", + "type": "text" + }, + { + "bbox": [ + 412, + 656, + 506, + 667 + ], + "score": 1.0, + "content": "We want to clarify that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "merely tuning the projector is insufficient for achieving the best performance. Notably, as shown in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "Fig. 10, significant performance gaps are observed between the “only linear” (green curve) and “train", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "from scratch” (orange curve) approaches for COCO caption and NoCaps. Therefore, if the goal is to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "build a multimodal conversation robot using carefully collected dialog data, training only the linear", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 711, + 328, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 328, + 723 + ], + "score": 1.0, + "content": "projector is insufficient to align with the provided data.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 742, + 311, + 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": "title", + "bbox": [ + 107, + 71, + 200, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 201, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 201, + 86 + ], + "score": 1.0, + "content": "A Related Work", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 98, + 255, + 110 + ], + "lines": [ + { + "bbox": [ + 106, + 98, + 257, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 98, + 257, + 112 + ], + "score": 1.0, + "content": "A.1 Vision and Language Models", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 120, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 120, + 504, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 504, + 132 + ], + "score": 1.0, + "content": "Vision and language models (VLMs) [50, 29, 32, 31] aim at understanding visual and textual", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 131, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 144 + ], + "score": 1.0, + "content": "information with a single model. Previously, the VLM mainly employs a pre-trained object detector", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "as its feature extractor and conducts unsupervised training on a huge amount of image-text pairs.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 167 + ], + "score": 1.0, + "content": "For example, VL-Bert [50], ViL-Bert [36] and Uniter [9] adopt Faster-RCNN [47] to extract image", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "information into object features and take advantage of the masked language model as their pre-training", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "task. Later, due to the prevalence of vision transformer (ViT), the VLM paradigm is turned into", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "score": 1.0, + "content": "end-to-end training with a ViT as the visual encoder. The representative works include ALBEF [28],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "score": 1.0, + "content": "BLIP [29], and BEIT-v3 [56], which show the state-of-the-arts supervised training performance on a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 239, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 239, + 220 + ], + "score": 1.0, + "content": "wide range of downstream tasks.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 120, + 506, + 220 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 224, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "Recently, LLMs have shown their remarkable capability as zero/few-shot learners [7] and a series", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "of emergent abilities [57] like in-context learning [7], and chain-of-thoughts reasoning [58]. A new", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "score": 1.0, + "content": "paradigm, i.e., MLLMs is created by associating the VLM or pure vision encoders with LLMs. As we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "illustrated before, the VLM or visual encoders are typically able to convert the input vision signals into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 266, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 506, + 281 + ], + "score": 1.0, + "content": "LLM-understandable soft prompts, and thus we call them VPG. The MLLMs advance in inheriting", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "the great potentials of the backbone LLMs, and thus are capable of achieving excellent zero/few-shot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "performances [2, 30] on downstream tasks or be equipped with visual planning ability [16]. However,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "connecting the VPG to the existing LLMs with further tuning is costly. Even the BLIP-2 [30], targeted", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "at efficient training, will take over 600 A100 GPU hours on over 100M image-text pairs for its largest", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "model. With this regard, our proposed VPGTrans can effectively reduce the cost of building new", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 333, + 265, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 265, + 346 + ], + "score": 1.0, + "content": "MLLMs with the help of existing ones.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 224, + 506, + 346 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 204, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 206, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 206, + 376 + ], + "score": 1.0, + "content": "A.2 Prompt Transfer", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 396 + ], + "score": 1.0, + "content": "In this paper, we investigate the VPG transfer, where the soft prompt is to represent the content of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "specific inputs like images and videos. In addition to the content prompt, the more explored soft", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "prompt is the task prompt [24, 61, 22], where a sequence of soft prompts are tuned to assist the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 430 + ], + "score": 1.0, + "content": "pre-trained models to achieve better performance on specific tasks. There have already been some", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "works exploring the transferability of task prompts. For example, Su et al. [51] conducts a series", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "of experiments to illustrate the transferability across tasks and models. Specifically, Su et al. [51]", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "find that the transfer between similar tasks is beneficial for training speed-up and better performance.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "Lester et al. [25] proposes to recycle soft prompts across models with vocab-to-vocab transformations,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "or linear-combination transformations. Note that our word converter initialization is similar to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 483, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 494 + ], + "score": 1.0, + "content": "idea of vocab-to-vocab transformations. However, we do not observe a zero-shot transfer in our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "experiments like them, which indicates a potential difference between the content prompts and task", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 504, + 507, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 507, + 518 + ], + "score": 1.0, + "content": "prompts. Another way of soft prompt transfer [14] is to conduct prompt tuning on discrete prompts,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "and thus the discrete prompts can be directly shared across models. Different from these task prompts", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "transfer works, our VPG transfer scenario actually suffers from fewer limitations. For example, the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "task soft prompts transfer suffers from the dimension change problem, where the main body of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "trainable parameters should be processed. However, our VPG (the main trainable parameters) can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 559, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 506, + 571 + ], + "score": 1.0, + "content": "naturally be shared among LLMs with different embedding dimensions and leave the dimension", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 570, + 367, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 367, + 583 + ], + "score": 1.0, + "content": "change problem to a simple projector with ignorable parameters.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 385, + 507, + 583 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 601, + 349, + 615 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 350, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 350, + 618 + ], + "score": 1.0, + "content": "B Extended Findings in Exploratory Analysis", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 411, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 412, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 412, + 642 + ], + "score": 1.0, + "content": "In this section, we show extended findings of exploratory analysis (cf. §3.1).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 627, + 412, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 405, + 668 + ], + "score": 1.0, + "content": "• 1. Merely tuning the projector can not achieve the best performance.", + "type": "text" + }, + { + "bbox": [ + 412, + 656, + 506, + 667 + ], + "score": 1.0, + "content": "We want to clarify that", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "merely tuning the projector is insufficient for achieving the best performance. Notably, as shown in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "Fig. 10, significant performance gaps are observed between the “only linear” (green curve) and “train", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "from scratch” (orange curve) approaches for COCO caption and NoCaps. Therefore, if the goal is to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "build a multimodal conversation robot using carefully collected dialog data, training only the linear", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 711, + 328, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 328, + 723 + ], + "score": 1.0, + "content": "projector is insufficient to align with the provided data.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 655, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 70, + 503, + 148 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 70, + 503, + 148 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 70, + 503, + 148 + ], + "spans": [ + { + "bbox": [ + 107, + 70, + 503, + 148 + 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training it with randomly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 167, + 504, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 199, + 182 + ], + "score": 1.0, + "content": "initialized projector for", + "type": "text" + }, + { + "bbox": [ + 199, + 169, + 236, + 180 + ], + "score": 0.57, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 167, + 466, + 182 + ], + "score": 1.0, + "content": "and ii) training VPG and randomly initialized projector for", + "type": "text" + }, + { + "bbox": [ + 467, + 169, + 504, + 180 + ], + "score": 0.3, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 178, + 285, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 285, + 192 + ], + "score": 1.0, + "content": "from scratch. iii) training only the projector.", + "type": "text" + } + ], + "index": 5 + } + ], 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"content": "• 2. Word embedding converter can not replace a trained linear projector. As demonstrated", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "in Fig. 11, we observe a common pattern: the last token of soft prompts is closest to EOS, while", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "the middle tokens represent the image content. Such a phenomenon indicates a similarity between", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "soft prompts and word embeddings. However, they are not identical. For instance, the norm of soft", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "prompts is typically around 10 times the average norm of word embeddings. It is important to note", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "that the linear projector initialization cannot replace the warm-up training. Using only the linear", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "projector initialization even yields a random performance. We believe that a better understanding of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 411, + 376, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 376, + 426 + ], + "score": 1.0, + "content": "how prompt works will further benefit the VPG’s transfer learning.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "• 3. The projector warm-up is robust to a larger learning rate, while VPG can not. The first", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "thing we want to clarify is that the 5 times normal learning rate will result in a training crash for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "VPG. Additionally, we find that although increasing the learning rate to 10 times in the projector", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "warm-up does not yield any additional acceleration, the projector can converge without crashing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 171, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 171, + 494 + ], + "score": 1.0, + "content": "during training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 268, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 269, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 269, + 523 + ], + "score": 1.0, + "content": "C Extended TaS Experiments", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 407, + 545 + ], + "score": 1.0, + "content": "In this section, we first illustrate extending findings of TaS experiments (cf.", + "type": "text" + }, + { + "bbox": [ + 407, + 533, + 419, + 544 + ], + "score": 0.4, + "content": "\\ S 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 532, + 505, + 545 + ], + "score": 1.0, + "content": ". Then, we introduce", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "the implementation details of scale-up experiments. 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However, we want to stress", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "that the retained data should be of high quality, which means that if the same number of SBU data", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 687, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 228, + 72, + 383, + 223 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 228, + 72, + 383, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 228, + 72, + 383, + 223 + ], + "spans": [ + { + "bbox": [ + 228, + 72, + 383, + 223 + ], + "score": 0.963, + "type": "image", + "image_path": "3e4dd23620bef89b02685655c6d2638d085099edb3dda84cf3da77d5e5a29fd1.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 228, + 72, + 383, + 85.72727272727272 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 228, + 85.72727272727272, + 383, + 99.45454545454544 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 228, + 99.45454545454544, + 383, + 113.18181818181816 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 228, + 113.18181818181816, + 383, + 126.90909090909088 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 228, + 126.90909090909088, + 383, + 140.6363636363636 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 228, + 140.6363636363636, + 383, + 154.36363636363632 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 228, + 154.36363636363632, + 383, + 168.09090909090904 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 228, + 168.09090909090904, + 383, + 181.81818181818176 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 228, + 181.81818181818176, + 383, + 195.54545454545448 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 228, + 195.54545454545448, + 383, + 209.2727272727272 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 228, + 209.2727272727272, + 383, + 222.99999999999991 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 163, + 234, + 447, + 247 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 233, + 448, + 248 + ], + "spans": [ + { + "bbox": [ + 163, + 233, + 332, + 248 + ], + "score": 1.0, + "content": "Figure 12: Comparison of training FlanT5", + "type": "text" + }, + { + "bbox": [ + 332, + 235, + 348, + 247 + ], + "score": 0.46, + "content": "\\mathrm { l a r g e }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 233, + 448, + 248 + ], + "score": 1.0, + "content": "using different methods.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + } + ], + "index": 8.0 + }, + { + "type": "table", + "bbox": [ + 146, + 290, + 464, + 392 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 262, + 504, + 285 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 261, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 276 + ], + "score": 1.0, + "content": "Table 6: The speed-up rate of our VPGTrans compared with VPG inherit. The symbol \"-\" means", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 366, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 366, + 285 + ], + "score": 1.0, + "content": "VPGTrans can not achieve better performance than VPG inherit.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "table_body", + "bbox": [ + 146, + 290, + 464, + 392 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 290, + 464, + 392 + ], + "spans": [ + { + "bbox": [ + 146, + 290, + 464, + 392 + ], + "score": 0.979, + "html": "
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.12.710.06.06.0
OPT125M→1.3B-2.71-1
OPT350M→1.3B11.71.02.01.0
OPT125M→2.7B3.03.03.01.03.0
OPT350M→2.7B3.34.51.01.01.0
OPT1.3B→2.7B2.33.0111
FlanT5base-→XL11.01.89.00.4
", + "type": "table", + "image_path": "a01109c17f0876641c1a64b088a94ea18d23d988fb0dbcc1a438ae02e0f7ba89.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 146, + 290, + 464, + 324.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 146, + 324.0, + 464, + 358.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 146, + 358.0, + 464, + 392.0 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 504, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "is retained, the performance especially for captioning will drop. The conclusion can also be found in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 429, + 380, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 380, + 441 + ], + "score": 1.0, + "content": "Table 4. We refer the readers to the next subsection for more details.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 106, + 460, + 345, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 347, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 347, + 474 + ], + "score": 1.0, + "content": "C.3 Comparison between VPGTrans and VPG Inherit", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "As shown in Fig. 13, the green line (our VPGTrans) can be higher than the orange line (VPG inherit)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "for the majority of various conditions. Especially when considering the best performance of different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 504, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 506, + 519 + ], + "score": 1.0, + "content": "tasks, our VPGTrans can achieve better performance than VPG inherit (non “-” in Table 6) for over", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 126, + 527 + ], + "score": 0.83, + "content": "74 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "Transfer-Task variants. Moreover, among the variants that our VPGTrans can achieve better", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 527, + 423, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 347, + 540 + ], + "score": 1.0, + "content": "performance, our VPGTrans can also achieve a speed-up on", + "type": "text" + }, + { + "bbox": [ + 348, + 527, + 375, + 538 + ], + "score": 0.85, + "content": "6 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 527, + 423, + 540 + ], + "score": 1.0, + "content": "conditions.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 325, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 326, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 326, + 573 + ], + "score": 1.0, + "content": "C.4 Scale-up Experiment Implementation Details", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "The scale-up experiment refers to the results in Table 4. We try to imitate BLIP-2’s pre-training data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "composition. First of all, two human-annotated datasets COCO and VG are used. SBU is also used.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "Then, BLIP-2 uses BLIP to generate captions for the 115M web images and rank them with CLIP", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "ViT-L/14. We also adopt similar synthetic data from Laion-COCO.8 We report the concrete number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "of data we use in Table 4. For the stage-1 training, we keep the same as the previous validation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "experiments where COCO and SBU are used for warm-up with a 5 times the learning rate. Then,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "we use COCO, VG, and Laion-COCO for the stage-2 training. 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The symbol \"-\" means", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 366, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 366, + 285 + ], + "score": 1.0, + "content": "VPGTrans can not achieve better performance than VPG inherit.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "table_body", + "bbox": [ + 146, + 290, + 464, + 392 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 290, + 464, + 392 + ], + "spans": [ + { + "bbox": [ + 146, + 290, + 464, + 392 + ], + "score": 0.979, + "html": "
TransferCOCO CaptionNoCapsVQAv2GQAOKVQA
OPT125M→350M1.12.710.06.06.0
OPT125M→1.3B-2.71-1
OPT350M→1.3B11.71.02.01.0
OPT125M→2.7B3.03.03.01.03.0
OPT350M→2.7B3.34.51.01.01.0
OPT1.3B→2.7B2.33.0111
FlanT5base-→XL11.01.89.00.4
", + "type": "table", + "image_path": "a01109c17f0876641c1a64b088a94ea18d23d988fb0dbcc1a438ae02e0f7ba89.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 146, + 290, + 464, + 324.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 146, + 324.0, + 464, + 358.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 146, + 358.0, + 464, + 392.0 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 504, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "is retained, the performance especially for captioning will drop. The conclusion can also be found in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 429, + 380, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 380, + 441 + ], + "score": 1.0, + "content": "Table 4. We refer the readers to the next subsection for more details.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 417, + 505, + 441 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 460, + 345, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 347, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 347, + 474 + ], + "score": 1.0, + "content": "C.3 Comparison between VPGTrans and VPG Inherit", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "As shown in Fig. 13, the green line (our VPGTrans) can be higher than the orange line (VPG inherit)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "for the majority of various conditions. 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We try to imitate BLIP-2’s pre-training data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "composition. First of all, two human-annotated datasets COCO and VG are used. SBU is also used.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "Then, BLIP-2 uses BLIP to generate captions for the 115M web images and rank them with CLIP", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "ViT-L/14. We also adopt similar synthetic data from Laion-COCO.8 We report the concrete number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "of data we use in Table 4. For the stage-1 training, we keep the same as the previous validation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "experiments where COCO and SBU are used for warm-up with a 5 times the learning rate. Then,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "we use COCO, VG, and Laion-COCO for the stage-2 training. 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FlanT5base → OPT350M110.8136.540.444.2
FlanT5xL →OPT2.7B132.1139.350.450.3
OPT350M→FlanT5base1.1122.134.049.9
OPT2.7B→FlanT5xL106.5133.251.353.5
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To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "better support our results, we compare the results of tuning only the projector (i.e. linear transfer) and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "the best results the VPGTrans can achieve. 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There is a 25.7 points gap between", + "type": "text" + }, + { + "bbox": [ + 448, + 337, + 505, + 348 + ], + "score": 0.7, + "content": "\\mathrm { F l a n T 5 _ { b a s e } }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 346, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 144, + 359 + ], + "score": 0.7, + "content": "\\mathrm { O P T } _ { 3 5 0 \\mathrm { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 346, + 488, + 362 + ], + "score": 1.0, + "content": "’s linear and best on COCO caption datasets but only 7.2 points gap between FlanT", + "type": "text" + }, + { + "bbox": [ + 489, + 349, + 504, + 359 + ], + "score": 0.43, + "content": "5 _ { \\mathrm { X L } }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 154, + 370 + ], + "score": 0.88, + "content": " \\mathrm { O P T } _ { 2 . 7 \\mathrm { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 357, + 506, + 372 + ], + "score": 1.0, + "content": "’s linear and best. 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We leave the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 275, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 275, + 393 + ], + "score": 1.0, + "content": "systematical exploration for future works.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 106, + 407, + 353, + 421 + ], + "lines": [ + { + "bbox": [ + 104, + 405, + 355, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 355, + 424 + ], + "score": 1.0, + "content": "E Extended Results for MLLM Customization", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "We show more comparisons between VL-Vicuna and MiniGPT-4 in Fig. 15. We can see that our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 456 + ], + "score": 1.0, + "content": "VL-Vicuna has better visual perception ability. For example, when MiniGPT-4 falsely recognizes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 455, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 504, + 466 + ], + "score": 1.0, + "content": "the three people in the image as two in the first example, our VL-Vicuna can not only recognize the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "number of people but also tell their roles. 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In the third example, our VL-Vicuna can recognize Leonardo and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 487, + 278, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 278, + 499 + ], + "score": 1.0, + "content": "link the content with his films like Titanic.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 513, + 297, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 298, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 298, + 529 + ], + "score": 1.0, + "content": "F Potential Impact and Limitations", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "Our VPGTrans is designed for building new MLLMs with lower computational cost, i.e., shorter", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "training time and less training data. With an already pre-trained MLLM, VPGTrans enables fast VPG", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "transfer to build either a larger MLLM or a MLLM with a different type of LLM. We hope VPGTrans", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "can facilitate teams in LLM communicty to customize their MLLMs with reduced cost. There are also", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "possible limitations of the current version of VPGTrans. 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TransferCOCO CaptionVQAv2
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FlanT5base → OPT350M110.8136.540.444.2
FlanT5xL →OPT2.7B132.1139.350.450.3
OPT350M→FlanT5base1.1122.134.049.9
OPT2.7B→FlanT5xL106.5133.251.353.5
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We plot a more", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 227, + 258, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 258, + 240 + ], + "score": 1.0, + "content": "complete version of Fig. 8 in Fig. 14.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 215, + 505, + 240 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 250, + 401, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 401, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 401, + 266 + ], + "score": 1.0, + "content": "D.1 Linear Transfer Gap between Different LLM’s Visual Prompts", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "As illustrated in Section 5.2, it is more difficult to transfer between two small LLMs with our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "VPGTrans due to the weaker linear transferability between two small LLMs’ visual prompts. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "better support our results, we compare the results of tuning only the projector (i.e. linear transfer) and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "the best results the VPGTrans can achieve. 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If considering the transfer between the small to large LLMs under", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "TaT, both the VPG’s visual perception ability and transfer gap should be considered. 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In the third example, our VL-Vicuna can recognize Leonardo and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 487, + 278, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 278, + 499 + ], + "score": 1.0, + "content": "link the content with his films like Titanic.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 431, + 506, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 513, + 297, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 298, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 298, + 529 + ], + "score": 1.0, + "content": "F Potential Impact and Limitations", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "Our VPGTrans is designed for building new MLLMs with lower computational cost, i.e., shorter", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "training time and less training data. 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"_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/dev/7UmjRGzp-A/7UmjRGzp-A.md b/parse/dev/7UmjRGzp-A/7UmjRGzp-A.md new file mode 100644 index 0000000000000000000000000000000000000000..21c843ca2dd7093bb7ebc81a9c0f38b0ef9510b4 --- /dev/null +++ b/parse/dev/7UmjRGzp-A/7UmjRGzp-A.md @@ -0,0 +1,1030 @@ +# UNDERSTANDING OVER-SQUASHING AND BOTTLENECKS ON GRAPHS VIA CURVATURE + +Jake Topping12†, Francesco Di Giovanni3†, Benjamin P. Chamberlain3, Xiaowen $\mathbf { D o n g } ^ { 1 }$ , and Michael M. Bronstein23 1University of Oxford 2Imperial College London 3Twitter + +# ABSTRACT + +Most graph neural networks (GNNs) use the message passing paradigm, in which node features are propagated on the input graph. Recent works pointed to the distortion of information flowing from distant nodes as a factor limiting the efficiency of message passing for tasks relying on long-distance interactions. This phenomenon, referred to as ‘over-squashing’, has been heuristically attributed to graph bottlenecks where the number of $k$ -hop neighbors grows rapidly with $k$ . We provide a precise description of the over-squashing phenomenon in GNNs and analyze how it arises from bottlenecks in the graph. For this purpose, we introduce a new edge-based combinatorial curvature and prove that negatively curved edges are responsible for the over-squashing issue. We also propose and experimentally test a curvature-based graph rewiring method to alleviate the over-squashing. + +# 1 INTRODUCTION + +In the past few years, deep learning on graphs and in particular graph neural networks (GNNs) (Sperduti, 1994; Goller & Kuchler, 1996; Sperduti & Starita, 1997; Frasconi et al., 1998; Gori et al., 2005; Scarselli et al., 2008; Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2017; Gilmer et al., 2017) have become very popular in the machine learning community due to their ability to deal with broad classes of systems of relations and interactions, ranging from social networks to particle physics (Shlomi et al., 2021). + +The vast majority of GNNs follow the message passing paradigm (Gilmer et al., + +![](images/437827f1c25f3a408f69b5456e32402e205c919e9d7a9e7a22fdc6c85a6a6f78.jpg) +Figure 1: Top: evolution of curvature on a surface may reduce the bottleneck. Bottom: this paper shows how the same may be done on graphs to improve GNN performance. Blue/red shows negative/positive curvature. + +2017), using learnable non-linear functions to diffuse information on the graph. Multiple popular GNN architectures such as GCN (Kipf & Welling, 2017) and GAT (Velickovi ˇ c et al., 2018) can be ´ posed as particular flavors of this scheme and considered instances of a more general framework of geometric deep learning (Bronstein et al., 2021). + +Some of the drawbacks of the message passing paradigm have now been identified and formalized, including the limits of expressive power (Xu et al., 2019; Morris et al., 2019; Maron et al., 2019) and the problem of over-smoothing (NT & Maehara, 2019; Oono & Suzuki, 2020). On the other hand, much less is known about the phenomenon of over-squashing, consisting in the distortion of messages being propagated from distant nodes. Alon & Yahav (2021) proposed rewiring the graph as a way of reducing the bottleneck, defined as those topological properties in the graph leading to over-squashing. This approach is in line with multiple other results e.g. using connectivity diffusion (Klicpera et al., 2019) as a preprocessing step to facilitate graph learning. Yet, the exact understanding of the over-squashing and how it originates from the bottlenecks in the topology of the underlying graph are still elusive. Consequently, there is currently no consensus on the right method (either based on graph rewiring or not) to address the bottleneck and hence alleviate the over-squashing. + +In this paper, we address these questions using tools from differential geometry, which traditionally is concerned with the study of manifolds. It offers an appealing framework to study the properties of graphs, in particular arguing that graphs, like manifolds, exhibit curvature that makes them more suitable to be realized in spaces with hyperbolic geometry (Liu et al., 2019; Chami et al., 2019; Boguna et al., 2021). One notion of curvature that has received attention for graph learning is Ricci curvature (Hamilton, 1988), also known in geometry for its use in Ricci flow and the subsequent proof of the Poincaré conjecture (Perelman, 2003). Certain graph analogues of the Ricci curvature (Forman, 2003; Ollivier, 2009; Sreejith et al., 2016) were used in Ni et al. (2018) for a discrete version of Ricci flow to construct a metric between graphs. Graph Ricci flow was also used in Ni et al. (2019) for community detection. Both of these methods use the edge weights as a substitute for the metric of a manifold, and do not change the topological structure of the graph. + +Contributions and Outline. This paper, to our knowledge, is the first theoretical study of the bottleneck and over-squashing phenomena in message passing neural networks from a geometric perspective. In Section 2, we propose the Jacobian of node representations as a formal way of measuring the over-squashing and we show that the graph topology may compromise message propagation in graph neural networks by creating a bottleneck. In Section 3, we investigate how such a bottleneck is induced which leads to the over-squashing of information. To this aim, we introduce a new combinatorial edge-based curvature called Balanced Forman curvature that constitutes a sharp lower bound to the standard Ollivier curvature on graphs, and prove that negatively curved edges are responsible for the formation of bottlenecks (and hence for over-squashing). In Section 4, we present a new curvature-based method for graph rewiring called Stochastic Discrete Ricci Flow. According to the theoretical results in Section 3, this rewiring method is suited to address the graph bottleneck and hence alleviate the over-squashing by surgically targeting the edges responsible for the issue. By contrast, we rigorously show that a recently introduced diffusion-based rewiring scheme might generally fail to reduce the bottleneck. Finally, in Section 5, we compare different rewiring strategies experimentally on several standard graph learning datasets. + +# 2 ANALYSIS OF THE OVER-SQUASHING PHENOMENON + +# 2.1 PRELIMINARIES + +Let $G = ( V , E )$ be a simple, undirected, and connected graph, where $( i , j ) \in E$ iff $i \sim j$ . We focus on the unweighted case, although the theory extends to the weighted setting as well. We denote the adjacency matrix by $A$ and let $\overset { \vartriangle } { \boldsymbol { \tilde { A } } } = \boldsymbol { A } + \boldsymbol { \bar { I } }$ be the adjacency matrix augmented with self-loops. Similarly we let $\tilde { D } = D + I$ , with $D$ the diagonal degree matrix, and let $\bar { { \cal A } } = \tilde { D } ^ { - \frac { 1 } { 2 } } \tilde { A } \tilde { D } ^ { - \frac { 1 } { 2 } }$ be the normalized augmented adjacency matrix (self-loops are commonly included in GNN architecture, and in Section 2 we formally explain why GNNs are expected to propagate information more reliably when self-loops are taken into account). Given $i \in V$ , we denote its degree by $d _ { i }$ and let + +$$ +S _ { r } ( i ) : = \{ j \in V : d _ { G } ( i , j ) = r \} , \quad B _ { r } ( i ) : = \{ j \in V : d _ { G } ( i , j ) \leq r \} , +$$ + +where $d _ { G }$ is the standard shortest-path distance on the graph and $r \in \mathbb { N }$ . The set $B _ { r } ( i )$ represents the receptive field of an $r$ -layer message passing neural network at node $i$ . + +Message passing neural networks (MPNNs). Assume that the graph $G$ is equipped with node features $X \in \mathbb { R } ^ { n \times p _ { 0 } }$ where $x _ { i } \in \mathbb { R } ^ { p _ { 0 } }$ is the feature vector at node $i = 1 , \ldots , n = | V |$ . We denote by $h _ { i } ^ { ( \ell ) } \in \mathbb { R } ^ { p _ { \ell } }$ the representation of node $i$ at layer $\ell \geq 0$ , with $h _ { i } ^ { ( 0 ) } = x _ { i }$ . Given a family of message functions $\psi _ { \ell } : \mathbb { R } ^ { p _ { \ell } } \times \mathbb { R } ^ { p _ { \ell } } \to \mathbb { R } ^ { p _ { \ell } ^ { \prime } }$ and update functions $\phi _ { \ell } : \mathbb { R } ^ { p _ { \ell } } \times \mathbb { R } ^ { p _ { \ell } ^ { \prime } } \to \mathbb { R } ^ { p _ { \ell + 1 } }$ , we can write the $( \ell + 1 )$ -st layer output of a generic MPNN as follows (Gilmer et al., 2017): + +$$ +h _ { i } ^ { ( \ell + 1 ) } = \phi _ { \ell } \left( h _ { i } ^ { ( \ell ) } , \sum _ { j = 1 } ^ { n } \hat { A } _ { i j } \psi _ { \ell } ( h _ { i } ^ { ( \ell ) } , h _ { j } ^ { ( \ell ) } ) \right) . +$$ + +Here we have used the augmented normalized adjacency matrix to propagate messages from each node to its neighbors, which simply leads to a degree normalization of the message functions $\psi _ { \ell }$ . To + +avoid heavy notations the node features and representations are assumed to be scalar from now on; +these assumptions simplify the discussion and the vector case leads to analogous results. + +# 2.2 THE OVER-SQUASHING PROBLEM + +Multiple recent papers observed that MPNNs tend to perform poorly in situations when the learned task requires long-range dependencies and at the same time the structure of the graph results in exponentially many long-range neighboring nodes. We say that a graph learning problem has long-range dependencies when the output of a MPNN depends on representations of distant nodes interacting with each other. If long-range dependencies are present, messages coming from nonadjacent nodes need to be propagated across the network without being too distorted. In many cases however (e.g. in ‘small-world’ graphs such as social networks), the size of the receptive field $B _ { r } ( i )$ grows exponentially with $r$ . If this occurs, representations of exponentially many neighboring nodes need to be compressed into fixed-size vectors to propagate messages to node $i$ , causing a phenomenon referred to as over-squashing of information (Alon & Yahav, 2021). In line with Alon & Yahav (2021), we refer to those structural properties of the graph that lead to over-squashing as a bottleneck1. + +Sensitivity analysis. The hidden feature $h _ { i } ^ { ( \ell ) } = h _ { i } ^ { ( \ell ) } ( x _ { 1 } , \dots , x _ { n } )$ computed by an MPNN with $\ell$ layers as in equation 1 is a differentiable function of the input node features $\{ x _ { 1 } , \ldots , x _ { n } \}$ as long as the update and message functions $\phi _ { \ell }$ and $\psi _ { \ell }$ are differentiable. The over-squashing of information can then be understood in terms of one node representation $h _ { i } ^ { ( \ell ) }$ failing to be affected by some input feature $x _ { s }$ of node $s$ at distance $r$ from node $i$ . Hence, we propose the Jacobian $\partial h _ { i } ^ { ( r ) } / \partial x _ { s }$ as an explicit and formal way of assessing the over-squashing effect2. + +Lemma 1. Assume an MPNN as in equation $I$ . Let $i , s \in V$ with $s \in S _ { r + 1 } ( i )$ . If $| \nabla \phi _ { \ell } | \le \alpha$ and $| \nabla \psi _ { \ell } | \le \beta$ for $0 \leq \ell \leq r$ , then + +$$ +\left| \frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } \right| \leq ( \alpha \beta ) ^ { r + 1 } ( \hat { A } ^ { r + 1 } ) _ { i s } . +$$ + +Lemma 1 states that if $\phi _ { \ell }$ and $\psi _ { \ell }$ have bounded derivatives, then the propagation of messages is controlled by a suitable power of $\hat { A }$ . For example, if $d _ { G } ( i , s ) = r + 1$ and the sub-graph induced on $B _ { r + 1 } ( i )$ is a binary tree, then $( \hat { A } ^ { r + 1 } ) _ { i s } = 2 ^ { - 1 } 3 ^ { - r }$ , which gives an exponential decay of the node dependence on input features at distance $r$ , as also heuristically argued by Alon & Yahav (2021). + +The sensitivity analysis in Lemma 1 relates the over-squashing – as measured by the Jacobian of the node representations – to the graph topology via powers of the augmented normalized adjacency matrix. In the next section we explore this connection further by analyzing which local properties of the graph structure affect the right hand side in equation 2, hence causing the bottleneck. We will address this problem by introducing a new combinatorial notion of edge-based curvature and showing that negatively curved edges are those responsible for the over-squashing phenomenon. + +# 3 GRAPH CURVATURE AND BOTTLENECK + +A natural object in Riemannian geometry is the Ricci curvature, a bilinear form determining the geodesic dispersion, i.e. whether geodesics starting at nearby points with ‘same’ velocity remain parallel (Euclidean space), converge (spherical space), or diverge (hyperbolic space). To motivate the introduction of a Ricci curvature for graphs, we focus on these three cases. Consider two nodes $i \sim j$ and two edges starting at $i$ and $j$ respectively. In a discrete spherical geometry (Figure 2a), the edges would meet at $k$ to form a triangle (complete graph). In a discrete Euclidean geometry (Figure 2b), the edges would stay parallel and form a 4-cycle based at $i \sim j$ (orthogonal grid). Finally, in a discrete hyperbolic geometry (Figure 2c), the mutual distance of the edge endpoints would have grown compared to that of $i$ and $j$ (tree). Therefore, a Ricci curvature for graphs should provide us with more sophisticated tools than the degree to analyze the neighborhood of an edge. + +Curvatures on graphs. The main examples of edge-based curvature are the Forman curvature $F ( i , j )$ (Forman, 2003) and the Ollivier curvature $\kappa ( i , j )$ in Ollivier (2007; 2009) (see Appendix). While $F ( i , j )$ is given in terms of combinatorial quantities (Sreejith et al., 2016), results are scarce and the definition is biased towards negative curvature. The theory on $\kappa ( i , j )$ instead is richer (Lin et al., 2011; Münch, 2019) but its formulation makes it hard to control local quantities. + +![](images/23259c93c37afcd782009cb348c4057564529808a5e2f4989beb67f4e8be2387.jpg) +Figure 2: Different regimes of curvatures on graphs analogous to spherical (a), planar (b), and hyperbolic (c) geometries in the continuous setting. + +Balanced Forman curvature. We propose a new curvature to address the shortcomings of the existing candidates. We use the following definitions to describe the neighborhood of an edge $i \sim j$ and we refer to the Appendix for a more complete discussion: + +(i) $\sharp _ { \Delta } ( i , j ) : = S _ { 1 } ( i ) \cap S _ { 1 } ( j )$ are the triangles based at $i \sim j$ . (ii) $\sharp _ { \bigtriangledown } ( i , j ) : = \{ k \in S _ { 1 } ( i ) \ \backslash \ S _ { 1 } ( j ) , k \neq j : \ \exists w \in ( S _ { 1 } ( k ) \cap S _ { 1 } ( j ) ) \setminus S _ { 1 } ( i ) \}$ are the neighbors of $i$ forming a 4-cycle based at the edge $i \sim j$ without diagonals inside. (iii) $\gamma _ { \operatorname* { m a x } } ( i , j )$ is the maximal number of n node (see Definition 4). $4 -$ cycles based at $i \sim j$ traversing + +In line with the discussion about geodesic dispersion, one expects $\sharp _ { \Delta }$ to be related to positive curvature (complete graph), $\sharp _ { \bigstar } ^ { i }$ to zero curvature (grid), and the remaining outgoing edges to negative curvature (tree). Our new curvature formulation reflects such an intuition and recovers the expected results in the classical cases. In the example in Figure 3 we have $\sharp _ { \bigstar } ^ { 0 } ( \bar { 0 } , 1 ) = \{ 2 , 3 \}$ while $\sharp _ { \bigstar } ^ { 1 } ( 0 , 1 ) = \{ 5 \}$ , both without 4,6 because of the triangle 1-6-0. The degeneracy factor $\gamma _ { \mathrm { m a x } } ( 0 , 1 ) = 2$ , as there exist two 4-cycles passing through node 5. + +![](images/a9d21074d19942603c3881fd9a01471a01eac7e7f8f372bacd0820ede9137be8.jpg) +Figure 3: 4-cycle contribution. + +Definition 1 (Balanced Forman curvature). For any edge $i \sim j$ in a simple, unweighted graph $G$ , we let $\operatorname { R i c } ( i , j )$ be zero $i f \operatorname* { m i n } \{ d _ { i } , d _ { j } \} = 1$ and otherwise + +$$ +\mathbf { R i c } ( i , j ) : = \frac { 2 } { d _ { i } } + \frac { 2 } { d _ { j } } - 2 + 2 \frac { \left| \sharp _ { \Delta } ( i , j ) \right| } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } + \frac { \left| \sharp _ { \Delta } ( i , j ) \right| } { \operatorname* { m i n } \{ d _ { i } , d _ { j } \} } + \frac { \left( \gamma _ { m a x } \right) ^ { - 1 } } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } ( | \sharp _ { \overline { { \Gamma } } } ^ { i } | + | \sharp _ { \overline { { \Gamma } } } ^ { j } | ) , +$$ + +where the last term is set to be zero $i f | \sharp _ { \bigstar } ^ { i } |$ (and hence $| \sharp _ { \bigstar } ^ { j } | \big )$ is zero. In particular $\operatorname { R i c } ( i , j ) > - 2$ + +The curvature is negative when $i \sim j$ behaves as a bridge between $S _ { 1 } ( i )$ and $S _ { 1 } ( j )$ , while it is positive when $S _ { 1 } ( i )$ and $S _ { 1 } ( j )$ stay connected after removing $i \sim j$ . We refer to Ric as Balanced Forman curvature. We can relate the Balanced Forman curvature to the Jacobian of hidden features, while also extending many results valid for the Ollivier curvature $\kappa ( i , j )$ thanks to our next theorem. + +Theorem 2. Given an unweighted graph $G$ , for any edge $i \sim j$ we have $\kappa ( i , j ) \geq \operatorname { R i c } ( i , j )$ + +Theorem 2 generalizes Jost & Liu (2014, Theorem 3) (see Appendix). We also note that the computational complexity for $\kappa$ scales as $O ( | E | d _ { \operatorname* { m a x } } ^ { 3 } )$ , while for our Ric we have $O ( | E | d _ { \operatorname* { m a x } } ^ { 2 } )$ , with $d _ { \mathrm { m a x } }$ the maximal degree. From Theorem 2 and Paeng (2012), we find: + +Corollary 3. If $\operatorname { R i c } ( i , j ) \geq k > 0$ for any edge $i \sim j$ , then there exists a polynomial $P$ such that + +$$ +| B _ { r } ( i ) | \leq P ( r ) , \forall i \in V . +$$ + +
Graph GRicG
C3 Cycles C4 Cn≥53
1
0
n
Complete Kn Grid Gnn-1 0
Tree Tr4 r+1-2
+ +Table 1: Examples of the Balanced Forman curvature. + +Therefore, when the curvature is positive everywhere, the bottleneck effect should not play a crucial role as the receptive field of each node will be polynomial in the hop-distance. This limit case shows how the curvature determines whether a learning task on a graph would suffer from over-squashing. + +Curvature and over-squashing. Thanks to our new combinatorial curvature and the sensitivity analysis in Lemma 1, we are able to relate local curvature properties to the Jacobian of the node representations. This leads to one of the main results of this paper: negatively curved edges are those causing the graph bottleneck and thus leading to the over-squashing phenomenon: + +Theorem 4. Consider a MPNN as in equation 1. Let $i \sim j$ with $d _ { i } \leq d _ { j }$ and assume that: + +(i) $| \nabla \phi _ { \ell } | \le \alpha$ and $| \nabla \psi _ { \ell } | \le \beta$ for each $0 \leq \ell \leq L - 1$ , with $L \geq 2$ the depth of the MPNN. + +(ii) There exists $\delta$ s.t. $0 < \delta < ( \operatorname* { m a x } \{ d _ { i } , d _ { j } \} ) ^ { - \frac { 1 } { 2 } }$ , $\delta < \gamma _ { m a x } ^ { - 1 } ,$ , and $\operatorname { R i c } ( i , j ) \leq - 2 + \delta$ . + +Then there exists $Q _ { j } \subset S _ { 2 } ( i )$ satisfying $| Q _ { j } | > \delta ^ { - 1 }$ and for $0 \leq \ell _ { 0 } \leq L - 2$ we have + +$$ +\frac { 1 } { | Q _ { j } | } \sum _ { k \in Q _ { j } } \left| \frac { \partial h _ { k } ^ { ( \ell _ { 0 } + 2 ) } } { \partial h _ { i } ^ { ( \ell _ { 0 } ) } } \right| < ( \alpha \beta ) ^ { 2 } \delta ^ { \frac { 1 } { 4 } } . +$$ + +Condition (i) is always satisfied and allows us to control the message passing functions. The requirement (ii) instead means that the curvature of $( i , j )$ is negative enough when compared to the degrees of $i$ and $j$ (recall that $\operatorname { R i c } ( i , j ) > - 2 )$ . The further condition on $\gamma _ { \mathrm { m a x } }$ is to avoid pathological cases where we have a large number of degenerate 4-cycles passing through the same three nodes. + +To understand the conclusions in Theorem 4, let us fix $\ell _ { 0 } = 0$ . The equation 4 shows that negatively curved edges are the ones causing bottlenecks, interpreted as how the graph topology prevents a representation h(2)k t o be affected by non-adjacent features $h _ { i } ^ { ( 0 ) } = x _ { i }$ . Theorem 4 implies that if we have a negatively curved edge as in (ii), then there exist a large number of nodes $k$ such that GNNs - on average - struggle to propagate messages from $i$ to $k$ in two layers despite these nodes $k$ being at distance 2 from i. In this case the over-squashing occurs as measured by the Jacobian in equation 4 and hence the propagation of information suffers. If the task at hand has long-range dependencies, then the over-squashing caused by the negatively curved edges may compromise the performance. + +Bottleneck via Cheeger constant. We now relate the previous discussion about bottlenecks and curvature to spectral properties of the graph. In particular, since the spectral gap of a graph can be interpreted as a topological obstruction to the graph being partitioned into two communities, we argue below that this quantity is related to the graph bottleneck and should hence be controllable by the curvature. We start with an intuitive explanation: suppose we are given a graph $G$ with two communities separated by few edges. In this case, we see that the graph can be easily disconnected. This property is encoded in the classical notion of the Cheeger constant (Chung & Graham, 1997) + +$$ +h _ { G } : = \operatorname* { m i n } _ { S \subset V } h _ { S } , \quad h _ { S } : = \operatorname* { m i n } _ { S \subset V } \frac { | \partial S | } { \operatorname* { m i n } \{ \mathrm { v o l } ( S ) , \mathrm { v o l } ( V \setminus S ) \} } +$$ + +where $\partial S = \{ ( i , j ) : i \in S , \ j \in V \setminus S \}$ and $\textstyle { \mathrm { v o l } } ( S ) = \sum _ { i \in S } d _ { i }$ . The main result about the Cheeger constant is the Cheeger inequality (Cheeger, 2015; Chung $\bar { \& }$ Graham, 1997): + +$$ +2 h _ { G } \geq \lambda _ { 1 } \geq \frac { h _ { G } ^ { 2 } } { 2 } +$$ + +where $\lambda _ { 1 }$ is the first non-zero eigenvalue of the normalized graph Laplacian, often referred to as the spectral gap. A graph with two tightly connected communities $S$ and $V \setminus S : = \bar { S } )$ and few inter-community edges has a small Cheeger constant $h _ { G }$ . For nodes in different communities to interact with each other, all messages need to go through the same few bridges hence leading to the over-squashing of information (a similar intuition was explored in Alon & Yahav (2021)). Therefore, $h _ { G }$ can be interpreted as a rough measure of graph ‘bottleneckedness’, in the sense that the smaller its value, the more likely the over-squashing is to occur across inter-community edges. Since Theorem 4 implies that negatively curved edges induce the bottleneck, we expect a relationship between $h _ { G }$ and the curvature of the graph. The next proposition follows from Theorem 2 and Lin et al. (2011): + +Proposition 5. If $\operatorname { R i c } ( i , j ) \geq k > 0$ for all $i \sim j$ , then $\textstyle \lambda _ { 1 } / 2 \geq h _ { G } \geq { \frac { k } { 2 } }$ . + +Therefore, a positive lower bound on the curvature gives us a control on $h _ { G }$ and hence on the spectral gap of the graph. In the next section, we show that diffusion-based graph-rewiring methods might fail to significantly alter $h _ { G }$ and hence correct the graph bottleneck potentially induced by inter-community edges. This will lead us to propose an alternative curvature-based graph rewiring. + +# 4 CURVATURE-BASED REWIRING METHODS + +The traditional paradigm of message passing graph neural networks assumes that messages are propagated on the input graph (Gilmer et al., 2017). More recently, there is a trend to decouple the input graph from the graph used for information propagation. This can take the form of graph subsampling or resampling to deal with scalability (Hamilton et al., 2017) or topological noise (Zhang et al., 2019), using larger motif-based (Monti et al., 2018) or multi-hop filters (Rossi et al., 2020), or changing the graph either as a preprocessing step (Klicpera et al., 2019; Alon & Yahav, 2021) or adaptively for the downstream task (Wang et al., 2019; Kazi et al., 2020). Such methods are often generically referred to as graph rewiring. + +In the context of this paper, we assume that graph rewiring attempts to produce a new graph $G ^ { \prime } = ( V , E ^ { \prime } )$ with a different edge structure that reduces the bottleneck and hence potentially alleviates the over-squashing of information. We propose a method that leverages the graph curvature to guide the rewiring steps in a surgical way by modifying the negatively-curved edges, so to decrease the bottleneck without significantly compromising the statistical properties of the input graph. We also rigorously show that a random-walk based rewiring method might generally fail to obtain an edge set $E ^ { \prime }$ with a significant improvement in its bottleneckedness as measured by the Cheeger constant. + +Curvature-based graph rewiring. Since according to Theorem 4 negatively curved edges induce a bottleneck and are hence responsible for over-squashing, a curvature-based rewiring method should attempt to alleviate a graph’s strongly-negatively curved edges. To this end we implement a simple rewiring method called Stochastic Discrete Ricci Flow (SDRF), described in Algorithm 1. + +# Algorithm 1: Stochastic Discrete Ricci Flow (SDRF) + +Input: graph $G$ , temperature $\tau > 0$ , max number of iterations, optional Ric upper-bound $C ^ { + }$ Repeat + +1) For edge $i \sim j$ with minimal Ricci curvature $\operatorname { R i c } ( i , j )$ + +Calculate vector $_ { \textbf { \em x } }$ where $x _ { k l } = \mathrm { R i c } _ { k l } ( i , j ) - \mathrm { R i c } ( i , j )$ , the improvement to $\operatorname { R i c } ( i , j )$ from adding edge $k \sim l$ where $k \in B _ { 1 } ( i )$ , $l \in B _ { 1 } ( j )$ ; Sample index $k , l$ with probability softmax $( \tau { \pmb x } ) _ { k l }$ and add edge $k \sim l$ to $G$ . + +2) Remove edge $i \sim j$ with maximal Ricci curvature $\operatorname { R i c } ( i , j )$ if $\operatorname { R i c } ( i , j ) > C ^ { + }$ + +Until convergence, or max iterations reached; + +At each iteration this preprocessing step adds an edge to ‘support’ the graph’s most negatively curved edge, and then removes the most positively curved edge. The requirement on the added edge $k \sim l$ that $k \in B _ { 1 } ( i )$ and $l \in B _ { 1 } ( j )$ ensures that we’re adding either an extra $3 -$ or 4-cycle around the negative edge $i \sim j$ so that this is a local modification. The graph edit distance between the original and preprocessed graph is bounded above by $2 ~ \times$ the max number of iterations. The temperature $\tau$ determines how stochastic the edge addition is, with $\tau = \infty$ being fully deterministic (the best edge is always added). At each step we remove the edge with most positive curvature to balance the distributions of curvature and node degrees. We use Balanced Forman curvature as in equation 3 for $\operatorname { R i c } ( i , j )$ . $C ^ { + }$ can be chosen to stop the method skewing the curvature distribution negative, including $C ^ { + } = \infty$ to not remove any edges. The method is inspired by the continuous (backwards) Ricci flow with the aim of homogenizing edge curvatures. This is different from more direct extensions of Ricci flow on graphs where it becomes increasingly expensive to propagate messages across negatively curved edges (as in other applications such as Ni et al. (2019)). An example alongside its continuous analogue can be seen in Figure 1. + +Can random-walk based rewiring address bottlenecks? A good way of understanding the effectiveness of SDRF in reducing the graph bottleneck is through comparison with random-walk based rewiring strategies. Recall that, as argued in Section 3, the Cheeger constant $h _ { G }$ of a graph constitutes a rough measure of its bottleneckedness as induced by the inter-community edges (a small $h _ { G }$ is indicative of a bottleneck). Suppose we are given a graph $G$ with a small $h _ { G }$ and wish to rewire it into a graph $G ^ { \prime }$ with a significantly improved Cheeger constant in order to reduce the inter-community bottleneck. A random-walk based rewiring method such as DIGL (Klicpera et al., 2019) acts by smoothing out the graph adjacency and hence tends to promote connections among nodes at short diffusion distance (Coifman & Lafon, 2006). Accordingly, such a rewiring method might fail to correct structural features like the bottleneck, which is instead more prominent for nodes that are at long diffusion distance.3 To emphasize this point, we consider a classic example: given $\alpha \in ( 0 , 1 )$ , the Personalized Page Rank (PPR) matrix is defined by (Brin & Page, 1998) as + +$$ +R _ { \alpha } : = \sum _ { k = 0 } ^ { \infty } \theta _ { k } ^ { P P R } ( D ^ { - 1 } A ) ^ { k } = \alpha \sum _ { k = 0 } ^ { \infty } \left( ( 1 - \alpha ) ( D ^ { - 1 } A ) \right) ^ { k } . +$$ + +Assume that we rewire the graph using $R _ { \alpha }$ as in Klicpera et al. (2019) with the PPR kernel, meaning that we replace the given adjacency $A$ with $R _ { \alpha }$ . Since $R _ { \alpha }$ is stochastic, the new Cheeger constant of the rewired graph can be computed as + +$$ +h _ { S , \alpha } = \frac { | \partial S | _ { \alpha } } { \mathrm { v o l } _ { \alpha } ( S ) } \equiv \frac { 1 } { | S | } \sum _ { i \in S } \sum _ { j \in \bar { S } } ( R _ { \alpha } ) _ { i j } . +$$ + +By applying (Chung, 2007, Lemma 5), we show that we cannot improve the Cheeger constant (and hence the bottleneck) arbitrarily well (in contrast to a curvature-based approach). We refer to Proposition 17 and Remark 18 in Appendix $\mathrm { E }$ for results that are more tailored to the actual strategy adopted in Klicpera et al. (2019) where we also take into account the effect of the sparsification. + +Theorem 6. Let $S \subset V$ with $\nu o l ( S ) \le \nu o l ( G ) / 2$ . Then $\begin{array} { r } { h _ { S , \alpha } \leq \left( \frac { 1 - \alpha } { \alpha } \right) \frac { d _ { \arg } ( S ) } { d _ { \operatorname* { m i n } } ( S ) } h _ { S } } \end{array}$ , where $d _ { \mathrm { a v g } } ( S )$ and $d _ { \mathrm { m i n } } ( S )$ are the average and minimum degree on $S$ , respectively. + +The property that the new Cheeger constant is directly controlled by the old one stems from the fact that a random-walk approach like in Klicpera et al. (2019) is meant to act more relevantly on intra-community edges rather than inter-community edges because it prioritizes short diffusion distance nodes. This is also why this method performs well on high-homophily datasets, as discussed below. In particular, for a fixed $\alpha \in ( 0 , 1 )$ , the bound in Theorem 6 can be very small. As a specific example, consider two complete graphs $K _ { n }$ joined by one bridge. Then $h _ { G } \overset { \cdot } { = } ( n ( n - 1 ) \overset { \cdot } { + } 1 ) ^ { - 1 }$ , which means that the bound on the right hand side is $\dot { O } ( n ^ { - 2 } )$ . + +Theorem 6 implies that a diffusion approach such as DIGL might fail to yield a new edge set $E ^ { \prime }$ with a sufficiently improved bottleneck. By contrast, from Theorem 4 and Proposition 5, we deduce that a curvature-based rewiring method such as SDRF properly addresses the edges that cause the bottleneck. + +Graph structure preservation. Although a graph-rewiring approach aims at providing a new edge set $E ^ { \prime }$ potentially more beneficial for the given learning task, it is still desirable to control how far $E ^ { \prime }$ is from $E$ . In this regard, we note that a curvature-based rewiring is surgical in nature and hence more likely to preserve the structure of the input graph better than a random-walk based approach. Consider, for example, that we are given $\rho > 0$ and wish to rewire the graph such that the new edge set $E ^ { \prime }$ is within graph-edit distance $\rho$ from the original $E$ . Theorem 4 tells us how to do the rewiring under such constraints in order to best address the over-squashing: the topological modifications need to be localized around the most negatively-curved edges. We can do this with SDRF, with the maximum number of iterations set to $\rho / 2$ . + +Secondly, we also point out that $\operatorname { R i c } ( i , j ) < - 2 + \delta$ implies that $\operatorname* { m i n } \{ d _ { i } , d _ { j } \} > 2 / \delta$ . Therefore, if we mostly modify the edge set at those nodes $i , j$ joined by an edge with large negative curvature, then we are perturbing nodes with high degree where such a change is relatively insignificant, and thus overall statistical properties of the rewired graph such as degree distribution are likely to be better preserved. Moreover, graph convolutional networks tend to be more stable to perturbations of high degree nodes (Zügner et al., 2020; Kenlay et al., 2021), making curvature-based rewiring more suitable for the downstream learning tasks with popular GNN architectures. + +Homophily and bottleneck. As a final remark, note that the graph rewiring techniques considered in this paper (both DIGL and SDRF) are based purely on the topological structure of the graph and completely agnostic to the node features and to whether the dataset is homophilic (adjacent nodes have same labels) or heterophilic. Nonetheless, the different nature of these rewiring methods allows us to draw a few broad conclusions about their suitability in each of these settings. A random-walk + +
H(G)Cornell 0.11Texas 0.06Wisconsin 0.16Chameleon 0.25Squirrel 0.22Actor 0.24Cora 0.83Citeseer 0.71Pubmed 0.79
None52.69 ± 0.2161.19 ± 0.4954.60±0.8641.80 ± 0.4139.83±0.1428.70±0.0981.89 ±0.7972.31 ±0.1778.16 ±0.23
Undirected53.20 ±0.5363.38 ±0.8751.37 ± 1.1542.63±0.3040.77 ± 0.1628.10 ±0.11=
+FA58.29±0.4964.82±0.2955.48±0.6242.33± 0.1740.74 ± 0.1328.68±0.1681.65±0.1870.47±0.18 79.48±0.12
DIGL (PPR)58.26±0.5062.03 ±0.4349.53±0.2742.02 ±0.1334.38 ±0.1130.79±0.10 83.21±0.27 73.29±0.17 78.84±0.08
DIGL + Undirected 59.54 ±0.6463.54±0.3852.23±0.5442.68± 0.1233.36±0.2129.71 ± 0.11=
SDRF54.60 ±0.3964.46±0.3855.51±0.27 43.75±0.31 40.97±0.14 29.70 ±0.13 82.76±0.23 72.58±0.20 79.10 ±0.11
SDRF+ Undirected 57.54±0.34 70.35±0.60 61.55±0.86 44.46±0.17 41.47±0.21 29.85±0.07
+ +Table 2: Experimental results on common node classification benchmarks. Top two in bold. + +approach such as DIGL tends to improve the connectivity among nodes that are at short diffusion distance; since for a high-homophily dataset these nodes often share the same label, a rewiring method like DIGL is likely to act as graph denoising and yield improved performance. On the other hand, for datasets with low homophily, nodes at short diffusion distance are more likely to belong to different label classes, meaning that a diffusion-based rewiring might inject noise and hence compromise performance as also noted in Klicpera et al. (2019). Conversely, on a low-homophily dataset, a curvature-based approach as SDRF modifies the edge set mainly around the most negatively curved edges, meaning that it decreases the bottleneck without significantly increasing the connectivity among nodes with different labels. In fact, long-range dependencies are often more relevant in low-homophily settings, where nodes sharing the same labels are in general not neighbors. This observation is largely confirmed by experimental results reported in the next section. + +# 5 EXPERIMENTAL RESULTS + +Experiment setup. To demonstrate the theoretical results in this paper we ran a suite of semisupervised node classification tasks comparing our curvature-based rewiring method SDRF to DIGL from Klicpera et al. (2019) (GDC with the PPR kernel) and the $+ \mathrm { F A }$ method from Alon & Yahav (2021), where the last layer of the GNN is made fully connected. We evaluate the methods on nine datasets: Cornell, Texas and Wisconsin from the WebKB dataset4; Chameleon and Squirrel (Rozemberczki et al., 2021) along with Actor (Tang et al., 2009); and Cora (McCallum et al., 2000), Citeseer (Sen et al., 2008) and Pubmed (Namata et al., 2012). Statistics for these datasets can be found in Appendix F.1. Our base model is a GCN (Kipf & Welling, 2017). Following Shchur et al. (2018) and Klicpera et al. (2019) we optimized hyperparameters for all dataset-preprocessing combinations separately by random search over 100 data splits. Results are reported as average accuracies on a test set used once with $9 5 \%$ confidence intervals calculated by bootstrapping. We compared the performance on graphs with no preprocessing, making the graph undirected, $+ \mathrm { F A }$ , DIGL, SDRF, and the given combinations. For DIGL $^ +$ Undirected we symmetrized the diffusion matrix as in Klicpera et al. (2019), and for SDRF $^ +$ Undirected we made the graph undirected before applying SDRF. For more details on the experiments and datasets see Appendix F, and for the hyperparameters used for each model and preprocessing see Appendix F.4. + +Node classification results. Table 2 shows the results of the experiments. As well as reporting results we give a measure of homophily $\mathcal { H } ( G )$ proposed by Pei et al. (2019) (restated in Appendix F, equation 21), by which we can see our experiment set is diverse with respect to homophily. We see that SDRF improves upon the baseline in all cases, and that the largest improvements are seen on the low-homophily datasets. We also see that SDRF matches or outperforms DIGL and $+ \mathrm { F A }$ on most datasets, supporting our argument that curvature-based rewiring is a viable candidate for improving GNN performance. + +Graph topology change. Furthermore, SDRF preserves the graph topology to a far greater extent than DIGL due to its surgical nature. Table 3 shows the number of edges added / removed by the two preprocessings on each dataset as a percentage of the original number of edges. We see that for optimal performance DIGL makes the graph much denser, which significantly affects the node degrees and may negatively impact the time and space complexity of the downstream GNN, which typically are $O ( E )$ . In comparison, SDRF adds and removes a similar number of edges and approximately preserves the degree distribution. The effect on the full degree distribution for three + +![](images/198001fa034307ccce83949e4016829a9a3ad6cc0c2500a49f6ecd4d0572ee0c.jpg) +Figure 4: Comparing the degree distribution of the original graphs to the preprocessed version. The xaxis is node degree in $\log _ { 2 }$ scale, and the plots are a kernel density estimate of the degree distribution. In the captions we see the Wasserstein distance $W _ { 1 }$ between the original and preprocessed graphs. + +datasets is shown in Figure 4. We see that the degree distribution following SDRF is close to, or in some cases indistinguishable from, the original distribution, whereas DIGL has a more noticeable effect. We also compute the Wasserstein distance between the degree distributions, denoted by $W _ { 1 }$ in the figure captions, to numerically confirm this observation. For this analysis extended to all nine datasets see Appendix F.2. + +This difference is also visually evident from Figure 6 in Appendix F.3, where we again observe that SDRF largely preserves the topology of the Cornell graph and that the curvature (encoded by the edge color) is homogenized across the graph. We also see that the entries of the trained + +
DIGLSDRF
Cornell351.1% /0.0%7.8% /33.3%
Texas483.3% /0.0%2.4% /10.4%
Wisconsin300.6%/0.0%1.4% / 7.5%
Chameleon336.1%/11.8%6.4% /6.4%
Squirrel228.8% /1.9%0.7%/0.7%
Actor2444.0% /2.3%5.4%/9.9%
Cora3038.0% /0.5%1.0% /1.0%
Citeseer2568.3%/ 0.0%1.1% / 1.1%
Pubmed2747.1% /0.1%0.2%/0.2%
+ +Table 3: $\%$ edges added / removed by method. + +network’s Jacobian between a node’s prediction and the features of its 2-hop neighbors (encoded as node colors) are increased over the graph, which we may attribute to both DIGL and SDRF’s ability to alleviate the upper bound presented in Theorem 4 and thus reduce over-squashing. + +# 6 CONCLUSION + +In this paper, we studied the graph bottleneck and the over-squashing phenomena limiting the performance of message passing graph neural networks from a geometric perspective. We started with a Jacobian approach to determine how the over-squashing phenomenon is dictated by the graph topology as in equation 2. We then investigated further how the topology induces the bottleneck and hence causes over-squashing. We introduced a new notion of edge-based Ricci curvature called Balanced Forman curvature, relating it to the classical Ollivier curvature (Theorem 2). We then proved in Theorem 4 that negatively-curved edges are responsible for over-squashing, calling for a possibility of curvature-based rewiring of the graph in order to improve its bottleneckedness. We show one such possibility (Algorithm 1), inspired by the classical Ricci flow and comment on the advantages of surgical method such as this. We show both theoretically and experimentally that the proposed method can be advantageous compared to a diffusion-based rewiring approach, opening the door for curvature-based rewiring methods for improving GNN performance going forward. + +Limitations and future directions. Our paper establishes a geometric perspective on the graph bottleneck and over-squashing, providing new tools to study and cope with these phenomena. One limitation of our work is that the theoretical results presented here do not currently extend to multigraphs. In addition, the current methodology is agnostic to information beyond the graph topology, such as node features. In future works, we will develop a notion of the curvature and the corresponding rewiring method that can take into account such information. + +Acknowledgements. This research was supported in part by the EPSRC CDT in Modern Statistics and Statistical Machine Learning (EP/S023151/1) and the ERC Consolidator Grant No. 724228 (LEMAN). X.D. gratefully acknowledges support from the Oxford-Man Institute of Quantitative Finance and the EPSRC (EP/T023333/1). + +# REFERENCES + +Uri Alon and Eran Yahav. On the bottleneck of graph neural networks and its practical implications. In International Conference on Learning Representations, 2021. URL https://openreview. net/forum?id=i80OPhOCVH2. + +Pablo Barceló, Egor V Kostylev, Mikael Monet, Jorge Pérez, Juan Reutter, and Juan Pablo Silva. The logical expressiveness of graph neural networks. 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In AAAI Conference on Artificial Intelligence, 2019. + +Daniel Zügner, Oliver Borchert, Amir Akbarnejad, and Stephan Günnemann. Adversarial attacks on graph neural networks: Perturbations and their patterns. ACM Transactions on Knowledge Discovery from Data (TKDD), 14(5):1–31, 2020. + +# APPENDIX + +The Appendix is structured as follows: + +(i) In Appendix A we prove Lemma 1 and a side-result about the role of self-loops. We also summarize how to extend the analysis in Lemma 1 to message passing models with sum aggregations (not average), meaning those architectures where we do not normalize the adjacency matrix. (ii) In Appendix B we introduce and describe different quantities we use to characterize the neighbourhood of a given edge $i \sim j$ . These objects are all essential to studying the new notion of balanced Forman curvature. The focus is on how we can distinguish 4-cycles in a computationally tractable way without losing too much accuracy. (iii) In Appendix C we provide a brief literature review about existing curvature candidates. In particular, we report the definitions of (modified) Ollivier curvature and Forman curvature. (iv) In Appendix D we prove the statements in Section 3, i.e. Theorem 2, Corollary 3, Theorem 4 and Proposition 5. We also comment on the role of the assumptions and compare the bound in Theorem 2 with the existing literature. Finally, we relate the classical notion of betweenness centrality to the over-squashing effect and the negatively curved edges in a graph. (v) In Appendix E we prove the results in Section 4, namely Theorem 6 and an analogous result. (vi) In Appendix F we describe more fully the experiments from Section 5, including a full analysis on degree distribution (Appendix F.2) and the hyperparameters used in our experiments (Appendix F.4). (vii) In Appendix G we comment on hardware specifications. + +# A PROOFS OF RESULTS IN SECTION 2 + +Lemma 1. Assume an MPNN as in equation $I$ . Let $i , s \in V$ with $s \in S _ { r + 1 } ( i )$ . If $| \nabla \phi _ { \ell } | \le \alpha$ and $| \nabla \psi _ { \ell } | \le \beta$ for $0 \leq \ell \leq r$ , then + +$$ +\left| \frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } \right| \leq ( \alpha \beta ) ^ { r + 1 } ( \hat { A } ^ { r + 1 } ) _ { i s } . +$$ + +Proof. Let $i \in V$ and $s \in S _ { r + 1 } ( i )$ . We recall that to ease the notations we assume that node features and hidden representations are scalar. The proof in the more general higher-dimensional case follows without any modification. We compute + +$$ +\begin{array} { r l } & { \displaystyle \frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } = \partial _ { 1 } \phi _ { r } ( . . . ) \partial _ { x _ { s } } h _ { i } ^ { ( r ) } } \\ & { \quad \quad \quad + \partial _ { 2 } \phi _ { r } ( . . . ) \displaystyle \sum _ { j _ { r } = 1 } ^ { n } \hat { a } _ { i j _ { r } } \left( \partial _ { 1 } \psi _ { r } ( h _ { i } ^ { ( r ) } , h _ { j _ { r } } ^ { ( r ) } ) \partial _ { x _ { s } } h _ { i } ^ { ( r ) } + \partial _ { 2 } \psi _ { r } ( h _ { i } ^ { ( r ) } , h _ { j _ { r } } ^ { ( r ) } ) \partial _ { x _ { s } } h _ { j _ { r } } ^ { ( r ) } \right) . } \end{array} +$$ + +We can iterate the computation above and see that the right hand side can be expanded as + +$$ +\frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } = \sum _ { j _ { r } , \dots , j _ { 0 } } \sum _ { k _ { r } \in \{ i , j _ { r } \} } \dots \sum _ { k _ { 1 } \in \{ i , j _ { r } , \dots , j _ { 1 } \} } \hat { a } _ { i j _ { r } } \hat { a } _ { k _ { r } j _ { r - 1 } } \dots \hat { a } _ { k _ { 1 } j _ { 0 } } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \dots k _ { 1 } j _ { 0 } } ( \boldsymbol { X } ) \partial _ { x _ { s } } h _ { j _ { 0 } } ^ { ( 0 ) } , +$$ + +for some functions $Z _ { i j _ { r } . . . k _ { 1 } j _ { 0 } }$ of the input features obtained as products of $r + 1$ partial derivatives of the maps $\phi _ { \ell }$ and $r + 1$ partial derivatives of the maps $\psi _ { \ell }$ . Since $H ^ { ( 0 ) } = X$ , we have + +$$ +\partial _ { x _ { s } } h _ { j _ { 0 } } ^ { ( 0 ) } = \delta _ { j _ { 0 } s } +$$ + +meaning that the previous sum becomes + +$$ +\frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } = \sum _ { j _ { r } , \dots , j _ { 1 } } \sum _ { k _ { r } \in \{ i , j _ { r } \} } \dots \sum _ { k _ { 1 } \in \{ i , j _ { r } , \dots , j _ { 1 } \} } \hat { a } _ { i j _ { r } } \hat { a } _ { k _ { r } j _ { r - 1 } } \dots \hat { a } _ { k _ { 1 } s } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \dots k _ { 1 } s } ( X ) . +$$ + +Since $d _ { G } ( i , s ) = r + 1$ , the only non-vanishing terms in the sum above are the minimal walks from $i$ to $s$ . In fact, if there existed a different choice of coefficients yielding a non-zero term then we would find a walk joining $i$ to $s$ of length lesser than $r + 1$ , which is in contradiction with the definition of geodesic distance. Since $Z _ { i \dots s } ( X )$ is a product of $r + 1$ -partial derivatives of the aggregation and update maps and by assumption their gradients are bounded by $\alpha$ and $\beta$ respectively, we conclude that + +$$ +\left. \frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } \right. \leq ( \alpha \beta ) ^ { r + 1 } \sum _ { j _ { r } , \ldots , j _ { 1 } } \hat { a } _ { i j _ { r } } \hat { a } _ { j _ { r } j _ { r - 1 } } \ldots \hat { a } _ { j _ { 1 } s } = \alpha ^ { r + 1 } ( \hat { A } ^ { r + 1 } ) _ { i s } +$$ + +which completes the proof of the Lemma. + +As a byproduct of this analysis, we can also provide a rigorous motivation for the role of self-loops in GNNs (see Appendix for details): + +Corollary 7. If $\begin{array} { r } { h _ { i } ^ { ( \ell + 1 ) } = \sum _ { j \sim i } \psi _ { \ell } ( h _ { j } ^ { ( \ell ) } ) } \end{array}$ , then $h _ { i } ^ { ( \ell + 1 ) }$ only depends on nodes that can be reached via walks of length exactly $\ell + 1$ . By adding self-loops, the GNN also takes into account nodes that can be reached via walks of length $r \leq \ell + 1$ . + +Proof. If $\begin{array} { r } { h _ { i } ^ { ( \ell + 1 ) } = \sum _ { j \sim i } \psi _ { \ell } ( h _ { j } ^ { ( \ell ) } ) } \end{array}$ for each $\ell \in [ 0 , L - 1 ]$ , then we can argue as in the proof of + +$$ +\frac { \partial h _ { i } ^ { ( \ell + 1 ) } } { \partial x _ { s } } = \sum _ { j _ { \ell } , \ldots , j _ { 1 } } a _ { i j _ { \ell } } \ldots a _ { j _ { 1 } s } \psi _ { \ell } ^ { \prime } ( h _ { j _ { \ell } } ^ { ( \ell ) } ) \cdot \cdot \cdot \psi _ { 0 } ^ { \prime } ( x _ { s } ) . +$$ + +The combinatorial coefficient $a _ { i j \ell } \ldots a _ { j _ { 1 } s }$ is non-zero iff there exists a walk from $i$ to $s$ of length exactly $\ell { + 1 }$ , since we are not taking into account the contribution coming from self-loops. Conversely, if each term $a _ { i j \ell }$ was replaced by $\hat { a } _ { i j \ell }$ then we would find that $\hat { a } _ { i j \ell } \ldots \hat { a } _ { j _ { 1 } s }$ is non-zero iff there exists a walk from $i$ to $s$ of length at most $r + 1$ , since the diagonal entries are now positive. □ + +Remark 8. As a specific instance of Corollary 7, we note that if we do not include self-loops in the adjacency matrix, then the output of a 2-layer simplified graph neural network at node i is independent of the features of neighbours $k$ that do not form a triangle with i. Once again here the dependence is precisely measured via the Jacobian of the hidden features with respect to the input features. + +Remark 9. We note that the role of self-loops has also implicitly been noted in Xu et al. (2018) where the analysis of the Jacobian of node representations on the graph augmented with self-loops has been related to lazy random-walks. + +GNNs with different aggregations We note that similar conclusions extend to message passing architectures where the aggregations are sums and not averages meaning that we take the augmented adjacency without normalizing by the degree matrices. Consistently with Lemma 1, we restrict to the setting where features and node representations at each layer are scalars to make the discussion simpler. In line with the $\mathrm { X u }$ et al. (2018) we consider a GNN-model of the form + +$$ +h _ { i } ^ { ( \ell + 1 ) } = \mathrm { R e L U } \left( \sum _ { j \in \tilde { \mathcal { N } } _ { i } } h _ { j } ^ { ( \ell ) } w _ { \ell } \right) . +$$ + +Note that the augmented neighbourhood $\tilde { \mathcal { N } } _ { i }$ is defined as $\mathcal { N } _ { i } \cup \{ i \}$ . Differently from the setting of Theorem 1 in $\mathrm { X u }$ et al. (2018), the aggregation here is not an average but a simple sum. Let us now take nodes $i$ and $s$ such that $s \in S _ { r + 1 } ( i )$ as in the statement of Lemma 1. In this case, instead of simply considering the quantity $| \partial h _ { i } ^ { ( r + 1 ) } / \partial x _ { s } |$ , we normalize the Jacobian entries - obtaining what is referred to as influence score in $\mathrm { X u }$ et al. (2018): + +$$ +\mathcal { T } _ { r + 1 } ( i , s ) : = \frac { \left| \frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { s } } \right| } { \sum _ { k } \left| \frac { \partial h _ { i } ^ { ( r + 1 ) } } { \partial x _ { k } } \right| } +$$ + +This of course represents now a relative importance of feature $x _ { s }$ on the representation of node $i$ at layer $r + 1$ . If - similarly to Theorem 1 in $\mathrm { X u }$ et al. (2018) - we assume that all paths in the computational graph of the model are activated with the same probability, then we obtain that on average + +$$ +\mathcal { T } _ { r + 1 } ( i , s ) = \frac { \tilde { A } _ { i s } ^ { r + 1 } } { \sum _ { k } \tilde { A } _ { i k } ^ { r + 1 } } \leq \frac { \tilde { A } _ { i s } ^ { r + 1 } } { \mathrm { V o l } ( B _ { r + 1 } ( i ) ) } , +$$ + +where ${ \tilde { A } } = A + I$ and $\begin{array} { r } { \mathrm { v o l } ( S ) = \sum _ { j \in S } d _ { j } } \end{array}$ . In particular, we again find that if we have a tree structure, then the right hand side decays exponentially as $2 ^ { - ( r + 1 ) }$ . + +# B PRELIMINARY ANALYSIS OF AN EDGE-NEIGHBORHOOD + +Given an edge $i \sim j$ , we introduce the sets below: + +(i) $\sharp _ { \Delta } ( i , j ) : = S _ { 1 } ( i ) \cap S _ { 1 } ( j )$ , the number of triangles based at the edge $i \sim j$ . +(ii) $\sharp _ { \bigtriangledown } ( i , j ) : = \{ k \in S _ { 1 } ( i ) \setminus S _ { 1 } ( j ) , k \neq j : \ \exists w \in ( S _ { 1 } ( k ) \cap S _ { 1 } ( j ) ) \setminus S _ { 1 } ( i ) \}$ , the number of nodes $k \in S _ { 1 } ( i )$ forming a 4-cycle based at $i \sim j$ without diagonals inside. +(iii) $Q _ { i } ( j ) : = S _ { 1 } ( i ) \setminus ( \{ j \} \cup \sharp _ { \Delta } ( i , j ) \cup \sharp _ { \Pi } ^ { i } ( i , j ) )$ , simply the complement of the neighbours of $i$ with respect to the sets introduced in (i) and (ii) once we also exclude $j$ . + +In the following we simply write $\sharp _ { \Delta } , \sharp _ { \bigstar } ^ { i }$ and $Q _ { i }$ when the edge $i \sim j$ is clear from the context. + +4-cycle contributions. In general the sets $\sharp _ { \bigstar } ^ { i }$ and $\sharp _ { \bigstar } ^ { j }$ may differ. This may occur when there exists a node $k$ belonging to $\sharp _ { \bigstar } ^ { i }$ that admits multiple solutions $w$ as in the definition of $\sharp _ { \bigstar } ^ { i }$ . This feature needs to be taken into account when comparing Ollivier’s Ricci curvature to the new notion we present below. We first introduce the following class to ease the notations. + +Definition 2. For any simple, undirected graph $G = ( V , E )$ , if $U \subset V$ , then we set + +$$ +\mathcal { D } ( U ) : = \left\{ \varphi : U \to V , \ | U | = | \varphi ( U ) | , \ ( z , \varphi ( z ) ) \in E , \ \forall z \in U \right\} . +$$ + +We note that any $\varphi \in { \mathcal { D } } ( U )$ is injective. + +We may now define a quantity which measures the maximal number of 1-1 pairings that can be performed from $\sharp _ { \bigstar } ^ { i }$ to $\sharp _ { \bigstar } ^ { j }$ . + +Definition 3. For any edge $i \sim j$ we let + +$$ +\sharp _ { \sharp } ( i , j ) : = \operatorname* { m a x } \left\{ | U | : U \subset \sharp _ { \sharp } ^ { i } , \ : \exists \varphi : U \to \sharp _ { \sharp } ^ { j } , \ : \varphi \in \mathcal { D } ( U ) \right\} . +$$ + +We often simply write $\sharp _ { \bigsqcup } ^ { \mathfrak { m } }$ . While the quantity $\sharp _ { \bigsqcup } ^ { \mathfrak { m } }$ plays a role in the derivation of the Ollivier curvature of $i \sim j$ it is not computationally-friendly, as to determine $\sharp _ { \bigsqcup } ^ { \mathfrak { m } }$ we need to identify and distinguish all possible 4-cycles based at $i \sim j$ and then choose a maximal pairing map. Accordingly, we consider a looser term which is easier to compute: + +Definition 4. For any pair of adjacent nodes $i \sim j$ we define + +$$ +\gamma _ { m a x } ( i , j ) : = \operatorname* { m a x } \left\{ \operatorname* { m a x } _ { k \in \sharp _ { \Sigma } ^ { k } } \{ \big ( A _ { k } \cdot ( A _ { j } - A _ { i } \odot A _ { j } ) \big ) - 1 \} , \operatorname* { m a x } _ { w \in \sharp _ { \Sigma } ^ { j } } \{ \big ( A _ { w } \cdot ( A _ { i } - A _ { j } \odot A _ { i } ) \big ) - 1 \} \right\} , +$$ + +where $A _ { s }$ denotes the s-th row of the adjacency matrix. We usually simply write $\gamma _ { m a x }$ + +Remark 10. We note that given $k \in S _ { 1 } ( i ) \setminus S _ { 1 } ( j )$ the term $( A _ { k } \cdot ( A _ { j } - A _ { i } \odot A _ { j } ) ) - 1$ yields the number of nodes w forming a 4-cycle of the form $i \sim k \sim w \sim j \sim i$ with no diagonals inside. The value $\gamma _ { m a x }$ measures the maximal degeneracy of edges forming $^ { 4 }$ -cycles, meaning that it is equal to $^ { l }$ iff for each $k \in \sharp _ { \bigstar } ^ { i }$ there exists a unique node $w \in \sharp _ { \bigstar } ^ { j }$ such that $i \sim k \sim w \sim j \sim i$ is a $^ { 4 }$ -cycle. + +We now end the discussion about 4-cycle contributions by proving the following inequality, which allows us to avoid to compute directly the term $\sharp _ { \bigsqcup } ^ { \mathfrak { m } }$ up to giving up some accuracy. + +Lemma 11. For any edge $i \sim j$ we have + +$$ +| \sharp _ { \sf H } ^ { \mathfrak { m } } | \geq \frac { \operatorname* { m a x } \{ | \sharp _ { \sf H } ^ { i } | , | \sharp _ { \sf H } ^ { j } | \} } { \gamma _ { m a x } } . +$$ + +Proof. The proof is based on a combinatorial argument. Let $\sharp _ { \sqcap } ^ { i } = \{ k _ { 1 } , \ldots , k _ { r } \}$ and let $\sharp _ { \bigcirc } ^ { \mathfrak { m } } =$ $\{ k _ { 1 } , \ldots , k _ { \ell } \}$ , with $\ell < r$ . By definition there exists $\varphi : \{ k _ { 1 } , \dots , k _ { \ell } \} \to \{ w _ { 1 } , \dots , w _ { \ell } \}$ , with $k _ { i } \sim w _ { i }$ and $w _ { i } \in \sharp _ { \bigstar } ^ { j }$ , for $1 \leq i \leq \ell$ . Given $k \in \sharp _ { \sqcap } ^ { i } \setminus \{ k _ { 1 } , . . . , k _ { \ell } \}$ , then there are no $w \in \sharp _ { \sqcap } ^ { j } \setminus \{ w _ { 1 } , . . . , w _ { \ell } \}$ such that $k \sim w$ , otherwise we could extend $\varphi$ by setting $k \mapsto \varphi ( k ) : = w$ and we would then get $| \mathtt { H } _ { \square } ^ { \mathfrak { m } } | = \ell + 1$ . Accordingly, we have + +$$ +\sum _ { s = 1 } ^ { \ell } ( A _ { w _ { s } } \cdot ( A _ { i } - A _ { j } \odot A _ { i } ) ) - 1 ) \geq | \sharp _ { \square } ^ { i } | , +$$ + +which implies + +$$ +\gamma _ { \operatorname* { m a x } } | \sharp _ { \sf D } ^ { \mathrm { m } } | \equiv \gamma _ { \operatorname* { m a x } } \ell \geq \sum _ { s = 1 } ^ { \ell } ( A _ { w _ { s } } \cdot ( A _ { i } - A _ { j } \odot A _ { i } ) ) - 1 ) \geq | \sharp _ { \sf D } ^ { i } | . +$$ + +# C EXISTING CURVATURE CANDIDATES + +Ollivier Ricci curvature For $i \in V$ and $\alpha \in [ 0 , 1 )$ we define a probability measure on $B _ { 1 } ( i )$ by: + +$$ +\mu _ { i } ^ { \alpha } : j \mapsto \left\{ \begin{array} { r l } & { \alpha , \ j = i } \\ & { \frac { 1 - \alpha } { d _ { i } } , \ j \in S _ { 1 } ( i ) , } \\ & { 0 , \ \mathrm { o t h e r w i s e } . } \end{array} \right. +$$ + +Before we introduce the Ollivier curvature, we recall that the transportation distance between two finitely supported probability measures as above can be computed as + +$$ +W _ { 1 } ( \mu _ { i } ^ { \alpha } , \mu _ { j } ^ { \alpha } ) : = \operatorname* { i n f } _ { M } \sum _ { k \in S _ { 1 } ( i ) } \sum _ { w \in S _ { 1 } ( j ) } M _ { k w } d _ { G } ( k , w ) , +$$ + +where $d _ { G } ( \cdot , \cdot )$ is the geodesic distance on the graph and the infimum is taken over all matrices $M$ satisfying the marginal constraints: + +$$ +\sum _ { k \in S _ { 1 } ( i ) } M _ { k w } = \mu _ { j } ^ { \alpha } ( w ) , \quad \sum _ { w \in S _ { 1 } ( j ) } M _ { k w } = \mu _ { i } ^ { \alpha } ( k ) . +$$ + +We are now ready to define the Ollivier Ricci curvature: the formulation below is due to Lin et al. +(2011). + +Definition 5. Given $i \sim j$ we define the $\alpha$ -Ollivier curvature by + +$$ +\kappa _ { \alpha } ( i , j ) : = 1 - W _ { 1 } ( \mu _ { i } ^ { \alpha } , \mu _ { j } ^ { \alpha } ) . +$$ + +Since $\kappa _ { \alpha } ( 1 - \alpha ) ^ { - 1 }$ is increasing and bounded the quantity below is well-defined: + +$$ +\kappa ( i , j ) : = \operatorname* { l i m } _ { \alpha \to 1 } \frac { 1 - W _ { 1 } ( \mu _ { i } ^ { \alpha } , \mu _ { j } ^ { \alpha } ) } { 1 - \alpha } . +$$ + +Forman Ricci curvature In the following we report a formula for the augmented Forman Ricci curvature on unweighted graphs Samal et al. (2018). We also note that Forman curvature on graphs has also been studied in Sreejith et al. (2016); Weber et al. (2018). + +Definition 6. For any edge $i \sim j$ the augmented Forman curvature is given by + +$$ +F ( i , j ) : = 4 - d _ { i } - d _ { j } + 3 | \sharp _ { \Delta } | . +$$ + +We note that such formulation of curvature does not distinguish contributions coming from 4-cycles. +In fact, for the orthogonal grid with degree $d \geq 4$ , Forman Ricci curvature is equal to $2 ( 2 - d ) < 0$ . +This does not reflect that the $r$ -hop neighbourhood for such a graph grows polynomially in $r$ . + +We conclude this appendix by reporting a lower bound for the Ollivier Ricci curvature derived in Jost & Liu (2014). We recall that $\kappa _ { \alpha }$ , with $\alpha \in [ 0 , 1 )$ was defined in equation 7. + +Theorem 12 (Jost & Liu (2014)). For any edge $i \sim j$ , with $d _ { i } \leq d _ { j }$ , the following bound is satisfied: + +$$ +\kappa _ { 0 } ( i , j ) \geq \Phi ( i , j ) : = - \left( 1 - \frac { 1 } { d _ { i } } - \frac { 1 } { d _ { j } } - \frac { | \sharp \Delta | } { d _ { j } } \right) _ { + } - \left( 1 - \frac { 1 } { d _ { i } } - \frac { 1 } { d _ { j } } - \frac { | \sharp \Delta | } { d _ { i } } \right) _ { + } + \frac { | \sharp \Delta | } { d _ { j } } . +$$ + +# D PROOFS OF RESULTS IN SECTION 3 + +We first recall our definition of Balanced Forman: + +Definition 7. For any edge $i \sim j$ we let $\operatorname { R i c } ( i , j )$ be zero $\gamma _ { \mathrm { m i n } \{ d _ { i } , d _ { j } \} } = 1$ , otherwise + +$$ +\mathbf { R i c } ( i , j ) : = \frac { 2 } { d _ { i } } + \frac { 2 } { d _ { j } } - 2 + 2 \frac { \lvert \sharp _ { \Delta } \rvert } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } + \frac { \lvert \sharp _ { \Delta } \rvert } { \operatorname* { m i n } \{ d _ { i } , d _ { j } \} } + \frac { ( \gamma _ { m a x } ) ^ { - 1 } } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } \big ( \lvert \sharp _ { \Xi } \rvert + \lvert \sharp _ { \Xi } ^ { j } \rvert \big ) +$$ + +where the last term is set to be zero $i f | \sharp _ { \bigstar } ^ { i } |$ (and hence $| \sharp _ { \bigstar } ^ { j } | \big )$ is zero. + +We also extend the previous definition to the weighted case. In this setting we let $G = ( V , E , \omega )$ be a simple, locally finite, undirected graph with normalized weights. We first report the formula for the augmented Forman in the weighted case Samal et al. (2018): + +$$ +\begin{array} { r l } & { \displaystyle F ( i , j ) = \omega ( i ) + \omega ( j ) + \sum _ { k \in S _ { 1 } ( i ) \cap S _ { 1 } ( j ) } \frac { \omega _ { i j } ^ { 2 } } { \omega _ { \Delta } } } \\ & { \displaystyle - \sum _ { k \in S _ { 1 } ( i ) \setminus S _ { 1 } ( j ) } \omega ( i ) \sqrt { \frac { \omega _ { i j } } { \omega _ { i k } } } - \sum _ { k \in S _ { 1 } ( j ) \setminus S _ { 1 } ( i ) } \omega ( j ) \sqrt { \frac { \omega _ { i j } } { \omega _ { j k } } } , } \end{array} +$$ + +where $\omega _ { \Delta }$ is taken to be the Heron formula for the area of a triangle while $\omega ( \cdot )$ denotes some weighting scheme for the nodes as well. We propose a similar definition for the weighted case, which reduces to the one discussed above in the combinatorial setting. We recall that $W$ is the weighted adjacency matrix while $A$ is the combinatorial one. Moreover, we write $A _ { j } ^ { i } = ( A _ { j } - A _ { i } ) _ { + } ^ { - }$ and similarly for $W _ { j } ^ { i }$ . + +Definition 8. For any pair of adjacent nodes $i , j \in V$ we define $\operatorname { R i c } ( i , j )$ to be $0 i f \operatorname* { m i n } \{ | d _ { i } | , | d _ { j } | \} =$ 1, otherwise we set + +$$ +\begin{array} { r l } & { \mathrm { R i c } ( i , j ) : = \cfrac { 1 } { d _ { i } } \left( 1 - \displaystyle \sum _ { k \in \mathcal { Q } _ { i } } { \sqrt { \frac { \omega _ { i j } } { \omega _ { i k } } } } \right) + \cfrac { 1 } { d _ { j } } \left( 1 - \displaystyle \sum _ { k \in \mathcal { Q } _ { i } } { \sqrt { \frac { \omega _ { i j } } { \omega _ { j k } } } } \right) } \\ & { \phantom { \frac { 1 } { d _ { i } } } + \cfrac { 1 } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } \displaystyle \sum _ { k \in \mathcal { S } _ { i } ( i ) \cap \mathcal { S } ( i , j ) } \frac { \omega _ { i j } ^ { 2 } } { \omega \Delta } } \\ & { \phantom { \frac { 1 } { d _ { i } } } + \displaystyle \sum _ { k \in \mathcal { Q } _ { i } ^ { * } } \frac { \omega _ { i j } } { \omega | } \left( \frac { \omega _ { i j } \left( \gamma _ { 0 n x } \right) ^ { - 1 } } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } - \frac { \sqrt { \omega _ { i j } \nu _ { \perp } } } { d _ { i } } \right) } \\ & { \phantom { \frac { 1 } { d _ { i } } } + \displaystyle \sum _ { k \in \mathcal { Q } _ { i } ^ { * } } \frac { \omega _ { i j } } { \omega | } \left( \frac { \omega _ { i j } \left( \gamma _ { 0 n x } \right) ^ { - 1 } } { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } - \frac { \sqrt { \omega _ { i j } \nu _ { \perp } } } { d _ { j } } \right) , } \end{array} +$$ + +with + +$$ +\omega _ { \Delta } : = \frac { 1 } { 3 } ( \omega _ { i j } ^ { 2 } + \omega _ { i k } ^ { 2 } + \omega _ { j k } ^ { 2 } ) , \quad z \in S _ { 1 } ( i ) \cap S _ { 1 } ( j ) , +$$ + +and, for a given $k \in \sharp _ { \bigstar } ^ { i }$ + +$$ +\nu _ { \perp } : = \frac { W _ { z } \cdot A _ { j } ^ { i } } { A _ { k } \cdot A _ { j } ^ { i } } , \omega _ { \perp } = \frac { 1 } { 4 } \left( \omega _ { i j } ^ { 2 } + \omega _ { i k } ^ { 2 } + \nu _ { \perp } ^ { 2 } + \frac { A _ { k } \cdot W _ { j } ^ { i } } { A _ { k } \cdot A _ { j } ^ { i } } \right) . +$$ + +Important convention. Without losing generality, in the following we always assume that $1 \leq d _ { i } \leq$ $d _ { j }$ . In particular, we write $d _ { i } \doteq d$ and $d _ { j } = d + s$ , for some $s \geq 0$ , omitting to specify that both $d$ and $s$ are of course depending on $i$ and $j$ . Moreover, from now on we only focus on the unweighted case. + +We can now prove our main comparison theorem. + +Theorem 2. Given an unweighted graph $G$ , for any edge $i \sim j$ we have $\kappa ( i , j ) \geq \operatorname { R i c } ( i , j )$ . + +Proof. We stick to the aforementioned convention: $d _ { i } : = d \le d _ { j } = d + s , s \ge 0 ,$ . The strategy of the proof amounts to finding a transportation plan providing an upper bound for $W _ { 1 } ( \mu _ { i } ^ { \alpha } , \mu _ { j } ^ { \alpha } )$ and hence a lower bound for the curvature $\kappa ( i , j )$ . In particular, we consider plans moving the mass $\mu _ { i } ^ { \alpha }$ from $B _ { 1 } ( i )$ to $B _ { 1 } ( j )$ . + +If $d = 1$ , then the optimal transport plan consists of moving the mass $\alpha$ from $i$ to $j$ and the remaining mass $1 - \alpha$ on $j$ to $\bar { S } _ { 1 } ( j )$ . This yields a unit Wasserstein distance between $\mu _ { i } ^ { \alpha }$ and $\mu _ { j } ^ { \alpha }$ and hence zero Ollivier curvature $\kappa ( i , j )$ , which coincides with the value of balanced Forman $\operatorname { R i c } ( i , j )$ . + +Assume now that $d \geq 2$ . A (possibly non-optimal) transport plan from $\mu _ { i } ^ { \alpha }$ to $\mu _ { j } ^ { \alpha }$ is given by: + +(i) Move mass $( 1 - \alpha ) / ( d + s )$ from each node $k \in \sharp _ { \bigstar } ^ { \mathfrak { m } } \subset \sharp _ { \bigstar } ^ { i }$ to its unique image in $\sharp _ { \bigstar } ^ { j }$ under a bijection $\varphi$ as per definition of $\sharp _ { \bigsqcup } ^ { \mathfrak { m } }$ . +(ii) The remaining mass on each node $k \in \sharp _ { \bigstar } ^ { \mathfrak { m } }$ will need to travel by at most distance 3 to $S _ { 1 } ( j )$ . +(iii) The extra-mass $( 1 - \alpha ) ( 1 / d - 1 / ( d + s ) )$ on each common neighbour $k \in \sharp _ { \Delta }$ will need to travel by at most distance 2 to $S _ { 1 } ( j )$ . +(iv) Move the mass $( 1 - \alpha ) / d$ from $j$ to $S _ { 1 } ( j )$ . +(v) Move the mass $\alpha$ from $i$ to $j$ . This leaves left-over mass $( 1 - \alpha ) / ( d + s )$ at $i$ from the distribution $\mu _ { j } ^ { \alpha }$ . This mass can be compensated from mass in $S _ { 1 } ( i )$ which is at distance one. +(vi) Finally, we move the mass $( 1 - \alpha ) / d$ of any untouched node in $S _ { 1 } ( i )$ to $S _ { 1 } ( j )$ along a path of length lesser or equal than three. Note that the remaining mass is equal to $( ( \bar { 1 } - \alpha ) / d ) ( \bar { d } - 1 - | \sharp _ { \Delta } | - | \bar { \sharp } _ { \varpi } ^ { \mathfrak { m } } | ) - ( 1 - \alpha ) ( d + s )$ , where the last terms comes from (v). + +If we sum all the contributions we find + +$$ +\begin{array} { l } { { V _ { 1 } ( \mu _ { i } ^ { \alpha } , \mu _ { j } ^ { \alpha } ) \leq ( 1 - \alpha ) \left( \displaystyle \frac { | \# _ { [ \alpha ] } ^ { \mathbb { m } } | } { d + s } + 3 | \# _ { \Sigma } ^ { \mathbb { m } } | \left( \displaystyle \frac { 1 } { d } - \displaystyle \frac { 1 } { d + s } \right) + 2 | \# _ { \Delta } | \left( \displaystyle \frac { 1 } { d } - \displaystyle \frac { 1 } { d + s } \right) + \displaystyle \frac { 1 } { d } \right) + \alpha + \displaystyle \frac { 1 - \alpha } { d + s } } } \\ { { \displaystyle ~ + 3 ( 1 - \alpha ) \left( \displaystyle \frac { 1 } { d } ( d - 1 - | \# _ { \Delta } | - | \# _ { \mathbb { m } } ^ { \mathbb { m } } | ) - \displaystyle \frac { 1 } { d + s } \right) } } \\ { { \displaystyle ~ = ( 1 - \alpha ) \left( \displaystyle \frac { 1 } { d + s } \left( - 2 | \# _ { \Delta } | - 2 | \# _ { \Xi } ^ { \mathbb { m } } | - 2 \right) + \displaystyle \frac { 1 } { d } \left( - | \# _ { \Delta } | - 2 \right) + 2 \right) + 1 } } \\ { { \displaystyle ~ = ( 1 - \alpha ) \left( \displaystyle \frac { 1 } { d + s } \left( - 3 | \# _ { \Delta } | - \displaystyle \frac { s } { d } | \# _ { \Delta } | - 2 | \# _ { \Xi } ^ { \mathbb { m } } | - 4 - 2 \displaystyle \frac { s } { d } + 2 ( d + s ) \right) \right) + 1 . } } \end{array} +$$ + +Therefore we have + +$$ +\kappa ( i , j ) = \operatorname* { l i m } _ { \alpha \to 1 } \frac { 1 - W ( \mu _ { i } ^ { \alpha } , \mu _ { j } ^ { \alpha } ) } { 1 - \alpha } \geq \left( \frac { 1 } { d + s } \left( 3 | \sharp _ { \Delta } | + \frac { s } { d } | \sharp _ { \Delta } | + 2 | \sharp _ { \overline { { \sharp } } } ^ { \mathrm { m } } | + 4 + 2 \frac { s } { d } - 2 ( d + s ) \right) \right) . +$$ + +By using Lemma 11, we can bound the right hand side as + +$$ +\kappa ( i , j ) \geq \left( \frac { 1 } { d + s } \left( 3 | \sharp _ { \Delta } | + \frac { s } { d } | \sharp _ { \Delta } | + ( \gamma _ { \operatorname* { m a x } } ) ^ { - 1 } \big ( | \sharp _ { \bigstar | } ^ { i } | + | \sharp _ { \bigstar | } ^ { j } | \big ) + 4 + 2 \frac { s } { d } - 2 ( d + s ) \right) \right) = \mathrm { R i c } ( i , j ) +$$ + +which completes the proof. + +Remark 13. By inspection $\operatorname { R i c } ( i , j ) \geq \Phi ( i , j )$ , with $\Phi ( i , j )$ as in Theorem 12. We have three cases: + +(i) $\Phi ( i , j ) = 2 / d + 2 / ( d + s ) - 2 + 2 | \sharp _ { \Delta } | / ( d + s ) + | \sharp _ { \Delta } | / d \leq \operatorname { R i c } ( i , j )$ , because $\operatorname { R i c } ( i , j )$ takes into account the positive contribution of 4-cycles as well. + +(ii) $\Phi ( i , j ) = - 1 + 1 / d + 1 / ( d + s ) + 2 | \sharp _ { \Delta } | / ( d + s )$ , which happens iff + +$$ +d + s - 2 - \frac { s } { d } - | \sharp _ { \Delta } | > 0 +$$ + +and + +$$ +d + s - 2 - \frac { s } { d } - | \sharp \triangle | - \frac { s } { d } | \sharp \triangle | \leq 0 . +$$ + +From the previous inequalities we derive + +$$ +\operatorname { R i c } ( i , j ) \geq \Phi ( i , j ) + \frac { 1 } { d + s } \left( 2 - d - s + | \sharp \Delta | + \frac { s } { d } | \sharp \Delta | + \frac { s } { d } \right) \geq \Phi ( i , j ) . +$$ + +(iii) $\Phi ( i , j ) = | \sharp _ { \Delta } | / ( d + s )$ which is equivalent to + +$$ +d + s - 2 - \frac { s } { d } - | \sharp _ { \Delta } | \leq 0 . +$$ + +In this case we have + +$$ +\operatorname { R i c } ( i , j ) \geq \Phi ( i , j ) + \frac { s } { d } \frac { | \sharp _ { \Delta } | } { d + s } +$$ + +Corollary 3. If $\operatorname { R i c } ( i , j ) \geq k > 0$ for any edge $i \sim j$ , then there exists a polynomial $P$ such that + +$$ +| B _ { r } ( i ) | \leq P ( r ) , \forall i \in V . +$$ + +Proof. This follows immediately from Theorem 2 and a Bishop-Gromov type of result for discrete Ollivier curvature on graphs Paeng (2012). □ + +To address the proof of Theorem 4, we first need the Lemma below. + +Lemma 14. Given $i \sim j$ , with $d _ { i } \leq d _ { j }$ , if $\operatorname { R i c } ( i , j ) \leq - 2 + \delta ,$ , for some $0 < \delta < ( 1 + \gamma _ { m a x } ) ^ { - 1 }$ , then + +$$ +\frac { | Q _ { j } | } { | \sharp _ { \Delta } | + 1 } > \delta ^ { - 1 } . +$$ + +Proof. According to our convention we let $d _ { i } = d$ and $d _ { j } = d + s$ , for $s \geq 0$ . We also recall that $Q _ { j } = S _ { 1 } ( j ) \backslash ( \sharp _ { \Delta } \cup \sharp _ { \Pi } ^ { j } \cup \{ i \} )$ . If we multiply equation 3 by $d _ { j } = d + s$ , we see that $\operatorname { R i c } ( i , j ) \leq - 2 + \delta$ iff + +$$ +4 + 2 \frac { s } { d } + 3 | \sharp _ { \Delta } | + \frac { s } { d } | \sharp _ { \Delta } | + \gamma _ { \operatorname* { m a x } } ^ { - 1 } ( | \sharp _ { \bigstar } ^ { i } | + | \sharp _ { \bigstar } ^ { j } | ) \leq \delta ( 1 + | \sharp _ { \Delta } | + | \sharp _ { \bigstar } ^ { j } | + | Q _ { j } | ) . +$$ + +By assumption $| \sharp _ { \sf D } ^ { j } | ( \gamma _ { \mathrm { m a x } } ^ { - 1 } - \delta ) \geq 0$ , meaning that + +$$ +\delta ( 1 + | \sharp \Delta | + | Q _ { j } | ) \geq 4 + 2 \frac { s } { d } + 3 | \sharp \Delta | + \frac { s } { d } | \sharp \Delta | \geq 4 + 3 | \sharp \Delta | . +$$ + +Therefore, we conclude + +$$ +\frac { | Q _ { j } | } { | \sharp _ { \Delta } | + 1 } \geq \frac { 3 } { \delta } - 1 . +$$ + +Theorem 4. Consider a MPNN as in equation $I$ . Let $i \sim j$ with $d _ { i } \leq d _ { j }$ and assume that: + +(i) $| \nabla \phi _ { \ell } | \le \alpha$ and $| \nabla \psi _ { \ell } | \le \beta$ for each $0 \leq \ell \leq L - 1$ , with $L \geq 2$ the depth of the MPNN. + +(ii) There exists $\delta$ s.t. $0 < \delta < ( \operatorname* { m a x } \{ d _ { i } , d _ { j } \} ) ^ { - \frac { 1 } { 2 } }$ , $\delta < \gamma _ { m a x } ^ { - 1 } ,$ , and $\operatorname { R i c } ( i , j ) \leq - 2 + \delta$ + +Then there exists $Q _ { j } \subset S _ { 2 } ( i )$ satisfying $| Q _ { j } | > \delta ^ { - 1 }$ and for $0 \leq \ell _ { 0 } \leq L - 2$ we have + +$$ +\frac { 1 } { | Q _ { j } | } \sum _ { k \in Q _ { j } } \left| \frac { \partial h _ { k } ^ { ( \ell _ { 0 } + 2 ) } } { \partial h _ { i } ^ { ( \ell _ { 0 } ) } } \right| < ( \alpha \beta ) ^ { 2 } \delta ^ { \frac { 1 } { 4 } } . +$$ + +Proof. As usual we let $d _ { i } : = d$ and $d _ { j } : = d + s$ , for some $s \geq 0$ . We first observe that from the requirement $\delta ^ { 2 } ( d + s ) \leq 1$ in (ii), we derive $\operatorname { R i c } ( i , j ) \leq - 2 + \delta$ iff + +$$ +4 + 2 \frac { s } { d } + 3 | \sharp \triangle | + \frac { s } { d } | \sharp \triangle | + \gamma _ { \mathrm { m a x } } ^ { - 1 } ( | \sharp \vert + | \sharp \vert ) \leq \delta ( d + s ) . +$$ + +Therefore we have + +$$ +\delta | \sharp _ { \Delta } | \left( 3 + \frac { s } { d } \right) \leq \delta ^ { 2 } ( d + s ) , +$$ + +meaning that + +$$ +\delta | \sharp _ { \Delta } | \leq 1 . +$$ + +From now on we let $Q _ { j }$ denote again the complement $S _ { 1 } ( j ) \setminus ( S _ { 1 } ( i ) \cup \sharp _ { \Pi } ^ { j } \cup \{ i \} )$ . Without loss of generality we set \`0 = 0 and hence h(0)i $h _ { i } ^ { ( 0 ) } = x _ { i }$ ; the very same proof applies to any other choice of $\ell _ { 0 }$ . , we can apply Lemma 1 and derive + +$$ +\left| \frac { \partial h _ { k } ^ { ( 2 ) } } { \partial x _ { i } } \right| \leq ( \alpha \beta ) ^ { 2 } ( \hat { A } ) _ { i k } ^ { 2 } . +$$ + +We may expand the power of the augmented normalized adjacency matrix as + +$$ +( \hat { A } ) _ { i k } ^ { 2 } = \frac { 1 } { \sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \sum _ { w \in S _ { 1 } ( k ) \cap S _ { 1 } ( i ) } \frac { 1 } { d _ { w } + 1 } . +$$ + +If we introduce the set $\hat { Q } _ { j } = \{ k \in Q _ { j } : \sigma _ { i k } > 1 \}$ , we can then write + +$$ +\begin{array} { l } { { \displaystyle \sum _ { k \in Q _ { j } } ( \hat { A } ) _ { i k } ^ { 2 } = \sum _ { k \in Q _ { j } } \frac { 1 } { \sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \sum _ { w \in S _ { 1 } ( k ) \cap S _ { 1 } ( i ) } \frac { 1 } { d _ { w } + 1 } = } } \\ { { = \displaystyle \frac { 1 } { \sqrt { d _ { i } + 1 } } \left( \sum _ { k \in Q _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \frac { 1 } { d _ { j } + 1 } + \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \sum _ { w \in S _ { 1 } ( k ) \cap S _ { 1 } ( i ) \cap S _ { 1 } ( j ) } \frac { 1 } { d _ { w } + 1 } \right) } } \end{array} +$$ + +where in the last equality we have again used the fact that $k \in \hat { Q } _ { j }$ iff there is $w \in S _ { 1 } ( k ) \cap S _ { 1 } ( i ) \cap S _ { 1 } ( j )$ To avoid heavy notations, we introduce $V _ { k } : = S _ { 1 } ( k ) \cap S _ { 1 } ( i ) \cap S _ { 1 } ( j )$ . Let us first focus on the first sum in equation 12. We have + +$$ +\frac { 1 } { \sqrt { d _ { i } + 1 } } \sum _ { k \in Q _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \frac { 1 } { d _ { j } + 1 } \leq \frac { 1 } { \sqrt { d _ { i } + 1 } } | Q _ { j } | \frac { 1 } { d _ { j } + 1 } \leq \frac { 1 } { \sqrt { d _ { i } + 1 } } \leq 1 . +$$ + +We now consider the second sum in equation 12. We assume $| \sharp _ { \Delta } | \geq 1$ , otherwise ${ \hat { Q } } _ { j } = \emptyset$ . We let + +$$ +\Omega : = \left\{ w \in \sharp _ { \Delta } : \ d _ { w } < \frac { 1 } { C } \frac { | \hat { Q } _ { j } | } { | \sharp _ { \Delta } | } + \frac { 2 } { C } \right\} +$$ + +for some $C > 0$ to be chosen below. Then, the second sum in equation 12 can be split as + +$$ +\sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \left( \sum _ { w \in V _ { k } \cap \Omega } \frac { 1 } { d _ { w } + 1 } + \sum _ { w \in V _ { k } \setminus \Omega } \frac { 1 } { d _ { w } + 1 } \right) . +$$ + +Since any $w \in V _ { k }$ has degree at least three, we can bound the first term in equation 14 as + +$$ +\sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \left( \sum _ { w \in V _ { k } \cap \Omega } \frac { 1 } { d _ { w } + 1 } \right) \leq \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \frac { | V _ { k } \cap \Omega | } { 4 } . +$$ + +We now observe that + +$$ +\sum _ { k \in \hat { \mathcal { Q } } _ { j } } \frac { \left| V _ { k } \cap \Omega \right| } { \sqrt { d _ { k } + 1 } } \leq \sum _ { k \in \hat { \mathcal { Q } } _ { j } } \left| V _ { k } \cap \Omega \right| = \left| \left\{ \left( k , w \right) \in E \colon k \in \hat { Q } _ { j } , w \in V _ { k } \cap \Omega \right\} \right| \leq \left( \operatorname* { m a x } _ { w \in \Omega } d _ { w } \right) \left| \Omega \right| . +$$ + +Since $d _ { w } \leq ( 1 / C ) | \hat { Q } _ { j } | / | \sharp _ { \Delta } | + 2 / C$ for any $w \in \Omega$ we see that the first term in equation 14 can be bounded by + +$$ +\begin{array} { r l r } { { \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \sum _ { w \in V _ { k } \cap \Omega } \frac { 1 } { d _ { w } + 1 } \le \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \frac { | V _ { k } \cap \Omega | } { 4 } } } \\ & { } & { \le ( \frac { 1 } { C } \frac { | \hat { Q } _ { j } | } { | \| \hat { { \boldsymbol { q } } } _ { \Delta } | } + \frac { 2 } { C } ) \frac { | \Omega | } { 4 } \le ( \frac { 1 } { C } \frac { | \hat { Q } _ { j } | } { | \| \hat { { \boldsymbol { q } } } _ { \Delta } | } + \frac { 2 } { C } ) \frac { | \| \hat { { \boldsymbol { q } } } _ { \Delta } | } { 4 } . } \end{array} +$$ + +by definition of $\Omega$ . We now bound the second term in equation 14 as + +$$ +\sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \sum _ { w \in V _ { k } \setminus \Omega } \frac { 1 } { d _ { w } + 1 } \leq \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \frac { C | \sharp \Delta | } { | \hat { Q } _ { j } | } | V _ { k } \setminus \Omega | +$$ + +where we have used that $d _ { w } ^ { - 1 } \leq C | \sharp _ { \Delta } | / | \hat { Q } _ { j } |$ if $w \in V _ { k } \setminus \Omega$ . Since + +$$ +\frac { | V _ { k } \setminus \Omega | } { \sqrt { d _ { k } + 1 } } \leq \frac { | V _ { k } | } { \sqrt { | S _ { 1 } ( k ) | } } \leq \frac { | V _ { k } | } { \sqrt { | V _ { k } | } } \leq \sqrt { | V _ { k } | } \leq \sqrt { | \sharp _ { \Delta } | } +$$ + +we see that + +$$ +\sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \frac { C | \sharp \Delta | } { | \hat { Q } _ { j } | } | V _ { k } \setminus \Omega | \leq \frac { C | \sharp \Delta | } { | \hat { Q } _ { j } | } \sqrt { | \sharp \Delta | } | \hat { Q } _ { j } | = C | \sharp \Delta | ^ { \frac { 3 } { 2 } } . +$$ + +We are now ready to complete the proof of the theorem. According to equation 11 it suffices to show that + +$$ +\frac { 1 } { | Q _ { j } | } \sum _ { k \in Q _ { j } } ( \hat { A } ^ { 2 } ) _ { i k } \leq \delta ^ { 1 / 4 } . +$$ + +From equation 12 and equation 13 we derive that the left hand side of the equation above is bounded by + +$$ +\begin{array} { r } { \frac { 1 } { | Q _ { j } | } \displaystyle \sum _ { k \in Q _ { j } } ( \hat { A } ^ { 2 } ) _ { i k } \leq \frac { 1 } { | Q _ { j } | } + \frac { 1 } { | Q _ { j } | } \left( \displaystyle \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \displaystyle \sum _ { w \in V _ { k } } \frac { 1 } { d _ { w } + 1 } \right) } \\ { \leq \delta + \frac { 1 } { | Q _ { j } | } \left( \displaystyle \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \displaystyle \sum _ { w \in V _ { k } } \frac { 1 } { d _ { w } + 1 } \right) } \end{array} +$$ + +where in the last inequality we have used Lemma 14 to bound $| Q _ { j } | ^ { - 1 }$ by $\delta$ . In particular we note that if $\sharp _ { \Delta } = \emptyset$ then ${ \hat { Q } } _ { j } = \emptyset$ , and the bound would be simply controlled by $\delta$ as claimed. When $\vert \sharp _ { \Delta } \vert > 0$ , we can use equation 15 and equation 16 to estimate the second term from above by + +$$ +\begin{array} { r l } & { \frac { 1 } { | Q _ { j } | } \left( \displaystyle \sum _ { k \in \hat { Q } _ { j } } \frac { 1 } { \sqrt { d _ { k } + 1 } } \displaystyle \sum _ { w \in V _ { k } } \frac { 1 } { d w + 1 } \right) \leq } \\ & { \frac { 1 } { | Q _ { j } | } \left( \left( \frac { 1 } { \bar { C } } \frac { | \hat { Q } _ { j } | } { | \sharp _ { \Delta } | } + \frac { 2 } { \bar { C } } \right) \frac { | \sharp _ { \Delta } | } { 4 } \right) + \frac { 1 } { | Q _ { j } | } \left( C | \sharp _ { \Delta } | ^ { \frac { 3 } { 2 } } \right) \leq \frac { 1 } { 4 } \left( \frac { 1 } { \bar { C } } + \frac { 2 | \sharp _ { \Delta } | } { \bar { C } | Q _ { j } | } \right) + \frac { C | \sharp _ { \Delta } | } { | Q _ { j } | } \sqrt { | \sharp _ { \Delta } | } . } \end{array} +$$ + +By applying Lemma 14 we get + +$$ +\frac { 1 } { 4 } \left( \frac { 1 } { C } + \frac { 2 | \sharp \Delta | } { C | Q _ { j } | } \right) + \frac { C | \sharp \Delta | } { | Q _ { j } | } \sqrt { | \sharp \Delta | } \leq \frac { 1 } { 4 } \left( \frac { 1 } { C } + 2 \frac { \delta } { C } \right) + C \delta \sqrt { | \sharp \Delta | } . +$$ + +We now choose $C = \delta ^ { - 1 / 4 }$ , so that the previous quantity can be bounded by + +$$ +\frac { 1 } { 4 } \left( \frac { 1 } { C } + 2 \frac { \delta } { C } \right) + C \delta \sqrt { | \sharp _ { \Delta } | } \leq \frac { 1 } { 4 } \left( \delta ^ { \frac { 1 } { 4 } } + 2 \delta ^ { \frac { 5 } { 4 } } \right) + \delta ^ { \frac { 1 } { 4 } } \sqrt { \delta | \sharp _ { \Delta } | } \leq \frac { 1 } { 4 } \left( \delta ^ { \frac { 1 } { 4 } } + 2 \delta ^ { \frac { 5 } { 4 } } \right) + \delta ^ { \frac { 1 } { 4 } } +$$ + +where in the last inequality we have used equation 10. Therefore, we have shown that + +$$ +\frac { 1 } { | Q _ { j } | } \sum _ { k \in Q _ { j } } ( \hat { A } ^ { 2 } ) _ { i k } \leq \delta + \frac { 1 } { 4 } \left( \delta ^ { \frac { 1 } { 4 } } + 2 \delta ^ { \frac { 5 } { 4 } } \right) + \delta ^ { \frac { 1 } { 4 } } \leq 3 \delta ^ { \frac { 1 } { 4 } } +$$ + +where we have used that $\delta < 1$ . This completes the proof (once we absorb the extra factor 3 in the constant $\alpha \beta$ in equation 11). □ + +Remark 15. The requirement $\delta \sqrt { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } \ < \ 1$ can be replaced by a more general bound $\delta \sqrt { \operatorname* { m a x } \{ d _ { i } , d _ { j } \} } < r$ . The argument above extends to this case up to renaming the constant $\alpha \beta$ in the statement so to include an extra factor $r$ . + +We note that the condition $\delta \operatorname* { m a x } \{ d _ { i } , d _ { j } \} < r$ would be stronger than the one appearing in (ii) of Theorem 4. In this regard, we recall that for a $d$ -tree the curvature satisfies $\begin{array} { r } { \operatorname { R i c } ( i , j ) = - 2 + \frac { 4 } { d } } \end{array}$ . + +We can also prove Proposition 5: + +Proposition 5. If $\operatorname { R i c } ( i , j ) \geq k > 0$ for all $i \sim j$ , then $\textstyle \lambda _ { 1 } / 2 \geq h _ { G } \geq { \frac { k } { 2 } }$ + +Proof. This follows as an immediate Corollary of Theorem 2 and (Lin et al., 2011, Theorem 4.2). + +Betweenness centrality to measure bottleneck. In equation 2 we have derived how the topology of the graph affects the dependence of the hidden node representation $h _ { i } ^ { ( r + 1 ) }$ on the input feature $x _ { s }$ , for nodes $i , s$ at distance $r + 1$ . We note that in this case $\hat { A } _ { i s } ^ { r + 1 }$ is exactly measuring the number of minimal paths from to $s$ . If the receptive field $B _ { r + 1 } ( i )$ is a binary tree, then we have seen that the entry of the power matrix decays exponentially. The reason for such decay stems from the existence of exponentially many nodes in the receptive field combined with the lack of multiple minimal paths (shortcuts). When such conditions hold, most of the minimal paths go through the same nodes, which is exactly what happens for the tree where each node is in the minimal paths between different branches. Since the frequency in which a node appears in the minimal path of distinct pairs of nodes is measured by the betweenness centrality Freeman (1977), we propose a topological characterization of the ‘bottleneckedness’ of a graph as follows: + +Definition 9 (bottleneck). The bottleneck-value of $G$ is $\begin{array} { r } { b _ { G } : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } b ( i ) } \end{array}$ , where $b ( i )$ denotes the betweenness centrality on node $i$ . + +From a standard combinatorial argument it follows that if $G$ is connected, then + +$$ +b _ { G } = \frac { 1 } { n } \sum _ { i , j } \left( d _ { G } ( i , j ) - 1 \right) . +$$ + +We note that $b _ { G } = 0$ iff $G$ is the complete graph $K _ { n }$ . Therefore, $b _ { G }$ determines how far the given topology is from $K _ { n }$ , with the latter representing the limit case of a fully connected layer Alon $\&$ Yahav (2021) where no bottleneck may occur as any pair of nodes would be neighbours. This further supports our intuition that the betweenness centrality is a good topological candidate for providing a global measurement of bottleneckedness in the graph. + +It also follows from equation 17 that any update to the graph topology consisting of edge additions would decrease $b _ { G }$ and thus reduce the bottleneck. The quantity $b _ { G }$ is global in nature though and hence lacks robustness. As a pedagogical example, consider a barbell $G ( m , 2 r + 1 )$ , with $m$ the size of the two cliques joined by a path of length $2 r + 1$ and focus on the edge $i \sim j$ in the middle of such path. Nodes $i$ and $j$ are central to the graph, in the sense that most minimal paths go through them and indeed their betweenness centrality is $\bar { b } ( i ) = b ( j ) = ( m + r ) ^ { 2 } + ( m + r )$ . If now we add a single edge joining the two cliques $K _ { m }$ , the values $b ( i )$ and $b ( j )$ decrease dramatically by $\Omega ( m ^ { 2 } + r )$ . Since the operation is non-local, we see that the representation $h _ { i } ^ { ( \ell ) }$ of any MPNN is unaffected by the edge addition for any $\ell \in ( 0 , r )$ , and similarly for $j$ . Eventually, if we keep adding edges, the receptive fields $B _ { s } ( i )$ will be affected for small values of $s$ as well: the drawback of such approach is that the resulting adjacency may be significantly different. Conversely, the curvature provides a more precise, local and hence robust way of controlling the bottleneck and hence the over-squashing problem. Nonetheless, we relate the betweenness centrality to the Jacobian of the hidden features. + +Theorem 16. Given $i \sim j$ , let $\Omega _ { j } : = S _ { 1 } ( i ) \cap S _ { 1 } ( j ) \cup \{ j \}$ . If $\operatorname { R i c } ( i , j ) \leq - 2 + \delta ,$ , for $0 < \delta <$ $( 1 + \gamma _ { m a x } ) ^ { - 1 }$ , then + +$$ +\frac { 1 } { | \Omega _ { j } | } \sum _ { k \in \Omega _ { j } } b ( k ) \geq \delta ^ { - 1 } . +$$ + +Proof. We rewrite the quantity in the statement as + +$$ +{ \frac { 1 } { \left| \sharp _ { \Delta } \right| + 1 } } \left( \sum _ { k \in \sharp _ { \Delta } } b ( k ) + b ( j ) \right) . +$$ + +By definition, given a node $k \in V$ , the betweenness centrality of $k$ is given by + +$$ +b ( k ) : = \sum _ { s , t \in V : s \neq k , t \neq k } \frac { \sigma _ { s t } ( k ) } { \sigma _ { s t } } +$$ + +where $\sigma _ { s t }$ is the number of minimal paths between $s$ and $t$ while $\sigma _ { s t } ( k )$ is the number of minimal paths from $s$ to $t$ passing through $k$ . For convenience, we introduce the set $\hat { Q _ { j } } \subset Q _ { j }$ defined by + +$$ +\hat { Q } _ { j } : = \{ w \in Q _ { j } : \ \sigma _ { i w } > 1 \} . +$$ + +Equivalently, $\hat { Q } _ { j }$ consists of those nodes in $S _ { 1 } ( j ) \setminus S _ { 1 } ( i )$ which form a 4-cycle based at $i \sim j$ with a diagonal inside. Indeed, if $w \in Q _ { j }$ and $\sigma _ { i w } > 1$ , then there exists more than one minimal path between $i$ and $w$ , in addition to the one passing through node $j$ . For any such path there exists $k \in S _ { 1 } ( i ) \cap S _ { 1 } ( w )$ . Since $w \in Q _ { j }$ and $Q _ { j } \cap { \sharp \underline { { \breve { \jmath } } } } = \emptyset$ , we derive that $k \in S _ { 1 } ( j )$ as well. We then get + +$$ +\sum _ { k \in \mathfrak { t } _ { \Delta } } b ( k ) = \sum _ { k \in \mathfrak { t } _ { \Delta } } \sum _ { s , t \in V : s \neq k , t \neq k } \frac { \sigma _ { s t } ( k ) } { \sigma _ { s t } } \geq \sum _ { k \in \mathfrak { t } _ { \Delta } } \sum _ { w \in \hat { Q } _ { j } } \frac { \sigma _ { i w } ( k ) } { \sigma _ { i w } } = \sum _ { w \in \hat { Q } _ { j } } \frac { 1 } { \sigma _ { i w } } \sum _ { k \in \mathfrak { t } _ { \Delta } } \sigma _ { i w } ( k ) . +$$ + +By summing $\sigma _ { i w } ( k )$ for all $k \in \sharp _ { \Delta }$ we obtain all the 2-long minimal paths between $i$ and $w$ with the exception of the one passing through $j$ : + +$$ +\sum _ { k \in \sharp _ { \Delta } } b ( k ) \geq \sum _ { w \in \hat { Q _ { j } } } \frac { 1 } { \sigma _ { i w } } ( \sigma _ { i w } - 1 ) . +$$ + +On the other hand we also have + +$$ +b ( j ) = \sum _ { s , t \in V : s \neq j , t \neq j } \frac { \sigma _ { s t } ( j ) } { \sigma _ { s t } } \geq \sum _ { z \in Q _ { j } } \frac { \sigma _ { i z } ( j ) } { \sigma _ { i z } } = \sum _ { z \in Q _ { j } } \frac { 1 } { \sigma _ { i z } } . +$$ + +By combining equation 18 and equation 19 we finally get + +$$ +\begin{array} { r l r } { { \frac { 1 } { | \sharp _ { \Delta } | + 1 } ( \sum _ { k \in \sharp _ { \Delta } } b ( k ) + b ( j ) ) \geq \frac { 1 } { | \sharp _ { \Delta } | + 1 } ( \sum _ { w \in \widehat { \mathcal { Q } } _ { j } } \frac { 1 } { \sigma _ { i w } } ( \sigma _ { i w } - 1 ) + \sum _ { z \in Q _ { j } } \frac { 1 } { \sigma _ { i z } } ) } } \\ & { } & { = \frac { 1 } { | \sharp _ { \Delta } | + 1 } ( | \widehat { Q } _ { j } | + \sum _ { z \in Q _ { j } \setminus \widehat { Q } _ { j } } \frac { 1 } { \sigma _ { i z } } ) } \\ & { } & { = \frac { | Q _ { j } | } { | \sharp _ { \Delta } | + 1 } , } \end{array} +$$ + +where in the last equality we have used that by definition $\sigma _ { i z } = 1$ for all $z \in Q _ { j } \backslash { \hat { Q } } _ { j }$ . By Lemma 14 the last quantity is larger than $\delta ^ { - 1 }$ . + +# E PROOFS OF RESULTS IN SECTION 4 + +Theorem 6. Let $S \subset V$ with $\nu o l ( S ) \le \nu o l ( G ) / 2$ . Then $\begin{array} { r } { h _ { S , \alpha } \leq \left( \frac { 1 - \alpha } { \alpha } \right) \frac { d _ { \arg } ( S ) } { d _ { \operatorname* { m i n } } ( S ) } h _ { S } } \end{array}$ , where $d _ { \mathrm { a v g } } ( S )$ and $d _ { \mathrm { m i n } } ( S )$ are the average and minimum degree on $S$ , respectively. + +Proof. Given a signal $f : V \to \mathbb { R }$ on the vertex set and $U \subset V$ , analogously to Chung (2007), we introduce the notation + +$$ +f ( U ) : = \sum _ { i \in U } f ( i ) . +$$ + +Let us rewrite the new Cheeger constant $h _ { S , \alpha }$ as follows: + +$$ +h _ { S , \alpha } = \frac { 1 } { | S | } \sum _ { i \in S , j \in \bar { S } } ( R _ { \alpha } ) _ { i j } = \frac { 1 } { | S | } \chi _ { S } R _ { \alpha } ( \bar { S } ) +$$ + +with $\chi _ { S }$ the characteristic function of the subset $S$ , i.e. $\chi _ { S } ( i ) = 1$ iff $i \in S$ . Since the graph $G$ is connected, we can bound $h _ { S , \alpha }$ from above as + +$$ +\frac { 1 } { | S | } \chi _ { S } R _ { \alpha } ( \bar { S } ) = \frac { 1 } { | S | } \sum _ { i \in S , j \in \bar { S } } ( R _ { \alpha } ) _ { i j } \le \frac { 1 } { | S | } \sum _ { i \in S , j \in \bar { S } } ( R _ { \alpha } ) _ { i j } \frac { d _ { i } } { d _ { \operatorname* { m i n } } ( S ) } = \frac { 1 } { | S | } \chi _ { S } D R _ { \alpha } ( \bar { S } ) \frac { 1 } { d _ { \operatorname* { m i n } } ( S ) } . +$$ + +It was proven in (Chung, 2007, Lemma 5) that + +$$ +\chi _ { S } D R _ { \alpha } ( { \bar { S } } ) \leq { \frac { 1 - \alpha } { \alpha } } | \partial S | . +$$ + +By applying equation equation 20 to the bound for the Cheeger constant $h _ { S , \alpha }$ we finally see that + +$$ +\begin{array} { l } { \displaystyle { h _ { S , \alpha } = \frac { 1 } { | S | } \chi _ { S } R _ { \alpha } ( \bar { S } ) \leq \frac { 1 } { | S | } \chi _ { S } D R _ { \alpha } ( \bar { S } ) \frac { 1 } { d _ { \operatorname* { m i n } } ( S ) } \leq \frac { 1 } { | S | } \frac { 1 - \alpha } { \alpha } | \partial S | \frac { 1 } { d _ { \operatorname* { m i n } } ( S ) } } } \\ { \displaystyle { \quad = \frac { 1 } { | S | } \frac { 1 - \alpha } { \alpha } h _ { S } \mathrm { v o l } ( S ) \frac { 1 } { d _ { \operatorname* { m i n } } ( S ) } = \frac { 1 - \alpha } { \alpha } h _ { S } \frac { d _ { \mathrm { a v g } } ( S ) } { d _ { \operatorname* { m i n } } ( S ) } . } } \end{array} +$$ + +We also report an equivalent result, again relying on (Chung, 2007, Lemma 5). + +Proposition 17. Let $S \subset V$ with $\nu o l ( S ) \le \nu o l ( G ) / 2$ . For any $k \in \mathbb N$ , there exists $S _ { k , \alpha } \subset S$ with $\nu o l ( S _ { k , \alpha } ) \geq \nu o l ( S ) ( 1 - ( 2 k ) ^ { - 1 } )$ such that + +$$ +\sum _ { j \in \bar { S } } ( R _ { \alpha } ) _ { i j } \le k \frac { 1 - \alpha } { \alpha } h _ { S } , +$$ + +for all $i \in S _ { k , \alpha }$ + +Proof. Let $k \in \mathbb N$ . By modifying slightly the argument in (Chung, 2007, Lemma 5), we derive that + +$$ +S _ { k , \alpha } ^ { \prime } : = \{ i \in S : \chi _ { i } R _ { \alpha } ( \bar { S } ) \geq k \frac { 1 - \alpha } { \alpha } h _ { S } \} +$$ + +satisfies + +$$ +\frac { 1 - \alpha } { 2 \alpha } | \partial S | \geq \mathrm { v o l } ( S _ { k , \alpha } ^ { \prime } ) k \frac { 1 - \alpha } { \alpha } h _ { S } . +$$ + +Therefore, we obtain + +$$ +\operatorname { v o l } ( S _ { k , \alpha } ^ { \prime } ) \leq { \frac { 1 } { k } } { \frac { \operatorname { v o l } ( S ) } { 2 } } . +$$ + +We then conclude that the complement of $S _ { k , \alpha } ^ { \prime }$ has volume greater or equal than $\operatorname { v o l } ( S ) ( 1 - ( 2 k ) ^ { - 1 } )$ , which completes the proof. + +Remark 18. The previous proposition shows that after sparsifying the personalized page rank operator $R _ { \alpha }$ as suggested in Klicpera et al. (2019) by setting entries below some threshold equal to zero, there will still be only few edges connecting different communities, once again highlighting that random-walk based methods are generally not suited to tackle the graph bottleneck. + +# F EXPERIMENTS + +Our experiments in this paper are semi-supervised node classification (semi-supervised in that the graph structure provides some unlabelled information) on nine common graph learning datasets. Cornell, Texas and Wisconsin are small heterophilic datasets based on webpage networks from the WebKB dataset. Chameleon and Squirrel (Rozemberczki et al., 2021) are medium heterophilic datasets based on Wikipedia networks, along with Actor, the actor-only induced subgraph of the filmdirector-actor-writer network (Tang et al., 2009). Cora (McCallum et al., 2000), Citeseer (Sen et al., 2008) and Pubmed (Namata et al., 2012) are medium homophilic datasets based on citation networks. As in Klicpera et al. (2019), for all experiments we consider the largest connected component of the graph. + +When splitting the data into train/validation/test sets, we first separate the data into a development set and the test set. This is done once to ensure the test set is not used for any training or hyperparameter fitting before the final evaluation. For each of the 100 random splits the development set is divided randomly into a train set and a validation set, where we train models on the train set and evaluate on the validation set. We fit hyperparameters by random search, maximising the mean accuracy across the validation sets. The accuracy then reported in Table 2 is the mean accuracy on the test set from models trained on the train sets with the chosen hyperparameters, along with a $9 5 \%$ confidence interval calculated by bootstrapping the test set accuracies with 1000 samples. For Cora, Citeseer and Pubmed the development set contains 1500 nodes with the rest kept for the test set, and for each random split the train set is chosen to contain 20 nodes of each class while the rest form the validation set. As this is the same method as Klicpera et al. (2019) and we use the same random seeds, we are using the same test set and expect to have comparable results. For the remaining datasets we perform a 60/20/20 split, with $20 \%$ of nodes set aside as the test set and then for each random split the remaining $80 \%$ is split into $60 \%$ training, $20 \%$ validation. + +The homophily index $\mathcal { H } ( G )$ proposed by Pei et al. (2019) is defined as + +$$ +\mathcal { H } ( G ) = \frac { 1 } { | V | } \sum _ { v \in V } \frac { \mathrm { N u m b e r ~ o f ~ } v \mathrm { ' s ~ n e i g h b o r s ~ w h o ~ h a v e ~ t h e ~ s a m e ~ l a b e l ~ a s ~ } v } { \mathrm { N u m b e r ~ o f ~ } v \mathrm { ' s ~ n e i g h b o r s } } . +$$ + +# F.1 DATASETS + +For datasets with disconnected graphs, the statistics shown here are for the largest connected component. + +
CornellTexasWisconsinChameleonSquirrelActorCoraCiteseerPubmed
H(G)0.110.060.160.250.220.240.830.720.79
Nodes140135184832218643882485212019717
Edges2192513621235565224219075069367944324
Features1703170317032323208993114333703500
Classes555555763
Directed?YesYesYesYesYesYesNoNoNo
+ +![](images/7e3012a9f8a829393e71b4047a8695aea3b326896fc24c8b02d1925a1a9c5594.jpg) +Figure 5: Comparing the degree distribution of the original graphs to the preprocessed version. The xaxis is node degree in $\log _ { 2 }$ scale, and the plots are a kernel density estimate of the degree distribution. In the captions we see the Wasserstein distance $W _ { 1 }$ between the original and preprocessed graphs. + +![](images/b9e9ad9e82055e01bd8d3c2a9fdf721719f922616fb3e5a8ed8f710f5b3ae409.jpg) +Figure 6: Rewiring of the Cornell graph. Left-to-right: original graph, DIGL, and SDRF rewiring. Edges are colored by curvature; nodes are colored by the maximum absolute entry of a trained 2-layer GCN’s Jacobian between the GCN’s prediction for that node and the features of the nodes 2 hops away in the original graph. SDRF homogenizes curvature and so lifts the upper bound on the Jacobian from Theorem 4. DIGL also does to an extent, though at the expense of preserving graph topology. + +# F.4 HYPERPARAMETERS + +Table 4: Hyperparameters for GCN with no preprocessing (None). + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
Cornell0.306011280.00820.1570
Texas0.234611280.00720.0037
Wisconsin0.28691640.02810.0113
Chameleon0.730431280.02480.0936
Squirrel0.59742640.01360.1346
Actor0.76051640.02900.0619
Cora0.41441640.00970.0639
Citeseer0.747711280.02510.4577
Pubmed0.40131640.00950.0448
+ +Table 5: Hyperparameters for GCN with the input graph made undirected (Undirected). + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
Cornell0.69101640.01850.0285
Texas0.266511280.00690.0035
Wisconsin0.289321280.01420.0001
Chameleon0.46573640.01890.0423
Squirrel0.59442640.00810.0309
Actor0.66262640.01950.0219
+ +Table 6: Hyperparameters for GCN with the last layer made fully connected ( $+ \mathrm { F A }$ from Alon & Yahav (2021)). + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
Cornell0.264311280.02160.0760
Texas0.220711280.01020.4450
Wisconsin0.26133640.00570.0131
Chameleon0.77833640.01560.0108
Squirrel0.36543640.00770.1922
Actor0.38241640.01650.1168
Cora0.784021280.01490.1429
Citeseer0.54602640.00660.0758
Pubmed0.337621280.02040.0215
+ +Table 7: Hyperparameters for GCN with DIGL preprocessing, or Graph Diffusion Convolution with PPR kernel (DIGL). Descriptions for $\alpha$ , $k$ and $\epsilon$ can be found in Klicpera et al. (2019). + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayαkE
Cornell0.62941640.01340.02580.179564-
Texas0.238221280.00630.01530.020632-
Wisconsin0.294111280.00830.02260.124610.0001
Chameleon0.419111280.00520.00010.0244641
Squirrel0.684411280.00560.45370.039532
Actor0.78201640.01700.01020.1584128、
Cora0.33151640.02840.05720.07731281
Citeseer0.55611640.00940.50130.1076-0.0008
Pubmed0.491521280.00570.05970.1155128-
+ +Table 8: Hyperparameters for GCN with DIGL preprocessing followed by symmetrizing the graph diffusion matrix (DIGL $^ +$ Undirected). + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayαkE
Cornell0.62941640.01340.02580.179564=
Texas0.238221280.00630.01530.0206321
Wisconsin0.294111280.00830.02260.124610.0001
Chameleon0.419111280.00520.00010.0244641
Squirrel0.70941640.01720.01920.1610641
Actor0.40121640.01610.01410.070610.0016
Cora0.33151640.02840.05720.0773128-
Citeseer0.55611640.00940.50130.1076-0.0008
Pubmed0.491521280.00570.05970.11551281
+ +Table 9: Hyperparameters for GCN with SDRF preprocessing (SDRF). Max iterations, $\tau$ and $C ^ { + }$ are the SDRF parameters described in Algorithm 1. + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayMax iterationsTC+
Cornell0.241111280.01720.01251351300.25
Texas0.595411280.02780.0623471722.25
Wisconsin0.603311280.02950.192027320.5
Chameleon0.726511280.01800.2101832773.35
Squirrel0.74012160.01890.225561571780.5
Actor0.688611280.00950.07271010691.22
Cora0.339611280.02440.10761001630.95
Citeseer0.41031640.01990.4551841800.22
Pubmed0.374931280.01120.013816611514.43
+ +Table 10: Hyperparameters for GCN with the input graph made undirected followed by SDRF preprocessing (SDRF $^ +$ Undirected). + +
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayMax iterationsTC+
Cornell0.291111280.00560.03361261450.88
Texas0.21601640.02290.013789221.64
Wisconsin0.24521640.01130.1559136127.95
Chameleon0.48861320.02680.405624412522.84
Squirrel0.42491640.02950.13977874317.19
Actor0.670511280.01150.044711414411.17
+ +# G HARDWARE SPECIFICATIONS + +Our experiments were performed on a server with the following specifications: + +
ComponentSpecification
Architecturex86_64
CPU40x Intel(R) Xeon(R) Silver 4210R CPU @ 2.40GHz
GPU4x GeForce RTX 3090 (24268MiB/GPU)
RAM126GB
OSUbuntu 20.04.2LTS
\ No newline at end of file diff --git a/parse/dev/7UmjRGzp-A/7UmjRGzp-A_content_list.json b/parse/dev/7UmjRGzp-A/7UmjRGzp-A_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..081d9fc1434c9e616c791a5418db3ef5928ca426 --- /dev/null +++ b/parse/dev/7UmjRGzp-A/7UmjRGzp-A_content_list.json @@ -0,0 +1,4789 @@ +[ + { + "type": "text", + "text": "UNDERSTANDING OVER-SQUASHING AND BOTTLENECKS ON GRAPHS VIA CURVATURE ", + "text_level": 1, + "bbox": [ + 174, + 101, + 696, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jake Topping12†, Francesco Di Giovanni3†, Benjamin P. Chamberlain3, Xiaowen $\\mathbf { D o n g } ^ { 1 }$ , and Michael M. Bronstein23 1University of Oxford 2Imperial College London 3Twitter ", + "bbox": [ + 183, + 167, + 683, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 252, + 544, + 267 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Most graph neural networks (GNNs) use the message passing paradigm, in which node features are propagated on the input graph. Recent works pointed to the distortion of information flowing from distant nodes as a factor limiting the efficiency of message passing for tasks relying on long-distance interactions. This phenomenon, referred to as ‘over-squashing’, has been heuristically attributed to graph bottlenecks where the number of $k$ -hop neighbors grows rapidly with $k$ . We provide a precise description of the over-squashing phenomenon in GNNs and analyze how it arises from bottlenecks in the graph. For this purpose, we introduce a new edge-based combinatorial curvature and prove that negatively curved edges are responsible for the over-squashing issue. We also propose and experimentally test a curvature-based graph rewiring method to alleviate the over-squashing. ", + "bbox": [ + 233, + 284, + 766, + 436 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 465, + 336, + 481 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In the past few years, deep learning on graphs and in particular graph neural networks (GNNs) (Sperduti, 1994; Goller & Kuchler, 1996; Sperduti & Starita, 1997; Frasconi et al., 1998; Gori et al., 2005; Scarselli et al., 2008; Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2017; Gilmer et al., 2017) have become very popular in the machine learning community due to their ability to deal with broad classes of systems of relations and interactions, ranging from social networks to particle physics (Shlomi et al., 2021). ", + "bbox": [ + 174, + 494, + 449, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The vast majority of GNNs follow the message passing paradigm (Gilmer et al., ", + "bbox": [ + 174, + 683, + 447, + 709 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/437827f1c25f3a408f69b5456e32402e205c919e9d7a9e7a22fdc6c85a6a6f78.jpg", + "image_caption": [ + "Figure 1: Top: evolution of curvature on a surface may reduce the bottleneck. Bottom: this paper shows how the same may be done on graphs to improve GNN performance. Blue/red shows negative/positive curvature. " + ], + "image_footnote": [], + "bbox": [ + 472, + 469, + 808, + 628 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2017), using learnable non-linear functions to diffuse information on the graph. Multiple popular GNN architectures such as GCN (Kipf & Welling, 2017) and GAT (Velickovi ˇ c et al., 2018) can be ´ posed as particular flavors of this scheme and considered instances of a more general framework of geometric deep learning (Bronstein et al., 2021). ", + "bbox": [ + 174, + 710, + 826, + 765 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Some of the drawbacks of the message passing paradigm have now been identified and formalized, including the limits of expressive power (Xu et al., 2019; Morris et al., 2019; Maron et al., 2019) and the problem of over-smoothing (NT & Maehara, 2019; Oono & Suzuki, 2020). On the other hand, much less is known about the phenomenon of over-squashing, consisting in the distortion of messages being propagated from distant nodes. Alon & Yahav (2021) proposed rewiring the graph as a way of reducing the bottleneck, defined as those topological properties in the graph leading to over-squashing. This approach is in line with multiple other results e.g. using connectivity diffusion (Klicpera et al., 2019) as a preprocessing step to facilitate graph learning. Yet, the exact understanding of the over-squashing and how it originates from the bottlenecks in the topology of the underlying graph are still elusive. Consequently, there is currently no consensus on the right method (either based on graph rewiring or not) to address the bottleneck and hence alleviate the over-squashing. ", + "bbox": [ + 174, + 772, + 825, + 897 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we address these questions using tools from differential geometry, which traditionally is concerned with the study of manifolds. It offers an appealing framework to study the properties of graphs, in particular arguing that graphs, like manifolds, exhibit curvature that makes them more suitable to be realized in spaces with hyperbolic geometry (Liu et al., 2019; Chami et al., 2019; Boguna et al., 2021). One notion of curvature that has received attention for graph learning is Ricci curvature (Hamilton, 1988), also known in geometry for its use in Ricci flow and the subsequent proof of the Poincaré conjecture (Perelman, 2003). Certain graph analogues of the Ricci curvature (Forman, 2003; Ollivier, 2009; Sreejith et al., 2016) were used in Ni et al. (2018) for a discrete version of Ricci flow to construct a metric between graphs. Graph Ricci flow was also used in Ni et al. (2019) for community detection. Both of these methods use the edge weights as a substitute for the metric of a manifold, and do not change the topological structure of the graph. ", + "bbox": [ + 174, + 138, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions and Outline. This paper, to our knowledge, is the first theoretical study of the bottleneck and over-squashing phenomena in message passing neural networks from a geometric perspective. In Section 2, we propose the Jacobian of node representations as a formal way of measuring the over-squashing and we show that the graph topology may compromise message propagation in graph neural networks by creating a bottleneck. In Section 3, we investigate how such a bottleneck is induced which leads to the over-squashing of information. To this aim, we introduce a new combinatorial edge-based curvature called Balanced Forman curvature that constitutes a sharp lower bound to the standard Ollivier curvature on graphs, and prove that negatively curved edges are responsible for the formation of bottlenecks (and hence for over-squashing). In Section 4, we present a new curvature-based method for graph rewiring called Stochastic Discrete Ricci Flow. According to the theoretical results in Section 3, this rewiring method is suited to address the graph bottleneck and hence alleviate the over-squashing by surgically targeting the edges responsible for the issue. By contrast, we rigorously show that a recently introduced diffusion-based rewiring scheme might generally fail to reduce the bottleneck. Finally, in Section 5, we compare different rewiring strategies experimentally on several standard graph learning datasets. ", + "bbox": [ + 174, + 306, + 825, + 515 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 ANALYSIS OF THE OVER-SQUASHING PHENOMENON ", + "text_level": 1, + "bbox": [ + 176, + 536, + 637, + 551 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 566, + 326, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Let $G = ( V , E )$ be a simple, undirected, and connected graph, where $( i , j ) \\in E$ iff $i \\sim j$ . We focus on the unweighted case, although the theory extends to the weighted setting as well. We denote the adjacency matrix by $A$ and let $\\overset { \\vartriangle } { \\boldsymbol { \\tilde { A } } } = \\boldsymbol { A } + \\boldsymbol { \\bar { I } }$ be the adjacency matrix augmented with self-loops. Similarly we let $\\tilde { D } = D + I$ , with $D$ the diagonal degree matrix, and let $\\bar { { \\cal A } } = \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }$ be the normalized augmented adjacency matrix (self-loops are commonly included in GNN architecture, and in Section 2 we formally explain why GNNs are expected to propagate information more reliably when self-loops are taken into account). Given $i \\in V$ , we denote its degree by $d _ { i }$ and let ", + "bbox": [ + 173, + 592, + 826, + 694 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/94ea58d66e8dd4f5769e5f0da089230511e450c7950d880ffecb2f6496b2ecf9.jpg", + "text": "$$\nS _ { r } ( i ) : = \\{ j \\in V : d _ { G } ( i , j ) = r \\} , \\quad B _ { r } ( i ) : = \\{ j \\in V : d _ { G } ( i , j ) \\leq r \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 266, + 699, + 730, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $d _ { G }$ is the standard shortest-path distance on the graph and $r \\in \\mathbb { N }$ . The set $B _ { r } ( i )$ represents the receptive field of an $r$ -layer message passing neural network at node $i$ . ", + "bbox": [ + 173, + 722, + 825, + 751 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Message passing neural networks (MPNNs). Assume that the graph $G$ is equipped with node features $X \\in \\mathbb { R } ^ { n \\times p _ { 0 } }$ where $x _ { i } \\in \\mathbb { R } ^ { p _ { 0 } }$ is the feature vector at node $i = 1 , \\ldots , n = | V |$ . We denote by $h _ { i } ^ { ( \\ell ) } \\in \\mathbb { R } ^ { p _ { \\ell } }$ the representation of node $i$ at layer $\\ell \\geq 0$ , with $h _ { i } ^ { ( 0 ) } = x _ { i }$ . Given a family of message functions $\\psi _ { \\ell } : \\mathbb { R } ^ { p _ { \\ell } } \\times \\mathbb { R } ^ { p _ { \\ell } } \\to \\mathbb { R } ^ { p _ { \\ell } ^ { \\prime } }$ and update functions $\\phi _ { \\ell } : \\mathbb { R } ^ { p _ { \\ell } } \\times \\mathbb { R } ^ { p _ { \\ell } ^ { \\prime } } \\to \\mathbb { R } ^ { p _ { \\ell + 1 } }$ , we can write the $( \\ell + 1 )$ -st layer output of a generic MPNN as follows (Gilmer et al., 2017): ", + "bbox": [ + 173, + 765, + 825, + 843 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/5acf9ad2264bec6c6abd19514311fa3932c1204ea0500ab6b7a1760d2f264ee1.jpg", + "text": "$$\nh _ { i } ^ { ( \\ell + 1 ) } = \\phi _ { \\ell } \\left( h _ { i } ^ { ( \\ell ) } , \\sum _ { j = 1 } ^ { n } \\hat { A } _ { i j } \\psi _ { \\ell } ( h _ { i } ^ { ( \\ell ) } , h _ { j } ^ { ( \\ell ) } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 844, + 647, + 895 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Here we have used the augmented normalized adjacency matrix to propagate messages from each node to its neighbors, which simply leads to a degree normalization of the message functions $\\psi _ { \\ell }$ . To ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "avoid heavy notations the node features and representations are assumed to be scalar from now on; \nthese assumptions simplify the discussion and the vector case leads to analogous results. ", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 THE OVER-SQUASHING PROBLEM ", + "text_level": 1, + "bbox": [ + 176, + 148, + 444, + 162 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Multiple recent papers observed that MPNNs tend to perform poorly in situations when the learned task requires long-range dependencies and at the same time the structure of the graph results in exponentially many long-range neighboring nodes. We say that a graph learning problem has long-range dependencies when the output of a MPNN depends on representations of distant nodes interacting with each other. If long-range dependencies are present, messages coming from nonadjacent nodes need to be propagated across the network without being too distorted. In many cases however (e.g. in ‘small-world’ graphs such as social networks), the size of the receptive field $B _ { r } ( i )$ grows exponentially with $r$ . If this occurs, representations of exponentially many neighboring nodes need to be compressed into fixed-size vectors to propagate messages to node $i$ , causing a phenomenon referred to as over-squashing of information (Alon & Yahav, 2021). In line with Alon & Yahav (2021), we refer to those structural properties of the graph that lead to over-squashing as a bottleneck1. ", + "bbox": [ + 174, + 174, + 825, + 328 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Sensitivity analysis. The hidden feature $h _ { i } ^ { ( \\ell ) } = h _ { i } ^ { ( \\ell ) } ( x _ { 1 } , \\dots , x _ { n } )$ computed by an MPNN with $\\ell$ layers as in equation 1 is a differentiable function of the input node features $\\{ x _ { 1 } , \\ldots , x _ { n } \\}$ as long as the update and message functions $\\phi _ { \\ell }$ and $\\psi _ { \\ell }$ are differentiable. The over-squashing of information can then be understood in terms of one node representation $h _ { i } ^ { ( \\ell ) }$ failing to be affected by some input feature $x _ { s }$ of node $s$ at distance $r$ from node $i$ . Hence, we propose the Jacobian $\\partial h _ { i } ^ { ( r ) } / \\partial x _ { s }$ as an explicit and formal way of assessing the over-squashing effect2. ", + "bbox": [ + 173, + 340, + 825, + 438 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lemma 1. Assume an MPNN as in equation $I$ . Let $i , s \\in V$ with $s \\in S _ { r + 1 } ( i )$ . If $| \\nabla \\phi _ { \\ell } | \\le \\alpha$ and $| \\nabla \\psi _ { \\ell } | \\le \\beta$ for $0 \\leq \\ell \\leq r$ , then ", + "bbox": [ + 178, + 438, + 821, + 465 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c7d0710d10f2d3fd55eeb92365c3f012e55435c1f716681a948bc9364112d343.jpg", + "text": "$$\n\\left| \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right| \\leq ( \\alpha \\beta ) ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 464, + 607, + 507 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lemma 1 states that if $\\phi _ { \\ell }$ and $\\psi _ { \\ell }$ have bounded derivatives, then the propagation of messages is controlled by a suitable power of $\\hat { A }$ . For example, if $d _ { G } ( i , s ) = r + 1$ and the sub-graph induced on $B _ { r + 1 } ( i )$ is a binary tree, then $( \\hat { A } ^ { r + 1 } ) _ { i s } = 2 ^ { - 1 } 3 ^ { - r }$ , which gives an exponential decay of the node dependence on input features at distance $r$ , as also heuristically argued by Alon & Yahav (2021). ", + "bbox": [ + 174, + 510, + 825, + 571 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The sensitivity analysis in Lemma 1 relates the over-squashing – as measured by the Jacobian of the node representations – to the graph topology via powers of the augmented normalized adjacency matrix. In the next section we explore this connection further by analyzing which local properties of the graph structure affect the right hand side in equation 2, hence causing the bottleneck. We will address this problem by introducing a new combinatorial notion of edge-based curvature and showing that negatively curved edges are those responsible for the over-squashing phenomenon. ", + "bbox": [ + 174, + 578, + 825, + 662 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 GRAPH CURVATURE AND BOTTLENECK ", + "text_level": 1, + "bbox": [ + 176, + 681, + 529, + 696 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A natural object in Riemannian geometry is the Ricci curvature, a bilinear form determining the geodesic dispersion, i.e. whether geodesics starting at nearby points with ‘same’ velocity remain parallel (Euclidean space), converge (spherical space), or diverge (hyperbolic space). To motivate the introduction of a Ricci curvature for graphs, we focus on these three cases. Consider two nodes $i \\sim j$ and two edges starting at $i$ and $j$ respectively. In a discrete spherical geometry (Figure 2a), the edges would meet at $k$ to form a triangle (complete graph). In a discrete Euclidean geometry (Figure 2b), the edges would stay parallel and form a 4-cycle based at $i \\sim j$ (orthogonal grid). Finally, in a discrete hyperbolic geometry (Figure 2c), the mutual distance of the edge endpoints would have grown compared to that of $i$ and $j$ (tree). Therefore, a Ricci curvature for graphs should provide us with more sophisticated tools than the degree to analyze the neighborhood of an edge. ", + "bbox": [ + 173, + 712, + 825, + 851 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Curvatures on graphs. The main examples of edge-based curvature are the Forman curvature $F ( i , j )$ (Forman, 2003) and the Ollivier curvature $\\kappa ( i , j )$ in Ollivier (2007; 2009) (see Appendix). While $F ( i , j )$ is given in terms of combinatorial quantities (Sreejith et al., 2016), results are scarce and the definition is biased towards negative curvature. The theory on $\\kappa ( i , j )$ instead is richer (Lin et al., 2011; Münch, 2019) but its formulation makes it hard to control local quantities. ", + "bbox": [ + 174, + 104, + 441, + 270 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/23259c93c37afcd782009cb348c4057564529808a5e2f4989beb67f4e8be2387.jpg", + "image_caption": [ + "Figure 2: Different regimes of curvatures on graphs analogous to spherical (a), planar (b), and hyperbolic (c) geometries in the continuous setting. " + ], + "image_footnote": [], + "bbox": [ + 457, + 104, + 818, + 213 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Balanced Forman curvature. We propose a new curvature to address the shortcomings of the existing candidates. We use the following definitions to describe the neighborhood of an edge $i \\sim j$ and we refer to the Appendix for a more complete discussion: ", + "bbox": [ + 174, + 285, + 825, + 328 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(i) $\\sharp _ { \\Delta } ( i , j ) : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j )$ are the triangles based at $i \\sim j$ . (ii) $\\sharp _ { \\bigtriangledown } ( i , j ) : = \\{ k \\in S _ { 1 } ( i ) \\ \\backslash \\ S _ { 1 } ( j ) , k \\neq j : \\ \\exists w \\in ( S _ { 1 } ( k ) \\cap S _ { 1 } ( j ) ) \\setminus S _ { 1 } ( i ) \\}$ are the neighbors of $i$ forming a 4-cycle based at the edge $i \\sim j$ without diagonals inside. (iii) $\\gamma _ { \\operatorname* { m a x } } ( i , j )$ is the maximal number of n node (see Definition 4). $4 -$ cycles based at $i \\sim j$ traversing ", + "bbox": [ + 200, + 338, + 825, + 422 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In line with the discussion about geodesic dispersion, one expects $\\sharp _ { \\Delta }$ to be related to positive curvature (complete graph), $\\sharp _ { \\bigstar } ^ { i }$ to zero curvature (grid), and the remaining outgoing edges to negative curvature (tree). Our new curvature formulation reflects such an intuition and recovers the expected results in the classical cases. In the example in Figure 3 we have $\\sharp _ { \\bigstar } ^ { 0 } ( \\bar { 0 } , 1 ) = \\{ 2 , 3 \\}$ while $\\sharp _ { \\bigstar } ^ { 1 } ( 0 , 1 ) = \\{ 5 \\}$ , both without 4,6 because of the triangle 1-6-0. The degeneracy factor $\\gamma _ { \\mathrm { m a x } } ( 0 , 1 ) = 2$ , as there exist two 4-cycles passing through node 5. ", + "bbox": [ + 173, + 429, + 692, + 530 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/a9d21074d19942603c3881fd9a01471a01eac7e7f8f372bacd0820ede9137be8.jpg", + "image_caption": [ + "Figure 3: 4-cycle contribution. " + ], + "image_footnote": [], + "bbox": [ + 712, + 402, + 816, + 494 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 1 (Balanced Forman curvature). For any edge $i \\sim j$ in a simple, unweighted graph $G$ , we let $\\operatorname { R i c } ( i , j )$ be zero $i f \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} = 1$ and otherwise ", + "bbox": [ + 174, + 540, + 825, + 568 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/70e1795908e7edf2901adf69d9dac73c9792d64aa8c51edd2742015e711d402a.jpg", + "text": "$$\n\\mathbf { R i c } ( i , j ) : = \\frac { 2 } { d _ { i } } + \\frac { 2 } { d _ { j } } - 2 + 2 \\frac { \\left| \\sharp _ { \\Delta } ( i , j ) \\right| } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\left| \\sharp _ { \\Delta } ( i , j ) \\right| } { \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\left( \\gamma _ { m a x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } ( | \\sharp _ { \\overline { { \\Gamma } } } ^ { i } | + | \\sharp _ { \\overline { { \\Gamma } } } ^ { j } | ) ,\n$$", + "text_format": "latex", + "bbox": [ + 194, + 571, + 781, + 606 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the last term is set to be zero $i f | \\sharp _ { \\bigstar } ^ { i } |$ (and hence $| \\sharp _ { \\bigstar } ^ { j } | \\big )$ is zero. In particular $\\operatorname { R i c } ( i , j ) > - 2$ ", + "bbox": [ + 173, + 609, + 803, + 627 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The curvature is negative when $i \\sim j$ behaves as a bridge between $S _ { 1 } ( i )$ and $S _ { 1 } ( j )$ , while it is positive when $S _ { 1 } ( i )$ and $S _ { 1 } ( j )$ stay connected after removing $i \\sim j$ . We refer to Ric as Balanced Forman curvature. We can relate the Balanced Forman curvature to the Jacobian of hidden features, while also extending many results valid for the Ollivier curvature $\\kappa ( i , j )$ thanks to our next theorem. ", + "bbox": [ + 173, + 640, + 826, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 2. Given an unweighted graph $G$ , for any edge $i \\sim j$ we have $\\kappa ( i , j ) \\geq \\operatorname { R i c } ( i , j )$ ", + "bbox": [ + 166, + 707, + 772, + 723 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 2 generalizes Jost & Liu (2014, Theorem 3) (see Appendix). We also note that the computational complexity for $\\kappa$ scales as $O ( | E | d _ { \\operatorname* { m a x } } ^ { 3 } )$ , while for our Ric we have $O ( | E | d _ { \\operatorname* { m a x } } ^ { 2 } )$ , with $d _ { \\mathrm { m a x } }$ the maximal degree. From Theorem 2 and Paeng (2012), we find: ", + "bbox": [ + 173, + 736, + 614, + 792 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Corollary 3. If $\\operatorname { R i c } ( i , j ) \\geq k > 0$ for any edge $i \\sim j$ , then there exists a polynomial $P$ such that ", + "bbox": [ + 173, + 803, + 612, + 832 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9fce9d515578118c2f889f2570948b44fcbba21a7e76bd4c4b6acd6e9d06f4d9.jpg", + "text": "$$\n| B _ { r } ( i ) | \\leq P ( r ) , \\forall i \\in V .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 837, + 482, + 854 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/8da8842d64206d3f3af8a03ff936001333c5ac6b8bea3abca30e94acf4c0360c.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Examples of the Balanced Forman curvature. " + ], + "table_body": "
Graph GRicG
C3 Cycles C4 Cn≥53
1
0
n
Complete Kn Grid Gnn-1 0
Tree Tr4 r+1-2
", + "bbox": [ + 625, + 738, + 815, + 862 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Therefore, when the curvature is positive everywhere, the bottleneck effect should not play a crucial role as the receptive field of each node will be polynomial in the hop-distance. This limit case shows how the curvature determines whether a learning task on a graph would suffer from over-squashing. ", + "bbox": [ + 173, + 867, + 614, + 910 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 910, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Curvature and over-squashing. Thanks to our new combinatorial curvature and the sensitivity analysis in Lemma 1, we are able to relate local curvature properties to the Jacobian of the node representations. This leads to one of the main results of this paper: negatively curved edges are those causing the graph bottleneck and thus leading to the over-squashing phenomenon: ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 4. Consider a MPNN as in equation 1. Let $i \\sim j$ with $d _ { i } \\leq d _ { j }$ and assume that: ", + "bbox": [ + 171, + 170, + 764, + 185 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(i) $| \\nabla \\phi _ { \\ell } | \\le \\alpha$ and $| \\nabla \\psi _ { \\ell } | \\le \\beta$ for each $0 \\leq \\ell \\leq L - 1$ , with $L \\geq 2$ the depth of the MPNN. ", + "bbox": [ + 200, + 194, + 805, + 212 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(ii) There exists $\\delta$ s.t. $0 < \\delta < ( \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} ) ^ { - \\frac { 1 } { 2 } }$ , $\\delta < \\gamma _ { m a x } ^ { - 1 } ,$ , and $\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta$ . ", + "bbox": [ + 204, + 219, + 761, + 239 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Then there exists $Q _ { j } \\subset S _ { 2 } ( i )$ satisfying $| Q _ { j } | > \\delta ^ { - 1 }$ and for $0 \\leq \\ell _ { 0 } \\leq L - 2$ we have ", + "bbox": [ + 173, + 247, + 730, + 265 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8058842145c59033fb6edaead13943e3b3d1288dde59ccf9db91c03ca0c5272e.jpg", + "text": "$$\n\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } \\left| \\frac { \\partial h _ { k } ^ { ( \\ell _ { 0 } + 2 ) } } { \\partial h _ { i } ^ { ( \\ell _ { 0 } ) } } \\right| < ( \\alpha \\beta ) ^ { 2 } \\delta ^ { \\frac { 1 } { 4 } } .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 266, + 614, + 310 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Condition (i) is always satisfied and allows us to control the message passing functions. The requirement (ii) instead means that the curvature of $( i , j )$ is negative enough when compared to the degrees of $i$ and $j$ (recall that $\\operatorname { R i c } ( i , j ) > - 2 )$ . The further condition on $\\gamma _ { \\mathrm { m a x } }$ is to avoid pathological cases where we have a large number of degenerate 4-cycles passing through the same three nodes. ", + "bbox": [ + 173, + 319, + 825, + 376 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To understand the conclusions in Theorem 4, let us fix $\\ell _ { 0 } = 0$ . The equation 4 shows that negatively curved edges are the ones causing bottlenecks, interpreted as how the graph topology prevents a representation h(2)k t o be affected by non-adjacent features $h _ { i } ^ { ( 0 ) } = x _ { i }$ . Theorem 4 implies that if we have a negatively curved edge as in (ii), then there exist a large number of nodes $k$ such that GNNs - on average - struggle to propagate messages from $i$ to $k$ in two layers despite these nodes $k$ being at distance 2 from i. In this case the over-squashing occurs as measured by the Jacobian in equation 4 and hence the propagation of information suffers. If the task at hand has long-range dependencies, then the over-squashing caused by the negatively curved edges may compromise the performance. ", + "bbox": [ + 173, + 381, + 825, + 497 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Bottleneck via Cheeger constant. We now relate the previous discussion about bottlenecks and curvature to spectral properties of the graph. In particular, since the spectral gap of a graph can be interpreted as a topological obstruction to the graph being partitioned into two communities, we argue below that this quantity is related to the graph bottleneck and should hence be controllable by the curvature. We start with an intuitive explanation: suppose we are given a graph $G$ with two communities separated by few edges. In this case, we see that the graph can be easily disconnected. This property is encoded in the classical notion of the Cheeger constant (Chung & Graham, 1997) ", + "bbox": [ + 173, + 511, + 825, + 609 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/46c4e336d84b9a71dfe2b039ff540b6da2a1dfbec9839bbdf1b33e0b1a2d846c.jpg", + "text": "$$\nh _ { G } : = \\operatorname* { m i n } _ { S \\subset V } h _ { S } , \\quad h _ { S } : = \\operatorname* { m i n } _ { S \\subset V } \\frac { | \\partial S | } { \\operatorname* { m i n } \\{ \\mathrm { v o l } ( S ) , \\mathrm { v o l } ( V \\setminus S ) \\} }\n$$", + "text_format": "latex", + "bbox": [ + 308, + 613, + 686, + 648 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\partial S = \\{ ( i , j ) : i \\in S , \\ j \\in V \\setminus S \\}$ and $\\textstyle { \\mathrm { v o l } } ( S ) = \\sum _ { i \\in S } d _ { i }$ . The main result about the Cheeger constant is the Cheeger inequality (Cheeger, 2015; Chung $\\bar { \\& }$ Graham, 1997): ", + "bbox": [ + 173, + 652, + 823, + 681 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/67c50aee7d647525c314748a9fd544c98f53ea9cd19f4bb2d88b2580005ad02b.jpg", + "text": "$$\n2 h _ { G } \\geq \\lambda _ { 1 } \\geq \\frac { h _ { G } ^ { 2 } } { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 441, + 683, + 558, + 714 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\lambda _ { 1 }$ is the first non-zero eigenvalue of the normalized graph Laplacian, often referred to as the spectral gap. A graph with two tightly connected communities $S$ and $V \\setminus S : = \\bar { S } )$ and few inter-community edges has a small Cheeger constant $h _ { G }$ . For nodes in different communities to interact with each other, all messages need to go through the same few bridges hence leading to the over-squashing of information (a similar intuition was explored in Alon & Yahav (2021)). Therefore, $h _ { G }$ can be interpreted as a rough measure of graph ‘bottleneckedness’, in the sense that the smaller its value, the more likely the over-squashing is to occur across inter-community edges. Since Theorem 4 implies that negatively curved edges induce the bottleneck, we expect a relationship between $h _ { G }$ and the curvature of the graph. The next proposition follows from Theorem 2 and Lin et al. (2011): ", + "bbox": [ + 173, + 714, + 825, + 840 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 5. If $\\operatorname { R i c } ( i , j ) \\geq k > 0$ for all $i \\sim j$ , then $\\textstyle \\lambda _ { 1 } / 2 \\geq h _ { G } \\geq { \\frac { k } { 2 } }$ . ", + "bbox": [ + 176, + 844, + 648, + 862 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Therefore, a positive lower bound on the curvature gives us a control on $h _ { G }$ and hence on the spectral gap of the graph. In the next section, we show that diffusion-based graph-rewiring methods might fail to significantly alter $h _ { G }$ and hence correct the graph bottleneck potentially induced by inter-community edges. This will lead us to propose an alternative curvature-based graph rewiring. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 CURVATURE-BASED REWIRING METHODS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 549, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The traditional paradigm of message passing graph neural networks assumes that messages are propagated on the input graph (Gilmer et al., 2017). More recently, there is a trend to decouple the input graph from the graph used for information propagation. This can take the form of graph subsampling or resampling to deal with scalability (Hamilton et al., 2017) or topological noise (Zhang et al., 2019), using larger motif-based (Monti et al., 2018) or multi-hop filters (Rossi et al., 2020), or changing the graph either as a preprocessing step (Klicpera et al., 2019; Alon & Yahav, 2021) or adaptively for the downstream task (Wang et al., 2019; Kazi et al., 2020). Such methods are often generically referred to as graph rewiring. ", + "bbox": [ + 174, + 132, + 825, + 244 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the context of this paper, we assume that graph rewiring attempts to produce a new graph $G ^ { \\prime } = ( V , E ^ { \\prime } )$ with a different edge structure that reduces the bottleneck and hence potentially alleviates the over-squashing of information. We propose a method that leverages the graph curvature to guide the rewiring steps in a surgical way by modifying the negatively-curved edges, so to decrease the bottleneck without significantly compromising the statistical properties of the input graph. We also rigorously show that a random-walk based rewiring method might generally fail to obtain an edge set $E ^ { \\prime }$ with a significant improvement in its bottleneckedness as measured by the Cheeger constant. ", + "bbox": [ + 174, + 251, + 825, + 348 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Curvature-based graph rewiring. Since according to Theorem 4 negatively curved edges induce a bottleneck and are hence responsible for over-squashing, a curvature-based rewiring method should attempt to alleviate a graph’s strongly-negatively curved edges. To this end we implement a simple rewiring method called Stochastic Discrete Ricci Flow (SDRF), described in Algorithm 1. ", + "bbox": [ + 174, + 363, + 825, + 419 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1: Stochastic Discrete Ricci Flow (SDRF) ", + "text_level": 1, + "bbox": [ + 173, + 436, + 526, + 452 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Input: graph $G$ , temperature $\\tau > 0$ , max number of iterations, optional Ric upper-bound $C ^ { + }$ Repeat ", + "bbox": [ + 176, + 457, + 782, + 484 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1) For edge $i \\sim j$ with minimal Ricci curvature $\\operatorname { R i c } ( i , j )$ ", + "bbox": [ + 202, + 486, + 578, + 500 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Calculate vector $_ { \\textbf { \\em x } }$ where $x _ { k l } = \\mathrm { R i c } _ { k l } ( i , j ) - \\mathrm { R i c } ( i , j )$ , the improvement to $\\operatorname { R i c } ( i , j )$ from adding edge $k \\sim l$ where $k \\in B _ { 1 } ( i )$ , $l \\in B _ { 1 } ( j )$ ; Sample index $k , l$ with probability softmax $( \\tau { \\pmb x } ) _ { k l }$ and add edge $k \\sim l$ to $G$ . ", + "bbox": [ + 210, + 501, + 764, + 549 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2) Remove edge $i \\sim j$ with maximal Ricci curvature $\\operatorname { R i c } ( i , j )$ if $\\operatorname { R i c } ( i , j ) > C ^ { + }$ ", + "bbox": [ + 187, + 551, + 725, + 566 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Until convergence, or max iterations reached; ", + "bbox": [ + 176, + 568, + 473, + 580 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "At each iteration this preprocessing step adds an edge to ‘support’ the graph’s most negatively curved edge, and then removes the most positively curved edge. The requirement on the added edge $k \\sim l$ that $k \\in B _ { 1 } ( i )$ and $l \\in B _ { 1 } ( j )$ ensures that we’re adding either an extra $3 -$ or 4-cycle around the negative edge $i \\sim j$ so that this is a local modification. The graph edit distance between the original and preprocessed graph is bounded above by $2 ~ \\times$ the max number of iterations. The temperature $\\tau$ determines how stochastic the edge addition is, with $\\tau = \\infty$ being fully deterministic (the best edge is always added). At each step we remove the edge with most positive curvature to balance the distributions of curvature and node degrees. We use Balanced Forman curvature as in equation 3 for $\\operatorname { R i c } ( i , j )$ . $C ^ { + }$ can be chosen to stop the method skewing the curvature distribution negative, including $C ^ { + } = \\infty$ to not remove any edges. The method is inspired by the continuous (backwards) Ricci flow with the aim of homogenizing edge curvatures. This is different from more direct extensions of Ricci flow on graphs where it becomes increasingly expensive to propagate messages across negatively curved edges (as in other applications such as Ni et al. (2019)). An example alongside its continuous analogue can be seen in Figure 1. ", + "bbox": [ + 174, + 603, + 825, + 797 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Can random-walk based rewiring address bottlenecks? A good way of understanding the effectiveness of SDRF in reducing the graph bottleneck is through comparison with random-walk based rewiring strategies. Recall that, as argued in Section 3, the Cheeger constant $h _ { G }$ of a graph constitutes a rough measure of its bottleneckedness as induced by the inter-community edges (a small $h _ { G }$ is indicative of a bottleneck). Suppose we are given a graph $G$ with a small $h _ { G }$ and wish to rewire it into a graph $G ^ { \\prime }$ with a significantly improved Cheeger constant in order to reduce the inter-community bottleneck. A random-walk based rewiring method such as DIGL (Klicpera et al., 2019) acts by smoothing out the graph adjacency and hence tends to promote connections among nodes at short diffusion distance (Coifman & Lafon, 2006). Accordingly, such a rewiring method might fail to correct structural features like the bottleneck, which is instead more prominent for nodes that are at long diffusion distance.3 To emphasize this point, we consider a classic example: given $\\alpha \\in ( 0 , 1 )$ , the Personalized Page Rank (PPR) matrix is defined by (Brin & Page, 1998) as ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/5416ff0352bb14371d74eb7ca20808279a57d3790066ef4dabc968e981bfa2ed.jpg", + "text": "$$\nR _ { \\alpha } : = \\sum _ { k = 0 } ^ { \\infty } \\theta _ { k } ^ { P P R } ( D ^ { - 1 } A ) ^ { k } = \\alpha \\sum _ { k = 0 } ^ { \\infty } \\left( ( 1 - \\alpha ) ( D ^ { - 1 } A ) \\right) ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 161, + 689, + 203 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Assume that we rewire the graph using $R _ { \\alpha }$ as in Klicpera et al. (2019) with the PPR kernel, meaning that we replace the given adjacency $A$ with $R _ { \\alpha }$ . Since $R _ { \\alpha }$ is stochastic, the new Cheeger constant of the rewired graph can be computed as ", + "bbox": [ + 174, + 204, + 821, + 246 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/d98d5f7d33063c1a63e99d2d7a053d9eb730bc5e31cc35bf16867fb4ab3d8558.jpg", + "text": "$$\nh _ { S , \\alpha } = \\frac { | \\partial S | _ { \\alpha } } { \\mathrm { v o l } _ { \\alpha } ( S ) } \\equiv \\frac { 1 } { | S | } \\sum _ { i \\in S } \\sum _ { j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 248, + 629, + 290 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "By applying (Chung, 2007, Lemma 5), we show that we cannot improve the Cheeger constant (and hence the bottleneck) arbitrarily well (in contrast to a curvature-based approach). We refer to Proposition 17 and Remark 18 in Appendix $\\mathrm { E }$ for results that are more tailored to the actual strategy adopted in Klicpera et al. (2019) where we also take into account the effect of the sparsification. ", + "bbox": [ + 174, + 290, + 825, + 347 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 6. Let $S \\subset V$ with $\\nu o l ( S ) \\le \\nu o l ( G ) / 2$ . Then $\\begin{array} { r } { h _ { S , \\alpha } \\leq \\left( \\frac { 1 - \\alpha } { \\alpha } \\right) \\frac { d _ { \\arg } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } h _ { S } } \\end{array}$ , where $d _ { \\mathrm { a v g } } ( S )$ and $d _ { \\mathrm { m i n } } ( S )$ are the average and minimum degree on $S$ , respectively. ", + "bbox": [ + 173, + 352, + 815, + 386 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The property that the new Cheeger constant is directly controlled by the old one stems from the fact that a random-walk approach like in Klicpera et al. (2019) is meant to act more relevantly on intra-community edges rather than inter-community edges because it prioritizes short diffusion distance nodes. This is also why this method performs well on high-homophily datasets, as discussed below. In particular, for a fixed $\\alpha \\in ( 0 , 1 )$ , the bound in Theorem 6 can be very small. As a specific example, consider two complete graphs $K _ { n }$ joined by one bridge. Then $h _ { G } \\overset { \\cdot } { = } ( n ( n - 1 ) \\overset { \\cdot } { + } 1 ) ^ { - 1 }$ , which means that the bound on the right hand side is $\\dot { O } ( n ^ { - 2 } )$ . ", + "bbox": [ + 173, + 396, + 825, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 6 implies that a diffusion approach such as DIGL might fail to yield a new edge set $E ^ { \\prime }$ with a sufficiently improved bottleneck. By contrast, from Theorem 4 and Proposition 5, we deduce that a curvature-based rewiring method such as SDRF properly addresses the edges that cause the bottleneck. ", + "bbox": [ + 173, + 501, + 825, + 558 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Graph structure preservation. Although a graph-rewiring approach aims at providing a new edge set $E ^ { \\prime }$ potentially more beneficial for the given learning task, it is still desirable to control how far $E ^ { \\prime }$ is from $E$ . In this regard, we note that a curvature-based rewiring is surgical in nature and hence more likely to preserve the structure of the input graph better than a random-walk based approach. Consider, for example, that we are given $\\rho > 0$ and wish to rewire the graph such that the new edge set $E ^ { \\prime }$ is within graph-edit distance $\\rho$ from the original $E$ . Theorem 4 tells us how to do the rewiring under such constraints in order to best address the over-squashing: the topological modifications need to be localized around the most negatively-curved edges. We can do this with SDRF, with the maximum number of iterations set to $\\rho / 2$ . ", + "bbox": [ + 173, + 571, + 825, + 699 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Secondly, we also point out that $\\operatorname { R i c } ( i , j ) < - 2 + \\delta$ implies that $\\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} > 2 / \\delta$ . Therefore, if we mostly modify the edge set at those nodes $i , j$ joined by an edge with large negative curvature, then we are perturbing nodes with high degree where such a change is relatively insignificant, and thus overall statistical properties of the rewired graph such as degree distribution are likely to be better preserved. Moreover, graph convolutional networks tend to be more stable to perturbations of high degree nodes (Zügner et al., 2020; Kenlay et al., 2021), making curvature-based rewiring more suitable for the downstream learning tasks with popular GNN architectures. ", + "bbox": [ + 173, + 704, + 825, + 803 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Homophily and bottleneck. As a final remark, note that the graph rewiring techniques considered in this paper (both DIGL and SDRF) are based purely on the topological structure of the graph and completely agnostic to the node features and to whether the dataset is homophilic (adjacent nodes have same labels) or heterophilic. Nonetheless, the different nature of these rewiring methods allows us to draw a few broad conclusions about their suitability in each of these settings. A random-walk ", + "bbox": [ + 174, + 818, + 825, + 887 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e34d427c4c0da221b6eb4532e7be80c0a5ee00795b09a2b562f073285c5fd73f.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
H(G)Cornell 0.11Texas 0.06Wisconsin 0.16Chameleon 0.25Squirrel 0.22Actor 0.24Cora 0.83Citeseer 0.71Pubmed 0.79
None52.69 ± 0.2161.19 ± 0.4954.60±0.8641.80 ± 0.4139.83±0.1428.70±0.0981.89 ±0.7972.31 ±0.1778.16 ±0.23
Undirected53.20 ±0.5363.38 ±0.8751.37 ± 1.1542.63±0.3040.77 ± 0.1628.10 ±0.11=
+FA58.29±0.4964.82±0.2955.48±0.6242.33± 0.1740.74 ± 0.1328.68±0.1681.65±0.1870.47±0.18 79.48±0.12
DIGL (PPR)58.26±0.5062.03 ±0.4349.53±0.2742.02 ±0.1334.38 ±0.1130.79±0.10 83.21±0.27 73.29±0.17 78.84±0.08
DIGL + Undirected 59.54 ±0.6463.54±0.3852.23±0.5442.68± 0.1233.36±0.2129.71 ± 0.11=
SDRF54.60 ±0.3964.46±0.3855.51±0.27 43.75±0.31 40.97±0.14 29.70 ±0.13 82.76±0.23 72.58±0.20 79.10 ±0.11
SDRF+ Undirected 57.54±0.34 70.35±0.60 61.55±0.86 44.46±0.17 41.47±0.21 29.85±0.07
", + "bbox": [ + 112, + 93, + 885, + 204 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2: Experimental results on common node classification benchmarks. Top two in bold. ", + "bbox": [ + 196, + 215, + 799, + 231 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "approach such as DIGL tends to improve the connectivity among nodes that are at short diffusion distance; since for a high-homophily dataset these nodes often share the same label, a rewiring method like DIGL is likely to act as graph denoising and yield improved performance. On the other hand, for datasets with low homophily, nodes at short diffusion distance are more likely to belong to different label classes, meaning that a diffusion-based rewiring might inject noise and hence compromise performance as also noted in Klicpera et al. (2019). Conversely, on a low-homophily dataset, a curvature-based approach as SDRF modifies the edge set mainly around the most negatively curved edges, meaning that it decreases the bottleneck without significantly increasing the connectivity among nodes with different labels. In fact, long-range dependencies are often more relevant in low-homophily settings, where nodes sharing the same labels are in general not neighbors. This observation is largely confirmed by experimental results reported in the next section. ", + "bbox": [ + 174, + 250, + 825, + 402 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 178, + 422, + 418, + 438 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experiment setup. To demonstrate the theoretical results in this paper we ran a suite of semisupervised node classification tasks comparing our curvature-based rewiring method SDRF to DIGL from Klicpera et al. (2019) (GDC with the PPR kernel) and the $+ \\mathrm { F A }$ method from Alon & Yahav (2021), where the last layer of the GNN is made fully connected. We evaluate the methods on nine datasets: Cornell, Texas and Wisconsin from the WebKB dataset4; Chameleon and Squirrel (Rozemberczki et al., 2021) along with Actor (Tang et al., 2009); and Cora (McCallum et al., 2000), Citeseer (Sen et al., 2008) and Pubmed (Namata et al., 2012). Statistics for these datasets can be found in Appendix F.1. Our base model is a GCN (Kipf & Welling, 2017). Following Shchur et al. (2018) and Klicpera et al. (2019) we optimized hyperparameters for all dataset-preprocessing combinations separately by random search over 100 data splits. Results are reported as average accuracies on a test set used once with $9 5 \\%$ confidence intervals calculated by bootstrapping. We compared the performance on graphs with no preprocessing, making the graph undirected, $+ \\mathrm { F A }$ , DIGL, SDRF, and the given combinations. For DIGL $^ +$ Undirected we symmetrized the diffusion matrix as in Klicpera et al. (2019), and for SDRF $^ +$ Undirected we made the graph undirected before applying SDRF. For more details on the experiments and datasets see Appendix F, and for the hyperparameters used for each model and preprocessing see Appendix F.4. ", + "bbox": [ + 173, + 449, + 825, + 670 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Node classification results. Table 2 shows the results of the experiments. As well as reporting results we give a measure of homophily $\\mathcal { H } ( G )$ proposed by Pei et al. (2019) (restated in Appendix F, equation 21), by which we can see our experiment set is diverse with respect to homophily. We see that SDRF improves upon the baseline in all cases, and that the largest improvements are seen on the low-homophily datasets. We also see that SDRF matches or outperforms DIGL and $+ \\mathrm { F A }$ on most datasets, supporting our argument that curvature-based rewiring is a viable candidate for improving GNN performance. ", + "bbox": [ + 174, + 689, + 825, + 786 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Graph topology change. Furthermore, SDRF preserves the graph topology to a far greater extent than DIGL due to its surgical nature. Table 3 shows the number of edges added / removed by the two preprocessings on each dataset as a percentage of the original number of edges. We see that for optimal performance DIGL makes the graph much denser, which significantly affects the node degrees and may negatively impact the time and space complexity of the downstream GNN, which typically are $O ( E )$ . In comparison, SDRF adds and removes a similar number of edges and approximately preserves the degree distribution. The effect on the full degree distribution for three ", + "bbox": [ + 174, + 804, + 825, + 902 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/198001fa034307ccce83949e4016829a9a3ad6cc0c2500a49f6ecd4d0572ee0c.jpg", + "image_caption": [ + "Figure 4: Comparing the degree distribution of the original graphs to the preprocessed version. The xaxis is node degree in $\\log _ { 2 }$ scale, and the plots are a kernel density estimate of the degree distribution. In the captions we see the Wasserstein distance $W _ { 1 }$ between the original and preprocessed graphs. " + ], + "image_footnote": [], + "bbox": [ + 178, + 75, + 815, + 247 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "datasets is shown in Figure 4. We see that the degree distribution following SDRF is close to, or in some cases indistinguishable from, the original distribution, whereas DIGL has a more noticeable effect. We also compute the Wasserstein distance between the degree distributions, denoted by $W _ { 1 }$ in the figure captions, to numerically confirm this observation. For this analysis extended to all nine datasets see Appendix F.2. ", + "bbox": [ + 174, + 308, + 485, + 433 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This difference is also visually evident from Figure 6 in Appendix F.3, where we again observe that SDRF largely preserves the topology of the Cornell graph and that the curvature (encoded by the edge color) is homogenized across the graph. We also see that the entries of the trained ", + "bbox": [ + 174, + 439, + 485, + 522 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/c8e72e4984ed2ac0211840aa3ca01af3a95dddb958aeac6ac1be5141d14342f9.jpg", + "table_caption": [], + "table_footnote": [ + "Table 3: $\\%$ edges added / removed by method. " + ], + "table_body": "
DIGLSDRF
Cornell351.1% /0.0%7.8% /33.3%
Texas483.3% /0.0%2.4% /10.4%
Wisconsin300.6%/0.0%1.4% / 7.5%
Chameleon336.1%/11.8%6.4% /6.4%
Squirrel228.8% /1.9%0.7%/0.7%
Actor2444.0% /2.3%5.4%/9.9%
Cora3038.0% /0.5%1.0% /1.0%
Citeseer2568.3%/ 0.0%1.1% / 1.1%
Pubmed2747.1% /0.1%0.2%/0.2%
", + "bbox": [ + 496, + 309, + 807, + 488 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "network’s Jacobian between a node’s prediction and the features of its 2-hop neighbors (encoded as node colors) are increased over the graph, which we may attribute to both DIGL and SDRF’s ability to alleviate the upper bound presented in Theorem 4 and thus reduce over-squashing. ", + "bbox": [ + 174, + 522, + 825, + 564 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 577, + 318, + 593 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we studied the graph bottleneck and the over-squashing phenomena limiting the performance of message passing graph neural networks from a geometric perspective. We started with a Jacobian approach to determine how the over-squashing phenomenon is dictated by the graph topology as in equation 2. We then investigated further how the topology induces the bottleneck and hence causes over-squashing. We introduced a new notion of edge-based Ricci curvature called Balanced Forman curvature, relating it to the classical Ollivier curvature (Theorem 2). We then proved in Theorem 4 that negatively-curved edges are responsible for over-squashing, calling for a possibility of curvature-based rewiring of the graph in order to improve its bottleneckedness. We show one such possibility (Algorithm 1), inspired by the classical Ricci flow and comment on the advantages of surgical method such as this. We show both theoretically and experimentally that the proposed method can be advantageous compared to a diffusion-based rewiring approach, opening the door for curvature-based rewiring methods for improving GNN performance going forward. ", + "bbox": [ + 173, + 602, + 825, + 768 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Limitations and future directions. Our paper establishes a geometric perspective on the graph bottleneck and over-squashing, providing new tools to study and cope with these phenomena. One limitation of our work is that the theoretical results presented here do not currently extend to multigraphs. In addition, the current methodology is agnostic to information beyond the graph topology, such as node features. In future works, we will develop a notion of the curvature and the corresponding rewiring method that can take into account such information. ", + "bbox": [ + 174, + 776, + 825, + 859 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Acknowledgements. This research was supported in part by the EPSRC CDT in Modern Statistics and Statistical Machine Learning (EP/S023151/1) and the ERC Consolidator Grant No. 724228 (LEMAN). X.D. gratefully acknowledges support from the Oxford-Man Institute of Quantitative Finance and the EPSRC (EP/T023333/1). 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", + "bbox": [ + 171, + 191, + 823, + 222 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Daniel Zügner, Oliver Borchert, Amir Akbarnejad, and Stephan Günnemann. Adversarial attacks on graph neural networks: Perturbations and their patterns. ACM Transactions on Knowledge Discovery from Data (TKDD), 14(5):1–31, 2020. ", + "bbox": [ + 176, + 231, + 823, + 272 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The Appendix is structured as follows: ", + "bbox": [ + 176, + 133, + 428, + 150 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(i) In Appendix A we prove Lemma 1 and a side-result about the role of self-loops. We also summarize how to extend the analysis in Lemma 1 to message passing models with sum aggregations (not average), meaning those architectures where we do not normalize the adjacency matrix. (ii) In Appendix B we introduce and describe different quantities we use to characterize the neighbourhood of a given edge $i \\sim j$ . These objects are all essential to studying the new notion of balanced Forman curvature. The focus is on how we can distinguish 4-cycles in a computationally tractable way without losing too much accuracy. (iii) In Appendix C we provide a brief literature review about existing curvature candidates. In particular, we report the definitions of (modified) Ollivier curvature and Forman curvature. (iv) In Appendix D we prove the statements in Section 3, i.e. Theorem 2, Corollary 3, Theorem 4 and Proposition 5. We also comment on the role of the assumptions and compare the bound in Theorem 2 with the existing literature. Finally, we relate the classical notion of betweenness centrality to the over-squashing effect and the negatively curved edges in a graph. (v) In Appendix E we prove the results in Section 4, namely Theorem 6 and an analogous result. (vi) In Appendix F we describe more fully the experiments from Section 5, including a full analysis on degree distribution (Appendix F.2) and the hyperparameters used in our experiments (Appendix F.4). (vii) In Appendix G we comment on hardware specifications. ", + "bbox": [ + 194, + 161, + 826, + 512 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A PROOFS OF RESULTS IN SECTION 2 ", + "text_level": 1, + "bbox": [ + 173, + 531, + 501, + 549 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma 1. Assume an MPNN as in equation $I$ . Let $i , s \\in V$ with $s \\in S _ { r + 1 } ( i )$ . If $| \\nabla \\phi _ { \\ell } | \\le \\alpha$ and $| \\nabla \\psi _ { \\ell } | \\le \\beta$ for $0 \\leq \\ell \\leq r$ , then ", + "bbox": [ + 176, + 564, + 825, + 593 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9f16d503d9eb9ff240ab5dee36df86033c8125a67a79da08b53151cfe3cd8c6e.jpg", + "text": "$$\n\\left| \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right| \\leq ( \\alpha \\beta ) ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 597, + 607, + 640 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. Let $i \\in V$ and $s \\in S _ { r + 1 } ( i )$ . We recall that to ease the notations we assume that node features and hidden representations are scalar. The proof in the more general higher-dimensional case follows without any modification. We compute ", + "bbox": [ + 174, + 654, + 825, + 696 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d1efbc8ffba481a5cd77a42d69a795c9439b8623f7402b0a57366f058b8ee9c2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\partial _ { 1 } \\phi _ { r } ( . . . ) \\partial _ { x _ { s } } h _ { i } ^ { ( r ) } } \\\\ & { \\quad \\quad \\quad + \\partial _ { 2 } \\phi _ { r } ( . . . ) \\displaystyle \\sum _ { j _ { r } = 1 } ^ { n } \\hat { a } _ { i j _ { r } } \\left( \\partial _ { 1 } \\psi _ { r } ( h _ { i } ^ { ( r ) } , h _ { j _ { r } } ^ { ( r ) } ) \\partial _ { x _ { s } } h _ { i } ^ { ( r ) } + \\partial _ { 2 } \\psi _ { r } ( h _ { i } ^ { ( r ) } , h _ { j _ { r } } ^ { ( r ) } ) \\partial _ { x _ { s } } h _ { j _ { r } } ^ { ( r ) } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 703, + 784, + 785 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We can iterate the computation above and see that the right hand side can be expanded as ", + "bbox": [ + 169, + 790, + 759, + 806 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/6680bb16a1c23d979b5d64c22939024c8ad89ca501c3eaf1599b0ce92a942541.jpg", + "text": "$$\n\\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { r } , \\dots , j _ { 0 } } \\sum _ { k _ { r } \\in \\{ i , j _ { r } \\} } \\dots \\sum _ { k _ { 1 } \\in \\{ i , j _ { r } , \\dots , j _ { 1 } \\} } \\hat { a } _ { i j _ { r } } \\hat { a } _ { k _ { r } j _ { r - 1 } } \\dots \\hat { a } _ { k _ { 1 } j _ { 0 } } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \\dots k _ { 1 } j _ { 0 } } ( \\boldsymbol { X } ) \\partial _ { x _ { s } } h _ { j _ { 0 } } ^ { ( 0 ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 189, + 813, + 808, + 859 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "for some functions $Z _ { i j _ { r } . . . k _ { 1 } j _ { 0 } }$ of the input features obtained as products of $r + 1$ partial derivatives of the maps $\\phi _ { \\ell }$ and $r + 1$ partial derivatives of the maps $\\psi _ { \\ell }$ . Since $H ^ { ( 0 ) } = X$ , we have ", + "bbox": [ + 171, + 866, + 825, + 897 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/be0f8474114d82184322e8f351a91924057ab1c64cf3182fdfe9af048f24223e.jpg", + "text": "$$\n\\partial _ { x _ { s } } h _ { j _ { 0 } } ^ { ( 0 ) } = \\delta _ { j _ { 0 } s }\n$$", + "text_format": "latex", + "bbox": [ + 449, + 905, + 549, + 928 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "meaning that the previous sum becomes ", + "bbox": [ + 174, + 103, + 436, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/edd5af0b1014fb301d3dba455ae6e0c7abff1ae973ba50579f6754264aa7cc1b.jpg", + "text": "$$\n\\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { r } , \\dots , j _ { 1 } } \\sum _ { k _ { r } \\in \\{ i , j _ { r } \\} } \\dots \\sum _ { k _ { 1 } \\in \\{ i , j _ { r } , \\dots , j _ { 1 } \\} } \\hat { a } _ { i j _ { r } } \\hat { a } _ { k _ { r } j _ { r - 1 } } \\dots \\hat { a } _ { k _ { 1 } s } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \\dots k _ { 1 } s } ( X ) .\n$$", + "text_format": "latex", + "bbox": [ + 218, + 126, + 779, + 171 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since $d _ { G } ( i , s ) = r + 1$ , the only non-vanishing terms in the sum above are the minimal walks from $i$ to $s$ . In fact, if there existed a different choice of coefficients yielding a non-zero term then we would find a walk joining $i$ to $s$ of length lesser than $r + 1$ , which is in contradiction with the definition of geodesic distance. Since $Z _ { i \\dots s } ( X )$ is a product of $r + 1$ -partial derivatives of the aggregation and update maps and by assumption their gradients are bounded by $\\alpha$ and $\\beta$ respectively, we conclude that ", + "bbox": [ + 173, + 179, + 826, + 262 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/96d2140cb525e2986e2b2b82f44a4416d5cb9cf2d70a9cf9142dd526c30ff11f.jpg", + "text": "$$\n\\left. \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right. \\leq ( \\alpha \\beta ) ^ { r + 1 } \\sum _ { j _ { r } , \\ldots , j _ { 1 } } \\hat { a } _ { i j _ { r } } \\hat { a } _ { j _ { r } j _ { r - 1 } } \\ldots \\hat { a } _ { j _ { 1 } s } = \\alpha ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 260, + 718, + 305 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "which completes the proof of the Lemma. ", + "bbox": [ + 174, + 309, + 447, + 324 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As a byproduct of this analysis, we can also provide a rigorous motivation for the role of self-loops in GNNs (see Appendix for details): ", + "bbox": [ + 171, + 343, + 825, + 372 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Corollary 7. If $\\begin{array} { r } { h _ { i } ^ { ( \\ell + 1 ) } = \\sum _ { j \\sim i } \\psi _ { \\ell } ( h _ { j } ^ { ( \\ell ) } ) } \\end{array}$ , then $h _ { i } ^ { ( \\ell + 1 ) }$ only depends on nodes that can be reached via walks of length exactly $\\ell + 1$ . By adding self-loops, the GNN also takes into account nodes that can be reached via walks of length $r \\leq \\ell + 1$ . ", + "bbox": [ + 173, + 376, + 825, + 425 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. If $\\begin{array} { r } { h _ { i } ^ { ( \\ell + 1 ) } = \\sum _ { j \\sim i } \\psi _ { \\ell } ( h _ { j } ^ { ( \\ell ) } ) } \\end{array}$ for each $\\ell \\in [ 0 , L - 1 ]$ , then we can argue as in the proof of ", + "bbox": [ + 174, + 444, + 823, + 477 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6d78f01d367a8a9fd82ee97cff12d8ded985b056b5b543d52e80bf12216ec8f7.jpg", + "text": "$$\n\\frac { \\partial h _ { i } ^ { ( \\ell + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { \\ell } , \\ldots , j _ { 1 } } a _ { i j _ { \\ell } } \\ldots a _ { j _ { 1 } s } \\psi _ { \\ell } ^ { \\prime } ( h _ { j _ { \\ell } } ^ { ( \\ell ) } ) \\cdot \\cdot \\cdot \\psi _ { 0 } ^ { \\prime } ( x _ { s } ) .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 474, + 666, + 518 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The combinatorial coefficient $a _ { i j \\ell } \\ldots a _ { j _ { 1 } s }$ is non-zero iff there exists a walk from $i$ to $s$ of length exactly $\\ell { + 1 }$ , since we are not taking into account the contribution coming from self-loops. Conversely, if each term $a _ { i j \\ell }$ was replaced by $\\hat { a } _ { i j \\ell }$ then we would find that $\\hat { a } _ { i j \\ell } \\ldots \\hat { a } _ { j _ { 1 } s }$ is non-zero iff there exists a walk from $i$ to $s$ of length at most $r + 1$ , since the diagonal entries are now positive. □ ", + "bbox": [ + 173, + 522, + 826, + 579 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Remark 8. As a specific instance of Corollary 7, we note that if we do not include self-loops in the adjacency matrix, then the output of a 2-layer simplified graph neural network at node i is independent of the features of neighbours $k$ that do not form a triangle with i. Once again here the dependence is precisely measured via the Jacobian of the hidden features with respect to the input features. ", + "bbox": [ + 173, + 590, + 825, + 661 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Remark 9. We note that the role of self-loops has also implicitly been noted in Xu et al. (2018) where the analysis of the Jacobian of node representations on the graph augmented with self-loops has been related to lazy random-walks. ", + "bbox": [ + 174, + 665, + 823, + 709 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "GNNs with different aggregations We note that similar conclusions extend to message passing architectures where the aggregations are sums and not averages meaning that we take the augmented adjacency without normalizing by the degree matrices. Consistently with Lemma 1, we restrict to the setting where features and node representations at each layer are scalars to make the discussion simpler. In line with the $\\mathrm { X u }$ et al. (2018) we consider a GNN-model of the form ", + "bbox": [ + 173, + 724, + 825, + 795 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cbfc7f5d734d7dff6902cb4c0df647e366ba9ff560dd05c75028a31a470b0b51.jpg", + "text": "$$\nh _ { i } ^ { ( \\ell + 1 ) } = \\mathrm { R e L U } \\left( \\sum _ { j \\in \\tilde { \\mathcal { N } } _ { i } } h _ { j } ^ { ( \\ell ) } w _ { \\ell } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 803, + 611, + 853 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Note that the augmented neighbourhood $\\tilde { \\mathcal { N } } _ { i }$ is defined as $\\mathcal { N } _ { i } \\cup \\{ i \\}$ . Differently from the setting of Theorem 1 in $\\mathrm { X u }$ et al. (2018), the aggregation here is not an average but a simple sum. Let us now take nodes $i$ and $s$ such that $s \\in S _ { r + 1 } ( i )$ as in the statement of Lemma 1. In this case, instead of simply considering the quantity $| \\partial h _ { i } ^ { ( r + 1 ) } / \\partial x _ { s } |$ , we normalize the Jacobian entries - obtaining what is referred to as influence score in $\\mathrm { X u }$ et al. (2018): ", + "bbox": [ + 173, + 862, + 825, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 508, + 118 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/ceee07bab3a5b10e26a20c1112761c4c410d7d88db47f928c88f9f3fd41e5ebf.jpg", + "text": "$$\n\\mathcal { T } _ { r + 1 } ( i , s ) : = \\frac { \\left| \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right| } { \\sum _ { k } \\left| \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { k } } \\right| }\n$$", + "text_format": "latex", + "bbox": [ + 403, + 126, + 594, + 195 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This of course represents now a relative importance of feature $x _ { s }$ on the representation of node $i$ at layer $r + 1$ . If - similarly to Theorem 1 in $\\mathrm { X u }$ et al. (2018) - we assume that all paths in the computational graph of the model are activated with the same probability, then we obtain that on average ", + "bbox": [ + 173, + 200, + 826, + 256 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/4a7f4b15f3bfc81ff6cab3b1b5345062f1a30c3b23873a5058327f01865b423c.jpg", + "text": "$$\n\\mathcal { T } _ { r + 1 } ( i , s ) = \\frac { \\tilde { A } _ { i s } ^ { r + 1 } } { \\sum _ { k } \\tilde { A } _ { i k } ^ { r + 1 } } \\leq \\frac { \\tilde { A } _ { i s } ^ { r + 1 } } { \\mathrm { V o l } ( B _ { r + 1 } ( i ) ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 357, + 255, + 638, + 295 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where ${ \\tilde { A } } = A + I$ and $\\begin{array} { r } { \\mathrm { v o l } ( S ) = \\sum _ { j \\in S } d _ { j } } \\end{array}$ . In particular, we again find that if we have a tree structure, then the right hand side decays exponentially as $2 ^ { - ( r + 1 ) }$ . ", + "bbox": [ + 173, + 299, + 826, + 334 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B PRELIMINARY ANALYSIS OF AN EDGE-NEIGHBORHOOD ", + "text_level": 1, + "bbox": [ + 174, + 354, + 668, + 371 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Given an edge $i \\sim j$ , we introduce the sets below: ", + "bbox": [ + 173, + 386, + 501, + 401 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(i) $\\sharp _ { \\Delta } ( i , j ) : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j )$ , the number of triangles based at the edge $i \\sim j$ . \n(ii) $\\sharp _ { \\bigtriangledown } ( i , j ) : = \\{ k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j ) , k \\neq j : \\ \\exists w \\in ( S _ { 1 } ( k ) \\cap S _ { 1 } ( j ) ) \\setminus S _ { 1 } ( i ) \\}$ , the number of nodes $k \\in S _ { 1 } ( i )$ forming a 4-cycle based at $i \\sim j$ without diagonals inside. \n(iii) $Q _ { i } ( j ) : = S _ { 1 } ( i ) \\setminus ( \\{ j \\} \\cup \\sharp _ { \\Delta } ( i , j ) \\cup \\sharp _ { \\Pi } ^ { i } ( i , j ) )$ , simply the complement of the neighbours of $i$ with respect to the sets introduced in (i) and (ii) once we also exclude $j$ . ", + "bbox": [ + 202, + 411, + 826, + 502 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In the following we simply write $\\sharp _ { \\Delta } , \\sharp _ { \\bigstar } ^ { i }$ and $Q _ { i }$ when the edge $i \\sim j$ is clear from the context. ", + "bbox": [ + 171, + 512, + 789, + 530 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "4-cycle contributions. In general the sets $\\sharp _ { \\bigstar } ^ { i }$ and $\\sharp _ { \\bigstar } ^ { j }$ may differ. This may occur when there exists a node $k$ belonging to $\\sharp _ { \\bigstar } ^ { i }$ that admits multiple solutions $w$ as in the definition of $\\sharp _ { \\bigstar } ^ { i }$ . This feature needs to be taken into account when comparing Ollivier’s Ricci curvature to the new notion we present below. We first introduce the following class to ease the notations. ", + "bbox": [ + 173, + 546, + 826, + 606 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Definition 2. For any simple, undirected graph $G = ( V , E )$ , if $U \\subset V$ , then we set ", + "bbox": [ + 173, + 609, + 718, + 626 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/1a61424f80d2ff1dcba047eb83d7803f4a76622a96014401a2002ef5b6d85748.jpg", + "text": "$$\n\\mathcal { D } ( U ) : = \\left\\{ \\varphi : U \\to V , \\ | U | = | \\varphi ( U ) | , \\ ( z , \\varphi ( z ) ) \\in E , \\ \\forall z \\in U \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 632, + 717, + 651 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We note that any $\\varphi \\in { \\mathcal { D } } ( U )$ is injective. ", + "bbox": [ + 174, + 665, + 436, + 680 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We may now define a quantity which measures the maximal number of 1-1 pairings that can be performed from $\\sharp _ { \\bigstar } ^ { i }$ to $\\sharp _ { \\bigstar } ^ { j }$ . ", + "bbox": [ + 173, + 680, + 823, + 712 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Definition 3. For any edge $i \\sim j$ we let ", + "bbox": [ + 174, + 714, + 437, + 729 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d9fd6b7a14a917ee307f5ab5fdd2cfd9f1d9546c651adcbb6b9665b287978d4f.jpg", + "text": "$$\n\\sharp _ { \\sharp } ( i , j ) : = \\operatorname* { m a x } \\left\\{ | U | : U \\subset \\sharp _ { \\sharp } ^ { i } , \\ : \\exists \\varphi : U \\to \\sharp _ { \\sharp } ^ { j } , \\ : \\varphi \\in \\mathcal { D } ( U ) \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 289, + 736, + 707, + 763 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We often simply write $\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }$ . While the quantity $\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }$ plays a role in the derivation of the Ollivier curvature of $i \\sim j$ it is not computationally-friendly, as to determine $\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }$ we need to identify and distinguish all possible 4-cycles based at $i \\sim j$ and then choose a maximal pairing map. Accordingly, we consider a looser term which is easier to compute: ", + "bbox": [ + 173, + 776, + 826, + 834 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Definition 4. For any pair of adjacent nodes $i \\sim j$ we define ", + "bbox": [ + 173, + 838, + 576, + 854 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/898d313ce04a3ff42ee4c397327ae65e9c4d5cd09222a0119a2daeeb9bf68d9f.jpg", + "text": "$$\n\\gamma _ { m a x } ( i , j ) : = \\operatorname* { m a x } \\left\\{ \\operatorname* { m a x } _ { k \\in \\sharp _ { \\Sigma } ^ { k } } \\{ \\big ( A _ { k } \\cdot ( A _ { j } - A _ { i } \\odot A _ { j } ) \\big ) - 1 \\} , \\operatorname* { m a x } _ { w \\in \\sharp _ { \\Sigma } ^ { j } } \\{ \\big ( A _ { w } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) \\big ) - 1 \\} \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 186, + 859, + 810, + 902 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $A _ { s }$ denotes the s-th row of the adjacency matrix. We usually simply write $\\gamma _ { m a x }$ ", + "bbox": [ + 171, + 909, + 730, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Remark 10. We note that given $k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j )$ the term $( A _ { k } \\cdot ( A _ { j } - A _ { i } \\odot A _ { j } ) ) - 1$ yields the number of nodes w forming a 4-cycle of the form $i \\sim k \\sim w \\sim j \\sim i$ with no diagonals inside. The value $\\gamma _ { m a x }$ measures the maximal degeneracy of edges forming $^ { 4 }$ -cycles, meaning that it is equal to $^ { l }$ iff for each $k \\in \\sharp _ { \\bigstar } ^ { i }$ there exists a unique node $w \\in \\sharp _ { \\bigstar } ^ { j }$ such that $i \\sim k \\sim w \\sim j \\sim i$ is a $^ { 4 }$ -cycle. ", + "bbox": [ + 173, + 102, + 825, + 164 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We now end the discussion about 4-cycle contributions by proving the following inequality, which allows us to avoid to compute directly the term $\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }$ up to giving up some accuracy. ", + "bbox": [ + 173, + 172, + 823, + 203 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma 11. For any edge $i \\sim j$ we have ", + "bbox": [ + 173, + 207, + 444, + 222 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/89bea760cd16cc49e7087d599acf6b3ca969b8894bfa47efa2d19f0a2c986f53.jpg", + "text": "$$\n| \\sharp _ { \\sf H } ^ { \\mathfrak { m } } | \\geq \\frac { \\operatorname* { m a x } \\{ | \\sharp _ { \\sf H } ^ { i } | , | \\sharp _ { \\sf H } ^ { j } | \\} } { \\gamma _ { m a x } } .\n$$", + "text_format": "latex", + "bbox": [ + 415, + 228, + 583, + 265 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. The proof is based on a combinatorial argument. Let $\\sharp _ { \\sqcap } ^ { i } = \\{ k _ { 1 } , \\ldots , k _ { r } \\}$ and let $\\sharp _ { \\bigcirc } ^ { \\mathfrak { m } } =$ $\\{ k _ { 1 } , \\ldots , k _ { \\ell } \\}$ , with $\\ell < r$ . By definition there exists $\\varphi : \\{ k _ { 1 } , \\dots , k _ { \\ell } \\} \\to \\{ w _ { 1 } , \\dots , w _ { \\ell } \\}$ , with $k _ { i } \\sim w _ { i }$ and $w _ { i } \\in \\sharp _ { \\bigstar } ^ { j }$ , for $1 \\leq i \\leq \\ell$ . Given $k \\in \\sharp _ { \\sqcap } ^ { i } \\setminus \\{ k _ { 1 } , . . . , k _ { \\ell } \\}$ , then there are no $w \\in \\sharp _ { \\sqcap } ^ { j } \\setminus \\{ w _ { 1 } , . . . , w _ { \\ell } \\}$ such that $k \\sim w$ , otherwise we could extend $\\varphi$ by setting $k \\mapsto \\varphi ( k ) : = w$ and we would then get $| \\mathtt { H } _ { \\square } ^ { \\mathfrak { m } } | = \\ell + 1$ . Accordingly, we have ", + "bbox": [ + 173, + 280, + 825, + 354 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/7529fcc278feaac0518c6fbd641d177105dffada53eefb337b64856a185dd038.jpg", + "text": "$$\n\\sum _ { s = 1 } ^ { \\ell } ( A _ { w _ { s } } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) ) - 1 ) \\geq | \\sharp _ { \\square } ^ { i } | ,\n$$", + "text_format": "latex", + "bbox": [ + 359, + 361, + 633, + 405 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "which implies ", + "bbox": [ + 173, + 411, + 267, + 425 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/06d40a0b2b39caa88d2abd80ab697f95b6e82628d65357f4ea7e0b12493ef0c2.jpg", + "text": "$$\n\\gamma _ { \\operatorname* { m a x } } | \\sharp _ { \\sf D } ^ { \\mathrm { m } } | \\equiv \\gamma _ { \\operatorname* { m a x } } \\ell \\geq \\sum _ { s = 1 } ^ { \\ell } ( A _ { w _ { s } } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) ) - 1 ) \\geq | \\sharp _ { \\sf D } ^ { i } | .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 431, + 707, + 476 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C EXISTING CURVATURE CANDIDATES ", + "text_level": 1, + "bbox": [ + 173, + 517, + 509, + 534 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Ollivier Ricci curvature For $i \\in V$ and $\\alpha \\in [ 0 , 1 )$ we define a probability measure on $B _ { 1 } ( i )$ by: ", + "bbox": [ + 174, + 547, + 818, + 564 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/7aa9e425bee708dea94fa37d93747a16569eddd36c9e02db4d32c03b031f412e.jpg", + "text": "$$\n\\mu _ { i } ^ { \\alpha } : j \\mapsto \\left\\{ \\begin{array} { r l } & { \\alpha , \\ j = i } \\\\ & { \\frac { 1 - \\alpha } { d _ { i } } , \\ j \\in S _ { 1 } ( i ) , } \\\\ & { 0 , \\ \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 392, + 570, + 602, + 628 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Before we introduce the Ollivier curvature, we recall that the transportation distance between two finitely supported probability measures as above can be computed as ", + "bbox": [ + 173, + 633, + 823, + 662 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/b963842df11bb6006c9305daafc642e9a7e950e52d21e8a67631e39dd52be7d7.jpg", + "text": "$$\nW _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) : = \\operatorname* { i n f } _ { M } \\sum _ { k \\in S _ { 1 } ( i ) } \\sum _ { w \\in S _ { 1 } ( j ) } M _ { k w } d _ { G } ( k , w ) ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 669, + 665, + 705 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $d _ { G } ( \\cdot , \\cdot )$ is the geodesic distance on the graph and the infimum is taken over all matrices $M$ satisfying the marginal constraints: ", + "bbox": [ + 174, + 713, + 821, + 742 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/35bd5ec6b11dc1457e6daa1ef3b68652590ec136dc5d29813fb7a55965c7d11b.jpg", + "text": "$$\n\\sum _ { k \\in S _ { 1 } ( i ) } M _ { k w } = \\mu _ { j } ^ { \\alpha } ( w ) , \\quad \\sum _ { w \\in S _ { 1 } ( j ) } M _ { k w } = \\mu _ { i } ^ { \\alpha } ( k ) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 748, + 661, + 785 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We are now ready to define the Ollivier Ricci curvature: the formulation below is due to Lin et al. \n(2011). ", + "bbox": [ + 174, + 791, + 825, + 820 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition 5. Given $i \\sim j$ we define the $\\alpha$ -Ollivier curvature by ", + "bbox": [ + 173, + 824, + 593, + 839 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d72597de2d5c7786cc275b8f8987765a0f4d37b4fd975b08b5e5d71dc553778a.jpg", + "text": "$$\n\\kappa _ { \\alpha } ( i , j ) : = 1 - W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 845, + 596, + 864 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since $\\kappa _ { \\alpha } ( 1 - \\alpha ) ^ { - 1 }$ is increasing and bounded the quantity below is well-defined: ", + "bbox": [ + 173, + 872, + 707, + 888 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/ba3d51ae9dcbbbd72f81491e29b7667fb9288933963bf5170264c57c58dc6e86.jpg", + "text": "$$\n\\kappa ( i , j ) : = \\operatorname* { l i m } _ { \\alpha \\to 1 } \\frac { 1 - W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) } { 1 - \\alpha } .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 895, + 609, + 928 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Forman Ricci curvature In the following we report a formula for the augmented Forman Ricci curvature on unweighted graphs Samal et al. (2018). We also note that Forman curvature on graphs has also been studied in Sreejith et al. (2016); Weber et al. (2018). ", + "bbox": [ + 176, + 103, + 825, + 146 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition 6. For any edge $i \\sim j$ the augmented Forman curvature is given by ", + "bbox": [ + 171, + 148, + 691, + 165 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d510b530852ebf41036a2503b8578df955ab7d843941af1c5dbc3992b109b14a.jpg", + "text": "$$\nF ( i , j ) : = 4 - d _ { i } - d _ { j } + 3 | \\sharp _ { \\Delta } | .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 169, + 604, + 186 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We note that such formulation of curvature does not distinguish contributions coming from 4-cycles. \nIn fact, for the orthogonal grid with degree $d \\geq 4$ , Forman Ricci curvature is equal to $2 ( 2 - d ) < 0$ . \nThis does not reflect that the $r$ -hop neighbourhood for such a graph grows polynomially in $r$ . ", + "bbox": [ + 173, + 198, + 828, + 241 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We conclude this appendix by reporting a lower bound for the Ollivier Ricci curvature derived in Jost & Liu (2014). We recall that $\\kappa _ { \\alpha }$ , with $\\alpha \\in [ 0 , 1 )$ was defined in equation 7. ", + "bbox": [ + 173, + 247, + 823, + 276 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem 12 (Jost & Liu (2014)). For any edge $i \\sim j$ , with $d _ { i } \\leq d _ { j }$ , the following bound is satisfied: ", + "bbox": [ + 174, + 279, + 825, + 295 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/0d69c532e3bea465e7a4b6047e49b3cf035cbc5fa2f750cd592052a967f6aab1.jpg", + "text": "$$\n\\kappa _ { 0 } ( i , j ) \\geq \\Phi ( i , j ) : = - \\left( 1 - \\frac { 1 } { d _ { i } } - \\frac { 1 } { d _ { j } } - \\frac { | \\sharp \\Delta | } { d _ { j } } \\right) _ { + } - \\left( 1 - \\frac { 1 } { d _ { i } } - \\frac { 1 } { d _ { j } } - \\frac { | \\sharp \\Delta | } { d _ { i } } \\right) _ { + } + \\frac { | \\sharp \\Delta | } { d _ { j } } .\n$$", + "text_format": "latex", + "bbox": [ + 209, + 299, + 787, + 337 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D PROOFS OF RESULTS IN SECTION 3 ", + "text_level": 1, + "bbox": [ + 173, + 351, + 501, + 368 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We first recall our definition of Balanced Forman: ", + "bbox": [ + 174, + 382, + 500, + 397 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition 7. For any edge $i \\sim j$ we let $\\operatorname { R i c } ( i , j )$ be zero $\\gamma _ { \\mathrm { m i n } \\{ d _ { i } , d _ { j } \\} } = 1$ , otherwise ", + "bbox": [ + 176, + 400, + 746, + 416 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ae55a567f24bbc66bb7a7a817a24f99dc2cb5733345957f274fe4c02e08617fd.jpg", + "text": "$$\n\\mathbf { R i c } ( i , j ) : = \\frac { 2 } { d _ { i } } + \\frac { 2 } { d _ { j } } - 2 + 2 \\frac { \\lvert \\sharp _ { \\Delta } \\rvert } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\lvert \\sharp _ { \\Delta } \\rvert } { \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} } + \\frac { ( \\gamma _ { m a x } ) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\big ( \\lvert \\sharp _ { \\Xi } \\rvert + \\lvert \\sharp _ { \\Xi } ^ { j } \\rvert \\big )\n$$", + "text_format": "latex", + "bbox": [ + 197, + 421, + 781, + 457 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the last term is set to be zero $i f | \\sharp _ { \\bigstar } ^ { i } |$ (and hence $| \\sharp _ { \\bigstar } ^ { j } | \\big )$ is zero. ", + "bbox": [ + 173, + 462, + 611, + 479 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We also extend the previous definition to the weighted case. In this setting we let $G = ( V , E , \\omega )$ be a simple, locally finite, undirected graph with normalized weights. We first report the formula for the augmented Forman in the weighted case Samal et al. (2018): ", + "bbox": [ + 173, + 488, + 823, + 531 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/f2fb232c0dd5c8fb2a4d8dbeb502734a366dccd7f700a1436375a62412a32e68.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle F ( i , j ) = \\omega ( i ) + \\omega ( j ) + \\sum _ { k \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) } \\frac { \\omega _ { i j } ^ { 2 } } { \\omega _ { \\Delta } } } \\\\ & { \\displaystyle - \\sum _ { k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j ) } \\omega ( i ) \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { i k } } } - \\sum _ { k \\in S _ { 1 } ( j ) \\setminus S _ { 1 } ( i ) } \\omega ( j ) \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { j k } } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 535, + 705, + 623 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\omega _ { \\Delta }$ is taken to be the Heron formula for the area of a triangle while $\\omega ( \\cdot )$ denotes some weighting scheme for the nodes as well. We propose a similar definition for the weighted case, which reduces to the one discussed above in the combinatorial setting. We recall that $W$ is the weighted adjacency matrix while $A$ is the combinatorial one. Moreover, we write $A _ { j } ^ { i } = ( A _ { j } - A _ { i } ) _ { + } ^ { - }$ and similarly for $W _ { j } ^ { i }$ . ", + "bbox": [ + 173, + 626, + 826, + 702 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition 8. For any pair of adjacent nodes $i , j \\in V$ we define $\\operatorname { R i c } ( i , j )$ to be $0 i f \\operatorname* { m i n } \\{ | d _ { i } | , | d _ { j } | \\} =$ 1, otherwise we set ", + "bbox": [ + 169, + 704, + 823, + 734 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/982772dcd4c03c70e55182cd9cf60af68d507c07bcb95fa71ad62ffab113eb0e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { R i c } ( i , j ) : = \\cfrac { 1 } { d _ { i } } \\left( 1 - \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } } { \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { i k } } } } \\right) + \\cfrac { 1 } { d _ { j } } \\left( 1 - \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } } { \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { j k } } } } \\right) } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\cfrac { 1 } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\displaystyle \\sum _ { k \\in \\mathcal { S } _ { i } ( i ) \\cap \\mathcal { S } ( i , j ) } \\frac { \\omega _ { i j } ^ { 2 } } { \\omega \\Delta } } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } ^ { * } } \\frac { \\omega _ { i j } } { \\omega | } \\left( \\frac { \\omega _ { i j } \\left( \\gamma _ { 0 n x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } - \\frac { \\sqrt { \\omega _ { i j } \\nu _ { \\perp } } } { d _ { i } } \\right) } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } ^ { * } } \\frac { \\omega _ { i j } } { \\omega | } \\left( \\frac { \\omega _ { i j } \\left( \\gamma _ { 0 n x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } - \\frac { \\sqrt { \\omega _ { i j } \\nu _ { \\perp } } } { d _ { j } } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 736, + 707, + 922 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "with ", + "bbox": [ + 173, + 103, + 205, + 117 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/9fd6963393dcd31feb8512f49fd9ace7dba0d35f905f17633364c28e9d78bda1.jpg", + "text": "$$\n\\omega _ { \\Delta } : = \\frac { 1 } { 3 } ( \\omega _ { i j } ^ { 2 } + \\omega _ { i k } ^ { 2 } + \\omega _ { j k } ^ { 2 } ) , \\quad z \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 114, + 661, + 143 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "and, for a given $k \\in \\sharp _ { \\bigstar } ^ { i }$ ", + "bbox": [ + 173, + 148, + 333, + 164 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/c544e00ab6a8b4ce85ca00d37fe41c53a27ebaa8796c2b04c779cc76738a2564.jpg", + "text": "$$\n\\nu _ { \\perp } : = \\frac { W _ { z } \\cdot A _ { j } ^ { i } } { A _ { k } \\cdot A _ { j } ^ { i } } , \\omega _ { \\perp } = \\frac { 1 } { 4 } \\left( \\omega _ { i j } ^ { 2 } + \\omega _ { i k } ^ { 2 } + \\nu _ { \\perp } ^ { 2 } + \\frac { A _ { k } \\cdot W _ { j } ^ { i } } { A _ { k } \\cdot A _ { j } ^ { i } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 171, + 699, + 214 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Important convention. Without losing generality, in the following we always assume that $1 \\leq d _ { i } \\leq$ $d _ { j }$ . In particular, we write $d _ { i } \\doteq d$ and $d _ { j } = d + s$ , for some $s \\geq 0$ , omitting to specify that both $d$ and $s$ are of course depending on $i$ and $j$ . Moreover, from now on we only focus on the unweighted case. ", + "bbox": [ + 174, + 227, + 825, + 271 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We can now prove our main comparison theorem. ", + "bbox": [ + 174, + 276, + 500, + 291 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Theorem 2. Given an unweighted graph $G$ , for any edge $i \\sim j$ we have $\\kappa ( i , j ) \\geq \\operatorname { R i c } ( i , j )$ . ", + "bbox": [ + 169, + 301, + 774, + 319 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. We stick to the aforementioned convention: $d _ { i } : = d \\le d _ { j } = d + s , s \\ge 0 ,$ . The strategy of the proof amounts to finding a transportation plan providing an upper bound for $W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } )$ and hence a lower bound for the curvature $\\kappa ( i , j )$ . In particular, we consider plans moving the mass $\\mu _ { i } ^ { \\alpha }$ from $B _ { 1 } ( i )$ to $B _ { 1 } ( j )$ . ", + "bbox": [ + 173, + 334, + 825, + 393 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "If $d = 1$ , then the optimal transport plan consists of moving the mass $\\alpha$ from $i$ to $j$ and the remaining mass $1 - \\alpha$ on $j$ to $\\bar { S } _ { 1 } ( j )$ . This yields a unit Wasserstein distance between $\\mu _ { i } ^ { \\alpha }$ and $\\mu _ { j } ^ { \\alpha }$ and hence zero Ollivier curvature $\\kappa ( i , j )$ , which coincides with the value of balanced Forman $\\operatorname { R i c } ( i , j )$ . ", + "bbox": [ + 173, + 400, + 825, + 445 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Assume now that $d \\geq 2$ . A (possibly non-optimal) transport plan from $\\mu _ { i } ^ { \\alpha }$ to $\\mu _ { j } ^ { \\alpha }$ is given by: ", + "bbox": [ + 171, + 450, + 777, + 467 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(i) Move mass $( 1 - \\alpha ) / ( d + s )$ from each node $k \\in \\sharp _ { \\bigstar } ^ { \\mathfrak { m } } \\subset \\sharp _ { \\bigstar } ^ { i }$ to its unique image in $\\sharp _ { \\bigstar } ^ { j }$ under a bijection $\\varphi$ as per definition of $\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }$ . \n(ii) The remaining mass on each node $k \\in \\sharp _ { \\bigstar } ^ { \\mathfrak { m } }$ will need to travel by at most distance 3 to $S _ { 1 } ( j )$ . \n(iii) The extra-mass $( 1 - \\alpha ) ( 1 / d - 1 / ( d + s ) )$ on each common neighbour $k \\in \\sharp _ { \\Delta }$ will need to travel by at most distance 2 to $S _ { 1 } ( j )$ . \n(iv) Move the mass $( 1 - \\alpha ) / d$ from $j$ to $S _ { 1 } ( j )$ . \n(v) Move the mass $\\alpha$ from $i$ to $j$ . This leaves left-over mass $( 1 - \\alpha ) / ( d + s )$ at $i$ from the distribution $\\mu _ { j } ^ { \\alpha }$ . This mass can be compensated from mass in $S _ { 1 } ( i )$ which is at distance one. \n(vi) Finally, we move the mass $( 1 - \\alpha ) / d$ of any untouched node in $S _ { 1 } ( i )$ to $S _ { 1 } ( j )$ along a path of length lesser or equal than three. Note that the remaining mass is equal to $( ( \\bar { 1 } - \\alpha ) / d ) ( \\bar { d } - 1 - | \\sharp _ { \\Delta } | - | \\bar { \\sharp } _ { \\varpi } ^ { \\mathfrak { m } } | ) - ( 1 - \\alpha ) ( d + s )$ , where the last terms comes from (v). ", + "bbox": [ + 200, + 484, + 826, + 694 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "If we sum all the contributions we find ", + "bbox": [ + 174, + 708, + 429, + 722 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/1d380fef1043fedd7f5e2c6067efa640e119c05dbdcf08fe4d6d820bdb1ac984.jpg", + "text": "$$\n\\begin{array} { l } { { V _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) \\leq ( 1 - \\alpha ) \\left( \\displaystyle \\frac { | \\# _ { [ \\alpha ] } ^ { \\mathbb { m } } | } { d + s } + 3 | \\# _ { \\Sigma } ^ { \\mathbb { m } } | \\left( \\displaystyle \\frac { 1 } { d } - \\displaystyle \\frac { 1 } { d + s } \\right) + 2 | \\# _ { \\Delta } | \\left( \\displaystyle \\frac { 1 } { d } - \\displaystyle \\frac { 1 } { d + s } \\right) + \\displaystyle \\frac { 1 } { d } \\right) + \\alpha + \\displaystyle \\frac { 1 - \\alpha } { d + s } } } \\\\ { { \\displaystyle ~ + 3 ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d } ( d - 1 - | \\# _ { \\Delta } | - | \\# _ { \\mathbb { m } } ^ { \\mathbb { m } } | ) - \\displaystyle \\frac { 1 } { d + s } \\right) } } \\\\ { { \\displaystyle ~ = ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d + s } \\left( - 2 | \\# _ { \\Delta } | - 2 | \\# _ { \\Xi } ^ { \\mathbb { m } } | - 2 \\right) + \\displaystyle \\frac { 1 } { d } \\left( - | \\# _ { \\Delta } | - 2 \\right) + 2 \\right) + 1 } } \\\\ { { \\displaystyle ~ = ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d + s } \\left( - 3 | \\# _ { \\Delta } | - \\displaystyle \\frac { s } { d } | \\# _ { \\Delta } | - 2 | \\# _ { \\Xi } ^ { \\mathbb { m } } | - 4 - 2 \\displaystyle \\frac { s } { d } + 2 ( d + s ) \\right) \\right) + 1 . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 726, + 831, + 869 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Therefore we have ", + "bbox": [ + 173, + 873, + 299, + 887 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/de7998cf6c0325ace6b5fc9fcc4d3409d524c2fe2defeb4b18cda88bdaa8420c.jpg", + "text": "$$\n\\kappa ( i , j ) = \\operatorname* { l i m } _ { \\alpha \\to 1 } \\frac { 1 - W ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) } { 1 - \\alpha } \\geq \\left( \\frac { 1 } { d + s } \\left( 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + 2 | \\sharp _ { \\overline { { \\sharp } } } ^ { \\mathrm { m } } | + 4 + 2 \\frac { s } { d } - 2 ( d + s ) \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 892, + 805, + 929 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "By using Lemma 11, we can bound the right hand side as ", + "bbox": [ + 173, + 103, + 552, + 119 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/91be38fd509a877a0a365b8ae6082f19c0e94bbf112a174df80de2ce2c52374e.jpg", + "text": "$$\n\\kappa ( i , j ) \\geq \\left( \\frac { 1 } { d + s } \\left( 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + ( \\gamma _ { \\operatorname* { m a x } } ) ^ { - 1 } \\big ( | \\sharp _ { \\bigstar | } ^ { i } | + | \\sharp _ { \\bigstar | } ^ { j } | \\big ) + 4 + 2 \\frac { s } { d } - 2 ( d + s ) \\right) \\right) = \\mathrm { R i c } ( i , j )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 125, + 818, + 160 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "which completes the proof. ", + "bbox": [ + 173, + 165, + 352, + 180 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Remark 13. By inspection $\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j )$ , with $\\Phi ( i , j )$ as in Theorem 12. We have three cases: ", + "bbox": [ + 174, + 186, + 825, + 204 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(i) $\\Phi ( i , j ) = 2 / d + 2 / ( d + s ) - 2 + 2 | \\sharp _ { \\Delta } | / ( d + s ) + | \\sharp _ { \\Delta } | / d \\leq \\operatorname { R i c } ( i , j )$ , because $\\operatorname { R i c } ( i , j )$ takes into account the positive contribution of 4-cycles as well. ", + "bbox": [ + 199, + 213, + 825, + 244 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(ii) $\\Phi ( i , j ) = - 1 + 1 / d + 1 / ( d + s ) + 2 | \\sharp _ { \\Delta } | / ( d + s )$ , which happens iff ", + "bbox": [ + 205, + 251, + 691, + 268 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/d1f6fe683a5e8ab6892853cc3eeb4f71a53108a37388e5f4b6c19191c5787c05.jpg", + "text": "$$\nd + s - 2 - \\frac { s } { d } - | \\sharp _ { \\Delta } | > 0\n$$", + "text_format": "latex", + "bbox": [ + 439, + 273, + 617, + 301 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 232, + 308, + 261, + 320 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/39a977af5a7f687001739dfcd4b0d8e5e8bc84659507af1e7282a2e9bc651ac9.jpg", + "text": "$$\nd + s - 2 - \\frac { s } { d } - | \\sharp \\triangle | - \\frac { s } { d } | \\sharp \\triangle | \\leq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 316, + 648, + 344 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "From the previous inequalities we derive ", + "bbox": [ + 233, + 345, + 500, + 361 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/7ea3fc0ff58772d69033be2421733b03ba1a14bc713273157bc473223b881c37.jpg", + "text": "$$\n\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j ) + \\frac { 1 } { d + s } \\left( 2 - d - s + | \\sharp \\Delta | + \\frac { s } { d } | \\sharp \\Delta | + \\frac { s } { d } \\right) \\geq \\Phi ( i , j ) .\n$$", + "text_format": "latex", + "bbox": [ + 285, + 366, + 767, + 398 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(iii) $\\Phi ( i , j ) = | \\sharp _ { \\Delta } | / ( d + s )$ which is equivalent to ", + "bbox": [ + 197, + 411, + 534, + 428 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f3f28b355adccd927babf4eb060d031a7b38bd86f816b38ce3f8d60bfc84563c.jpg", + "text": "$$\nd + s - 2 - \\frac { s } { d } - | \\sharp _ { \\Delta } | \\leq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 434, + 619, + 460 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this case we have ", + "bbox": [ + 232, + 467, + 367, + 481 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f85aba5e484a965f85b7a1ab932934768e48a9af575bd2c353f03a3a8d0dcb34.jpg", + "text": "$$\n\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j ) + \\frac { s } { d } \\frac { | \\sharp _ { \\Delta } | } { d + s }\n$$", + "text_format": "latex", + "bbox": [ + 428, + 477, + 627, + 508 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Corollary 3. If $\\operatorname { R i c } ( i , j ) \\geq k > 0$ for any edge $i \\sim j$ , then there exists a polynomial $P$ such that ", + "bbox": [ + 171, + 511, + 803, + 526 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/cf30c4a55a758afb35f09043f25c1633cfbd18a9da8e16c72144668acf051f7d.jpg", + "text": "$$\n| B _ { r } ( i ) | \\leq P ( r ) , \\forall i \\in V .\n$$", + "text_format": "latex", + "bbox": [ + 410, + 532, + 586, + 550 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. This follows immediately from Theorem 2 and a Bishop-Gromov type of result for discrete Ollivier curvature on graphs Paeng (2012). □ ", + "bbox": [ + 173, + 564, + 825, + 594 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To address the proof of Theorem 4, we first need the Lemma below. ", + "bbox": [ + 173, + 608, + 629, + 623 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma 14. Given $i \\sim j$ , with $d _ { i } \\leq d _ { j }$ , if $\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta ,$ , for some $0 < \\delta < ( 1 + \\gamma _ { m a x } ) ^ { - 1 }$ , then ", + "bbox": [ + 176, + 627, + 826, + 643 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/5135b0ee892c9176b3ec4a57eb864382f806c30eedb72ba52e77ddbab6b16c12.jpg", + "text": "$$\n\\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } > \\delta ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 441, + 650, + 555, + 684 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. According to our convention we let $d _ { i } = d$ and $d _ { j } = d + s$ , for $s \\geq 0$ . We also recall that $Q _ { j } = S _ { 1 } ( j ) \\backslash ( \\sharp _ { \\Delta } \\cup \\sharp _ { \\Pi } ^ { j } \\cup \\{ i \\} )$ . If we multiply equation 3 by $d _ { j } = d + s$ , we see that $\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta$ iff ", + "bbox": [ + 173, + 696, + 826, + 739 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/b0c98e02d75dd29ed2bf4912063bdf321a7e4afced87e8999a755bc6e86c94f5.jpg", + "text": "$$\n4 + 2 \\frac { s } { d } + 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + \\gamma _ { \\operatorname* { m a x } } ^ { - 1 } ( | \\sharp _ { \\bigstar } ^ { i } | + | \\sharp _ { \\bigstar } ^ { j } | ) \\leq \\delta ( 1 + | \\sharp _ { \\Delta } | + | \\sharp _ { \\bigstar } ^ { j } | + | Q _ { j } | ) .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 738, + 746, + 765 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By assumption $| \\sharp _ { \\sf D } ^ { j } | ( \\gamma _ { \\mathrm { m a x } } ^ { - 1 } - \\delta ) \\geq 0$ , meaning that ", + "bbox": [ + 174, + 768, + 496, + 787 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/50d912e56e079e68a06055f817c960dacd85092ac8ad62ad2f59e2571ec4940c.jpg", + "text": "$$\n\\delta ( 1 + | \\sharp \\Delta | + | Q _ { j } | ) \\geq 4 + 2 \\frac { s } { d } + 3 | \\sharp \\Delta | + \\frac { s } { d } | \\sharp \\Delta | \\geq 4 + 3 | \\sharp \\Delta | .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 792, + 699, + 821 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Therefore, we conclude ", + "bbox": [ + 174, + 827, + 331, + 840 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/df913a23ab92bc7d988438a9a5c7d99dd1a4944c87b0ecd42c28459e137e58e5.jpg", + "text": "$$\n\\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } \\geq \\frac { 3 } { \\delta } - 1 .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 838, + 562, + 872 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Theorem 4. Consider a MPNN as in equation $I$ . Let $i \\sim j$ with $d _ { i } \\leq d _ { j }$ and assume that: ", + "bbox": [ + 168, + 909, + 764, + 925 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(i) $| \\nabla \\phi _ { \\ell } | \\le \\alpha$ and $| \\nabla \\psi _ { \\ell } | \\le \\beta$ for each $0 \\leq \\ell \\leq L - 1$ , with $L \\geq 2$ the depth of the MPNN. ", + "bbox": [ + 200, + 102, + 808, + 119 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "(ii) There exists $\\delta$ s.t. $0 < \\delta < ( \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} ) ^ { - \\frac { 1 } { 2 } }$ , $\\delta < \\gamma _ { m a x } ^ { - 1 } ,$ , and $\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta$ ", + "bbox": [ + 202, + 127, + 759, + 147 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Then there exists $Q _ { j } \\subset S _ { 2 } ( i )$ satisfying $| Q _ { j } | > \\delta ^ { - 1 }$ and for $0 \\leq \\ell _ { 0 } \\leq L - 2$ we have ", + "bbox": [ + 173, + 156, + 730, + 174 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/31785543f89ea4e42799e24dd197e94beed501078acb90789194d0b4ed233bdd.jpg", + "text": "$$\n\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } \\left| \\frac { \\partial h _ { k } ^ { ( \\ell _ { 0 } + 2 ) } } { \\partial h _ { i } ^ { ( \\ell _ { 0 } ) } } \\right| < ( \\alpha \\beta ) ^ { 2 } \\delta ^ { \\frac { 1 } { 4 } } .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 178, + 614, + 222 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. As usual we let $d _ { i } : = d$ and $d _ { j } : = d + s$ , for some $s \\geq 0$ . We first observe that from the requirement $\\delta ^ { 2 } ( d + s ) \\leq 1$ in (ii), we derive $\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta$ iff ", + "bbox": [ + 173, + 234, + 825, + 266 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/64422b369dc6fc6343a3b6bfd1f160892077fc2a790da1dc7b6f3bc550125654.jpg", + "text": "$$\n4 + 2 \\frac { s } { d } + 3 | \\sharp \\triangle | + \\frac { s } { d } | \\sharp \\triangle | + \\gamma _ { \\mathrm { m a x } } ^ { - 1 } ( | \\sharp \\vert + | \\sharp \\vert ) \\leq \\delta ( d + s ) .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 272, + 691, + 300 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Therefore we have ", + "bbox": [ + 174, + 306, + 299, + 320 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/e7ce9fe92ab4bccc6f4e81da6845c373f0711d71ab5138c7b8888554efb5cf8f.jpg", + "text": "$$\n\\delta | \\sharp _ { \\Delta } | \\left( 3 + \\frac { s } { d } \\right) \\leq \\delta ^ { 2 } ( d + s ) ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 316, + 593, + 344 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "meaning that ", + "bbox": [ + 173, + 347, + 263, + 362 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/2632797d0107ac8cf203aab845cc4563cd6937309c808454b492904a2b51990f.jpg", + "text": "$$\n\\delta | \\sharp _ { \\Delta } | \\leq 1 .\n$$", + "text_format": "latex", + "bbox": [ + 460, + 361, + 534, + 378 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "From now on we let $Q _ { j }$ denote again the complement $S _ { 1 } ( j ) \\setminus ( S _ { 1 } ( i ) \\cup \\sharp _ { \\Pi } ^ { j } \\cup \\{ i \\} )$ . Without loss of generality we set \\`0 = 0 and hence h(0)i $h _ { i } ^ { ( 0 ) } = x _ { i }$ ; the very same proof applies to any other choice of $\\ell _ { 0 }$ . , we can apply Lemma 1 and derive ", + "bbox": [ + 173, + 383, + 826, + 433 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/5972ec1c965c6f4868788dfc728268866190a343f71f7785295cdcfd8554142d.jpg", + "text": "$$\n\\left| \\frac { \\partial h _ { k } ^ { ( 2 ) } } { \\partial x _ { i } } \\right| \\leq ( \\alpha \\beta ) ^ { 2 } ( \\hat { A } ) _ { i k } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 439, + 581, + 482 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We may expand the power of the augmented normalized adjacency matrix as ", + "bbox": [ + 173, + 488, + 676, + 503 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/4cb7b53c92ecb7797648b1dd887f2a4c08c543a80dee2c7de9c098d608afb1a3.jpg", + "text": "$$\n( \\hat { A } ) _ { i k } ^ { 2 } = \\frac { 1 } { \\sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) } \\frac { 1 } { d _ { w } + 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 508, + 671, + 549 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "If we introduce the set $\\hat { Q } _ { j } = \\{ k \\in Q _ { j } : \\sigma _ { i k } > 1 \\}$ , we can then write ", + "bbox": [ + 173, + 558, + 624, + 577 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/3f511b5735d645f8188f68708495e0ce8ce32454dd6527389c26f6f5a941e46f.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\sum _ { k \\in Q _ { j } } ( \\hat { A } ) _ { i k } ^ { 2 } = \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) } \\frac { 1 } { d _ { w } + 1 } = } } \\\\ { { = \\displaystyle \\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\left( \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { 1 } { d _ { j } + 1 } + \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) } \\frac { 1 } { d _ { w } + 1 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 582, + 767, + 675 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where in the last equality we have again used the fact that $k \\in \\hat { Q } _ { j }$ iff there is $w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j )$ To avoid heavy notations, we introduce $V _ { k } : = S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j )$ . Let us first focus on the first sum in equation 12. We have ", + "bbox": [ + 174, + 684, + 823, + 727 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/39f8d11fceab864d8dd50f0c09555ded49db5f2cd172567f3bdc7a2ba51402cd.jpg", + "text": "$$\n\\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { 1 } { d _ { j } + 1 } \\leq \\frac { 1 } { \\sqrt { d _ { i } + 1 } } | Q _ { j } | \\frac { 1 } { d _ { j } + 1 } \\leq \\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\leq 1 .\n$$", + "text_format": "latex", + "bbox": [ + 258, + 733, + 740, + 773 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We now consider the second sum in equation 12. We assume $| \\sharp _ { \\Delta } | \\geq 1$ , otherwise ${ \\hat { Q } } _ { j } = \\emptyset$ . We let ", + "bbox": [ + 173, + 782, + 808, + 799 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/c1a3c559129021b37f42842438d2c9982f76a7a9729b0d9af1681c11fbd9ced2.jpg", + "text": "$$\n\\Omega : = \\left\\{ w \\in \\sharp _ { \\Delta } : \\ d _ { w } < \\frac { 1 } { C } \\frac { | \\hat { Q } _ { j } | } { | \\sharp _ { \\Delta } | } + \\frac { 2 } { C } \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 806, + 632, + 849 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "for some $C > 0$ to be chosen below. Then, the second sum in equation 12 can be split as ", + "bbox": [ + 171, + 854, + 751, + 871 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/8f9783c339c6a69de25be4c7913ddc8bf319e11eeb5e35da0fffd0d590ee5640.jpg", + "text": "$$\n\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\left( \\sum _ { w \\in V _ { k } \\cap \\Omega } \\frac { 1 } { d _ { w } + 1 } + \\sum _ { w \\in V _ { k } \\setminus \\Omega } \\frac { 1 } { d _ { w } + 1 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 876, + 684, + 928 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Since any $w \\in V _ { k }$ has degree at least three, we can bound the first term in equation 14 as ", + "bbox": [ + 169, + 103, + 754, + 119 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/f26a843d3e539d2dadf803cd16e729b431a20854f19a2736b3b5c18e55ff3d9d.jpg", + "text": "$$\n\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\left( \\sum _ { w \\in V _ { k } \\cap \\Omega } \\frac { 1 } { d _ { w } + 1 } \\right) \\leq \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { | V _ { k } \\cap \\Omega | } { 4 } .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 123, + 709, + 171 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We now observe that ", + "bbox": [ + 173, + 178, + 313, + 191 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/50f29712816c6ae9056f945fba1258f458957c70b257f05141809e729e35181c.jpg", + "text": "$$\n\\sum _ { k \\in \\hat { \\mathcal { Q } } _ { j } } \\frac { \\left| V _ { k } \\cap \\Omega \\right| } { \\sqrt { d _ { k } + 1 } } \\leq \\sum _ { k \\in \\hat { \\mathcal { Q } } _ { j } } \\left| V _ { k } \\cap \\Omega \\right| = \\left| \\left\\{ \\left( k , w \\right) \\in E \\colon k \\in \\hat { Q } _ { j } , w \\in V _ { k } \\cap \\Omega \\right\\} \\right| \\leq \\left( \\operatorname* { m a x } _ { w \\in \\Omega } d _ { w } \\right) \\left| \\Omega \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 187, + 194, + 808, + 238 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Since $d _ { w } \\leq ( 1 / C ) | \\hat { Q } _ { j } | / | \\sharp _ { \\Delta } | + 2 / C$ for any $w \\in \\Omega$ we see that the first term in equation 14 can be bounded by ", + "bbox": [ + 173, + 246, + 823, + 277 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/8c36feaad1d7c8048f1c2d71aabb62318de2190b1204aa4f24af08f3eafd987d.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in V _ { k } \\cap \\Omega } \\frac { 1 } { d _ { w } + 1 } \\le \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { | V _ { k } \\cap \\Omega | } { 4 } } } \\\\ & { } & { \\le ( \\frac { 1 } { C } \\frac { | \\hat { Q } _ { j } | } { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } + \\frac { 2 } { C } ) \\frac { | \\Omega | } { 4 } \\le ( \\frac { 1 } { C } \\frac { | \\hat { Q } _ { j } | } { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } + \\frac { 2 } { C } ) \\frac { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } { 4 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 281, + 769, + 371 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "by definition of $\\Omega$ . We now bound the second term in equation 14 as ", + "bbox": [ + 173, + 375, + 622, + 390 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/27847d1c42582677aa26c7195856d42ddf3e64d957b4388725272affc400669d.jpg", + "text": "$$\n\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in V _ { k } \\setminus \\Omega } \\frac { 1 } { d _ { w } + 1 } \\leq \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } | V _ { k } \\setminus \\Omega |\n$$", + "text_format": "latex", + "bbox": [ + 284, + 395, + 712, + 438 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where we have used that $d _ { w } ^ { - 1 } \\leq C | \\sharp _ { \\Delta } | / | \\hat { Q } _ { j } |$ if $w \\in V _ { k } \\setminus \\Omega$ . Since ", + "bbox": [ + 173, + 445, + 606, + 464 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/b560d20e6d08863233482ae5f9c2e58e35b263065b263e465c9dbb89969b904d.jpg", + "text": "$$\n\\frac { | V _ { k } \\setminus \\Omega | } { \\sqrt { d _ { k } + 1 } } \\leq \\frac { | V _ { k } | } { \\sqrt { | S _ { 1 } ( k ) | } } \\leq \\frac { | V _ { k } | } { \\sqrt { | V _ { k } | } } \\leq \\sqrt { | V _ { k } | } \\leq \\sqrt { | \\sharp _ { \\Delta } | }\n$$", + "text_format": "latex", + "bbox": [ + 323, + 468, + 676, + 506 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "we see that ", + "bbox": [ + 173, + 512, + 250, + 525 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/621604bc78a9a4832cc147688ecfa1038a795003f62cc92ad854103326517814.jpg", + "text": "$$\n\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } | V _ { k } \\setminus \\Omega | \\leq \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } \\sqrt { | \\sharp \\Delta | } | \\hat { Q } _ { j } | = C | \\sharp \\Delta | ^ { \\frac { 3 } { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 520, + 707, + 564 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We are now ready to complete the proof of the theorem. According to equation 11 it suffices to show that ", + "bbox": [ + 176, + 565, + 823, + 593 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/cf4a4b5b6a10b3ce955d57a38d5b77d865ecb8f2d9ef4f221364051eab389b30.jpg", + "text": "$$\n\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\delta ^ { 1 / 4 } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 589, + 584, + 630 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "From equation 12 and equation 13 we derive that the left hand side of the equation above is bounded by ", + "bbox": [ + 171, + 631, + 823, + 661 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/04e94f2c52890b4fd31f0b7e770bc146ceb3b9ea27a5967cb99952b5a01ddd5d.jpg", + "text": "$$\n\\begin{array} { r } { \\frac { 1 } { | Q _ { j } | } \\displaystyle \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\frac { 1 } { | Q _ { j } | } + \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d _ { w } + 1 } \\right) } \\\\ { \\leq \\delta + \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d _ { w } + 1 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 277, + 665, + 720, + 770 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where in the last inequality we have used Lemma 14 to bound $| Q _ { j } | ^ { - 1 }$ by $\\delta$ . In particular we note that if $\\sharp _ { \\Delta } = \\emptyset$ then ${ \\hat { Q } } _ { j } = \\emptyset$ , and the bound would be simply controlled by $\\delta$ as claimed. When $\\vert \\sharp _ { \\Delta } \\vert > 0$ , we can use equation 15 and equation 16 to estimate the second term from above by ", + "bbox": [ + 173, + 776, + 828, + 823 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/78d3bb8ebc7cde2d93ab31d327be997585260efb55bc68db7ff4a41e8622500b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d w + 1 } \\right) \\leq } \\\\ & { \\frac { 1 } { | Q _ { j } | } \\left( \\left( \\frac { 1 } { \\bar { C } } \\frac { | \\hat { Q } _ { j } | } { | \\sharp _ { \\Delta } | } + \\frac { 2 } { \\bar { C } } \\right) \\frac { | \\sharp _ { \\Delta } | } { 4 } \\right) + \\frac { 1 } { | Q _ { j } | } \\left( C | \\sharp _ { \\Delta } | ^ { \\frac { 3 } { 2 } } \\right) \\leq \\frac { 1 } { 4 } \\left( \\frac { 1 } { \\bar { C } } + \\frac { 2 | \\sharp _ { \\Delta } | } { \\bar { C } | Q _ { j } | } \\right) + \\frac { C | \\sharp _ { \\Delta } | } { | Q _ { j } | } \\sqrt { | \\sharp _ { \\Delta } | } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 827, + 794, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "By applying Lemma 14 we get ", + "bbox": [ + 174, + 103, + 379, + 118 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/f4bca66dbb8a95e63a1446d43c3ccb00c4f4b9bcd53bd6c7d80cbc445091cc2f.jpg", + "text": "$$\n\\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + \\frac { 2 | \\sharp \\Delta | } { C | Q _ { j } | } \\right) + \\frac { C | \\sharp \\Delta | } { | Q _ { j } | } \\sqrt { | \\sharp \\Delta | } \\leq \\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + 2 \\frac { \\delta } { C } \\right) + C \\delta \\sqrt { | \\sharp \\Delta | } .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 125, + 720, + 160 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We now choose $C = \\delta ^ { - 1 / 4 }$ , so that the previous quantity can be bounded by ", + "bbox": [ + 171, + 166, + 674, + 183 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/27166968dce570051905b32bb49e0ad01f70b1cdc6ce88093ae8ba6f991608c2.jpg", + "text": "$$\n\\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + 2 \\frac { \\delta } { C } \\right) + C \\delta \\sqrt { | \\sharp _ { \\Delta } | } \\leq \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } } \\sqrt { \\delta | \\sharp _ { \\Delta } | } \\leq \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } }\n$$", + "text_format": "latex", + "bbox": [ + 222, + 188, + 776, + 223 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where in the last inequality we have used equation 10. Therefore, we have shown that ", + "bbox": [ + 173, + 227, + 735, + 243 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/c31ddcaa5dda7a9fe4cf8d3b6228abd5da10869a3cbad58f4af7f345d018094d.jpg", + "text": "$$\n\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\delta + \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } } \\leq 3 \\delta ^ { \\frac { 1 } { 4 } }\n$$", + "text_format": "latex", + "bbox": [ + 323, + 248, + 673, + 289 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where we have used that $\\delta < 1$ . This completes the proof (once we absorb the extra factor 3 in the constant $\\alpha \\beta$ in equation 11). □ ", + "bbox": [ + 174, + 295, + 823, + 324 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Remark 15. The requirement $\\delta \\sqrt { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\ < \\ 1$ can be replaced by a more general bound $\\delta \\sqrt { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } < r$ . The argument above extends to this case up to renaming the constant $\\alpha \\beta$ in the statement so to include an extra factor $r$ . ", + "bbox": [ + 173, + 330, + 826, + 377 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We note that the condition $\\delta \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} < r$ would be stronger than the one appearing in (ii) of Theorem 4. In this regard, we recall that for a $d$ -tree the curvature satisfies $\\begin{array} { r } { \\operatorname { R i c } ( i , j ) = - 2 + \\frac { 4 } { d } } \\end{array}$ . ", + "bbox": [ + 173, + 378, + 825, + 409 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We can also prove Proposition 5: ", + "bbox": [ + 174, + 419, + 390, + 434 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proposition 5. If $\\operatorname { R i c } ( i , j ) \\geq k > 0$ for all $i \\sim j$ , then $\\textstyle \\lambda _ { 1 } / 2 \\geq h _ { G } \\geq { \\frac { k } { 2 } }$ ", + "bbox": [ + 176, + 436, + 647, + 454 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. This follows as an immediate Corollary of Theorem 2 and (Lin et al., 2011, Theorem 4.2). ", + "bbox": [ + 171, + 463, + 794, + 479 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Betweenness centrality to measure bottleneck. In equation 2 we have derived how the topology of the graph affects the dependence of the hidden node representation $h _ { i } ^ { ( r + 1 ) }$ on the input feature $x _ { s }$ , for nodes $i , s$ at distance $r + 1$ . We note that in this case $\\hat { A } _ { i s } ^ { r + 1 }$ is exactly measuring the number of minimal paths from to $s$ . If the receptive field $B _ { r + 1 } ( i )$ is a binary tree, then we have seen that the entry of the power matrix decays exponentially. The reason for such decay stems from the existence of exponentially many nodes in the receptive field combined with the lack of multiple minimal paths (shortcuts). When such conditions hold, most of the minimal paths go through the same nodes, which is exactly what happens for the tree where each node is in the minimal paths between different branches. Since the frequency in which a node appears in the minimal path of distinct pairs of nodes is measured by the betweenness centrality Freeman (1977), we propose a topological characterization of the ‘bottleneckedness’ of a graph as follows: ", + "bbox": [ + 173, + 493, + 826, + 654 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition 9 (bottleneck). The bottleneck-value of $G$ is $\\begin{array} { r } { b _ { G } : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } b ( i ) } \\end{array}$ , where $b ( i )$ denotes the betweenness centrality on node $i$ . ", + "bbox": [ + 176, + 656, + 821, + 686 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "From a standard combinatorial argument it follows that if $G$ is connected, then ", + "bbox": [ + 173, + 696, + 689, + 713 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/7c8b7f038bc6553e8a24c7dd9cf4a88d5a51dbfe47af33c0675480105d6a1ebc.jpg", + "text": "$$\nb _ { G } = \\frac { 1 } { n } \\sum _ { i , j } \\left( d _ { G } ( i , j ) - 1 \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 718, + 593, + 757 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We note that $b _ { G } = 0$ iff $G$ is the complete graph $K _ { n }$ . Therefore, $b _ { G }$ determines how far the given topology is from $K _ { n }$ , with the latter representing the limit case of a fully connected layer Alon $\\&$ Yahav (2021) where no bottleneck may occur as any pair of nodes would be neighbours. This further supports our intuition that the betweenness centrality is a good topological candidate for providing a global measurement of bottleneckedness in the graph. ", + "bbox": [ + 173, + 762, + 825, + 833 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "It also follows from equation 17 that any update to the graph topology consisting of edge additions would decrease $b _ { G }$ and thus reduce the bottleneck. The quantity $b _ { G }$ is global in nature though and hence lacks robustness. As a pedagogical example, consider a barbell $G ( m , 2 r + 1 )$ , with $m$ the size of the two cliques joined by a path of length $2 r + 1$ and focus on the edge $i \\sim j$ in the middle of such path. Nodes $i$ and $j$ are central to the graph, in the sense that most minimal paths go through them and indeed their betweenness centrality is $\\bar { b } ( i ) = b ( j ) = ( m + r ) ^ { 2 } + ( m + r )$ . If now we add a single edge joining the two cliques $K _ { m }$ , the values $b ( i )$ and $b ( j )$ decrease dramatically by $\\Omega ( m ^ { 2 } + r )$ . Since the operation is non-local, we see that the representation $h _ { i } ^ { ( \\ell ) }$ of any MPNN is unaffected by the edge addition for any $\\ell \\in ( 0 , r )$ , and similarly for $j$ . Eventually, if we keep adding edges, the receptive fields $B _ { s } ( i )$ will be affected for small values of $s$ as well: the drawback of such approach is that the resulting adjacency may be significantly different. Conversely, the curvature provides a more precise, local and hence robust way of controlling the bottleneck and hence the over-squashing problem. Nonetheless, we relate the betweenness centrality to the Jacobian of the hidden features. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 826, + 205 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Theorem 16. Given $i \\sim j$ , let $\\Omega _ { j } : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) \\cup \\{ j \\}$ . If $\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta ,$ , for $0 < \\delta <$ $( 1 + \\gamma _ { m a x } ) ^ { - 1 }$ , then ", + "bbox": [ + 173, + 207, + 823, + 238 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/a8f5ff6b2f238f70b9a64016cfa2a257a7a7872ab9a33f24c01c2be3070ff645.jpg", + "text": "$$\n\\frac { 1 } { | \\Omega _ { j } | } \\sum _ { k \\in \\Omega _ { j } } b ( k ) \\geq \\delta ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 236, + 575, + 275 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof. We rewrite the quantity in the statement as ", + "bbox": [ + 173, + 289, + 504, + 304 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/5e40ccf8c483923eeefe1d34016ed7cc28596d0320649039656ad9418faa100b.jpg", + "text": "$$\n{ \\frac { 1 } { \\left| \\sharp _ { \\Delta } \\right| + 1 } } \\left( \\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) + b ( j ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 306, + 604, + 357 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "By definition, given a node $k \\in V$ , the betweenness centrality of $k$ is given by ", + "bbox": [ + 173, + 359, + 684, + 375 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/71d9f578b9baf810e917a3064a8a8d03d72552e7edcdc83144d6227eca2ee13d.jpg", + "text": "$$\nb ( k ) : = \\sum _ { s , t \\in V : s \\neq k , t \\neq k } \\frac { \\sigma _ { s t } ( k ) } { \\sigma _ { s t } }\n$$", + "text_format": "latex", + "bbox": [ + 400, + 378, + 598, + 419 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $\\sigma _ { s t }$ is the number of minimal paths between $s$ and $t$ while $\\sigma _ { s t } ( k )$ is the number of minimal paths from $s$ to $t$ passing through $k$ . For convenience, we introduce the set $\\hat { Q _ { j } } \\subset Q _ { j }$ defined by ", + "bbox": [ + 173, + 422, + 826, + 454 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/21dfbb34539e61be528d5cda9f9df399831f7335e95cb5c40ba42c5fe5ba78b2.jpg", + "text": "$$\n\\hat { Q } _ { j } : = \\{ w \\in Q _ { j } : \\ \\sigma _ { i w } > 1 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 458, + 596, + 478 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Equivalently, $\\hat { Q } _ { j }$ consists of those nodes in $S _ { 1 } ( j ) \\setminus S _ { 1 } ( i )$ which form a 4-cycle based at $i \\sim j$ with a diagonal inside. Indeed, if $w \\in Q _ { j }$ and $\\sigma _ { i w } > 1$ , then there exists more than one minimal path between $i$ and $w$ , in addition to the one passing through node $j$ . For any such path there exists $k \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( w )$ . Since $w \\in Q _ { j }$ and $Q _ { j } \\cap { \\sharp \\underline { { \\breve { \\jmath } } } } = \\emptyset$ , we derive that $k \\in S _ { 1 } ( j )$ as well. We then get ", + "bbox": [ + 174, + 483, + 826, + 544 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/bb76815faac1f0782a381fac11cb58fd6ee7c5ce0fd7df976464c29ce492e922.jpg", + "text": "$$\n\\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } b ( k ) = \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sum _ { s , t \\in V : s \\neq k , t \\neq k } \\frac { \\sigma _ { s t } ( k ) } { \\sigma _ { s t } } \\geq \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sum _ { w \\in \\hat { Q } _ { j } } \\frac { \\sigma _ { i w } ( k ) } { \\sigma _ { i w } } = \\sum _ { w \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sigma _ { i w } } \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sigma _ { i w } ( k ) .\n$$", + "text_format": "latex", + "bbox": [ + 209, + 547, + 785, + 590 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "By summing $\\sigma _ { i w } ( k )$ for all $k \\in \\sharp _ { \\Delta }$ we obtain all the 2-long minimal paths between $i$ and $w$ with the exception of the one passing through $j$ : ", + "bbox": [ + 174, + 594, + 821, + 623 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/e1a36dedc62b5370d407420da24454eb1551c56fc39d524dceeeafd28b446b5c.jpg", + "text": "$$\n\\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) \\geq \\sum _ { w \\in \\hat { Q _ { j } } } \\frac { 1 } { \\sigma _ { i w } } ( \\sigma _ { i w } - 1 ) .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 626, + 612, + 669 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "On the other hand we also have ", + "bbox": [ + 173, + 671, + 382, + 685 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/cc77537ee3e7c98e0df51067884bffd1ecd457396b01af498457903e19433892.jpg", + "text": "$$\nb ( j ) = \\sum _ { s , t \\in V : s \\neq j , t \\neq j } \\frac { \\sigma _ { s t } ( j ) } { \\sigma _ { s t } } \\geq \\sum _ { z \\in Q _ { j } } \\frac { \\sigma _ { i z } ( j ) } { \\sigma _ { i z } } = \\sum _ { z \\in Q _ { j } } \\frac { 1 } { \\sigma _ { i z } } .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 688, + 686, + 728 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "By combining equation 18 and equation 19 we finally get ", + "bbox": [ + 174, + 732, + 552, + 747 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/1af42c25efb7b219bb1220f64cca319a02ecf901f42bb2c7e767a9d668c971c4.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( \\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) + b ( j ) ) \\geq \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( \\sum _ { w \\in \\widehat { \\mathcal { Q } } _ { j } } \\frac { 1 } { \\sigma _ { i w } } ( \\sigma _ { i w } - 1 ) + \\sum _ { z \\in Q _ { j } } \\frac { 1 } { \\sigma _ { i z } } ) } } \\\\ & { } & { = \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( | \\widehat { Q } _ { j } | + \\sum _ { z \\in Q _ { j } \\setminus \\widehat { Q } _ { j } } \\frac { 1 } { \\sigma _ { i z } } ) } \\\\ & { } & { = \\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 750, + 761, + 890 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where in the last equality we have used that by definition $\\sigma _ { i z } = 1$ for all $z \\in Q _ { j } \\backslash { \\hat { Q } } _ { j }$ . By Lemma 14 the last quantity is larger than $\\delta ^ { - 1 }$ . ", + "bbox": [ + 173, + 893, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E PROOFS OF RESULTS IN SECTION 4 ", + "text_level": 1, + "bbox": [ + 173, + 101, + 500, + 118 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Theorem 6. Let $S \\subset V$ with $\\nu o l ( S ) \\le \\nu o l ( G ) / 2$ . Then $\\begin{array} { r } { h _ { S , \\alpha } \\leq \\left( \\frac { 1 - \\alpha } { \\alpha } \\right) \\frac { d _ { \\arg } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } h _ { S } } \\end{array}$ , where $d _ { \\mathrm { a v g } } ( S )$ and $d _ { \\mathrm { m i n } } ( S )$ are the average and minimum degree on $S$ , respectively. ", + "bbox": [ + 173, + 130, + 821, + 166 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Given a signal $f : V \\to \\mathbb { R }$ on the vertex set and $U \\subset V$ , analogously to Chung (2007), we introduce the notation ", + "bbox": [ + 168, + 179, + 825, + 207 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/6e6d448086dabe8e193aa4d7cca544cd3b2f04b3a98e6783b42e5926df9cdb83.jpg", + "text": "$$\nf ( U ) : = \\sum _ { i \\in U } f ( i ) .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 203, + 560, + 237 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Let us rewrite the new Cheeger constant $h _ { S , \\alpha }$ as follows: ", + "bbox": [ + 173, + 239, + 547, + 255 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/0bd47bf2694fb8c6aaba7f19ca7384952b2acd58e199a707e29ace4a3217cc5a.jpg", + "text": "$$\nh _ { S , \\alpha } = \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } = \\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } )\n$$", + "text_format": "latex", + "bbox": [ + 354, + 258, + 643, + 300 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "with $\\chi _ { S }$ the characteristic function of the subset $S$ , i.e. $\\chi _ { S } ( i ) = 1$ iff $i \\in S$ . Since the graph $G$ is connected, we can bound $h _ { S , \\alpha }$ from above as ", + "bbox": [ + 174, + 305, + 825, + 334 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/9e977a7e91afca18aa536aac501f0755aeef5919118e7104f9ce85aa3d779620.jpg", + "text": "$$\n\\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } \\le \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } \\frac { d _ { i } } { d _ { \\operatorname* { m i n } } ( S ) } = \\frac { 1 } { | S | } \\chi _ { S } D R _ { \\alpha } ( \\bar { S } ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 339, + 807, + 381 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "It was proven in (Chung, 2007, Lemma 5) that ", + "bbox": [ + 173, + 385, + 478, + 400 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/875a4317dcf9b55a862e2e4c3b60965a9bbc026b0ad2aa1e073eb2ce9bf66797.jpg", + "text": "$$\n\\chi _ { S } D R _ { \\alpha } ( { \\bar { S } } ) \\leq { \\frac { 1 - \\alpha } { \\alpha } } | \\partial S | .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 405, + 589, + 435 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "By applying equation equation 20 to the bound for the Cheeger constant $h _ { S , \\alpha }$ we finally see that ", + "bbox": [ + 171, + 439, + 803, + 455 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/5cf5a7f1e6da2fe054d67396942aab7784699787d96a9ebc36fa27a7fe85d8a4.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { h _ { S , \\alpha } = \\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } ) \\leq \\frac { 1 } { | S | } \\chi _ { S } D R _ { \\alpha } ( \\bar { S } ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } \\leq \\frac { 1 } { | S | } \\frac { 1 - \\alpha } { \\alpha } | \\partial S | \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } } } \\\\ { \\displaystyle { \\quad = \\frac { 1 } { | S | } \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\mathrm { v o l } ( S ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } = \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\frac { d _ { \\mathrm { a v g } } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 459, + 748, + 531 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We also report an equivalent result, again relying on (Chung, 2007, Lemma 5). ", + "bbox": [ + 171, + 563, + 689, + 578 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition 17. Let $S \\subset V$ with $\\nu o l ( S ) \\le \\nu o l ( G ) / 2$ . For any $k \\in \\mathbb N$ , there exists $S _ { k , \\alpha } \\subset S$ with $\\nu o l ( S _ { k , \\alpha } ) \\geq \\nu o l ( S ) ( 1 - ( 2 k ) ^ { - 1 } )$ such that ", + "bbox": [ + 173, + 580, + 820, + 611 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/8e1ea37a27a397897ab7b1e2958104c957ea65035a65b83b20ef147f9f6bb01c.jpg", + "text": "$$\n\\sum _ { j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } \\le k \\frac { 1 - \\alpha } { \\alpha } h _ { S } ,\n$$", + "text_format": "latex", + "bbox": [ + 411, + 616, + 583, + 656 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "for all $i \\in S _ { k , \\alpha }$ ", + "bbox": [ + 173, + 661, + 279, + 676 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Let $k \\in \\mathbb N$ . By modifying slightly the argument in (Chung, 2007, Lemma 5), we derive that ", + "bbox": [ + 174, + 690, + 820, + 707 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/bb70030d6858dc6927b26ec824be3af52ac59b4637404c40078e4ad882c49253.jpg", + "text": "$$\nS _ { k , \\alpha } ^ { \\prime } : = \\{ i \\in S : \\chi _ { i } R _ { \\alpha } ( \\bar { S } ) \\geq k \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 710, + 640, + 741 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "satisfies ", + "bbox": [ + 173, + 746, + 228, + 760 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/a1d9525e32659b6a5f5e9d8766a58c3b7f5020995e7d44034a8c284c2d255cb8.jpg", + "text": "$$\n\\frac { 1 - \\alpha } { 2 \\alpha } | \\partial S | \\geq \\mathrm { v o l } ( S _ { k , \\alpha } ^ { \\prime } ) k \\frac { 1 - \\alpha } { \\alpha } h _ { S } .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 755, + 617, + 785 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Therefore, we obtain ", + "bbox": [ + 174, + 785, + 312, + 800 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/98154137e559de0a55680dbfb3d32b63cc3896a92a27d929da2caf8663d72c64.jpg", + "text": "$$\n\\operatorname { v o l } ( S _ { k , \\alpha } ^ { \\prime } ) \\leq { \\frac { 1 } { k } } { \\frac { \\operatorname { v o l } ( S ) } { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 797, + 575, + 829 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We then conclude that the complement of $S _ { k , \\alpha } ^ { \\prime }$ has volume greater or equal than $\\operatorname { v o l } ( S ) ( 1 - ( 2 k ) ^ { - 1 } )$ , which completes the proof. ", + "bbox": [ + 173, + 830, + 826, + 861 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Remark 18. The previous proposition shows that after sparsifying the personalized page rank operator $R _ { \\alpha }$ as suggested in Klicpera et al. (2019) by setting entries below some threshold equal to zero, there will still be only few edges connecting different communities, once again highlighting that random-walk based methods are generally not suited to tackle the graph bottleneck. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "F EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 328, + 117 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Our experiments in this paper are semi-supervised node classification (semi-supervised in that the graph structure provides some unlabelled information) on nine common graph learning datasets. Cornell, Texas and Wisconsin are small heterophilic datasets based on webpage networks from the WebKB dataset. Chameleon and Squirrel (Rozemberczki et al., 2021) are medium heterophilic datasets based on Wikipedia networks, along with Actor, the actor-only induced subgraph of the filmdirector-actor-writer network (Tang et al., 2009). Cora (McCallum et al., 2000), Citeseer (Sen et al., 2008) and Pubmed (Namata et al., 2012) are medium homophilic datasets based on citation networks. As in Klicpera et al. (2019), for all experiments we consider the largest connected component of the graph. ", + "bbox": [ + 174, + 133, + 826, + 258 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "When splitting the data into train/validation/test sets, we first separate the data into a development set and the test set. This is done once to ensure the test set is not used for any training or hyperparameter fitting before the final evaluation. For each of the 100 random splits the development set is divided randomly into a train set and a validation set, where we train models on the train set and evaluate on the validation set. We fit hyperparameters by random search, maximising the mean accuracy across the validation sets. The accuracy then reported in Table 2 is the mean accuracy on the test set from models trained on the train sets with the chosen hyperparameters, along with a $9 5 \\%$ confidence interval calculated by bootstrapping the test set accuracies with 1000 samples. For Cora, Citeseer and Pubmed the development set contains 1500 nodes with the rest kept for the test set, and for each random split the train set is chosen to contain 20 nodes of each class while the rest form the validation set. As this is the same method as Klicpera et al. (2019) and we use the same random seeds, we are using the same test set and expect to have comparable results. For the remaining datasets we perform a 60/20/20 split, with $20 \\%$ of nodes set aside as the test set and then for each random split the remaining $80 \\%$ is split into $60 \\%$ training, $20 \\%$ validation. ", + "bbox": [ + 173, + 265, + 826, + 460 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The homophily index $\\mathcal { H } ( G )$ proposed by Pei et al. (2019) is defined as ", + "bbox": [ + 174, + 467, + 635, + 482 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/54112c1ada2b2e184522e173a738622a1263f5983dccd16331a07e6c564e0967.jpg", + "text": "$$\n\\mathcal { H } ( G ) = \\frac { 1 } { | V | } \\sum _ { v \\in V } \\frac { \\mathrm { N u m b e r ~ o f ~ } v \\mathrm { ' s ~ n e i g h b o r s ~ w h o ~ h a v e ~ t h e ~ s a m e ~ l a b e l ~ a s ~ } v } { \\mathrm { N u m b e r ~ o f ~ } v \\mathrm { ' s ~ n e i g h b o r s } } .\n$$", + "text_format": "latex", + "bbox": [ + 253, + 488, + 743, + 527 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "F.1 DATASETS ", + "text_level": 1, + "bbox": [ + 173, + 541, + 287, + 556 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/106d1eb0ea6c73f1d9a69a02d2b0b3d1fcaa9de3b9a40920e5523e1ee7d7e61f.jpg", + "table_caption": [ + "For datasets with disconnected graphs, the statistics shown here are for the largest connected component. " + ], + "table_footnote": [], + "table_body": "
CornellTexasWisconsinChameleonSquirrelActorCoraCiteseerPubmed
H(G)0.110.060.160.250.220.240.830.720.79
Nodes140135184832218643882485212019717
Edges2192513621235565224219075069367944324
Features1703170317032323208993114333703500
Classes555555763
Directed?YesYesYesYesYesYesNoNoNo
", + "bbox": [ + 163, + 599, + 833, + 703 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/7e3012a9f8a829393e71b4047a8695aea3b326896fc24c8b02d1925a1a9c5594.jpg", + "image_caption": [ + "Figure 5: Comparing the degree distribution of the original graphs to the preprocessed version. The xaxis is node degree in $\\log _ { 2 }$ scale, and the plots are a kernel density estimate of the degree distribution. In the captions we see the Wasserstein distance $W _ { 1 }$ between the original and preprocessed graphs. " + ], + "image_footnote": [], + "bbox": [ + 179, + 127, + 818, + 691 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/b9e9ad9e82055e01bd8d3c2a9fdf721719f922616fb3e5a8ed8f710f5b3ae409.jpg", + "image_caption": [ + "Figure 6: Rewiring of the Cornell graph. Left-to-right: original graph, DIGL, and SDRF rewiring. Edges are colored by curvature; nodes are colored by the maximum absolute entry of a trained 2-layer GCN’s Jacobian between the GCN’s prediction for that node and the features of the nodes 2 hops away in the original graph. SDRF homogenizes curvature and so lifts the upper bound on the Jacobian from Theorem 4. DIGL also does to an extent, though at the expense of preserving graph topology. " + ], + "image_footnote": [], + "bbox": [ + 173, + 133, + 825, + 275 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "F.4 HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 383, + 356, + 397 + ], + "page_idx": 27 + }, + { + "type": "table", + "img_path": "images/58322d9e5130cf23a5f6b67a134305df1a9280cd2e69f75a66ca0b4ba72fd1ff.jpg", + "table_caption": [ + "Table 4: Hyperparameters for GCN with no preprocessing (None). " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
Cornell0.306011280.00820.1570
Texas0.234611280.00720.0037
Wisconsin0.28691640.02810.0113
Chameleon0.730431280.02480.0936
Squirrel0.59742640.01360.1346
Actor0.76051640.02900.0619
Cora0.41441640.00970.0639
Citeseer0.747711280.02510.4577
Pubmed0.40131640.00950.0448
", + "bbox": [ + 281, + 439, + 717, + 593 + ], + "page_idx": 27 + }, + { + "type": "table", + "img_path": "images/1d72f7010b29995509484ec6002efc1532d0d97fa997b30b12a9a8de59c6413e.jpg", + "table_caption": [ + "Table 5: Hyperparameters for GCN with the input graph made undirected (Undirected). " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
Cornell0.69101640.01850.0285
Texas0.266511280.00690.0035
Wisconsin0.289321280.01420.0001
Chameleon0.46573640.01890.0423
Squirrel0.59442640.00810.0309
Actor0.66262640.01950.0219
", + "bbox": [ + 281, + 674, + 717, + 791 + ], + "page_idx": 27 + }, + { + "type": "table", + "img_path": "images/99cc3426a3e7f162e2188f8fa18f1fa5f81c395668a74355a70b6d8ca5b38003.jpg", + "table_caption": [ + "Table 6: Hyperparameters for GCN with the last layer made fully connected ( $+ \\mathrm { F A }$ from Alon & Yahav (2021)). " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
Cornell0.264311280.02160.0760
Texas0.220711280.01020.4450
Wisconsin0.26133640.00570.0131
Chameleon0.77833640.01560.0108
Squirrel0.36543640.00770.1922
Actor0.38241640.01650.1168
Cora0.784021280.01490.1429
Citeseer0.54602640.00660.0758
Pubmed0.337621280.02040.0215
", + "bbox": [ + 281, + 141, + 717, + 296 + ], + "page_idx": 28 + }, + { + "type": "table", + "img_path": "images/3bdd04df1f80ef3fe57fae9bb6607b9ffc6a3b6318cb949b0a6d6800952f8ed3.jpg", + "table_caption": [ + "Table 7: Hyperparameters for GCN with DIGL preprocessing, or Graph Diffusion Convolution with PPR kernel (DIGL). Descriptions for $\\alpha$ , $k$ and $\\epsilon$ can be found in Klicpera et al. (2019). " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayαkE
Cornell0.62941640.01340.02580.179564-
Texas0.238221280.00630.01530.020632-
Wisconsin0.294111280.00830.02260.124610.0001
Chameleon0.419111280.00520.00010.0244641
Squirrel0.684411280.00560.45370.039532
Actor0.78201640.01700.01020.1584128、
Cora0.33151640.02840.05720.07731281
Citeseer0.55611640.00940.50130.1076-0.0008
Pubmed0.491521280.00570.05970.1155128-
", + "bbox": [ + 199, + 392, + 797, + 546 + ], + "page_idx": 28 + }, + { + "type": "table", + "img_path": "images/203d499c46c2cd277e65196c4ad72dcd74dd1b6a294d51785e2b73380bc09a3a.jpg", + "table_caption": [ + "Table 8: Hyperparameters for GCN with DIGL preprocessing followed by symmetrizing the graph diffusion matrix (DIGL $^ +$ Undirected). " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayαkE
Cornell0.62941640.01340.02580.179564=
Texas0.238221280.00630.01530.0206321
Wisconsin0.294111280.00830.02260.124610.0001
Chameleon0.419111280.00520.00010.0244641
Squirrel0.70941640.01720.01920.1610641
Actor0.40121640.01610.01410.070610.0016
Cora0.33151640.02840.05720.0773128-
Citeseer0.55611640.00940.50130.1076-0.0008
Pubmed0.491521280.00570.05970.11551281
", + "bbox": [ + 199, + 642, + 797, + 795 + ], + "page_idx": 28 + }, + { + "type": "table", + "img_path": "images/4923af42370bc40c5af95f7b4ef2cf3df63a99782bf1bb2afae4521fa5b01292.jpg", + "table_caption": [ + "Table 9: Hyperparameters for GCN with SDRF preprocessing (SDRF). Max iterations, $\\tau$ and $C ^ { + }$ are the SDRF parameters described in Algorithm 1. " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayMax iterationsTC+
Cornell0.241111280.01720.01251351300.25
Texas0.595411280.02780.0623471722.25
Wisconsin0.603311280.02950.192027320.5
Chameleon0.726511280.01800.2101832773.35
Squirrel0.74012160.01890.225561571780.5
Actor0.688611280.00950.07271010691.22
Cora0.339611280.02440.10761001630.95
Citeseer0.41031640.01990.4551841800.22
Pubmed0.374931280.01120.013816611514.43
", + "bbox": [ + 196, + 141, + 802, + 295 + ], + "page_idx": 29 + }, + { + "type": "table", + "img_path": "images/876261df6066c0aeb457d0eabbcc5b731b0897aed91fdb252da3a33044ebb3a9.jpg", + "table_caption": [ + "Table 10: Hyperparameters for GCN with the input graph made undirected followed by SDRF preprocessing (SDRF $^ +$ Undirected). " + ], + "table_footnote": [], + "table_body": "
DatasetDropoutHidden depthHidden dimensionLearning rateWeight decayMax iterationsTC+
Cornell0.291111280.00560.03361261450.88
Texas0.21601640.02290.013789221.64
Wisconsin0.24521640.01130.1559136127.95
Chameleon0.48861320.02680.405624412522.84
Squirrel0.42491640.02950.13977874317.19
Actor0.670511280.01150.044711414411.17
", + "bbox": [ + 196, + 348, + 800, + 463 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "G HARDWARE SPECIFICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 487, + 447, + 503 + ], + "page_idx": 29 + }, + { + "type": "table", + "img_path": "images/f2507b8262efb7105b2042c26d045ecbe39fe2363ef6d189e8ceb596ea568208.jpg", + "table_caption": [ + "Our experiments were performed on a server with the following specifications: " + ], + "table_footnote": [], + "table_body": "
ComponentSpecification
Architecturex86_64
CPU40x Intel(R) Xeon(R) Silver 4210R CPU @ 2.40GHz
GPU4x GeForce RTX 3090 (24268MiB/GPU)
RAM126GB
OSUbuntu 20.04.2LTS
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Consequently, there is currently no consensus on the right method (either", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 494, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 494, + 108 + ], + "score": 1.0, + "content": "based on graph rewiring or not) to address the bottleneck and hence alleviate the over-squashing.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 54, + "bbox_fs": [ + 105, + 611, + 506, + 714 + ] + } + ] + }, + { + "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": "graph are still elusive. Consequently, there is currently no consensus on the right method (either", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 494, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 494, + 108 + ], + "score": 1.0, + "content": "based on graph rewiring or not) to address the bottleneck and hence alleviate the over-squashing.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "In this paper, we address these questions using tools from differential geometry, which traditionally", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "is concerned with the study of manifolds. 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One notion of curvature that has received attention for graph learning is Ricci", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "curvature (Hamilton, 1988), also known in geometry for its use in Ricci flow and the subsequent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "proof of the Poincaré conjecture (Perelman, 2003). Certain graph analogues of the Ricci curvature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "(Forman, 2003; Ollivier, 2009; Sreejith et al., 2016) were used in Ni et al. 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This paper, to our knowledge, is the first theoretical study of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "bottleneck and over-squashing phenomena in message passing neural networks from a geometric", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "perspective. In Section 2, we propose the Jacobian of node representations as a formal way of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "measuring the over-squashing and we show that the graph topology may compromise message", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "propagation in graph neural networks by creating a bottleneck. In Section 3, we investigate how such", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "a bottleneck is induced which leads to the over-squashing of information. To this aim, we introduce a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 308, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 505, + 323 + ], + "score": 1.0, + "content": "new combinatorial edge-based curvature called Balanced Forman curvature that constitutes a sharp", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "lower bound to the standard Ollivier curvature on graphs, and prove that negatively curved edges are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "responsible for the formation of bottlenecks (and hence for over-squashing). In Section 4, we present", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "a new curvature-based method for graph rewiring called Stochastic Discrete Ricci Flow. According", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "to the theoretical results in Section 3, this rewiring method is suited to address the graph bottleneck", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "and hence alleviate the over-squashing by surgically targeting the edges responsible for the issue.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "By contrast, we rigorously show that a recently introduced diffusion-based rewiring scheme might", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "generally fail to reduce the bottleneck. Finally, in Section 5, we compare different rewiring strategies", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 344, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 344, + 410 + ], + "score": 1.0, + "content": "experimentally on several standard graph learning datasets.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 390, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 392, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 392, + 439 + ], + "score": 1.0, + "content": "2 ANALYSIS OF THE OVER-SQUASHING PHENOMENON", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 449, + 200, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 201, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 201, + 462 + ], + "score": 1.0, + "content": "2.1 PRELIMINARIES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 469, + 506, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 122, + 483 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 470, + 171, + 482 + ], + "score": 0.92, + "content": "G = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 470, + 384, + 483 + ], + "score": 1.0, + "content": "be a simple, undirected, and connected graph, where", + "type": "text" + }, + { + "bbox": [ + 384, + 470, + 426, + 482 + ], + "score": 0.93, + "content": "( i , j ) \\in E", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 470, + 438, + 483 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 439, + 470, + 461, + 481 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 470, + 505, + 483 + ], + "score": 1.0, + "content": ". We focus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "on the unweighted case, although the theory extends to the weighted setting as well. We denote", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 206, + 506 + ], + "score": 1.0, + "content": "the adjacency matrix by", + "type": "text" + }, + { + "bbox": [ + 207, + 493, + 215, + 503 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 492, + 246, + 506 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 246, + 491, + 294, + 504 + ], + "score": 0.94, + "content": "\\overset { \\vartriangle } { \\boldsymbol { \\tilde { A } } } = \\boldsymbol { A } + \\boldsymbol { \\bar { I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "be the adjacency matrix augmented with self-loops.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 174, + 518 + ], + "score": 1.0, + "content": "Similarly we let", + "type": "text" + }, + { + "bbox": [ + 174, + 504, + 224, + 516 + ], + "score": 0.92, + "content": "\\tilde { D } = D + I", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 504, + 249, + 518 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 249, + 506, + 259, + 515 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 504, + 404, + 518 + ], + "score": 1.0, + "content": "the diagonal degree matrix, and let", + "type": "text" + }, + { + "bbox": [ + 405, + 504, + 476, + 516 + ], + "score": 0.92, + "content": "\\bar { { \\cal A } } = \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 507, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 507, + 529 + ], + "score": 1.0, + "content": "normalized augmented adjacency matrix (self-loops are commonly included in GNN architecture,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "and in Section 2 we formally explain why GNNs are expected to propagate information more reliably", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 538, + 461, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 295, + 551 + ], + "score": 1.0, + "content": "when self-loops are taken into account). Given", + "type": "text" + }, + { + "bbox": [ + 295, + 539, + 320, + 549 + ], + "score": 0.9, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 538, + 420, + 551 + ], + "score": 1.0, + "content": ", we denote its degree by", + "type": "text" + }, + { + "bbox": [ + 420, + 539, + 430, + 550 + ], + "score": 0.87, + "content": "d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 538, + 461, + 551 + ], + "score": 1.0, + "content": "and let", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 554, + 447, + 568 + ], + "lines": [ + { + "bbox": [ + 163, + 554, + 447, + 568 + ], + "spans": [ + { + "bbox": [ + 163, + 554, + 447, + 568 + ], + "score": 0.88, + "content": "S _ { r } ( i ) : = \\{ j \\in V : d _ { G } ( i , j ) = r \\} , \\quad B _ { r } ( i ) : = \\{ j \\in V : d _ { G } ( i , j ) \\leq r \\} ,", + "type": "interline_equation", + "image_path": "94ea58d66e8dd4f5769e5f0da089230511e450c7950d880ffecb2f6496b2ecf9.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 163, + 554, + 447, + 568 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 132, + 585 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 573, + 145, + 583 + ], + "score": 0.88, + "content": "d _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 572, + 362, + 585 + ], + "score": 1.0, + "content": "is the standard shortest-path distance on the graph and", + "type": "text" + }, + { + "bbox": [ + 362, + 573, + 387, + 583 + ], + "score": 0.9, + "content": "r \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 572, + 423, + 585 + ], + "score": 1.0, + "content": ". The set", + "type": "text" + }, + { + "bbox": [ + 423, + 572, + 447, + 585 + ], + "score": 0.92, + "content": "B _ { r } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "represents the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 584, + 389, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 187, + 596 + ], + "score": 1.0, + "content": "receptive field of an", + "type": "text" + }, + { + "bbox": [ + 188, + 586, + 194, + 594 + ], + "score": 0.79, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 584, + 380, + 596 + ], + "score": 1.0, + "content": "-layer message passing neural network at node", + "type": "text" + }, + { + "bbox": [ + 381, + 585, + 385, + 594 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 584, + 389, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 401, + 619 + ], + "score": 1.0, + "content": "Message passing neural networks (MPNNs). Assume that the graph", + "type": "text" + }, + { + "bbox": [ + 401, + 608, + 410, + 617 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "is equipped with node", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 616, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 140, + 631 + ], + "score": 1.0, + "content": "features", + "type": "text" + }, + { + "bbox": [ + 140, + 618, + 189, + 628 + ], + "score": 0.92, + "content": "X \\in \\mathbb { R } ^ { n \\times p _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 616, + 217, + 631 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 217, + 618, + 254, + 629 + ], + "score": 0.92, + "content": "x _ { i } \\in \\mathbb { R } ^ { p _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 616, + 367, + 631 + ], + "score": 1.0, + "content": "is the feature vector at node", + "type": "text" + }, + { + "bbox": [ + 367, + 618, + 444, + 630 + ], + "score": 0.92, + "content": "i = 1 , \\ldots , n = | V |", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 616, + 505, + 631 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 626, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 151, + 644 + ], + "score": 0.93, + "content": "h _ { i } ^ { ( \\ell ) } \\in \\mathbb { R } ^ { p _ { \\ell } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 626, + 261, + 648 + ], + "score": 1.0, + "content": "the representation of node", + "type": "text" + }, + { + "bbox": [ + 261, + 633, + 266, + 642 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 626, + 299, + 648 + ], + "score": 1.0, + "content": "at layer", + "type": "text" + }, + { + "bbox": [ + 300, + 631, + 324, + 643 + ], + "score": 0.89, + "content": "\\ell \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 626, + 349, + 648 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 349, + 629, + 389, + 644 + ], + "score": 0.91, + "content": "h _ { i } ^ { ( 0 ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 626, + 507, + 648 + ], + "score": 1.0, + "content": ". Given a family of message", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 147, + 657 + ], + "score": 1.0, + "content": "functions", + "type": "text" + }, + { + "bbox": [ + 147, + 643, + 240, + 657 + ], + "score": 0.9, + "content": "\\psi _ { \\ell } : \\mathbb { R } ^ { p _ { \\ell } } \\times \\mathbb { R } ^ { p _ { \\ell } } \\to \\mathbb { R } ^ { p _ { \\ell } ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 644, + 330, + 657 + ], + "score": 1.0, + "content": "and update functions", + "type": "text" + }, + { + "bbox": [ + 330, + 644, + 431, + 657 + ], + "score": 0.9, + "content": "\\phi _ { \\ell } : \\mathbb { R } ^ { p _ { \\ell } } \\times \\mathbb { R } ^ { p _ { \\ell } ^ { \\prime } } \\to \\mathbb { R } ^ { p _ { \\ell + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", we can write the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 656, + 408, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 135, + 668 + ], + "score": 0.91, + "content": "( \\ell + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 656, + 408, + 668 + ], + "score": 1.0, + "content": "-st layer output of a generic MPNN as follows (Gilmer et al., 2017):", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 669, + 396, + 709 + ], + "lines": [ + { + "bbox": [ + 216, + 669, + 396, + 709 + ], + "spans": [ + { + "bbox": [ + 216, + 669, + 396, + 709 + ], + "score": 0.95, + "content": "h _ { i } ^ { ( \\ell + 1 ) } = \\phi _ { \\ell } \\left( h _ { i } ^ { ( \\ell ) } , \\sum _ { j = 1 } ^ { n } \\hat { A } _ { i j } \\psi _ { \\ell } ( h _ { i } ^ { ( \\ell ) } , h _ { j } ^ { ( \\ell ) } ) \\right) .", + "type": "interline_equation", + "image_path": "5acf9ad2264bec6c6abd19514311fa3932c1204ea0500ab6b7a1760d2f264ee1.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 216, + 669, + 396, + 682.3333333333334 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 216, + 682.3333333333334, + 396, + 695.6666666666667 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 216, + 695.6666666666667, + 396, + 709.0000000000001 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "Here we have used the augmented normalized adjacency matrix to propagate messages from each", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 477, + 734 + ], + "score": 1.0, + "content": "node to its neighbors, which simply leads to a degree normalization of the message functions", + "type": "text" + }, + { + "bbox": [ + 477, + 722, + 488, + 732 + ], + "score": 0.88, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 721, + 505, + 734 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "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": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 108 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "In this paper, we address these questions using tools from differential geometry, which traditionally", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "is concerned with the study of manifolds. It offers an appealing framework to study the properties", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "of graphs, in particular arguing that graphs, like manifolds, exhibit curvature that makes them more", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "suitable to be realized in spaces with hyperbolic geometry (Liu et al., 2019; Chami et al., 2019;", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "Boguna et al., 2021). One notion of curvature that has received attention for graph learning is Ricci", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "curvature (Hamilton, 1988), also known in geometry for its use in Ricci flow and the subsequent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "proof of the Poincaré conjecture (Perelman, 2003). Certain graph analogues of the Ricci curvature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "(Forman, 2003; Ollivier, 2009; Sreejith et al., 2016) were used in Ni et al. (2018) for a discrete version", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "of Ricci flow to construct a metric between graphs. Graph Ricci flow was also used in Ni et al. (2019)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "for community detection. Both of these methods use the edge weights as a substitute for the metric of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 381, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 381, + 233 + ], + "score": 1.0, + "content": "a manifold, and do not change the topological structure of the graph.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 111, + 506, + 233 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 242, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 257 + ], + "score": 1.0, + "content": "Contributions and Outline. This paper, to our knowledge, is the first theoretical study of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "bottleneck and over-squashing phenomena in message passing neural networks from a geometric", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "perspective. In Section 2, we propose the Jacobian of node representations as a formal way of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "measuring the over-squashing and we show that the graph topology may compromise message", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "propagation in graph neural networks by creating a bottleneck. In Section 3, we investigate how such", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "a bottleneck is induced which leads to the over-squashing of information. To this aim, we introduce a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 308, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 505, + 323 + ], + "score": 1.0, + "content": "new combinatorial edge-based curvature called Balanced Forman curvature that constitutes a sharp", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "lower bound to the standard Ollivier curvature on graphs, and prove that negatively curved edges are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "responsible for the formation of bottlenecks (and hence for over-squashing). In Section 4, we present", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "a new curvature-based method for graph rewiring called Stochastic Discrete Ricci Flow. According", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "to the theoretical results in Section 3, this rewiring method is suited to address the graph bottleneck", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "and hence alleviate the over-squashing by surgically targeting the edges responsible for the issue.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "By contrast, we rigorously show that a recently introduced diffusion-based rewiring scheme might", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 399 + ], + "score": 1.0, + "content": "generally fail to reduce the bottleneck. Finally, in Section 5, we compare different rewiring strategies", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 344, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 344, + 410 + ], + "score": 1.0, + "content": "experimentally on several standard graph learning datasets.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 242, + 506, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 390, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 392, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 392, + 439 + ], + "score": 1.0, + "content": "2 ANALYSIS OF THE OVER-SQUASHING PHENOMENON", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 449, + 200, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 201, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 201, + 462 + ], + "score": 1.0, + "content": "2.1 PRELIMINARIES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 469, + 506, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 122, + 483 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 470, + 171, + 482 + ], + "score": 0.92, + "content": "G = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 470, + 384, + 483 + ], + "score": 1.0, + "content": "be a simple, undirected, and connected graph, where", + "type": "text" + }, + { + "bbox": [ + 384, + 470, + 426, + 482 + ], + "score": 0.93, + "content": "( i , j ) \\in E", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 470, + 438, + 483 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 439, + 470, + 461, + 481 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 470, + 505, + 483 + ], + "score": 1.0, + "content": ". We focus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "on the unweighted case, although the theory extends to the weighted setting as well. We denote", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 206, + 506 + ], + "score": 1.0, + "content": "the adjacency matrix by", + "type": "text" + }, + { + "bbox": [ + 207, + 493, + 215, + 503 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 492, + 246, + 506 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 246, + 491, + 294, + 504 + ], + "score": 0.94, + "content": "\\overset { \\vartriangle } { \\boldsymbol { \\tilde { A } } } = \\boldsymbol { A } + \\boldsymbol { \\bar { I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "be the adjacency matrix augmented with self-loops.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 174, + 518 + ], + "score": 1.0, + "content": "Similarly we let", + "type": "text" + }, + { + "bbox": [ + 174, + 504, + 224, + 516 + ], + "score": 0.92, + "content": "\\tilde { D } = D + I", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 504, + 249, + 518 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 249, + 506, + 259, + 515 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 504, + 404, + 518 + ], + "score": 1.0, + "content": "the diagonal degree matrix, and let", + "type": "text" + }, + { + "bbox": [ + 405, + 504, + 476, + 516 + ], + "score": 0.92, + "content": "\\bar { { \\cal A } } = \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 507, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 507, + 529 + ], + "score": 1.0, + "content": "normalized augmented adjacency matrix (self-loops are commonly included in GNN architecture,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "and in Section 2 we formally explain why GNNs are expected to propagate information more reliably", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 538, + 461, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 295, + 551 + ], + "score": 1.0, + "content": "when self-loops are taken into account). Given", + "type": "text" + }, + { + "bbox": [ + 295, + 539, + 320, + 549 + ], + "score": 0.9, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 538, + 420, + 551 + ], + "score": 1.0, + "content": ", we denote its degree by", + "type": "text" + }, + { + "bbox": [ + 420, + 539, + 430, + 550 + ], + "score": 0.87, + "content": "d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 538, + 461, + 551 + ], + "score": 1.0, + "content": "and let", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 470, + 507, + 551 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 554, + 447, + 568 + ], + "lines": [ + { + "bbox": [ + 163, + 554, + 447, + 568 + ], + "spans": [ + { + "bbox": [ + 163, + 554, + 447, + 568 + ], + "score": 0.88, + "content": "S _ { r } ( i ) : = \\{ j \\in V : d _ { G } ( i , j ) = r \\} , \\quad B _ { r } ( i ) : = \\{ j \\in V : d _ { G } ( i , j ) \\leq r \\} ,", + "type": "interline_equation", + "image_path": "94ea58d66e8dd4f5769e5f0da089230511e450c7950d880ffecb2f6496b2ecf9.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 163, + 554, + 447, + 568 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 132, + 585 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 573, + 145, + 583 + ], + "score": 0.88, + "content": "d _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 572, + 362, + 585 + ], + "score": 1.0, + "content": "is the standard shortest-path distance on the graph and", + "type": "text" + }, + { + "bbox": [ + 362, + 573, + 387, + 583 + ], + "score": 0.9, + "content": "r \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 572, + 423, + 585 + ], + "score": 1.0, + "content": ". 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Assume that the graph", + "type": "text" + }, + { + "bbox": [ + 401, + 608, + 410, + 617 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "is equipped with node", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 616, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 140, + 631 + ], + "score": 1.0, + "content": "features", + "type": "text" + }, + { + "bbox": [ + 140, + 618, + 189, + 628 + ], + "score": 0.92, + "content": "X \\in \\mathbb { R } ^ { n \\times p _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 616, + 217, + 631 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 217, + 618, + 254, + 629 + ], + "score": 0.92, + "content": "x _ { i } \\in \\mathbb { R } ^ { p _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 616, + 367, + 631 + ], + "score": 1.0, + "content": "is the feature vector at node", + "type": "text" + }, + { + "bbox": [ + 367, + 618, + 444, + 630 + ], + "score": 0.92, + "content": "i = 1 , \\ldots , n = | V |", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 616, + 505, + 631 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 626, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 107, + 629, + 151, + 644 + ], + "score": 0.93, + "content": "h _ { i } ^ { ( \\ell ) } \\in \\mathbb { R } ^ { p _ { \\ell } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 626, + 261, + 648 + ], + "score": 1.0, + "content": "the representation of node", + "type": "text" + }, + { + "bbox": [ + 261, + 633, + 266, + 642 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 626, + 299, + 648 + ], + "score": 1.0, + "content": "at layer", + "type": "text" + }, + { + "bbox": [ + 300, + 631, + 324, + 643 + ], + "score": 0.89, + "content": "\\ell \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 626, + 349, + 648 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 349, + 629, + 389, + 644 + ], + "score": 0.91, + "content": "h _ { i } ^ { ( 0 ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 626, + 507, + 648 + ], + "score": 1.0, + "content": ". Given a family of message", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 147, + 657 + ], + "score": 1.0, + "content": "functions", + "type": "text" + }, + { + "bbox": [ + 147, + 643, + 240, + 657 + ], + "score": 0.9, + "content": "\\psi _ { \\ell } : \\mathbb { R } ^ { p _ { \\ell } } \\times \\mathbb { R } ^ { p _ { \\ell } } \\to \\mathbb { R } ^ { p _ { \\ell } ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 644, + 330, + 657 + ], + "score": 1.0, + "content": "and update functions", + "type": "text" + }, + { + "bbox": [ + 330, + 644, + 431, + 657 + ], + "score": 0.9, + "content": "\\phi _ { \\ell } : \\mathbb { R } ^ { p _ { \\ell } } \\times \\mathbb { R } ^ { p _ { \\ell } ^ { \\prime } } \\to \\mathbb { R } ^ { p _ { \\ell + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", we can write the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 656, + 408, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 135, + 668 + ], + "score": 0.91, + "content": "( \\ell + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 656, + 408, + 668 + ], + "score": 1.0, + "content": "-st layer output of a generic MPNN as follows (Gilmer et al., 2017):", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 606, + 507, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 669, + 396, + 709 + ], + "lines": [ + { + "bbox": [ + 216, + 669, + 396, + 709 + ], + "spans": [ + { + "bbox": [ + 216, + 669, + 396, + 709 + ], + "score": 0.95, + "content": "h _ { i } ^ { ( \\ell + 1 ) } = \\phi _ { \\ell } \\left( h _ { i } ^ { ( \\ell ) } , \\sum _ { j = 1 } ^ { n } \\hat { A } _ { i j } \\psi _ { \\ell } ( h _ { i } ^ { ( \\ell ) } , h _ { j } ^ { ( \\ell ) } ) \\right) .", + "type": "interline_equation", + "image_path": "5acf9ad2264bec6c6abd19514311fa3932c1204ea0500ab6b7a1760d2f264ee1.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 216, + 669, + 396, + 682.3333333333334 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 216, + 682.3333333333334, + 396, + 695.6666666666667 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 216, + 695.6666666666667, + 396, + 709.0000000000001 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "Here we have used the augmented normalized adjacency matrix to propagate messages from each", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 477, + 734 + ], + "score": 1.0, + "content": "node to its neighbors, which simply leads to a degree normalization of the message functions", + "type": "text" + }, + { + "bbox": [ + 477, + 722, + 488, + 732 + ], + "score": 0.88, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 721, + 505, + 734 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 708, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "avoid heavy notations the node features and representations are assumed to be scalar from now on;", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 461, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 461, + 106 + ], + "score": 1.0, + "content": "these assumptions simplify the discussion and the vector case leads to analogous results.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 118, + 272, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 274, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 274, + 130 + ], + "score": 1.0, + "content": "2.2 THE OVER-SQUASHING PROBLEM", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "Multiple recent papers observed that MPNNs tend to perform poorly in situations when the learned", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "task requires long-range dependencies and at the same time the structure of the graph results in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "exponentially many long-range neighboring nodes. We say that a graph learning problem has", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "long-range dependencies when the output of a MPNN depends on representations of distant nodes", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "interacting with each other. If long-range dependencies are present, messages coming from non-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 506, + 206 + ], + "score": 1.0, + "content": "adjacent nodes need to be propagated across the network without being too distorted. In many cases", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 480, + 217 + ], + "score": 1.0, + "content": "however (e.g. in ‘small-world’ graphs such as social networks), the size of the receptive field", + "type": "text" + }, + { + "bbox": [ + 480, + 204, + 505, + 216 + ], + "score": 0.92, + "content": "B _ { r } ( i )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 209, + 228 + ], + "score": 1.0, + "content": "grows exponentially with", + "type": "text" + }, + { + "bbox": [ + 209, + 217, + 215, + 225 + ], + "score": 0.64, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 215, + 506, + 228 + ], + "score": 1.0, + "content": ". If this occurs, representations of exponentially many neighboring nodes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 405, + 239 + ], + "score": 1.0, + "content": "need to be compressed into fixed-size vectors to propagate messages to node", + "type": "text" + }, + { + "bbox": [ + 405, + 227, + 410, + 236 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 226, + 505, + 239 + ], + "score": 1.0, + "content": ", causing a phenomenon", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "referred to as over-squashing of information (Alon & Yahav, 2021). In line with Alon & Yahav", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "(2021), we refer to those structural properties of the graph that lead to over-squashing as a bottleneck1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 104, + 268, + 504, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 279, + 288 + ], + "score": 1.0, + "content": "Sensitivity analysis. The hidden feature", + "type": "text" + }, + { + "bbox": [ + 279, + 270, + 376, + 285 + ], + "score": 0.94, + "content": "h _ { i } ^ { ( \\ell ) } = h _ { i } ^ { ( \\ell ) } ( x _ { 1 } , \\dots , x _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 268, + 497, + 288 + ], + "score": 1.0, + "content": "computed by an MPNN with", + "type": "text" + }, + { + "bbox": [ + 498, + 273, + 504, + 283 + ], + "score": 0.7, + "content": "\\ell", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 408, + 298 + ], + "score": 1.0, + "content": "layers as in equation 1 is a differentiable function of the input node features", + "type": "text" + }, + { + "bbox": [ + 408, + 284, + 462, + 296 + ], + "score": 0.94, + "content": "\\{ x _ { 1 } , \\ldots , x _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 282, + 506, + 298 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 244, + 307 + ], + "score": 1.0, + "content": "the update and message functions", + "type": "text" + }, + { + "bbox": [ + 245, + 295, + 256, + 307 + ], + "score": 0.89, + "content": "\\phi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 295, + 274, + 307 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 295, + 286, + 307 + ], + "score": 0.88, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "are differentiable. The over-squashing of information", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 306, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 343, + 322 + ], + "score": 1.0, + "content": "can then be understood in terms of one node representation", + "type": "text" + }, + { + "bbox": [ + 344, + 306, + 360, + 321 + ], + "score": 0.92, + "content": "h _ { i } ^ { ( \\ell ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 306, + 506, + 322 + ], + "score": 1.0, + "content": "failing to be affected by some input", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 320, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 137, + 336 + ], + "score": 1.0, + "content": "feature", + "type": "text" + }, + { + "bbox": [ + 137, + 325, + 149, + 334 + ], + "score": 0.85, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 321, + 183, + 336 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 184, + 325, + 190, + 333 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 321, + 237, + 336 + ], + "score": 1.0, + "content": "at distance", + "type": "text" + }, + { + "bbox": [ + 237, + 325, + 244, + 333 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 321, + 290, + 336 + ], + "score": 1.0, + "content": "from node", + "type": "text" + }, + { + "bbox": [ + 290, + 324, + 295, + 333 + ], + "score": 0.65, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 321, + 436, + 336 + ], + "score": 1.0, + "content": ". Hence, we propose the Jacobian", + "type": "text" + }, + { + "bbox": [ + 436, + 320, + 480, + 335 + ], + "score": 0.93, + "content": "\\partial h _ { i } ^ { ( r ) } / \\partial x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "as an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 364, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 364, + 347 + ], + "score": 1.0, + "content": "explicit and formal way of assessing the over-squashing effect2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 109, + 347, + 503, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 293, + 361 + ], + "score": 1.0, + "content": "Lemma 1. Assume an MPNN as in equation", + "type": "text" + }, + { + "bbox": [ + 293, + 349, + 298, + 358 + ], + "score": 0.44, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 347, + 319, + 361 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 320, + 348, + 354, + 359 + ], + "score": 0.89, + "content": "i , s \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 347, + 376, + 361 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 376, + 348, + 426, + 360 + ], + "score": 0.92, + "content": "s \\in S _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 347, + 441, + 361 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 441, + 348, + 486, + 360 + ], + "score": 0.91, + "content": "| \\nabla \\phi _ { \\ell } | \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 359, + 232, + 371 + ], + "spans": [ + { + "bbox": [ + 107, + 359, + 151, + 371 + ], + "score": 0.92, + "content": "| \\nabla \\psi _ { \\ell } | \\le \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 359, + 167, + 370 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 167, + 359, + 208, + 370 + ], + "score": 0.9, + "content": "0 \\leq \\ell \\leq r", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 359, + 232, + 370 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 368, + 372, + 402 + ], + "lines": [ + { + "bbox": [ + 239, + 368, + 372, + 402 + ], + "spans": [ + { + "bbox": [ + 239, + 368, + 372, + 402 + ], + "score": 0.94, + "content": "\\left| \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right| \\leq ( \\alpha \\beta ) ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s } .", + "type": "interline_equation", + "image_path": "c7d0710d10f2d3fd55eeb92365c3f012e55435c1f716681a948bc9364112d343.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 368, + 372, + 385.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 239, + 385.0, + 372, + 402.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 201, + 417 + ], + "score": 1.0, + "content": "Lemma 1 states that if", + "type": "text" + }, + { + "bbox": [ + 202, + 406, + 213, + 416 + ], + "score": 0.89, + "content": "\\phi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 405, + 232, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 405, + 244, + 416 + ], + "score": 0.87, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "have bounded derivatives, then the propagation of messages is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 239, + 430 + ], + "score": 1.0, + "content": "controlled by a suitable power of", + "type": "text" + }, + { + "bbox": [ + 240, + 416, + 248, + 428 + ], + "score": 0.85, + "content": "\\hat { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 416, + 316, + 430 + ], + "score": 1.0, + "content": ". For example, if", + "type": "text" + }, + { + "bbox": [ + 316, + 417, + 384, + 430 + ], + "score": 0.93, + "content": "d _ { G } ( i , s ) = r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "and the sub-graph induced on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 429, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 141, + 443 + ], + "score": 0.92, + "content": "B _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 429, + 229, + 444 + ], + "score": 1.0, + "content": "is a binary tree, then", + "type": "text" + }, + { + "bbox": [ + 229, + 429, + 312, + 443 + ], + "score": 0.92, + "content": "( \\hat { A } ^ { r + 1 } ) _ { i s } = 2 ^ { - 1 } 3 ^ { - r }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 429, + 506, + 444 + ], + "score": 1.0, + "content": ", which gives an exponential decay of the node", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 442, + 495, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 270, + 454 + ], + "score": 1.0, + "content": "dependence on input features at distance", + "type": "text" + }, + { + "bbox": [ + 271, + 444, + 276, + 451 + ], + "score": 0.78, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 442, + 495, + 454 + ], + "score": 1.0, + "content": ", as also heuristically argued by Alon & Yahav (2021).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "The sensitivity analysis in Lemma 1 relates the over-squashing – as measured by the Jacobian of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "the node representations – to the graph topology via powers of the augmented normalized adjacency", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 481, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "matrix. In the next section we explore this connection further by analyzing which local properties of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "the graph structure affect the right hand side in equation 2, hence causing the bottleneck. We will", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "score": 1.0, + "content": "address this problem by introducing a new combinatorial notion of edge-based curvature and showing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 513, + 457, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 457, + 527 + ], + "score": 1.0, + "content": "that negatively curved edges are those responsible for the over-squashing phenomenon.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 540, + 324, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 538, + 326, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 326, + 555 + ], + "score": 1.0, + "content": "3 GRAPH CURVATURE AND BOTTLENECK", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "A natural object in Riemannian geometry is the Ricci curvature, a bilinear form determining the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "geodesic dispersion, i.e. whether geodesics starting at nearby points with ‘same’ velocity remain", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "parallel (Euclidean space), converge (spherical space), or diverge (hyperbolic space). To motivate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "the introduction of a Ricci curvature for graphs, we focus on these three cases. Consider two nodes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 130, + 619 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 608, + 236, + 621 + ], + "score": 1.0, + "content": "and two edges starting at", + "type": "text" + }, + { + "bbox": [ + 236, + 609, + 241, + 618 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 608, + 259, + 621 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 609, + 266, + 619 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "respectively. In a discrete spherical geometry (Figure 2a),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 208, + 633 + ], + "score": 1.0, + "content": "the edges would meet at", + "type": "text" + }, + { + "bbox": [ + 209, + 619, + 216, + 629 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 618, + 506, + 633 + ], + "score": 1.0, + "content": "to form a triangle (complete graph). In a discrete Euclidean geometry", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 378, + 643 + ], + "score": 1.0, + "content": "(Figure 2b), the edges would stay parallel and form a 4-cycle based at", + "type": "text" + }, + { + "bbox": [ + 379, + 630, + 401, + 641 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "(orthogonal grid). Finally,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 641, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 654 + ], + "score": 1.0, + "content": "in a discrete hyperbolic geometry (Figure 2c), the mutual distance of the edge endpoints would have", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 651, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 215, + 665 + ], + "score": 1.0, + "content": "grown compared to that of", + "type": "text" + }, + { + "bbox": [ + 215, + 653, + 220, + 662 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 651, + 238, + 665 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 238, + 653, + 244, + 663 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 651, + 506, + 665 + ], + "score": 1.0, + "content": "(tree). Therefore, a Ricci curvature for graphs should provide us", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 662, + 451, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 451, + 677 + ], + "score": 1.0, + "content": "with more sophisticated tools than the degree to analyze the neighborhood of an edge.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 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": "3", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 680, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 119, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "1We note that the over-squashing issue is different from the problem of under-reaching; the latter simply", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "amounts to a MPNN failing to fully explore a graph when the depth is smaller than the diameter (Barceló et al.,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "2019). The over-squashing phenomenon instead may occur even in deep GNNs with the number of layers larger", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 709, + 376, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 314, + 722 + ], + "score": 1.0, + "content": "than the graph diameter, as tested experimentally in Alon", + "type": "text" + }, + { + "bbox": [ + 314, + 712, + 322, + 720 + ], + "score": 0.26, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 709, + 376, + 722 + ], + "score": 1.0, + "content": "Yahav (2021).", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 719, + 495, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 495, + 734 + ], + "score": 1.0, + "content": "2The Jacobian of a GNN-output was also used by Xu et al. (2018) to set a similarity score among nodes.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "avoid heavy notations the node features and representations are assumed to be scalar from now on;", + "type": "text" + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 93, + 461, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 461, + 106 + ], + "score": 1.0, + "content": "these assumptions simplify the discussion and the vector case leads to analogous results.", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 106 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 118, + 272, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 274, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 274, + 130 + ], + "score": 1.0, + "content": "2.2 THE OVER-SQUASHING PROBLEM", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "Multiple recent papers observed that MPNNs tend to perform poorly in situations when the learned", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "task requires long-range dependencies and at the same time the structure of the graph results in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "exponentially many long-range neighboring nodes. We say that a graph learning problem has", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "long-range dependencies when the output of a MPNN depends on representations of distant nodes", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "interacting with each other. If long-range dependencies are present, messages coming from non-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 506, + 206 + ], + "score": 1.0, + "content": "adjacent nodes need to be propagated across the network without being too distorted. In many cases", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 480, + 217 + ], + "score": 1.0, + "content": "however (e.g. in ‘small-world’ graphs such as social networks), the size of the receptive field", + "type": "text" + }, + { + "bbox": [ + 480, + 204, + 505, + 216 + ], + "score": 0.92, + "content": "B _ { r } ( i )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 209, + 228 + ], + "score": 1.0, + "content": "grows exponentially with", + "type": "text" + }, + { + "bbox": [ + 209, + 217, + 215, + 225 + ], + "score": 0.64, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 215, + 506, + 228 + ], + "score": 1.0, + "content": ". If this occurs, representations of exponentially many neighboring nodes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 405, + 239 + ], + "score": 1.0, + "content": "need to be compressed into fixed-size vectors to propagate messages to node", + "type": "text" + }, + { + "bbox": [ + 405, + 227, + 410, + 236 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 226, + 505, + 239 + ], + "score": 1.0, + "content": ", causing a phenomenon", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "referred to as over-squashing of information (Alon & Yahav, 2021). In line with Alon & Yahav", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "(2021), we refer to those structural properties of the graph that lead to over-squashing as a bottleneck1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 138, + 506, + 260 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 104, + 268, + 504, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 279, + 288 + ], + "score": 1.0, + "content": "Sensitivity analysis. The hidden feature", + "type": "text" + }, + { + "bbox": [ + 279, + 270, + 376, + 285 + ], + "score": 0.94, + "content": "h _ { i } ^ { ( \\ell ) } = h _ { i } ^ { ( \\ell ) } ( x _ { 1 } , \\dots , x _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 268, + 497, + 288 + ], + "score": 1.0, + "content": "computed by an MPNN with", + "type": "text" + }, + { + "bbox": [ + 498, + 273, + 504, + 283 + ], + "score": 0.7, + "content": "\\ell", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 408, + 298 + ], + "score": 1.0, + "content": "layers as in equation 1 is a differentiable function of the input node features", + "type": "text" + }, + { + "bbox": [ + 408, + 284, + 462, + 296 + ], + "score": 0.94, + "content": "\\{ x _ { 1 } , \\ldots , x _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 282, + 506, + 298 + ], + "score": 1.0, + "content": "as long as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 244, + 307 + ], + "score": 1.0, + "content": "the update and message functions", + "type": "text" + }, + { + "bbox": [ + 245, + 295, + 256, + 307 + ], + "score": 0.89, + "content": "\\phi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 295, + 274, + 307 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 295, + 286, + 307 + ], + "score": 0.88, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "are differentiable. The over-squashing of information", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 306, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 343, + 322 + ], + "score": 1.0, + "content": "can then be understood in terms of one node representation", + "type": "text" + }, + { + "bbox": [ + 344, + 306, + 360, + 321 + ], + "score": 0.92, + "content": "h _ { i } ^ { ( \\ell ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 306, + 506, + 322 + ], + "score": 1.0, + "content": "failing to be affected by some input", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 320, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 137, + 336 + ], + "score": 1.0, + "content": "feature", + "type": "text" + }, + { + "bbox": [ + 137, + 325, + 149, + 334 + ], + "score": 0.85, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 321, + 183, + 336 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 184, + 325, + 190, + 333 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 321, + 237, + 336 + ], + "score": 1.0, + "content": "at distance", + "type": "text" + }, + { + "bbox": [ + 237, + 325, + 244, + 333 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 321, + 290, + 336 + ], + "score": 1.0, + "content": "from node", + "type": "text" + }, + { + "bbox": [ + 290, + 324, + 295, + 333 + ], + "score": 0.65, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 321, + 436, + 336 + ], + "score": 1.0, + "content": ". Hence, we propose the Jacobian", + "type": "text" + }, + { + "bbox": [ + 436, + 320, + 480, + 335 + ], + "score": 0.93, + "content": "\\partial h _ { i } ^ { ( r ) } / \\partial x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "as an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 364, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 364, + 347 + ], + "score": 1.0, + "content": "explicit and formal way of assessing the over-squashing effect2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 268, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 347, + 503, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 293, + 361 + ], + "score": 1.0, + "content": "Lemma 1. Assume an MPNN as in equation", + "type": "text" + }, + { + "bbox": [ + 293, + 349, + 298, + 358 + ], + "score": 0.44, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 347, + 319, + 361 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 320, + 348, + 354, + 359 + ], + "score": 0.89, + "content": "i , s \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 347, + 376, + 361 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 376, + 348, + 426, + 360 + ], + "score": 0.92, + "content": "s \\in S _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 347, + 441, + 361 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 441, + 348, + 486, + 360 + ], + "score": 0.91, + "content": "| \\nabla \\phi _ { \\ell } | \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 359, + 232, + 371 + ], + "spans": [ + { + "bbox": [ + 107, + 359, + 151, + 371 + ], + "score": 0.92, + "content": "| \\nabla \\psi _ { \\ell } | \\le \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 359, + 167, + 370 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 167, + 359, + 208, + 370 + ], + "score": 0.9, + "content": "0 \\leq \\ell \\leq r", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 359, + 232, + 370 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 347, + 505, + 371 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 368, + 372, + 402 + ], + "lines": [ + { + "bbox": [ + 239, + 368, + 372, + 402 + ], + "spans": [ + { + "bbox": [ + 239, + 368, + 372, + 402 + ], + "score": 0.94, + "content": "\\left| \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right| \\leq ( \\alpha \\beta ) ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s } .", + "type": "interline_equation", + "image_path": "c7d0710d10f2d3fd55eeb92365c3f012e55435c1f716681a948bc9364112d343.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 368, + 372, + 385.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 239, + 385.0, + 372, + 402.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 201, + 417 + ], + "score": 1.0, + "content": "Lemma 1 states that if", + "type": "text" + }, + { + "bbox": [ + 202, + 406, + 213, + 416 + ], + "score": 0.89, + "content": "\\phi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 405, + 232, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 405, + 244, + 416 + ], + "score": 0.87, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "have bounded derivatives, then the propagation of messages is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 239, + 430 + ], + "score": 1.0, + "content": "controlled by a suitable power of", + "type": "text" + }, + { + "bbox": [ + 240, + 416, + 248, + 428 + ], + "score": 0.85, + "content": "\\hat { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 416, + 316, + 430 + ], + "score": 1.0, + "content": ". For example, if", + "type": "text" + }, + { + "bbox": [ + 316, + 417, + 384, + 430 + ], + "score": 0.93, + "content": "d _ { G } ( i , s ) = r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "and the sub-graph induced on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 429, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 141, + 443 + ], + "score": 0.92, + "content": "B _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 429, + 229, + 444 + ], + "score": 1.0, + "content": "is a binary tree, then", + "type": "text" + }, + { + "bbox": [ + 229, + 429, + 312, + 443 + ], + "score": 0.92, + "content": "( \\hat { A } ^ { r + 1 } ) _ { i s } = 2 ^ { - 1 } 3 ^ { - r }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 429, + 506, + 444 + ], + "score": 1.0, + "content": ", which gives an exponential decay of the node", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 442, + 495, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 270, + 454 + ], + "score": 1.0, + "content": "dependence on input features at distance", + "type": "text" + }, + { + "bbox": [ + 271, + 444, + 276, + 451 + ], + "score": 0.78, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 442, + 495, + 454 + ], + "score": 1.0, + "content": ", as also heuristically argued by Alon & Yahav (2021).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 405, + 506, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "The sensitivity analysis in Lemma 1 relates the over-squashing – as measured by the Jacobian of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "the node representations – to the graph topology via powers of the augmented normalized adjacency", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 481, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "matrix. In the next section we explore this connection further by analyzing which local properties of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "the graph structure affect the right hand side in equation 2, hence causing the bottleneck. We will", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "score": 1.0, + "content": "address this problem by introducing a new combinatorial notion of edge-based curvature and showing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 513, + 457, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 457, + 527 + ], + "score": 1.0, + "content": "that negatively curved edges are those responsible for the over-squashing phenomenon.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 458, + 506, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 540, + 324, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 538, + 326, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 326, + 555 + ], + "score": 1.0, + "content": "3 GRAPH CURVATURE AND BOTTLENECK", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "A natural object in Riemannian geometry is the Ricci curvature, a bilinear form determining the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "geodesic dispersion, i.e. whether geodesics starting at nearby points with ‘same’ velocity remain", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "parallel (Euclidean space), converge (spherical space), or diverge (hyperbolic space). To motivate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "the introduction of a Ricci curvature for graphs, we focus on these three cases. Consider two nodes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 107, + 608, + 130, + 619 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 608, + 236, + 621 + ], + "score": 1.0, + "content": "and two edges starting at", + "type": "text" + }, + { + "bbox": [ + 236, + 609, + 241, + 618 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 608, + 259, + 621 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 609, + 266, + 619 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "respectively. In a discrete spherical geometry (Figure 2a),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 208, + 633 + ], + "score": 1.0, + "content": "the edges would meet at", + "type": "text" + }, + { + "bbox": [ + 209, + 619, + 216, + 629 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 618, + 506, + 633 + ], + "score": 1.0, + "content": "to form a triangle (complete graph). In a discrete Euclidean geometry", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 378, + 643 + ], + "score": 1.0, + "content": "(Figure 2b), the edges would stay parallel and form a 4-cycle based at", + "type": "text" + }, + { + "bbox": [ + 379, + 630, + 401, + 641 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "(orthogonal grid). Finally,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 641, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 654 + ], + "score": 1.0, + "content": "in a discrete hyperbolic geometry (Figure 2c), the mutual distance of the edge endpoints would have", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 651, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 104, + 651, + 215, + 665 + ], + "score": 1.0, + "content": "grown compared to that of", + "type": "text" + }, + { + "bbox": [ + 215, + 653, + 220, + 662 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 651, + 238, + 665 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 238, + 653, + 244, + 663 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 651, + 506, + 665 + ], + "score": 1.0, + "content": "(tree). Therefore, a Ricci curvature for graphs should provide us", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 662, + 451, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 451, + 677 + ], + "score": 1.0, + "content": "with more sophisticated tools than the degree to analyze the neighborhood of an edge.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 564, + 506, + 677 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 270, + 214 + ], + "lines": [ + { + "bbox": [ + 107, + 82, + 271, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 271, + 94 + ], + "score": 1.0, + "content": "Curvatures on graphs. The main ex-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 270, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 270, + 105 + ], + "score": 1.0, + "content": "amples of edge-based curvature are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 270, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 199, + 116 + ], + "score": 1.0, + "content": "the Forman curvature", + "type": "text" + }, + { + "bbox": [ + 200, + 104, + 228, + 117 + ], + "score": 0.93, + "content": "F ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 104, + 270, + 116 + ], + "score": 1.0, + "content": "(Forman,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 269, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 241, + 128 + ], + "score": 1.0, + "content": "2003) and the Ollivier curvature", + "type": "text" + }, + { + "bbox": [ + 242, + 115, + 269, + 127 + ], + "score": 0.91, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 271, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 271, + 138 + ], + "score": 1.0, + "content": "in Ollivier (2007; 2009) (see Appendix).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 271, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 132, + 149 + ], + "score": 1.0, + "content": "While", + "type": "text" + }, + { + "bbox": [ + 132, + 137, + 162, + 149 + ], + "score": 0.93, + "content": "F ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 137, + 271, + 149 + ], + "score": 1.0, + "content": "is given in terms of combi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 271, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 271, + 160 + ], + "score": 1.0, + "content": "natorial quantities (Sreejith et al., 2016),", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 271, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 271, + 170 + ], + "score": 1.0, + "content": "results are scarce and the definition is bi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 270, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 270, + 181 + ], + "score": 1.0, + "content": "ased towards negative curvature. 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We propose a new curvature to address the shortcomings of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 237, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 481, + 251 + ], + "score": 1.0, + "content": "existing candidates. We use the following definitions to describe the neighborhood of an edge", + "type": "text" + }, + { + "bbox": [ + 481, + 238, + 504, + 249 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 248, + 354, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 354, + 261 + ], + "score": 1.0, + "content": "and we refer to the Appendix for a more complete discussion:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 123, + 268, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 127, + 268, + 375, + 282 + ], + "spans": [ + { + "bbox": [ + 127, + 268, + 141, + 282 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 268, + 246, + 281 + ], + "score": 0.9, + "content": "\\sharp _ { \\Delta } ( i , j ) : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 268, + 349, + 282 + ], + "score": 1.0, + "content": "are the triangles based at", + "type": "text" + }, + { + "bbox": [ + 349, + 270, + 371, + 281 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 268, + 375, + 282 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 124, + 285, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 124, + 285, + 141, + 301 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 142, + 286, + 434, + 299 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigtriangledown } ( i , j ) : = \\{ k \\in S _ { 1 } ( i ) \\ \\backslash \\ S _ { 1 } ( j ) , k \\neq j : \\ \\exists w \\in ( S _ { 1 } ( k ) \\cap S _ { 1 } ( j ) ) \\setminus S _ { 1 } ( i ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 285, + 505, + 301 + ], + "score": 1.0, + "content": "are the neighbors", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 297, + 429, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 153, + 311 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 153, + 299, + 158, + 307 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 297, + 303, + 311 + ], + "score": 1.0, + "content": "forming a 4-cycle based at the edge", + "type": "text" + }, + { + "bbox": [ + 303, + 299, + 326, + 309 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 297, + 429, + 311 + ], + "score": 1.0, + "content": "without diagonals inside.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 122, + 311, + 425, + 335 + ], + "spans": [ + { + "bbox": [ + 122, + 311, + 141, + 335 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 142, + 312, + 181, + 325 + ], + "score": 0.93, + "content": "\\gamma _ { \\operatorname* { m a x } } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 311, + 285, + 335 + ], + "score": 1.0, + "content": "is the maximal number of n node (see Definition 4).", + "type": "text" + }, + { + "bbox": [ + 285, + 313, + 299, + 324 + ], + "score": 0.55, + "content": "4 -", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 311, + 358, + 335 + ], + "score": 1.0, + "content": "cycles based at", + "type": "text" + }, + { + "bbox": [ + 359, + 313, + 381, + 324 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 311, + 425, + 335 + ], + "score": 1.0, + "content": "traversing", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 424, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 424, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 385, + 354 + ], + "score": 1.0, + "content": "In line with the discussion about geodesic dispersion, one expects", + "type": "text" + }, + { + "bbox": [ + 385, + 341, + 398, + 352 + ], + "score": 0.88, + "content": "\\sharp _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 339, + 424, + 354 + ], + "score": 1.0, + "content": "to be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 352, + 424, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 295, + 365 + ], + "score": 1.0, + "content": "related to positive curvature (complete graph),", + "type": "text" + }, + { + "bbox": [ + 295, + 352, + 307, + 364 + ], + "score": 0.87, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 352, + 424, + 365 + ], + "score": 1.0, + "content": "to zero curvature (grid), and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 363, + 424, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 424, + 375 + ], + "score": 1.0, + "content": "the remaining outgoing edges to negative curvature (tree). Our new curvature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 372, + 424, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 424, + 385 + ], + "score": 1.0, + "content": "formulation reflects such an intuition and recovers the expected results in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 383, + 424, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 323, + 397 + ], + "score": 1.0, + "content": "classical cases. In the example in Figure 3 we have", + "type": "text" + }, + { + "bbox": [ + 323, + 384, + 397, + 397 + ], + "score": 0.92, + "content": "\\sharp _ { \\bigstar } ^ { 0 } ( \\bar { 0 } , 1 ) = \\{ 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 383, + 424, + 397 + ], + "score": 1.0, + "content": "while", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 395, + 425, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 168, + 409 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigstar } ^ { 1 } ( 0 , 1 ) = \\{ 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 395, + 425, + 411 + ], + "score": 1.0, + "content": ", both without 4,6 because of the triangle 1-6-0. The degeneracy", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 407, + 401, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 132, + 421 + ], + "score": 1.0, + "content": "factor", + "type": "text" + }, + { + "bbox": [ + 132, + 408, + 191, + 420 + ], + "score": 0.92, + "content": "\\gamma _ { \\mathrm { m a x } } ( 0 , 1 ) = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 407, + 401, + 421 + ], + "score": 1.0, + "content": ", as there exist two 4-cycles passing through node 5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "image", + "bbox": [ + 436, + 319, + 500, + 392 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 436, + 319, + 500, + 392 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 436, + 319, + 500, + 392 + ], + "spans": [ + { + "bbox": [ + 436, + 319, + 500, + 392 + ], + "score": 0.936, + "type": "image", + "image_path": "a9d21074d19942603c3881fd9a01471a01eac7e7f8f372bacd0820ede9137be8.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 436, + 319, + 500, + 355.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 436, + 355.5, + 500, + 392.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 430, + 395, + 504, + 417 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 431, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 431, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "Figure 3: 4-cycle", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 431, + 406, + 485, + 417 + ], + "spans": [ + { + "bbox": [ + 431, + 406, + 485, + 417 + ], + "score": 1.0, + "content": "contribution.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 347, + 441 + ], + "score": 1.0, + "content": "Definition 1 (Balanced Forman curvature). For any edge", + "type": "text" + }, + { + "bbox": [ + 347, + 429, + 370, + 440 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 426, + 494, + 441 + ], + "score": 1.0, + "content": "in a simple, unweighted graph", + "type": "text" + }, + { + "bbox": [ + 494, + 428, + 503, + 438 + ], + "score": 0.73, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 426, + 506, + 441 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 439, + 336, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 132, + 452 + ], + "score": 1.0, + "content": "we let", + "type": "text" + }, + { + "bbox": [ + 132, + 439, + 168, + 451 + ], + "score": 0.68, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 439, + 200, + 452 + ], + "score": 1.0, + "content": "be zero", + "type": "text" + }, + { + "bbox": [ + 200, + 439, + 277, + 451 + ], + "score": 0.91, + "content": "i f \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 439, + 336, + 452 + ], + "score": 1.0, + "content": "and otherwise", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 453, + 478, + 480 + ], + "lines": [ + { + "bbox": [ + 119, + 453, + 478, + 480 + ], + "spans": [ + { + "bbox": [ + 119, + 453, + 478, + 480 + ], + "score": 0.91, + "content": "\\mathbf { R i c } ( i , j ) : = \\frac { 2 } { d _ { i } } + \\frac { 2 } { d _ { j } } - 2 + 2 \\frac { \\left| \\sharp _ { \\Delta } ( i , j ) \\right| } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\left| \\sharp _ { \\Delta } ( i , j ) \\right| } { \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\left( \\gamma _ { m a x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } ( | \\sharp _ { \\overline { { \\Gamma } } } ^ { i } | + | \\sharp _ { \\overline { { \\Gamma } } } ^ { j } | ) ,", + "type": "interline_equation", + "image_path": "70e1795908e7edf2901adf69d9dac73c9792d64aa8c51edd2742015e711d402a.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 119, + 453, + 478, + 462.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 119, + 462.0, + 478, + 471.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 119, + 471.0, + 478, + 480.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 492, + 497 + ], + "lines": [ + { + "bbox": [ + 104, + 482, + 491, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 249, + 498 + ], + "score": 1.0, + "content": "where the last term is set to be zero", + "type": "text" + }, + { + "bbox": [ + 249, + 483, + 275, + 497 + ], + "score": 0.56, + "content": "i f | \\sharp _ { \\bigstar } ^ { i } |", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 482, + 322, + 498 + ], + "score": 1.0, + "content": "(and hence", + "type": "text" + }, + { + "bbox": [ + 323, + 483, + 342, + 497 + ], + "score": 0.8, + "content": "| \\sharp _ { \\bigstar } ^ { j } | \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 482, + 429, + 498 + ], + "score": 1.0, + "content": "is zero. In particular", + "type": "text" + }, + { + "bbox": [ + 430, + 484, + 491, + 496 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) > - 2", + "type": "inline_equation" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 506, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 239, + 520 + ], + "score": 1.0, + "content": "The curvature is negative when", + "type": "text" + }, + { + "bbox": [ + 239, + 508, + 265, + 519 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 507, + 389, + 520 + ], + "score": 1.0, + "content": "behaves as a bridge between", + "type": "text" + }, + { + "bbox": [ + 389, + 507, + 412, + 519 + ], + "score": 0.92, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 507, + 432, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 507, + 456, + 520 + ], + "score": 0.91, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 507, + 505, + 520 + ], + "score": 1.0, + "content": ", while it is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 165, + 531 + ], + "score": 1.0, + "content": "positive when", + "type": "text" + }, + { + "bbox": [ + 166, + 518, + 189, + 531 + ], + "score": 0.94, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 518, + 207, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 519, + 232, + 531 + ], + "score": 0.93, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 518, + 359, + 531 + ], + "score": 1.0, + "content": "stay connected after removing", + "type": "text" + }, + { + "bbox": [ + 359, + 519, + 383, + 530 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 518, + 506, + 531 + ], + "score": 1.0, + "content": ". We refer to Ric as Balanced", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 529, + 507, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 507, + 542 + ], + "score": 1.0, + "content": "Forman curvature. We can relate the Balanced Forman curvature to the Jacobian of hidden features,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 540, + 503, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 363, + 553 + ], + "score": 1.0, + "content": "while also extending many results valid for the Ollivier curvature", + "type": "text" + }, + { + "bbox": [ + 364, + 540, + 390, + 552 + ], + "score": 0.93, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 540, + 503, + 553 + ], + "score": 1.0, + "content": "thanks to our next theorem.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 102, + 560, + 473, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 471, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 273, + 574 + ], + "score": 1.0, + "content": "Theorem 2. Given an unweighted graph", + "type": "text" + }, + { + "bbox": [ + 273, + 561, + 282, + 571 + ], + "score": 0.69, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 559, + 338, + 574 + ], + "score": 1.0, + "content": ", for any edge", + "type": "text" + }, + { + "bbox": [ + 338, + 561, + 361, + 573 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 559, + 397, + 574 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 397, + 560, + 471, + 573 + ], + "score": 0.9, + "content": "\\kappa ( i , j ) \\geq \\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 376, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 378, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 378, + 596 + ], + "score": 1.0, + "content": "Theorem 2 generalizes Jost & Liu (2014, Theorem 3) (see Appendix).", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 593, + 376, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 327, + 607 + ], + "score": 1.0, + "content": "We also note that the computational complexity for", + "type": "text" + }, + { + "bbox": [ + 327, + 597, + 335, + 604 + ], + "score": 0.73, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 593, + 376, + 607 + ], + "score": 1.0, + "content": "scales as", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 107, + 604, + 377, + 620 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 154, + 618 + ], + "score": 0.93, + "content": "O ( | E | d _ { \\operatorname* { m a x } } ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 604, + 268, + 620 + ], + "score": 1.0, + "content": ", while for our Ric we have", + "type": "text" + }, + { + "bbox": [ + 268, + 605, + 315, + 618 + ], + "score": 0.92, + "content": "O ( | E | d _ { \\operatorname* { m a x } } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 604, + 340, + 620 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 341, + 606, + 360, + 617 + ], + "score": 0.9, + "content": "d _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 604, + 377, + 620 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 616, + 356, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 356, + 629 + ], + "score": 1.0, + "content": "maximal degree. 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We propose a new curvature to address the shortcomings of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 237, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 481, + 251 + ], + "score": 1.0, + "content": "existing candidates. 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the discussion about geodesic dispersion, one expects", + "type": "text" + }, + { + "bbox": [ + 385, + 341, + 398, + 352 + ], + "score": 0.88, + "content": "\\sharp _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 339, + 424, + 354 + ], + "score": 1.0, + "content": "to be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 352, + 424, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 295, + 365 + ], + "score": 1.0, + "content": "related to positive curvature (complete graph),", + "type": "text" + }, + { + "bbox": [ + 295, + 352, + 307, + 364 + ], + "score": 0.87, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 352, + 424, + 365 + ], + "score": 1.0, + "content": "to zero curvature (grid), and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 363, + 424, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 424, + 375 + ], + "score": 1.0, + "content": "the 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Our new curvature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 372, + 424, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 424, + 385 + ], + "score": 1.0, + "content": "formulation reflects such an intuition and recovers the expected results in the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 383, + 424, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 323, + 397 + ], + "score": 1.0, + "content": "classical cases. In the example in Figure 3 we have", + "type": "text" + }, + { + "bbox": [ + 323, + 384, + 397, + 397 + ], + "score": 0.92, + "content": "\\sharp _ { \\bigstar } ^ { 0 } ( \\bar { 0 } , 1 ) = \\{ 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 383, + 424, + 397 + ], + "score": 1.0, + "content": "while", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 395, + 425, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 168, + 409 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigstar } ^ { 1 } ( 0 , 1 ) = \\{ 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 395, + 425, + 411 + ], + "score": 1.0, + "content": ", both without 4,6 because of the triangle 1-6-0. The degeneracy", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 407, + 401, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 132, + 421 + ], + "score": 1.0, + "content": "factor", + "type": "text" + }, + { + "bbox": [ + 132, + 408, + 191, + 420 + ], + "score": 0.92, + "content": "\\gamma _ { \\mathrm { m a x } } ( 0 , 1 ) = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 407, + 401, + 421 + ], + "score": 1.0, + "content": ", as there exist two 4-cycles passing through node 5.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 339, + 425, + 421 + ] + }, + { + "type": "image", + "bbox": [ + 436, + 319, + 500, + 392 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 436, + 319, + 500, + 392 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 436, + 319, + 500, + 392 + ], + "spans": [ + { + "bbox": [ + 436, + 319, + 500, + 392 + ], + "score": 0.936, + "type": "image", + "image_path": "a9d21074d19942603c3881fd9a01471a01eac7e7f8f372bacd0820ede9137be8.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 436, + 319, + 500, + 355.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 436, + 355.5, + 500, + 392.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 430, + 395, + 504, + 417 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 431, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 431, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "Figure 3: 4-cycle", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 431, + 406, + 485, + 417 + ], + "spans": [ + { + "bbox": [ + 431, + 406, + 485, + 417 + ], + "score": 1.0, + "content": "contribution.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 347, + 441 + ], + "score": 1.0, + "content": "Definition 1 (Balanced Forman curvature). For any edge", + "type": "text" + }, + { + "bbox": [ + 347, + 429, + 370, + 440 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 426, + 494, + 441 + ], + "score": 1.0, + "content": "in a simple, unweighted graph", + "type": "text" + }, + { + "bbox": [ + 494, + 428, + 503, + 438 + ], + "score": 0.73, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 426, + 506, + 441 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 439, + 336, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 132, + 452 + ], + "score": 1.0, + "content": "we let", + "type": "text" + }, + { + "bbox": [ + 132, + 439, + 168, + 451 + ], + "score": 0.68, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 439, + 200, + 452 + ], + "score": 1.0, + "content": "be zero", + "type": "text" + }, + { + "bbox": [ + 200, + 439, + 277, + 451 + ], + "score": 0.91, + "content": "i f \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 439, + 336, + 452 + ], + "score": 1.0, + "content": "and otherwise", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 426, + 506, + 452 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 453, + 478, + 480 + ], + "lines": [ + { + "bbox": [ + 119, + 453, + 478, + 480 + ], + "spans": [ + { + "bbox": [ + 119, + 453, + 478, + 480 + ], + "score": 0.91, + "content": "\\mathbf { R i c } ( i , j ) : = \\frac { 2 } { d _ { i } } + \\frac { 2 } { d _ { j } } - 2 + 2 \\frac { \\left| \\sharp _ { \\Delta } ( i , j ) \\right| } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\left| \\sharp _ { \\Delta } ( i , j ) \\right| } { \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\left( \\gamma _ { m a x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } ( | \\sharp _ { \\overline { { \\Gamma } } } ^ { i } | + | \\sharp _ { \\overline { { \\Gamma } } } ^ { j } | ) ,", + "type": "interline_equation", + "image_path": "70e1795908e7edf2901adf69d9dac73c9792d64aa8c51edd2742015e711d402a.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 119, + 453, + 478, + 462.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 119, + 462.0, + 478, + 471.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 119, + 471.0, + 478, + 480.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 492, + 497 + ], + "lines": [ + { + "bbox": [ + 104, + 482, + 491, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 249, + 498 + ], + "score": 1.0, + "content": "where the last term is set to be zero", + "type": "text" + }, + { + "bbox": [ + 249, + 483, + 275, + 497 + ], + "score": 0.56, + "content": "i f | \\sharp _ { \\bigstar } ^ { i } |", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 482, + 322, + 498 + ], + "score": 1.0, + "content": "(and hence", + "type": "text" + }, + { + "bbox": [ + 323, + 483, + 342, + 497 + ], + "score": 0.8, + "content": "| \\sharp _ { \\bigstar } ^ { j } | \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 482, + 429, + 498 + ], + "score": 1.0, + "content": "is zero. In particular", + "type": "text" + }, + { + "bbox": [ + 430, + 484, + 491, + 496 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) > - 2", + "type": "inline_equation" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 104, + 482, + 491, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 506, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 239, + 520 + ], + "score": 1.0, + "content": "The curvature is negative when", + "type": "text" + }, + { + "bbox": [ + 239, + 508, + 265, + 519 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 507, + 389, + 520 + ], + "score": 1.0, + "content": "behaves as a bridge between", + "type": "text" + }, + { + "bbox": [ + 389, + 507, + 412, + 519 + ], + "score": 0.92, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 507, + 432, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 507, + 456, + 520 + ], + "score": 0.91, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 507, + 505, + 520 + ], + "score": 1.0, + "content": ", while it is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 165, + 531 + ], + "score": 1.0, + "content": "positive when", + "type": "text" + }, + { + "bbox": [ + 166, + 518, + 189, + 531 + ], + "score": 0.94, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 518, + 207, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 519, + 232, + 531 + ], + "score": 0.93, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 518, + 359, + 531 + ], + "score": 1.0, + "content": "stay connected after removing", + "type": "text" + }, + { + "bbox": [ + 359, + 519, + 383, + 530 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 518, + 506, + 531 + ], + "score": 1.0, + "content": ". We refer to Ric as Balanced", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 529, + 507, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 507, + 542 + ], + "score": 1.0, + "content": "Forman curvature. We can relate the Balanced Forman curvature to the Jacobian of hidden features,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 540, + 503, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 363, + 553 + ], + "score": 1.0, + "content": "while also extending many results valid for the Ollivier curvature", + "type": "text" + }, + { + "bbox": [ + 364, + 540, + 390, + 552 + ], + "score": 0.93, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 540, + 503, + 553 + ], + "score": 1.0, + "content": "thanks to our next theorem.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 507, + 507, + 553 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 560, + 473, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 471, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 273, + 574 + ], + "score": 1.0, + "content": "Theorem 2. Given an unweighted graph", + "type": "text" + }, + { + "bbox": [ + 273, + 561, + 282, + 571 + ], + "score": 0.69, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 559, + 338, + 574 + ], + "score": 1.0, + "content": ", for any edge", + "type": "text" + }, + { + "bbox": [ + 338, + 561, + 361, + 573 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 559, + 397, + 574 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 397, + 560, + 471, + 573 + ], + "score": 0.9, + "content": "\\kappa ( i , j ) \\geq \\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 49 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 559, + 471, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 376, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 378, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 378, + 596 + ], + "score": 1.0, + "content": "Theorem 2 generalizes Jost & Liu (2014, Theorem 3) (see Appendix).", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 593, + 376, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 327, + 607 + ], + "score": 1.0, + "content": "We also note that the computational complexity for", + "type": "text" + }, + { + "bbox": [ + 327, + 597, + 335, + 604 + ], + "score": 0.73, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 593, + 376, + 607 + ], + "score": 1.0, + "content": "scales as", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 107, + 604, + 377, + 620 + ], + "spans": [ + { + "bbox": [ + 107, + 605, + 154, + 618 + ], + "score": 0.93, + "content": "O ( | E | d _ { \\operatorname* { m a x } } ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 604, + 268, + 620 + ], + "score": 1.0, + "content": ", while for our Ric we have", + "type": "text" + }, + { + "bbox": [ + 268, + 605, + 315, + 618 + ], + "score": 0.92, + "content": "O ( | E | d _ { \\operatorname* { m a x } } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 604, + 340, + 620 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 341, + 606, + 360, + 617 + ], + "score": 0.9, + "content": "d _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 604, + 377, + 620 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 616, + 356, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 356, + 629 + ], + "score": 1.0, + "content": "maximal degree. From Theorem 2 and Paeng (2012), we find:", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 582, + 378, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 375, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 376, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 172, + 649 + ], + "score": 1.0, + "content": "Corollary 3. If", + "type": "text" + }, + { + "bbox": [ + 173, + 636, + 249, + 648 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 636, + 304, + 649 + ], + "score": 1.0, + "content": "for any edge", + "type": "text" + }, + { + "bbox": [ + 304, + 637, + 328, + 648 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 636, + 376, + 649 + ], + "score": 1.0, + "content": ", then there", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 648, + 235, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 186, + 659 + ], + "score": 1.0, + "content": "exists a polynomial", + "type": "text" + }, + { + "bbox": [ + 186, + 649, + 195, + 657 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 648, + 235, + 659 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 55.5, + "bbox_fs": [ + 106, + 636, + 376, + 659 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 663, + 295, + 677 + ], + "lines": [ + { + "bbox": [ + 187, + 663, + 295, + 677 + ], + "spans": [ + { + "bbox": [ + 187, + 663, + 295, + 677 + ], + "score": 0.9, + "content": "| B _ { r } ( i ) | \\leq P ( r ) , \\forall i \\in V .", + "type": "interline_equation", + "image_path": "9fce9d515578118c2f889f2570948b44fcbba21a7e76bd4c4b6acd6e9d06f4d9.jpg" + } + ] + } + ], + "index": 57, + "virtual_lines": [ + { + "bbox": [ + 187, + 663, + 295, + 677 + ], + "spans": [], + "index": 57 + } + ] + }, + { + "type": "table", + "bbox": [ + 383, + 585, + 499, + 683 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 383, + 585, + 499, + 683 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 585, + 499, + 683 + ], + "spans": [ + { + "bbox": [ + 383, + 585, + 499, + 683 + ], + "score": 0.968, + "html": "
Graph GRicG
C3 Cycles C4 Cn≥53
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", + "type": "table", + "image_path": "8da8842d64206d3f3af8a03ff936001333c5ac6b8bea3abca30e94acf4c0360c.jpg" + } + ] + } + ], + "index": 56.0, + "virtual_lines": [ + { + "bbox": [ + 383, + 585, + 499, + 634.0 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 383, + 634.0, + 499, + 683.0 + ], + "spans": [], + "index": 58 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 384, + 688, + 497, + 711 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 688, + 486, + 700 + ], + "spans": [ + { + "bbox": [ + 383, + 688, + 486, + 700 + ], + "score": 1.0, + "content": "Table 1: Examples of the", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 383, + 699, + 498, + 711 + ], + "spans": [ + { + "bbox": [ + 383, + 699, + 498, + 711 + ], + "score": 1.0, + "content": "Balanced Forman curvature.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 62.5 + } + ], + "index": 59.25 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 376, + 721 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 377, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 377, + 700 + ], + "score": 1.0, + "content": "Therefore, when the curvature is positive everywhere, the bottleneck", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 699, + 376, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 376, + 710 + ], + "score": 1.0, + "content": "effect should not play a crucial role as the receptive field of each", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 709, + 377, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 377, + 722 + ], + "score": 1.0, + "content": "node will be polynomial in the hop-distance. This limit case shows", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 104, + 717, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 506, + 735 + ], + "score": 1.0, + "content": "how the curvature determines whether a learning task on a graph would suffer from over-squashing.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 60, + "bbox_fs": [ + 105, + 687, + 377, + 722 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 721, + 505, + 732 + ], + "lines": [], + "index": 64, + "bbox_fs": [ + 104, + 717, + 506, + 735 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Curvature and over-squashing. Thanks to our new combinatorial curvature and the sensitivity", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "analysis in Lemma 1, we are able to relate local curvature properties to the Jacobian of the node", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "representations. This leads to one of the main results of this paper: negatively curved edges are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 483, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 483, + 128 + ], + "score": 1.0, + "content": "those causing the graph bottleneck and thus leading to the over-squashing phenomenon:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 105, + 135, + 468, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 469, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 321, + 149 + ], + "score": 1.0, + "content": "Theorem 4. Consider a MPNN as in equation 1. Let", + "type": "text" + }, + { + "bbox": [ + 321, + 136, + 344, + 147 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 134, + 364, + 149 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 365, + 136, + 397, + 148 + ], + "score": 0.94, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 134, + 469, + 149 + ], + "score": 1.0, + "content": "and assume that:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 123, + 154, + 493, + 168 + ], + "lines": [ + { + "bbox": [ + 126, + 155, + 496, + 169 + ], + "spans": [ + { + "bbox": [ + 126, + 155, + 142, + 169 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 156, + 187, + 168 + ], + "score": 0.9, + "content": "| \\nabla \\phi _ { \\ell } | \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 155, + 205, + 169 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 156, + 250, + 168 + ], + "score": 0.92, + "content": "| \\nabla \\psi _ { \\ell } | \\le \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 155, + 287, + 169 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 287, + 156, + 348, + 167 + ], + "score": 0.9, + "content": "0 \\leq \\ell \\leq L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 155, + 371, + 169 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 371, + 156, + 398, + 167 + ], + "score": 0.9, + "content": "L \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 155, + 496, + 169 + ], + "score": 1.0, + "content": "the depth of the MPNN.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 125, + 174, + 466, + 190 + ], + "lines": [ + { + "bbox": [ + 123, + 173, + 468, + 191 + ], + "spans": [ + { + "bbox": [ + 123, + 173, + 192, + 191 + ], + "score": 1.0, + "content": "(ii) There exists", + "type": "text" + }, + { + "bbox": [ + 192, + 177, + 198, + 186 + ], + "score": 0.46, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 173, + 214, + 191 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 215, + 174, + 322, + 189 + ], + "score": 0.75, + "content": "0 < \\delta < ( \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} ) ^ { - \\frac { 1 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 173, + 326, + 191 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 326, + 175, + 363, + 189 + ], + "score": 0.78, + "content": "\\delta < \\gamma _ { m a x } ^ { - 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 173, + 384, + 191 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 385, + 176, + 463, + 189 + ], + "score": 0.88, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 173, + 468, + 191 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 447, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 448, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 175, + 212 + ], + "score": 1.0, + "content": "Then there exists", + "type": "text" + }, + { + "bbox": [ + 176, + 198, + 223, + 210 + ], + "score": 0.93, + "content": "Q _ { j } \\subset S _ { 2 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 197, + 266, + 212 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 266, + 197, + 313, + 210 + ], + "score": 0.92, + "content": "| Q _ { j } | > \\delta ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 197, + 346, + 212 + ], + "score": 1.0, + "content": "and for", + "type": "text" + }, + { + "bbox": [ + 346, + 198, + 412, + 209 + ], + "score": 0.91, + "content": "0 \\leq \\ell _ { 0 } \\leq L - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 197, + 448, + 212 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 211, + 376, + 246 + ], + "lines": [ + { + "bbox": [ + 234, + 211, + 376, + 246 + ], + "spans": [ + { + "bbox": [ + 234, + 211, + 376, + 246 + ], + "score": 0.94, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } \\left| \\frac { \\partial h _ { k } ^ { ( \\ell _ { 0 } + 2 ) } } { \\partial h _ { i } ^ { ( \\ell _ { 0 } ) } } \\right| < ( \\alpha \\beta ) ^ { 2 } \\delta ^ { \\frac { 1 } { 4 } } .", + "type": "interline_equation", + "image_path": "8058842145c59033fb6edaead13943e3b3d1288dde59ccf9db91c03ca0c5272e.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 211, + 376, + 228.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 234, + 228.5, + 376, + 246.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "Condition (i) is always satisfied and allows us to control the message passing functions. The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 315, + 277 + ], + "score": 1.0, + "content": "requirement (ii) instead means that the curvature of", + "type": "text" + }, + { + "bbox": [ + 315, + 264, + 335, + 276 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "is negative enough when compared to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 149, + 288 + ], + "score": 1.0, + "content": "degrees of", + "type": "text" + }, + { + "bbox": [ + 149, + 276, + 154, + 285 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 275, + 172, + 288 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 276, + 178, + 286 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 275, + 224, + 288 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 224, + 275, + 288, + 287 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) > - 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 275, + 390, + 288 + ], + "score": 1.0, + "content": ". The further condition on", + "type": "text" + }, + { + "bbox": [ + 391, + 277, + 410, + 286 + ], + "score": 0.87, + "content": "\\gamma _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "is to avoid pathological", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 287, + 500, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 500, + 299 + ], + "score": 1.0, + "content": "cases where we have a large number of degenerate 4-cycles passing through the same three nodes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 323, + 316 + ], + "score": 1.0, + "content": "To understand the conclusions in Theorem 4, let us fix", + "type": "text" + }, + { + "bbox": [ + 324, + 303, + 352, + 314 + ], + "score": 0.91, + "content": "\\ell _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 302, + 506, + 316 + ], + "score": 1.0, + "content": ". The equation 4 shows that negatively", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "curved edges are the ones causing bottlenecks, interpreted as how the graph topology prevents a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 101, + 323, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 101, + 323, + 189, + 348 + ], + "score": 1.0, + "content": "representation h(2)k t", + "type": "text" + }, + { + "bbox": [ + 183, + 324, + 342, + 343 + ], + "score": 1.0, + "content": "o be affected by non-adjacent features", + "type": "text" + }, + { + "bbox": [ + 342, + 325, + 382, + 340 + ], + "score": 0.93, + "content": "h _ { i } ^ { ( 0 ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 327, + 506, + 340 + ], + "score": 1.0, + "content": ". Theorem 4 implies that if we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 338, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 426, + 351 + ], + "score": 1.0, + "content": "have a negatively curved edge as in (ii), then there exist a large number of nodes", + "type": "text" + }, + { + "bbox": [ + 427, + 339, + 434, + 348 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 338, + 507, + 351 + ], + "score": 1.0, + "content": "such that GNNs -", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 306, + 363 + ], + "score": 1.0, + "content": "on average - struggle to propagate messages from", + "type": "text" + }, + { + "bbox": [ + 306, + 351, + 311, + 360 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 349, + 322, + 363 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 322, + 350, + 329, + 360 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 349, + 462, + 363 + ], + "score": 1.0, + "content": "in two layers despite these nodes", + "type": "text" + }, + { + "bbox": [ + 463, + 350, + 470, + 360 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "being at", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "distance 2 from i. In this case the over-squashing occurs as measured by the Jacobian in equation 4", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "and hence the propagation of information suffers. If the task at hand has long-range dependencies,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 383, + 500, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 500, + 395 + ], + "score": 1.0, + "content": "then the over-squashing caused by the negatively curved edges may compromise the performance.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "Bottleneck via Cheeger constant. We now relate the previous discussion about bottlenecks and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "curvature to spectral properties of the graph. In particular, since the spectral gap of a graph can be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "interpreted as a topological obstruction to the graph being partitioned into two communities, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "argue below that this quantity is related to the graph bottleneck and should hence be controllable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 456, + 463 + ], + "score": 1.0, + "content": "by the curvature. We start with an intuitive explanation: suppose we are given a graph", + "type": "text" + }, + { + "bbox": [ + 457, + 451, + 466, + 460 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "with two", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "communities separated by few edges. In this case, we see that the graph can be easily disconnected.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 500, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 500, + 484 + ], + "score": 1.0, + "content": "This property is encoded in the classical notion of the Cheeger constant (Chung & Graham, 1997)", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 486, + 420, + 514 + ], + "lines": [ + { + "bbox": [ + 189, + 486, + 420, + 514 + ], + "spans": [ + { + "bbox": [ + 189, + 486, + 420, + 514 + ], + "score": 0.93, + "content": "h _ { G } : = \\operatorname* { m i n } _ { S \\subset V } h _ { S } , \\quad h _ { S } : = \\operatorname* { m i n } _ { S \\subset V } \\frac { | \\partial S | } { \\operatorname* { m i n } \\{ \\mathrm { v o l } ( S ) , \\mathrm { v o l } ( V \\setminus S ) \\} }", + "type": "interline_equation", + "image_path": "46c4e336d84b9a71dfe2b039ff540b6da2a1dfbec9839bbdf1b33e0b1a2d846c.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 486, + 420, + 500.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 189, + 500.0, + 420, + 514.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 504, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 132, + 532 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 517, + 268, + 530 + ], + "score": 0.89, + "content": "\\partial S = \\{ ( i , j ) : i \\in S , \\ j \\in V \\setminus S \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 515, + 286, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 287, + 517, + 363, + 531 + ], + "score": 0.91, + "content": "\\textstyle { \\mathrm { v o l } } ( S ) = \\sum _ { i \\in S } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 515, + 506, + 532 + ], + "score": 1.0, + "content": ". The main result about the Cheeger", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 528, + 415, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 339, + 541 + ], + "score": 1.0, + "content": "constant is the Cheeger inequality (Cheeger, 2015; Chung", + "type": "text" + }, + { + "bbox": [ + 340, + 530, + 349, + 538 + ], + "score": 0.65, + "content": "\\bar { \\& }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 528, + 415, + 541 + ], + "score": 1.0, + "content": "Graham, 1997):", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 541, + 342, + 566 + ], + "lines": [ + { + "bbox": [ + 270, + 541, + 342, + 566 + ], + "spans": [ + { + "bbox": [ + 270, + 541, + 342, + 566 + ], + "score": 0.93, + "content": "2 h _ { G } \\geq \\lambda _ { 1 } \\geq \\frac { h _ { G } ^ { 2 } } { 2 }", + "type": "interline_equation", + "image_path": "67c50aee7d647525c314748a9fd544c98f53ea9cd19f4bb2d88b2580005ad02b.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 270, + 541, + 342, + 566 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 133, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 567, + 145, + 578 + ], + "score": 0.88, + "content": "\\lambda _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "is the first non-zero eigenvalue of the normalized graph Laplacian, often referred to as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 387, + 590 + ], + "score": 1.0, + "content": "the spectral gap. A graph with two tightly connected communities", + "type": "text" + }, + { + "bbox": [ + 388, + 578, + 396, + 588 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 577, + 415, + 590 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 415, + 577, + 467, + 590 + ], + "score": 0.91, + "content": "V \\setminus S : = \\bar { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "and few", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 326, + 601 + ], + "score": 1.0, + "content": "inter-community edges has a small Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 327, + 589, + 340, + 600 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 588, + 506, + 601 + ], + "score": 1.0, + "content": ". For nodes in different communities to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "interact with each other, all messages need to go through the same few bridges hence leading to the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "over-squashing of information (a similar intuition was explored in Alon & Yahav (2021)). Therefore,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 120, + 633 + ], + "score": 0.87, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "can be interpreted as a rough measure of graph ‘bottleneckedness’, in the sense that the smaller its", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "value, the more likely the over-squashing is to occur across inter-community edges. Since Theorem 4", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 473, + 657 + ], + "score": 1.0, + "content": "implies that negatively curved edges induce the bottleneck, we expect a relationship between", + "type": "text" + }, + { + "bbox": [ + 473, + 644, + 487, + 654 + ], + "score": 0.88, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 655, + 486, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 486, + 667 + ], + "score": 1.0, + "content": "the curvature of the graph. The next proposition follows from Theorem 2 and Lin et al. (2011):", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 669, + 397, + 683 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 398, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 178, + 686 + ], + "score": 1.0, + "content": "Proposition 5. If", + "type": "text" + }, + { + "bbox": [ + 179, + 669, + 252, + 682 + ], + "score": 0.87, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 666, + 280, + 686 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 280, + 670, + 302, + 682 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 666, + 326, + 686 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 326, + 669, + 393, + 684 + ], + "score": 0.92, + "content": "\\textstyle \\lambda _ { 1 } / 2 \\geq h _ { G } \\geq { \\frac { k } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 666, + 398, + 686 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 414, + 700 + ], + "score": 1.0, + "content": "Therefore, a positive lower bound on the curvature gives us a control on", + "type": "text" + }, + { + "bbox": [ + 415, + 688, + 429, + 699 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "and hence on the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "spectral gap of the graph. In the next section, we show that diffusion-based graph-rewiring methods", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 234, + 722 + ], + "score": 1.0, + "content": "might fail to significantly alter", + "type": "text" + }, + { + "bbox": [ + 234, + 710, + 248, + 721 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "and hence correct the graph bottleneck potentially induced by", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 503, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 503, + 734 + ], + "score": 1.0, + "content": "inter-community edges. This will lead us to propose an alternative curvature-based graph rewiring.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5 + } + ], + "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, + 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": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Curvature and over-squashing. Thanks to our new combinatorial curvature and the sensitivity", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "analysis in Lemma 1, we are able to relate local curvature properties to the Jacobian of the node", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "representations. This leads to one of the main results of this paper: negatively curved edges are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 483, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 483, + 128 + ], + "score": 1.0, + "content": "those causing the graph bottleneck and thus leading to the over-squashing phenomenon:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 135, + 468, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 134, + 469, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 321, + 149 + ], + "score": 1.0, + "content": "Theorem 4. Consider a MPNN as in equation 1. Let", + "type": "text" + }, + { + "bbox": [ + 321, + 136, + 344, + 147 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 134, + 364, + 149 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 365, + 136, + 397, + 148 + ], + "score": 0.94, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 134, + 469, + 149 + ], + "score": 1.0, + "content": "and assume that:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 134, + 469, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 123, + 154, + 493, + 168 + ], + "lines": [ + { + "bbox": [ + 126, + 155, + 496, + 169 + ], + "spans": [ + { + "bbox": [ + 126, + 155, + 142, + 169 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 156, + 187, + 168 + ], + "score": 0.9, + "content": "| \\nabla \\phi _ { \\ell } | \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 155, + 205, + 169 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 156, + 250, + 168 + ], + "score": 0.92, + "content": "| \\nabla \\psi _ { \\ell } | \\le \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 155, + 287, + 169 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 287, + 156, + 348, + 167 + ], + "score": 0.9, + "content": "0 \\leq \\ell \\leq L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 155, + 371, + 169 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 371, + 156, + 398, + 167 + ], + "score": 0.9, + "content": "L \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 155, + 496, + 169 + ], + "score": 1.0, + "content": "the depth of the MPNN.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 126, + 155, + 496, + 169 + ] + }, + { + "type": "text", + "bbox": [ + 125, + 174, + 466, + 190 + ], + "lines": [ + { + "bbox": [ + 123, + 173, + 468, + 191 + ], + "spans": [ + { + "bbox": [ + 123, + 173, + 192, + 191 + ], + "score": 1.0, + "content": "(ii) There exists", + "type": "text" + }, + { + "bbox": [ + 192, + 177, + 198, + 186 + ], + "score": 0.46, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 173, + 214, + 191 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 215, + 174, + 322, + 189 + ], + "score": 0.75, + "content": "0 < \\delta < ( \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} ) ^ { - \\frac { 1 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 173, + 326, + 191 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 326, + 175, + 363, + 189 + ], + "score": 0.78, + "content": "\\delta < \\gamma _ { m a x } ^ { - 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 173, + 384, + 191 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 385, + 176, + 463, + 189 + ], + "score": 0.88, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 173, + 468, + 191 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 123, + 173, + 468, + 191 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 447, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 448, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 175, + 212 + ], + "score": 1.0, + "content": "Then there exists", + "type": "text" + }, + { + "bbox": [ + 176, + 198, + 223, + 210 + ], + "score": 0.93, + "content": "Q _ { j } \\subset S _ { 2 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 197, + 266, + 212 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 266, + 197, + 313, + 210 + ], + "score": 0.92, + "content": "| Q _ { j } | > \\delta ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 197, + 346, + 212 + ], + "score": 1.0, + "content": "and for", + "type": "text" + }, + { + "bbox": [ + 346, + 198, + 412, + 209 + ], + "score": 0.91, + "content": "0 \\leq \\ell _ { 0 } \\leq L - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 197, + 448, + 212 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 197, + 448, + 212 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 211, + 376, + 246 + ], + "lines": [ + { + "bbox": [ + 234, + 211, + 376, + 246 + ], + "spans": [ + { + "bbox": [ + 234, + 211, + 376, + 246 + ], + "score": 0.94, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } \\left| \\frac { \\partial h _ { k } ^ { ( \\ell _ { 0 } + 2 ) } } { \\partial h _ { i } ^ { ( \\ell _ { 0 } ) } } \\right| < ( \\alpha \\beta ) ^ { 2 } \\delta ^ { \\frac { 1 } { 4 } } .", + "type": "interline_equation", + "image_path": "8058842145c59033fb6edaead13943e3b3d1288dde59ccf9db91c03ca0c5272e.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 211, + 376, + 228.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 234, + 228.5, + 376, + 246.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "Condition (i) is always satisfied and allows us to control the message passing functions. The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 315, + 277 + ], + "score": 1.0, + "content": "requirement (ii) instead means that the curvature of", + "type": "text" + }, + { + "bbox": [ + 315, + 264, + 335, + 276 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "is negative enough when compared to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 149, + 288 + ], + "score": 1.0, + "content": "degrees of", + "type": "text" + }, + { + "bbox": [ + 149, + 276, + 154, + 285 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 275, + 172, + 288 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 276, + 178, + 286 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 275, + 224, + 288 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 224, + 275, + 288, + 287 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) > - 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 275, + 390, + 288 + ], + "score": 1.0, + "content": ". The further condition on", + "type": "text" + }, + { + "bbox": [ + 391, + 277, + 410, + 286 + ], + "score": 0.87, + "content": "\\gamma _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "is to avoid pathological", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 287, + 500, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 500, + 299 + ], + "score": 1.0, + "content": "cases where we have a large number of degenerate 4-cycles passing through the same three nodes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 253, + 505, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 323, + 316 + ], + "score": 1.0, + "content": "To understand the conclusions in Theorem 4, let us fix", + "type": "text" + }, + { + "bbox": [ + 324, + 303, + 352, + 314 + ], + "score": 0.91, + "content": "\\ell _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 302, + 506, + 316 + ], + "score": 1.0, + "content": ". The equation 4 shows that negatively", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "curved edges are the ones causing bottlenecks, interpreted as how the graph topology prevents a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 101, + 323, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 101, + 323, + 189, + 348 + ], + "score": 1.0, + "content": "representation h(2)k t", + "type": "text" + }, + { + "bbox": [ + 183, + 324, + 342, + 343 + ], + "score": 1.0, + "content": "o be affected by non-adjacent features", + "type": "text" + }, + { + "bbox": [ + 342, + 325, + 382, + 340 + ], + "score": 0.93, + "content": "h _ { i } ^ { ( 0 ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 327, + 506, + 340 + ], + "score": 1.0, + "content": ". Theorem 4 implies that if we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 338, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 426, + 351 + ], + "score": 1.0, + "content": "have a negatively curved edge as in (ii), then there exist a large number of nodes", + "type": "text" + }, + { + "bbox": [ + 427, + 339, + 434, + 348 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 338, + 507, + 351 + ], + "score": 1.0, + "content": "such that GNNs -", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 306, + 363 + ], + "score": 1.0, + "content": "on average - struggle to propagate messages from", + "type": "text" + }, + { + "bbox": [ + 306, + 351, + 311, + 360 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 349, + 322, + 363 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 322, + 350, + 329, + 360 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 349, + 462, + 363 + ], + "score": 1.0, + "content": "in two layers despite these nodes", + "type": "text" + }, + { + "bbox": [ + 463, + 350, + 470, + 360 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "being at", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "distance 2 from i. In this case the over-squashing occurs as measured by the Jacobian in equation 4", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "and hence the propagation of information suffers. If the task at hand has long-range dependencies,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 383, + 500, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 500, + 395 + ], + "score": 1.0, + "content": "then the over-squashing caused by the negatively curved edges may compromise the performance.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 101, + 302, + 507, + 395 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "Bottleneck via Cheeger constant. We now relate the previous discussion about bottlenecks and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "curvature to spectral properties of the graph. In particular, since the spectral gap of a graph can be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "interpreted as a topological obstruction to the graph being partitioned into two communities, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "argue below that this quantity is related to the graph bottleneck and should hence be controllable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 456, + 463 + ], + "score": 1.0, + "content": "by the curvature. We start with an intuitive explanation: suppose we are given a graph", + "type": "text" + }, + { + "bbox": [ + 457, + 451, + 466, + 460 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "with two", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "communities separated by few edges. In this case, we see that the graph can be easily disconnected.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 500, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 500, + 484 + ], + "score": 1.0, + "content": "This property is encoded in the classical notion of the Cheeger constant (Chung & Graham, 1997)", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 405, + 506, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 486, + 420, + 514 + ], + "lines": [ + { + "bbox": [ + 189, + 486, + 420, + 514 + ], + "spans": [ + { + "bbox": [ + 189, + 486, + 420, + 514 + ], + "score": 0.93, + "content": "h _ { G } : = \\operatorname* { m i n } _ { S \\subset V } h _ { S } , \\quad h _ { S } : = \\operatorname* { m i n } _ { S \\subset V } \\frac { | \\partial S | } { \\operatorname* { m i n } \\{ \\mathrm { v o l } ( S ) , \\mathrm { v o l } ( V \\setminus S ) \\} }", + "type": "interline_equation", + "image_path": "46c4e336d84b9a71dfe2b039ff540b6da2a1dfbec9839bbdf1b33e0b1a2d846c.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 486, + 420, + 500.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 189, + 500.0, + 420, + 514.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 504, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 132, + 532 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 517, + 268, + 530 + ], + "score": 0.89, + "content": "\\partial S = \\{ ( i , j ) : i \\in S , \\ j \\in V \\setminus S \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 515, + 286, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 287, + 517, + 363, + 531 + ], + "score": 0.91, + "content": "\\textstyle { \\mathrm { v o l } } ( S ) = \\sum _ { i \\in S } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 515, + 506, + 532 + ], + "score": 1.0, + "content": ". The main result about the Cheeger", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 528, + 415, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 339, + 541 + ], + "score": 1.0, + "content": "constant is the Cheeger inequality (Cheeger, 2015; Chung", + "type": "text" + }, + { + "bbox": [ + 340, + 530, + 349, + 538 + ], + "score": 0.65, + "content": "\\bar { \\& }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 528, + 415, + 541 + ], + "score": 1.0, + "content": "Graham, 1997):", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 515, + 506, + 541 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 541, + 342, + 566 + ], + "lines": [ + { + "bbox": [ + 270, + 541, + 342, + 566 + ], + "spans": [ + { + "bbox": [ + 270, + 541, + 342, + 566 + ], + "score": 0.93, + "content": "2 h _ { G } \\geq \\lambda _ { 1 } \\geq \\frac { h _ { G } ^ { 2 } } { 2 }", + "type": "interline_equation", + "image_path": "67c50aee7d647525c314748a9fd544c98f53ea9cd19f4bb2d88b2580005ad02b.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 270, + 541, + 342, + 566 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 133, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 567, + 145, + 578 + ], + "score": 0.88, + "content": "\\lambda _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "is the first non-zero eigenvalue of the normalized graph Laplacian, often referred to as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 387, + 590 + ], + "score": 1.0, + "content": "the spectral gap. A graph with two tightly connected communities", + "type": "text" + }, + { + "bbox": [ + 388, + 578, + 396, + 588 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 577, + 415, + 590 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 415, + 577, + 467, + 590 + ], + "score": 0.91, + "content": "V \\setminus S : = \\bar { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "and few", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 326, + 601 + ], + "score": 1.0, + "content": "inter-community edges has a small Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 327, + 589, + 340, + 600 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 588, + 506, + 601 + ], + "score": 1.0, + "content": ". For nodes in different communities to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "interact with each other, all messages need to go through the same few bridges hence leading to the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "over-squashing of information (a similar intuition was explored in Alon & Yahav (2021)). Therefore,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 120, + 633 + ], + "score": 0.87, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "can be interpreted as a rough measure of graph ‘bottleneckedness’, in the sense that the smaller its", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "value, the more likely the over-squashing is to occur across inter-community edges. Since Theorem 4", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 473, + 657 + ], + "score": 1.0, + "content": "implies that negatively curved edges induce the bottleneck, we expect a relationship between", + "type": "text" + }, + { + "bbox": [ + 473, + 644, + 487, + 654 + ], + "score": 0.88, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 655, + 486, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 486, + 667 + ], + "score": 1.0, + "content": "the curvature of the graph. The next proposition follows from Theorem 2 and Lin et al. (2011):", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 566, + 506, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 669, + 397, + 683 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 398, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 178, + 686 + ], + "score": 1.0, + "content": "Proposition 5. If", + "type": "text" + }, + { + "bbox": [ + 179, + 669, + 252, + 682 + ], + "score": 0.87, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 666, + 280, + 686 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 280, + 670, + 302, + 682 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 666, + 326, + 686 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 326, + 669, + 393, + 684 + ], + "score": 0.92, + "content": "\\textstyle \\lambda _ { 1 } / 2 \\geq h _ { G } \\geq { \\frac { k } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 666, + 398, + 686 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 666, + 398, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 414, + 700 + ], + "score": 1.0, + "content": "Therefore, a positive lower bound on the curvature gives us a control on", + "type": "text" + }, + { + "bbox": [ + 415, + 688, + 429, + 699 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "and hence on the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "spectral gap of the graph. In the next section, we show that diffusion-based graph-rewiring methods", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 234, + 722 + ], + "score": 1.0, + "content": "might fail to significantly alter", + "type": "text" + }, + { + "bbox": [ + 234, + 710, + 248, + 721 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "and hence correct the graph bottleneck potentially induced by", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 503, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 503, + 734 + ], + "score": 1.0, + "content": "inter-community edges. This will lead us to propose an alternative curvature-based graph rewiring.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 336, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 337, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 337, + 96 + ], + "score": 1.0, + "content": "4 CURVATURE-BASED REWIRING METHODS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "The traditional paradigm of message passing graph neural networks assumes that messages are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "propagated on the input graph (Gilmer et al., 2017). More recently, there is a trend to decouple", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "the input graph from the graph used for information propagation. This can take the form of graph", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 152 + ], + "score": 1.0, + "content": "subsampling or resampling to deal with scalability (Hamilton et al., 2017) or topological noise (Zhang", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 162 + ], + "score": 1.0, + "content": "et al., 2019), using larger motif-based (Monti et al., 2018) or multi-hop filters (Rossi et al., 2020),", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "or changing the graph either as a preprocessing step (Klicpera et al., 2019; Alon & Yahav, 2021) or", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "adaptively for the downstream task (Wang et al., 2019; Kazi et al., 2020). Such methods are often", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 273, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 273, + 196 + ], + "score": 1.0, + "content": "generically referred to as graph rewiring.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 199, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "In the context of this paper, we assume that graph rewiring attempts to produce a new graph", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 165, + 222 + ], + "score": 0.94, + "content": "G ^ { \\prime } = ( V , E ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "with a different edge structure that reduces the bottleneck and hence potentially", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 235 + ], + "score": 1.0, + "content": "alleviates the over-squashing of information. We propose a method that leverages the graph curvature", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "to guide the rewiring steps in a surgical way by modifying the negatively-curved edges, so to decrease", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "the bottleneck without significantly compromising the statistical properties of the input graph. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "also rigorously show that a random-walk based rewiring method might generally fail to obtain an edge", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 120, + 278 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 265, + 132, + 275 + ], + "score": 0.86, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "with a significant improvement in its bottleneckedness as measured by the Cheeger constant.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 505, + 332 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "Curvature-based graph rewiring. Since according to Theorem 4 negatively curved edges induce", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "a bottleneck and are hence responsible for over-squashing, a curvature-based rewiring method should", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "attempt to alleviate a graph’s strongly-negatively curved edges. To this end we implement a simple", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 321, + 468, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 468, + 333 + ], + "score": 1.0, + "content": "rewiring method called Stochastic Discrete Ricci Flow (SDRF), described in Algorithm 1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 106, + 346, + 322, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 323, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 323, + 361 + ], + "score": 1.0, + "content": "Algorithm 1: Stochastic Discrete Ricci Flow (SDRF)", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 362, + 479, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 479, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 162, + 376 + ], + "score": 1.0, + "content": "Input: graph", + "type": "text" + }, + { + "bbox": [ + 162, + 363, + 171, + 373 + ], + "score": 0.72, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 360, + 225, + 376 + ], + "score": 1.0, + "content": ", temperature", + "type": "text" + }, + { + "bbox": [ + 225, + 363, + 250, + 373 + ], + "score": 0.87, + "content": "\\tau > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 360, + 464, + 376 + ], + "score": 1.0, + "content": ", max number of iterations, optional Ric upper-bound", + "type": "text" + }, + { + "bbox": [ + 464, + 362, + 479, + 373 + ], + "score": 0.77, + "content": "C ^ { + }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 141, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 141, + 387 + ], + "score": 1.0, + "content": "Repeat", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 124, + 385, + 354, + 396 + ], + "lines": [ + { + "bbox": [ + 124, + 383, + 351, + 398 + ], + "spans": [ + { + "bbox": [ + 124, + 383, + 172, + 398 + ], + "score": 1.0, + "content": "1) For edge", + "type": "text" + }, + { + "bbox": [ + 172, + 385, + 195, + 396 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 383, + 315, + 398 + ], + "score": 1.0, + "content": "with minimal Ricci curvature", + "type": "text" + }, + { + "bbox": [ + 316, + 384, + 351, + 397 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 129, + 397, + 468, + 435 + ], + "lines": [ + { + "bbox": [ + 135, + 396, + 475, + 412 + ], + "spans": [ + { + "bbox": [ + 135, + 396, + 204, + 412 + ], + "score": 1.0, + "content": "Calculate vector", + "type": "text" + }, + { + "bbox": [ + 205, + 400, + 213, + 408 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 396, + 240, + 412 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 240, + 398, + 356, + 410 + ], + "score": 0.9, + "content": "x _ { k l } = \\mathrm { R i c } _ { k l } ( i , j ) - \\mathrm { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 396, + 439, + 412 + ], + "score": 1.0, + "content": ", the improvement to", + "type": "text" + }, + { + "bbox": [ + 440, + 398, + 475, + 410 + ], + "score": 0.82, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 408, + 358, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 214, + 422 + ], + "score": 1.0, + "content": "from adding edge", + "type": "text" + }, + { + "bbox": [ + 215, + 409, + 238, + 419 + ], + "score": 0.89, + "content": "k \\sim l", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 408, + 266, + 422 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 266, + 410, + 308, + 421 + ], + "score": 0.86, + "content": "k \\in B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 408, + 312, + 422 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 312, + 410, + 353, + 421 + ], + "score": 0.86, + "content": "l \\in B _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 408, + 358, + 422 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 134, + 422, + 440, + 435 + ], + "spans": [ + { + "bbox": [ + 134, + 422, + 194, + 435 + ], + "score": 1.0, + "content": "Sample index", + "type": "text" + }, + { + "bbox": [ + 194, + 423, + 208, + 434 + ], + "score": 0.85, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 422, + 309, + 435 + ], + "score": 1.0, + "content": "with probability softmax", + "type": "text" + }, + { + "bbox": [ + 309, + 423, + 336, + 435 + ], + "score": 0.87, + "content": "( \\tau { \\pmb x } ) _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 422, + 393, + 435 + ], + "score": 1.0, + "content": "and add edge", + "type": "text" + }, + { + "bbox": [ + 393, + 423, + 416, + 433 + ], + "score": 0.88, + "content": "k \\sim l", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 422, + 428, + 435 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 428, + 423, + 437, + 433 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 422, + 440, + 435 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 115, + 437, + 444, + 449 + ], + "lines": [ + { + "bbox": [ + 120, + 436, + 442, + 450 + ], + "spans": [ + { + "bbox": [ + 120, + 436, + 189, + 450 + ], + "score": 1.0, + "content": "2) Remove edge", + "type": "text" + }, + { + "bbox": [ + 189, + 438, + 212, + 449 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 436, + 334, + 450 + ], + "score": 1.0, + "content": "with maximal Ricci curvature", + "type": "text" + }, + { + "bbox": [ + 334, + 437, + 369, + 449 + ], + "score": 0.84, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 436, + 379, + 450 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 380, + 437, + 442, + 449 + ], + "score": 0.89, + "content": "\\operatorname { R i c } ( i , j ) > C ^ { + }", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 450, + 290, + 460 + ], + "lines": [ + { + "bbox": [ + 107, + 448, + 291, + 461 + ], + "spans": [ + { + "bbox": [ + 107, + 448, + 291, + 461 + ], + "score": 1.0, + "content": "Until convergence, or max iterations reached;", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "At each iteration this preprocessing step adds an edge to ‘support’ the graph’s most negatively curved", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 480, + 502 + ], + "score": 1.0, + "content": "edge, and then removes the most positively curved edge. The requirement on the added edge", + "type": "text" + }, + { + "bbox": [ + 480, + 489, + 504, + 500 + ], + "score": 0.89, + "content": "k \\sim l", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 124, + 514 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 500, + 168, + 512 + ], + "score": 0.93, + "content": "k \\in B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 499, + 188, + 514 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 189, + 500, + 231, + 512 + ], + "score": 0.93, + "content": "l \\in B _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 499, + 403, + 514 + ], + "score": 1.0, + "content": "ensures that we’re adding either an extra", + "type": "text" + }, + { + "bbox": [ + 403, + 501, + 414, + 510 + ], + "score": 0.26, + "content": "3 -", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "or 4-cycle around the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 163, + 524 + ], + "score": 1.0, + "content": "negative edge", + "type": "text" + }, + { + "bbox": [ + 163, + 512, + 186, + 523 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "so that this is a local modification. The graph edit distance between the original", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 292, + 534 + ], + "score": 1.0, + "content": "and preprocessed graph is bounded above by", + "type": "text" + }, + { + "bbox": [ + 292, + 522, + 310, + 533 + ], + "score": 0.83, + "content": "2 ~ \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "the max number of iterations. The temperature", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 114, + 543 + ], + "score": 0.69, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 533, + 331, + 546 + ], + "score": 1.0, + "content": "determines how stochastic the edge addition is, with", + "type": "text" + }, + { + "bbox": [ + 331, + 534, + 363, + 543 + ], + "score": 0.89, + "content": "\\tau = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "being fully deterministic (the best", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "edge is always added). At each step we remove the edge with most positive curvature to balance the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "distributions of curvature and node degrees. We use Balanced Forman curvature as in equation 3 for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 564, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 141, + 578 + ], + "score": 0.86, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 564, + 146, + 580 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 146, + 566, + 162, + 577 + ], + "score": 0.86, + "content": "C ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 564, + 506, + 580 + ], + "score": 1.0, + "content": "can be chosen to stop the method skewing the curvature distribution negative, including", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 575, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 107, + 577, + 146, + 587 + ], + "score": 0.88, + "content": "C ^ { + } = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 575, + 506, + 590 + ], + "score": 1.0, + "content": "to not remove any edges. The method is inspired by the continuous (backwards) Ricci flow", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "with the aim of homogenizing edge curvatures. This is different from more direct extensions of Ricci", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "flow on graphs where it becomes increasingly expensive to propagate messages across negatively", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "curved edges (as in other applications such as Ni et al. (2019)). An example alongside its continuous", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 621, + 243, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 243, + 633 + ], + "score": 1.0, + "content": "analogue can be seen in Figure 1.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "Can random-walk based rewiring address bottlenecks? A good way of understanding the effec-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "tiveness of SDRF in reducing the graph bottleneck is through comparison with random-walk based", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 404, + 678 + ], + "score": 1.0, + "content": "rewiring strategies. Recall that, as argued in Section 3, the Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 405, + 666, + 418, + 677 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "of a graph constitutes", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 480, + 690 + ], + "score": 1.0, + "content": "a rough measure of its bottleneckedness as induced by the inter-community edges (a small", + "type": "text" + }, + { + "bbox": [ + 480, + 677, + 494, + 688 + ], + "score": 0.88, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 331, + 701 + ], + "score": 1.0, + "content": "indicative of a bottleneck). Suppose we are given a graph", + "type": "text" + }, + { + "bbox": [ + 331, + 689, + 340, + 698 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 687, + 391, + 701 + ], + "score": 1.0, + "content": "with a small", + "type": "text" + }, + { + "bbox": [ + 391, + 688, + 405, + 699 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "and wish to rewire it into", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 139, + 712 + ], + "score": 1.0, + "content": "a graph", + "type": "text" + }, + { + "bbox": [ + 139, + 699, + 151, + 709 + ], + "score": 0.86, + "content": "G ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "with a significantly improved Cheeger constant in order to reduce the inter-community", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "bottleneck. A random-walk based rewiring method such as DIGL (Klicpera et al., 2019) acts by", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "smoothing out the graph adjacency and hence tends to promote connections among nodes at short", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 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": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 336, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 337, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 337, + 96 + ], + "score": 1.0, + "content": "4 CURVATURE-BASED REWIRING METHODS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "The traditional paradigm of message passing graph neural networks assumes that messages are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "propagated on the input graph (Gilmer et al., 2017). More recently, there is a trend to decouple", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "the input graph from the graph used for information propagation. This can take the form of graph", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 152 + ], + "score": 1.0, + "content": "subsampling or resampling to deal with scalability (Hamilton et al., 2017) or topological noise (Zhang", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 162 + ], + "score": 1.0, + "content": "et al., 2019), using larger motif-based (Monti et al., 2018) or multi-hop filters (Rossi et al., 2020),", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "or changing the graph either as a preprocessing step (Klicpera et al., 2019; Alon & Yahav, 2021) or", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "adaptively for the downstream task (Wang et al., 2019; Kazi et al., 2020). Such methods are often", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 273, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 273, + 196 + ], + "score": 1.0, + "content": "generically referred to as graph rewiring.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 105, + 506, + 196 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 199, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "In the context of this paper, we assume that graph rewiring attempts to produce a new graph", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 165, + 222 + ], + "score": 0.94, + "content": "G ^ { \\prime } = ( V , E ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "with a different edge structure that reduces the bottleneck and hence potentially", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 235 + ], + "score": 1.0, + "content": "alleviates the over-squashing of information. We propose a method that leverages the graph curvature", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "to guide the rewiring steps in a surgical way by modifying the negatively-curved edges, so to decrease", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "the bottleneck without significantly compromising the statistical properties of the input graph. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "also rigorously show that a random-walk based rewiring method might generally fail to obtain an edge", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 120, + 278 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 265, + 132, + 275 + ], + "score": 0.86, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "with a significant improvement in its bottleneckedness as measured by the Cheeger constant.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 198, + 506, + 278 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 288, + 505, + 332 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "Curvature-based graph rewiring. Since according to Theorem 4 negatively curved edges induce", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "a bottleneck and are hence responsible for over-squashing, a curvature-based rewiring method should", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "attempt to alleviate a graph’s strongly-negatively curved edges. To this end we implement a simple", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 321, + 468, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 468, + 333 + ], + "score": 1.0, + "content": "rewiring method called Stochastic Discrete Ricci Flow (SDRF), described in Algorithm 1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 288, + 506, + 333 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 346, + 322, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 323, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 323, + 361 + ], + "score": 1.0, + "content": "Algorithm 1: Stochastic Discrete Ricci Flow (SDRF)", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 362, + 479, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 479, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 162, + 376 + ], + "score": 1.0, + "content": "Input: graph", + "type": "text" + }, + { + "bbox": [ + 162, + 363, + 171, + 373 + ], + "score": 0.72, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 360, + 225, + 376 + ], + "score": 1.0, + "content": ", temperature", + "type": "text" + }, + { + "bbox": [ + 225, + 363, + 250, + 373 + ], + "score": 0.87, + "content": "\\tau > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 360, + 464, + 376 + ], + "score": 1.0, + "content": ", max number of iterations, optional Ric upper-bound", + "type": "text" + }, + { + "bbox": [ + 464, + 362, + 479, + 373 + ], + "score": 0.77, + "content": "C ^ { + }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 141, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 141, + 387 + ], + "score": 1.0, + "content": "Repeat", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 360, + 479, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 124, + 385, + 354, + 396 + ], + "lines": [ + { + "bbox": [ + 124, + 383, + 351, + 398 + ], + "spans": [ + { + "bbox": [ + 124, + 383, + 172, + 398 + ], + "score": 1.0, + "content": "1) For edge", + "type": "text" + }, + { + "bbox": [ + 172, + 385, + 195, + 396 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 383, + 315, + 398 + ], + "score": 1.0, + "content": "with minimal Ricci curvature", + "type": "text" + }, + { + "bbox": [ + 316, + 384, + 351, + 397 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 124, + 383, + 351, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 397, + 468, + 435 + ], + "lines": [ + { + "bbox": [ + 135, + 396, + 475, + 412 + ], + "spans": [ + { + "bbox": [ + 135, + 396, + 204, + 412 + ], + "score": 1.0, + "content": "Calculate vector", + "type": "text" + }, + { + "bbox": [ + 205, + 400, + 213, + 408 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 396, + 240, + 412 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 240, + 398, + 356, + 410 + ], + "score": 0.9, + "content": "x _ { k l } = \\mathrm { R i c } _ { k l } ( i , j ) - \\mathrm { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 396, + 439, + 412 + ], + "score": 1.0, + "content": ", the improvement to", + "type": "text" + }, + { + "bbox": [ + 440, + 398, + 475, + 410 + ], + "score": 0.82, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 408, + 358, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 214, + 422 + ], + "score": 1.0, + "content": "from adding edge", + "type": "text" + }, + { + "bbox": [ + 215, + 409, + 238, + 419 + ], + "score": 0.89, + "content": "k \\sim l", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 408, + 266, + 422 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 266, + 410, + 308, + 421 + ], + "score": 0.86, + "content": "k \\in B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 408, + 312, + 422 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 312, + 410, + 353, + 421 + ], + "score": 0.86, + "content": "l \\in B _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 408, + 358, + 422 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 134, + 422, + 440, + 435 + ], + "spans": [ + { + "bbox": [ + 134, + 422, + 194, + 435 + ], + "score": 1.0, + "content": "Sample index", + "type": "text" + }, + { + "bbox": [ + 194, + 423, + 208, + 434 + ], + "score": 0.85, + "content": "k , l", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 422, + 309, + 435 + ], + "score": 1.0, + "content": "with probability softmax", + "type": "text" + }, + { + "bbox": [ + 309, + 423, + 336, + 435 + ], + "score": 0.87, + "content": "( \\tau { \\pmb x } ) _ { k l }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 422, + 393, + 435 + ], + "score": 1.0, + "content": "and add edge", + "type": "text" + }, + { + "bbox": [ + 393, + 423, + 416, + 433 + ], + "score": 0.88, + "content": "k \\sim l", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 422, + 428, + 435 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 428, + 423, + 437, + 433 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 422, + 440, + 435 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 134, + 396, + 475, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 115, + 437, + 444, + 449 + ], + "lines": [ + { + "bbox": [ + 120, + 436, + 442, + 450 + ], + "spans": [ + { + "bbox": [ + 120, + 436, + 189, + 450 + ], + "score": 1.0, + "content": "2) Remove edge", + "type": "text" + }, + { + "bbox": [ + 189, + 438, + 212, + 449 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 436, + 334, + 450 + ], + "score": 1.0, + "content": "with maximal Ricci curvature", + "type": "text" + }, + { + "bbox": [ + 334, + 437, + 369, + 449 + ], + "score": 0.84, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 436, + 379, + 450 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 380, + 437, + 442, + 449 + ], + "score": 0.89, + "content": "\\operatorname { R i c } ( i , j ) > C ^ { + }", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 120, + 436, + 442, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 450, + 290, + 460 + ], + "lines": [ + { + "bbox": [ + 107, + 448, + 291, + 461 + ], + "spans": [ + { + "bbox": [ + 107, + 448, + 291, + 461 + ], + "score": 1.0, + "content": "Until convergence, or max iterations reached;", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 107, + 448, + 291, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "At each iteration this preprocessing step adds an edge to ‘support’ the graph’s most negatively curved", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 480, + 502 + ], + "score": 1.0, + "content": "edge, and then removes the most positively curved edge. The requirement on the added edge", + "type": "text" + }, + { + "bbox": [ + 480, + 489, + 504, + 500 + ], + "score": 0.89, + "content": "k \\sim l", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 124, + 514 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 500, + 168, + 512 + ], + "score": 0.93, + "content": "k \\in B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 499, + 188, + 514 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 189, + 500, + 231, + 512 + ], + "score": 0.93, + "content": "l \\in B _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 499, + 403, + 514 + ], + "score": 1.0, + "content": "ensures that we’re adding either an extra", + "type": "text" + }, + { + "bbox": [ + 403, + 501, + 414, + 510 + ], + "score": 0.26, + "content": "3 -", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "or 4-cycle around the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 163, + 524 + ], + "score": 1.0, + "content": "negative edge", + "type": "text" + }, + { + "bbox": [ + 163, + 512, + 186, + 523 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "so that this is a local modification. The graph edit distance between the original", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 292, + 534 + ], + "score": 1.0, + "content": "and preprocessed graph is bounded above by", + "type": "text" + }, + { + "bbox": [ + 292, + 522, + 310, + 533 + ], + "score": 0.83, + "content": "2 ~ \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "the max number of iterations. The temperature", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 114, + 543 + ], + "score": 0.69, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 533, + 331, + 546 + ], + "score": 1.0, + "content": "determines how stochastic the edge addition is, with", + "type": "text" + }, + { + "bbox": [ + 331, + 534, + 363, + 543 + ], + "score": 0.89, + "content": "\\tau = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "being fully deterministic (the best", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "edge is always added). At each step we remove the edge with most positive curvature to balance the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "distributions of curvature and node degrees. We use Balanced Forman curvature as in equation 3 for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 564, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 141, + 578 + ], + "score": 0.86, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 564, + 146, + 580 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 146, + 566, + 162, + 577 + ], + "score": 0.86, + "content": "C ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 564, + 506, + 580 + ], + "score": 1.0, + "content": "can be chosen to stop the method skewing the curvature distribution negative, including", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 575, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 107, + 577, + 146, + 587 + ], + "score": 0.88, + "content": "C ^ { + } = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 575, + 506, + 590 + ], + "score": 1.0, + "content": "to not remove any edges. The method is inspired by the continuous (backwards) Ricci flow", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "with the aim of homogenizing edge curvatures. This is different from more direct extensions of Ricci", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "flow on graphs where it becomes increasingly expensive to propagate messages across negatively", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "curved edges (as in other applications such as Ni et al. (2019)). An example alongside its continuous", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 621, + 243, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 243, + 633 + ], + "score": 1.0, + "content": "analogue can be seen in Figure 1.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 477, + 506, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "Can random-walk based rewiring address bottlenecks? A good way of understanding the effec-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "tiveness of SDRF in reducing the graph bottleneck is through comparison with random-walk based", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 404, + 678 + ], + "score": 1.0, + "content": "rewiring strategies. Recall that, as argued in Section 3, the Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 405, + 666, + 418, + 677 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "of a graph constitutes", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 480, + 690 + ], + "score": 1.0, + "content": "a rough measure of its bottleneckedness as induced by the inter-community edges (a small", + "type": "text" + }, + { + "bbox": [ + 480, + 677, + 494, + 688 + ], + "score": 0.88, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 331, + 701 + ], + "score": 1.0, + "content": "indicative of a bottleneck). Suppose we are given a graph", + "type": "text" + }, + { + "bbox": [ + 331, + 689, + 340, + 698 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 687, + 391, + 701 + ], + "score": 1.0, + "content": "with a small", + "type": "text" + }, + { + "bbox": [ + 391, + 688, + 405, + 699 + ], + "score": 0.89, + "content": "h _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "and wish to rewire it into", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 139, + 712 + ], + "score": 1.0, + "content": "a graph", + "type": "text" + }, + { + "bbox": [ + 139, + 699, + 151, + 709 + ], + "score": 0.86, + "content": "G ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "with a significantly improved Cheeger constant in order to reduce the inter-community", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "bottleneck. A random-walk based rewiring method such as DIGL (Klicpera et al., 2019) acts by", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "smoothing out the graph adjacency and hence tends to promote connections among nodes at short", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "diffusion distance (Coifman & Lafon, 2006). Accordingly, such a rewiring method might fail to", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "correct structural features like the bottleneck, which is instead more prominent for nodes that are at", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 460, + 118 + ], + "score": 1.0, + "content": "long diffusion distance.3 To emphasize this point, we consider a classic example: given", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 461, + 104, + 502, + 117 + ], + "score": 0.93, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 502, + 104, + 507, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 424, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 424, + 128 + ], + "score": 1.0, + "content": "the Personalized Page Rank (PPR) matrix is defined by (Brin & Page, 1998) as", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "diffusion distance (Coifman & Lafon, 2006). Accordingly, such a rewiring method might fail to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "correct structural features like the bottleneck, which is instead more prominent for nodes that are at", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 460, + 118 + ], + "score": 1.0, + "content": "long diffusion distance.3 To emphasize this point, we consider a classic example: given", + "type": "text" + }, + { + "bbox": [ + 461, + 104, + 502, + 117 + ], + "score": 0.93, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 104, + 507, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 424, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 424, + 128 + ], + "score": 1.0, + "content": "the Personalized Page Rank (PPR) matrix is defined by (Brin & Page, 1998) as", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 128, + 422, + 161 + ], + "lines": [ + { + "bbox": [ + 189, + 128, + 422, + 161 + ], + "spans": [ + { + "bbox": [ + 189, + 128, + 422, + 161 + ], + "score": 0.94, + "content": "R _ { \\alpha } : = \\sum _ { k = 0 } ^ { \\infty } \\theta _ { k } ^ { P P R } ( D ^ { - 1 } A ) ^ { k } = \\alpha \\sum _ { k = 0 } ^ { \\infty } \\left( ( 1 - \\alpha ) ( D ^ { - 1 } A ) \\right) ^ { k } .", + "type": "interline_equation", + "image_path": "5416ff0352bb14371d74eb7ca20808279a57d3790066ef4dabc968e981bfa2ed.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 128, + 422, + 144.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 189, + 144.5, + 422, + 161.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 503, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 262, + 176 + ], + "score": 1.0, + "content": "Assume that we rewire the graph using", + "type": "text" + }, + { + "bbox": [ + 262, + 163, + 276, + 173 + ], + "score": 0.9, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 160, + 505, + 176 + ], + "score": 1.0, + "content": "as in Klicpera et al. (2019) with the PPR kernel, meaning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 249, + 185 + ], + "score": 1.0, + "content": "that we replace the given adjacency", + "type": "text" + }, + { + "bbox": [ + 249, + 174, + 258, + 183 + ], + "score": 0.8, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 173, + 279, + 185 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 279, + 174, + 293, + 184 + ], + "score": 0.89, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 173, + 322, + 185 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 322, + 174, + 336, + 184 + ], + "score": 0.9, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "is stochastic, the new Cheeger constant of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 260, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 260, + 196 + ], + "score": 1.0, + "content": "the rewired graph can be computed as", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 197, + 385, + 230 + ], + "lines": [ + { + "bbox": [ + 225, + 197, + 385, + 230 + ], + "spans": [ + { + "bbox": [ + 225, + 197, + 385, + 230 + ], + "score": 0.95, + "content": "h _ { S , \\alpha } = \\frac { | \\partial S | _ { \\alpha } } { \\mathrm { v o l } _ { \\alpha } ( S ) } \\equiv \\frac { 1 } { | S | } \\sum _ { i \\in S } \\sum _ { j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } .", + "type": "interline_equation", + "image_path": "d98d5f7d33063c1a63e99d2d7a053d9eb730bc5e31cc35bf16867fb4ab3d8558.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 197, + 385, + 213.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 225, + 213.5, + 385, + 230.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "By applying (Chung, 2007, Lemma 5), we show that we cannot improve the Cheeger constant", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "(and hence the bottleneck) arbitrarily well (in contrast to a curvature-based approach). We refer to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 281, + 266 + ], + "score": 1.0, + "content": "Proposition 17 and Remark 18 in Appendix", + "type": "text" + }, + { + "bbox": [ + 282, + 253, + 290, + 263 + ], + "score": 0.31, + "content": "\\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "for results that are more tailored to the actual strategy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 491, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 491, + 277 + ], + "score": 1.0, + "content": "adopted in Klicpera et al. (2019) where we also take into account the effect of the sparsification.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 499, + 306 + ], + "lines": [ + { + "bbox": [ + 103, + 277, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 103, + 277, + 176, + 299 + ], + "score": 1.0, + "content": "Theorem 6. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 281, + 208, + 291 + ], + "score": 0.88, + "content": "S \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 277, + 229, + 299 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 230, + 280, + 309, + 293 + ], + "score": 0.9, + "content": "\\nu o l ( S ) \\le \\nu o l ( G ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 277, + 339, + 299 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 339, + 278, + 442, + 296 + ], + "score": 0.88, + "content": "\\begin{array} { r } { h _ { S , \\alpha } \\leq \\left( \\frac { 1 - \\alpha } { \\alpha } \\right) \\frac { d _ { \\arg } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } h _ { S } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 277, + 473, + 299 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 280, + 505, + 294 + ], + "score": 0.92, + "content": "d _ { \\mathrm { a v g } } ( S )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 383, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 124, + 308 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 294, + 156, + 307 + ], + "score": 0.92, + "content": "d _ { \\mathrm { m i n } } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 293, + 320, + 308 + ], + "score": 1.0, + "content": "are the average and minimum degree on", + "type": "text" + }, + { + "bbox": [ + 320, + 295, + 328, + 304 + ], + "score": 0.72, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 293, + 383, + 308 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "The property that the new Cheeger constant is directly controlled by the old one stems from the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "fact that a random-walk approach like in Klicpera et al. (2019) is meant to act more relevantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "on intra-community edges rather than inter-community edges because it prioritizes short diffusion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "distance nodes. This is also why this method performs well on high-homophily datasets, as discussed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 232, + 371 + ], + "score": 1.0, + "content": "below. In particular, for a fixed", + "type": "text" + }, + { + "bbox": [ + 232, + 359, + 274, + 371 + ], + "score": 0.93, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 359, + 505, + 371 + ], + "score": 1.0, + "content": ", the bound in Theorem 6 can be very small. As a specific", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 271, + 383 + ], + "score": 1.0, + "content": "example, consider two complete graphs", + "type": "text" + }, + { + "bbox": [ + 271, + 371, + 286, + 381 + ], + "score": 0.89, + "content": "K _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 369, + 401, + 383 + ], + "score": 1.0, + "content": "joined by one bridge. Then", + "type": "text" + }, + { + "bbox": [ + 402, + 370, + 502, + 382 + ], + "score": 0.91, + "content": "h _ { G } \\overset { \\cdot } { = } ( n ( n - 1 ) \\overset { \\cdot } { + } 1 ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 369, + 506, + 383 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 356, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 318, + 394 + ], + "score": 1.0, + "content": "which means that the bound on the right hand side is", + "type": "text" + }, + { + "bbox": [ + 318, + 380, + 352, + 393 + ], + "score": 0.93, + "content": "\\dot { O } ( n ^ { - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 379, + 356, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 491, + 410 + ], + "score": 1.0, + "content": "Theorem 6 implies that a diffusion approach such as DIGL might fail to yield a new edge set", + "type": "text" + }, + { + "bbox": [ + 492, + 398, + 504, + 408 + ], + "score": 0.85, + "content": "E ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "with a sufficiently improved bottleneck. By contrast, from Theorem 4 and Proposition 5, we deduce", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "that a curvature-based rewiring method such as SDRF properly addresses the edges that cause the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 152, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 152, + 441 + ], + "score": 1.0, + "content": "bottleneck.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "Graph structure preservation. Although a graph-rewiring approach aims at providing a new edge", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 120, + 478 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 465, + 132, + 475 + ], + "score": 0.85, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "potentially more beneficial for the given learning task, it is still desirable to control how far", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 118, + 486 + ], + "score": 0.84, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 475, + 150, + 489 + ], + "score": 1.0, + "content": "is from", + "type": "text" + }, + { + "bbox": [ + 151, + 477, + 160, + 486 + ], + "score": 0.77, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 475, + 506, + 489 + ], + "score": 1.0, + "content": ". In this regard, we note that a curvature-based rewiring is surgical in nature and hence", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "more likely to preserve the structure of the input graph better than a random-walk based approach.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 270, + 511 + ], + "score": 1.0, + "content": "Consider, for example, that we are given", + "type": "text" + }, + { + "bbox": [ + 270, + 499, + 295, + 510 + ], + "score": 0.91, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "and wish to rewire the graph such that the new edge", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 508, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 120, + 523 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 509, + 132, + 519 + ], + "score": 0.86, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 508, + 247, + 523 + ], + "score": 1.0, + "content": "is within graph-edit distance", + "type": "text" + }, + { + "bbox": [ + 247, + 511, + 254, + 521 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 508, + 324, + 523 + ], + "score": 1.0, + "content": "from the original", + "type": "text" + }, + { + "bbox": [ + 324, + 510, + 333, + 519 + ], + "score": 0.79, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 508, + 506, + 523 + ], + "score": 1.0, + "content": ". Theorem 4 tells us how to do the rewiring", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "under such constraints in order to best address the over-squashing: the topological modifications", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "need to be localized around the most negatively-curved edges. We can do this with SDRF, with the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 542, + 276, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 256, + 554 + ], + "score": 1.0, + "content": "maximum number of iterations set to", + "type": "text" + }, + { + "bbox": [ + 257, + 542, + 273, + 554 + ], + "score": 0.89, + "content": "\\rho / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 542, + 276, + 554 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 239, + 573 + ], + "score": 1.0, + "content": "Secondly, we also point out that", + "type": "text" + }, + { + "bbox": [ + 239, + 559, + 317, + 571 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) < - 2 + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 557, + 369, + 573 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 369, + 559, + 447, + 571 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} > 2 / \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 557, + 506, + 573 + ], + "score": 1.0, + "content": ". Therefore, if", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 293, + 583 + ], + "score": 1.0, + "content": "we mostly modify the edge set at those nodes", + "type": "text" + }, + { + "bbox": [ + 294, + 571, + 307, + 581 + ], + "score": 0.85, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "joined by an edge with large negative curvature,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "then we are perturbing nodes with high degree where such a change is relatively insignificant, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "thus overall statistical properties of the rewired graph such as degree distribution are likely to be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 603, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 614 + ], + "score": 1.0, + "content": "better preserved. Moreover, graph convolutional networks tend to be more stable to perturbations of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "high degree nodes (Zügner et al., 2020; Kenlay et al., 2021), making curvature-based rewiring more", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 625, + 410, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 410, + 637 + ], + "score": 1.0, + "content": "suitable for the downstream learning tasks with popular GNN architectures.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "Homophily and bottleneck. As a final remark, note that the graph rewiring techniques considered", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "in this paper (both DIGL and SDRF) are based purely on the topological structure of the graph and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "completely agnostic to the node features and to whether the dataset is homophilic (adjacent nodes", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "have same labels) or heterophilic. Nonetheless, the different nature of these rewiring methods allows", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "us to draw a few broad conclusions about their suitability in each of these settings. A random-walk", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "3We refer to the right hand side of equation 2 where the power of the normalized augmented adjacency is", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 314, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 263, + 732 + ], + "score": 1.0, + "content": "measuring the number of walks of distance", + "type": "text" + }, + { + "bbox": [ + 263, + 724, + 268, + 730 + ], + "score": 0.71, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 721, + 289, + 732 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 290, + 723, + 293, + 730 + ], + "score": 0.81, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 721, + 304, + 732 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 304, + 725, + 309, + 730 + ], + "score": 0.74, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 721, + 314, + 732 + ], + "score": 1.0, + "content": ".", + "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, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 507, + 128 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 128, + 422, + 161 + ], + "lines": [ + { + "bbox": [ + 189, + 128, + 422, + 161 + ], + "spans": [ + { + "bbox": [ + 189, + 128, + 422, + 161 + ], + "score": 0.94, + "content": "R _ { \\alpha } : = \\sum _ { k = 0 } ^ { \\infty } \\theta _ { k } ^ { P P R } ( D ^ { - 1 } A ) ^ { k } = \\alpha \\sum _ { k = 0 } ^ { \\infty } \\left( ( 1 - \\alpha ) ( D ^ { - 1 } A ) \\right) ^ { k } .", + "type": "interline_equation", + "image_path": "5416ff0352bb14371d74eb7ca20808279a57d3790066ef4dabc968e981bfa2ed.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 128, + 422, + 144.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 189, + 144.5, + 422, + 161.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 503, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 262, + 176 + ], + "score": 1.0, + "content": "Assume that we rewire the graph using", + "type": "text" + }, + { + "bbox": [ + 262, + 163, + 276, + 173 + ], + "score": 0.9, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 160, + 505, + 176 + ], + "score": 1.0, + "content": "as in Klicpera et al. (2019) with the PPR kernel, meaning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 249, + 185 + ], + "score": 1.0, + "content": "that we replace the given adjacency", + "type": "text" + }, + { + "bbox": [ + 249, + 174, + 258, + 183 + ], + "score": 0.8, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 173, + 279, + 185 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 279, + 174, + 293, + 184 + ], + "score": 0.89, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 173, + 322, + 185 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 322, + 174, + 336, + 184 + ], + "score": 0.9, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "is stochastic, the new Cheeger constant of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 260, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 260, + 196 + ], + "score": 1.0, + "content": "the rewired graph can be computed as", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 160, + 505, + 196 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 197, + 385, + 230 + ], + "lines": [ + { + "bbox": [ + 225, + 197, + 385, + 230 + ], + "spans": [ + { + "bbox": [ + 225, + 197, + 385, + 230 + ], + "score": 0.95, + "content": "h _ { S , \\alpha } = \\frac { | \\partial S | _ { \\alpha } } { \\mathrm { v o l } _ { \\alpha } ( S ) } \\equiv \\frac { 1 } { | S | } \\sum _ { i \\in S } \\sum _ { j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } .", + "type": "interline_equation", + "image_path": "d98d5f7d33063c1a63e99d2d7a053d9eb730bc5e31cc35bf16867fb4ab3d8558.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 197, + 385, + 213.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 225, + 213.5, + 385, + 230.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "By applying (Chung, 2007, Lemma 5), we show that we cannot improve the Cheeger constant", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "(and hence the bottleneck) arbitrarily well (in contrast to a curvature-based approach). We refer to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 281, + 266 + ], + "score": 1.0, + "content": "Proposition 17 and Remark 18 in Appendix", + "type": "text" + }, + { + "bbox": [ + 282, + 253, + 290, + 263 + ], + "score": 0.31, + "content": "\\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "for results that are more tailored to the actual strategy", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 491, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 491, + 277 + ], + "score": 1.0, + "content": "adopted in Klicpera et al. (2019) where we also take into account the effect of the sparsification.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 230, + 506, + 277 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 499, + 306 + ], + "lines": [ + { + "bbox": [ + 103, + 277, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 103, + 277, + 176, + 299 + ], + "score": 1.0, + "content": "Theorem 6. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 281, + 208, + 291 + ], + "score": 0.88, + "content": "S \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 277, + 229, + 299 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 230, + 280, + 309, + 293 + ], + "score": 0.9, + "content": "\\nu o l ( S ) \\le \\nu o l ( G ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 277, + 339, + 299 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 339, + 278, + 442, + 296 + ], + "score": 0.88, + "content": "\\begin{array} { r } { h _ { S , \\alpha } \\leq \\left( \\frac { 1 - \\alpha } { \\alpha } \\right) \\frac { d _ { \\arg } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } h _ { S } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 277, + 473, + 299 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 280, + 505, + 294 + ], + "score": 0.92, + "content": "d _ { \\mathrm { a v g } } ( S )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 383, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 124, + 308 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 294, + 156, + 307 + ], + "score": 0.92, + "content": "d _ { \\mathrm { m i n } } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 293, + 320, + 308 + ], + "score": 1.0, + "content": "are the average and minimum degree on", + "type": "text" + }, + { + "bbox": [ + 320, + 295, + 328, + 304 + ], + "score": 0.72, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 293, + 383, + 308 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 103, + 277, + 505, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "The property that the new Cheeger constant is directly controlled by the old one stems from the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "fact that a random-walk approach like in Klicpera et al. (2019) is meant to act more relevantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "on intra-community edges rather than inter-community edges because it prioritizes short diffusion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "distance nodes. This is also why this method performs well on high-homophily datasets, as discussed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 232, + 371 + ], + "score": 1.0, + "content": "below. In particular, for a fixed", + "type": "text" + }, + { + "bbox": [ + 232, + 359, + 274, + 371 + ], + "score": 0.93, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 359, + 505, + 371 + ], + "score": 1.0, + "content": ", the bound in Theorem 6 can be very small. As a specific", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 271, + 383 + ], + "score": 1.0, + "content": "example, consider two complete graphs", + "type": "text" + }, + { + "bbox": [ + 271, + 371, + 286, + 381 + ], + "score": 0.89, + "content": "K _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 369, + 401, + 383 + ], + "score": 1.0, + "content": "joined by one bridge. Then", + "type": "text" + }, + { + "bbox": [ + 402, + 370, + 502, + 382 + ], + "score": 0.91, + "content": "h _ { G } \\overset { \\cdot } { = } ( n ( n - 1 ) \\overset { \\cdot } { + } 1 ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 369, + 506, + 383 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 356, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 318, + 394 + ], + "score": 1.0, + "content": "which means that the bound on the right hand side is", + "type": "text" + }, + { + "bbox": [ + 318, + 380, + 352, + 393 + ], + "score": 0.93, + "content": "\\dot { O } ( n ^ { - 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 379, + 356, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 315, + 506, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 491, + 410 + ], + "score": 1.0, + "content": "Theorem 6 implies that a diffusion approach such as DIGL might fail to yield a new edge set", + "type": "text" + }, + { + "bbox": [ + 492, + 398, + 504, + 408 + ], + "score": 0.85, + "content": "E ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "with a sufficiently improved bottleneck. By contrast, from Theorem 4 and Proposition 5, we deduce", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "that a curvature-based rewiring method such as SDRF properly addresses the edges that cause the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 152, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 152, + 441 + ], + "score": 1.0, + "content": "bottleneck.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 397, + 505, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "Graph structure preservation. Although a graph-rewiring approach aims at providing a new edge", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 120, + 478 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 465, + 132, + 475 + ], + "score": 0.85, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "potentially more beneficial for the given learning task, it is still desirable to control how far", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 118, + 486 + ], + "score": 0.84, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 475, + 150, + 489 + ], + "score": 1.0, + "content": "is from", + "type": "text" + }, + { + "bbox": [ + 151, + 477, + 160, + 486 + ], + "score": 0.77, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 475, + 506, + 489 + ], + "score": 1.0, + "content": ". In this regard, we note that a curvature-based rewiring is surgical in nature and hence", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "more likely to preserve the structure of the input graph better than a random-walk based approach.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 270, + 511 + ], + "score": 1.0, + "content": "Consider, for example, that we are given", + "type": "text" + }, + { + "bbox": [ + 270, + 499, + 295, + 510 + ], + "score": 0.91, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "and wish to rewire the graph such that the new edge", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 508, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 120, + 523 + ], + "score": 1.0, + "content": "set", + "type": "text" + }, + { + "bbox": [ + 120, + 509, + 132, + 519 + ], + "score": 0.86, + "content": "E ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 508, + 247, + 523 + ], + "score": 1.0, + "content": "is within graph-edit distance", + "type": "text" + }, + { + "bbox": [ + 247, + 511, + 254, + 521 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 508, + 324, + 523 + ], + "score": 1.0, + "content": "from the original", + "type": "text" + }, + { + "bbox": [ + 324, + 510, + 333, + 519 + ], + "score": 0.79, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 508, + 506, + 523 + ], + "score": 1.0, + "content": ". Theorem 4 tells us how to do the rewiring", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "under such constraints in order to best address the over-squashing: the topological modifications", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "need to be localized around the most negatively-curved edges. We can do this with SDRF, with the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 542, + 276, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 256, + 554 + ], + "score": 1.0, + "content": "maximum number of iterations set to", + "type": "text" + }, + { + "bbox": [ + 257, + 542, + 273, + 554 + ], + "score": 0.89, + "content": "\\rho / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 542, + 276, + 554 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 454, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 239, + 573 + ], + "score": 1.0, + "content": "Secondly, we also point out that", + "type": "text" + }, + { + "bbox": [ + 239, + 559, + 317, + 571 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) < - 2 + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 557, + 369, + 573 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 369, + 559, + 447, + 571 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} > 2 / \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 557, + 506, + 573 + ], + "score": 1.0, + "content": ". Therefore, if", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 293, + 583 + ], + "score": 1.0, + "content": "we mostly modify the edge set at those nodes", + "type": "text" + }, + { + "bbox": [ + 294, + 571, + 307, + 581 + ], + "score": 0.85, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "joined by an edge with large negative curvature,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "then we are perturbing nodes with high degree where such a change is relatively insignificant, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "thus overall statistical properties of the rewired graph such as degree distribution are likely to be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 603, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 614 + ], + "score": 1.0, + "content": "better preserved. Moreover, graph convolutional networks tend to be more stable to perturbations of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "high degree nodes (Zügner et al., 2020; Kenlay et al., 2021), making curvature-based rewiring more", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 625, + 410, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 410, + 637 + ], + "score": 1.0, + "content": "suitable for the downstream learning tasks with popular GNN architectures.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 557, + 506, + 637 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "Homophily and bottleneck. As a final remark, note that the graph rewiring techniques considered", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "in this paper (both DIGL and SDRF) are based purely on the topological structure of the graph and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "completely agnostic to the node features and to whether the dataset is homophilic (adjacent nodes", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "have same labels) or heterophilic. Nonetheless, the different nature of these rewiring methods allows", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "us to draw a few broad conclusions about their suitability in each of these settings. A random-walk", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 648, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 69, + 74, + 542, + 162 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 69, + 74, + 542, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 69, + 74, + 542, + 162 + ], + "spans": [ + { + "bbox": [ + 69, + 74, + 542, + 162 + ], + "score": 0.975, + "html": "
H(G)Cornell 0.11Texas 0.06Wisconsin 0.16Chameleon 0.25Squirrel 0.22Actor 0.24Cora 0.83Citeseer 0.71Pubmed 0.79
None52.69 ± 0.2161.19 ± 0.4954.60±0.8641.80 ± 0.4139.83±0.1428.70±0.0981.89 ±0.7972.31 ±0.1778.16 ±0.23
Undirected53.20 ±0.5363.38 ±0.8751.37 ± 1.1542.63±0.3040.77 ± 0.1628.10 ±0.11=
+FA58.29±0.4964.82±0.2955.48±0.6242.33± 0.1740.74 ± 0.1328.68±0.1681.65±0.1870.47±0.18 79.48±0.12
DIGL (PPR)58.26±0.5062.03 ±0.4349.53±0.2742.02 ±0.1334.38 ±0.1130.79±0.10 83.21±0.27 73.29±0.17 78.84±0.08
DIGL + Undirected 59.54 ±0.6463.54±0.3852.23±0.5442.68± 0.1233.36±0.2129.71 ± 0.11=
SDRF54.60 ±0.3964.46±0.3855.51±0.27 43.75±0.31 40.97±0.14 29.70 ±0.13 82.76±0.23 72.58±0.20 79.10 ±0.11
SDRF+ Undirected 57.54±0.34 70.35±0.60 61.55±0.86 44.46±0.17 41.47±0.21 29.85±0.07
", + "type": "table", + "image_path": "e34d427c4c0da221b6eb4532e7be80c0a5ee00795b09a2b562f073285c5fd73f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 69, + 74, + 542, + 103.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 69, + 103.33333333333333, + 542, + 132.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 69, + 132.66666666666666, + 542, + 162.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 120, + 171, + 489, + 183 + ], + "lines": [ + { + "bbox": [ + 121, + 171, + 489, + 184 + ], + "spans": [ + { + "bbox": [ + 121, + 171, + 489, + 184 + ], + "score": 1.0, + "content": "Table 2: Experimental results on common node classification benchmarks. Top two in bold.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "approach such as DIGL tends to improve the connectivity among nodes that are at short diffusion", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "distance; since for a high-homophily dataset these nodes often share the same label, a rewiring method", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "like DIGL is likely to act as graph denoising and yield improved performance. On the other hand, for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "datasets with low homophily, nodes at short diffusion distance are more likely to belong to different", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "label classes, meaning that a diffusion-based rewiring might inject noise and hence compromise", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "performance as also noted in Klicpera et al. (2019). Conversely, on a low-homophily dataset, a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "curvature-based approach as SDRF modifies the edge set mainly around the most negatively curved", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "edges, meaning that it decreases the bottleneck without significantly increasing the connectivity", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "among nodes with different labels. In fact, long-range dependencies are often more relevant in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "low-homophily settings, where nodes sharing the same labels are in general not neighbors. This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 308, + 446, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 446, + 321 + ], + "score": 1.0, + "content": "observation is largely confirmed by experimental results reported in the next section.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 109, + 335, + 256, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 257, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 257, + 349 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "Experiment setup. To demonstrate the theoretical results in this paper we ran a suite of semi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "supervised node classification tasks comparing our curvature-based rewiring method SDRF to DIGL", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 367, + 389 + ], + "score": 1.0, + "content": "from Klicpera et al. (2019) (GDC with the PPR kernel) and the", + "type": "text" + }, + { + "bbox": [ + 367, + 378, + 387, + 388 + ], + "score": 0.8, + "content": "+ \\mathrm { F A }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "method from Alon & Yahav", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 389, + 504, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 504, + 400 + ], + "score": 1.0, + "content": "(2021), where the last layer of the GNN is made fully connected. We evaluate the methods on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "nine datasets: Cornell, Texas and Wisconsin from the WebKB dataset4; Chameleon and Squirrel", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "(Rozemberczki et al., 2021) along with Actor (Tang et al., 2009); and Cora (McCallum et al., 2000),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "Citeseer (Sen et al., 2008) and Pubmed (Namata et al., 2012). Statistics for these datasets can be found", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "in Appendix F.1. Our base model is a GCN (Kipf & Welling, 2017). Following Shchur et al. (2018)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "and Klicpera et al. (2019) we optimized hyperparameters for all dataset-preprocessing combinations", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "separately by random search over 100 data splits. Results are reported as average accuracies on a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 203, + 478 + ], + "score": 1.0, + "content": "test set used once with", + "type": "text" + }, + { + "bbox": [ + 203, + 465, + 223, + 476 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "confidence intervals calculated by bootstrapping. We compared the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 408, + 488 + ], + "score": 1.0, + "content": "performance on graphs with no preprocessing, making the graph undirected,", + "type": "text" + }, + { + "bbox": [ + 409, + 477, + 428, + 487 + ], + "score": 0.84, + "content": "+ \\mathrm { F A }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 476, + 506, + 488 + ], + "score": 1.0, + "content": ", DIGL, SDRF, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 244, + 500 + ], + "score": 1.0, + "content": "the given combinations. For DIGL", + "type": "text" + }, + { + "bbox": [ + 245, + 488, + 254, + 497 + ], + "score": 0.7, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Undirected we symmetrized the diffusion matrix as in Klicpera", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 217, + 511 + ], + "score": 1.0, + "content": "et al. (2019), and for SDRF", + "type": "text" + }, + { + "bbox": [ + 218, + 499, + 226, + 508 + ], + "score": 0.61, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "Undirected we made the graph undirected before applying SDRF. For", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "more details on the experiments and datasets see Appendix F, and for the hyperparameters used for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 304, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 304, + 533 + ], + "score": 1.0, + "content": "each model and preprocessing see Appendix F.4.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "Node classification results. Table 2 shows the results of the experiments. As well as reporting", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 265, + 569 + ], + "score": 1.0, + "content": "results we give a measure of homophily", + "type": "text" + }, + { + "bbox": [ + 266, + 557, + 290, + 569 + ], + "score": 0.9, + "content": "\\mathcal { H } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "proposed by Pei et al. (2019) (restated in Appendix F,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "equation 21), by which we can see our experiment set is diverse with respect to homophily. We see", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "that SDRF improves upon the baseline in all cases, and that the largest improvements are seen on the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 449, + 602 + ], + "score": 1.0, + "content": "low-homophily datasets. 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H(G)Cornell 0.11Texas 0.06Wisconsin 0.16Chameleon 0.25Squirrel 0.22Actor 0.24Cora 0.83Citeseer 0.71Pubmed 0.79
None52.69 ± 0.2161.19 ± 0.4954.60±0.8641.80 ± 0.4139.83±0.1428.70±0.0981.89 ±0.7972.31 ±0.1778.16 ±0.23
Undirected53.20 ±0.5363.38 ±0.8751.37 ± 1.1542.63±0.3040.77 ± 0.1628.10 ±0.11=
+FA58.29±0.4964.82±0.2955.48±0.6242.33± 0.1740.74 ± 0.1328.68±0.1681.65±0.1870.47±0.18 79.48±0.12
DIGL (PPR)58.26±0.5062.03 ±0.4349.53±0.2742.02 ±0.1334.38 ±0.1130.79±0.10 83.21±0.27 73.29±0.17 78.84±0.08
DIGL + Undirected 59.54 ±0.6463.54±0.3852.23±0.5442.68± 0.1233.36±0.2129.71 ± 0.11=
SDRF54.60 ±0.3964.46±0.3855.51±0.27 43.75±0.31 40.97±0.14 29.70 ±0.13 82.76±0.23 72.58±0.20 79.10 ±0.11
SDRF+ Undirected 57.54±0.34 70.35±0.60 61.55±0.86 44.46±0.17 41.47±0.21 29.85±0.07
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Top two in bold.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 121, + 171, + 489, + 184 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "approach such as DIGL tends to improve the connectivity among nodes that are at short diffusion", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "distance; since for a high-homophily dataset these nodes often share the same label, a rewiring method", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "like DIGL is likely to act as graph denoising and yield improved performance. 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To demonstrate the theoretical results in this paper we ran a suite of semi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "supervised node classification tasks comparing our curvature-based rewiring method SDRF to DIGL", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 367, + 389 + ], + "score": 1.0, + "content": "from Klicpera et al. (2019) (GDC with the PPR kernel) and the", + "type": "text" + }, + { + "bbox": [ + 367, + 378, + 387, + 388 + ], + "score": 0.8, + "content": "+ \\mathrm { F A }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "method from Alon & Yahav", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 389, + 504, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 504, + 400 + ], + "score": 1.0, + "content": "(2021), where the last layer of the GNN is made fully connected. We evaluate the methods on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "nine datasets: Cornell, Texas and Wisconsin from the WebKB dataset4; Chameleon and Squirrel", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "(Rozemberczki et al., 2021) along with Actor (Tang et al., 2009); and Cora (McCallum et al., 2000),", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "Citeseer (Sen et al., 2008) and Pubmed (Namata et al., 2012). Statistics for these datasets can be found", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "in Appendix F.1. Our base model is a GCN (Kipf & Welling, 2017). Following Shchur et al. (2018)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "and Klicpera et al. (2019) we optimized hyperparameters for all dataset-preprocessing combinations", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "separately by random search over 100 data splits. Results are reported as average accuracies on a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 203, + 478 + ], + "score": 1.0, + "content": "test set used once with", + "type": "text" + }, + { + "bbox": [ + 203, + 465, + 223, + 476 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "confidence intervals calculated by bootstrapping. We compared the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 476, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 408, + 488 + ], + "score": 1.0, + "content": "performance on graphs with no preprocessing, making the graph undirected,", + "type": "text" + }, + { + "bbox": [ + 409, + 477, + 428, + 487 + ], + "score": 0.84, + "content": "+ \\mathrm { F A }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 476, + 506, + 488 + ], + "score": 1.0, + "content": ", DIGL, SDRF, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 244, + 500 + ], + "score": 1.0, + "content": "the given combinations. For DIGL", + "type": "text" + }, + { + "bbox": [ + 245, + 488, + 254, + 497 + ], + "score": 0.7, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Undirected we symmetrized the diffusion matrix as in Klicpera", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 217, + 511 + ], + "score": 1.0, + "content": "et al. (2019), and for SDRF", + "type": "text" + }, + { + "bbox": [ + 218, + 499, + 226, + 508 + ], + "score": 0.61, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "Undirected we made the graph undirected before applying SDRF. For", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "more details on the experiments and datasets see Appendix F, and for the hyperparameters used for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 304, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 304, + 533 + ], + "score": 1.0, + "content": "each model and preprocessing see Appendix F.4.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 355, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "Node classification results. Table 2 shows the results of the experiments. As well as reporting", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 265, + 569 + ], + "score": 1.0, + "content": "results we give a measure of homophily", + "type": "text" + }, + { + "bbox": [ + 266, + 557, + 290, + 569 + ], + "score": 0.9, + "content": "\\mathcal { H } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "proposed by Pei et al. (2019) (restated in Appendix F,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "equation 21), by which we can see our experiment set is diverse with respect to homophily. We see", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "that SDRF improves upon the baseline in all cases, and that the largest improvements are seen on the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 449, + 602 + ], + "score": 1.0, + "content": "low-homophily datasets. We also see that SDRF matches or outperforms DIGL and", + "type": "text" + }, + { + "bbox": [ + 449, + 590, + 469, + 600 + ], + "score": 0.86, + "content": "+ \\mathrm { F A }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "on most", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "datasets, supporting our argument that curvature-based rewiring is a viable candidate for improving", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 611, + 186, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 186, + 625 + ], + "score": 1.0, + "content": "GNN performance.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 544, + 506, + 625 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "Graph topology change. Furthermore, SDRF preserves the graph topology to a far greater extent", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "than DIGL due to its surgical nature. Table 3 shows the number of edges added / removed by", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "the two preprocessings on each dataset as a percentage of the original number of edges. We see", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "that for optimal performance DIGL makes the graph much denser, which significantly affects the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "node degrees and may negatively impact the time and space complexity of the downstream GNN,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 186, + 705 + ], + "score": 1.0, + "content": "which typically are", + "type": "text" + }, + { + "bbox": [ + 186, + 693, + 210, + 704 + ], + "score": 0.92, + "content": "O ( E )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 693, + 505, + 705 + ], + "score": 1.0, + "content": ". In comparison, SDRF adds and removes a similar number of edges and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "score": 1.0, + "content": "approximately preserves the degree distribution. 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The x-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 213, + 507, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 193, + 227 + ], + "score": 1.0, + "content": "axis is node degree in", + "type": "text" + }, + { + "bbox": [ + 193, + 214, + 212, + 226 + ], + "score": 0.9, + "content": "\\log _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 213, + 507, + 227 + ], + "score": 1.0, + "content": "scale, and the plots are a kernel density estimate of the degree distribution.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 224, + 498, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 296, + 237 + ], + "score": 1.0, + "content": "In the captions we see the Wasserstein distance", + "type": "text" + }, + { + "bbox": [ + 297, + 225, + 312, + 236 + ], + "score": 0.88, + "content": "W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 224, + 498, + 237 + ], + "score": 1.0, + "content": "between the original and preprocessed graphs.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 297, + 343 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 297, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 297, + 255 + ], + "score": 1.0, + "content": "datasets is shown in Figure 4. We see that the de-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 254, + 298, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 298, + 267 + ], + "score": 1.0, + "content": "gree distribution following SDRF is close to, or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 265, + 298, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 298, + 278 + ], + "score": 1.0, + "content": "in some cases indistinguishable from, the orig-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 276, + 298, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 298, + 288 + ], + "score": 1.0, + "content": "inal distribution, whereas DIGL has a more no-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 287, + 298, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 298, + 299 + ], + "score": 1.0, + "content": "ticeable effect. We also compute the Wasser-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 298, + 299, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 299, + 311 + ], + "score": 1.0, + "content": "stein distance between the degree distributions,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 309, + 299, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 153, + 321 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 153, + 310, + 168, + 321 + ], + "score": 0.89, + "content": "W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 309, + 299, + 321 + ], + "score": 1.0, + "content": "in the figure captions, to numer-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 320, + 297, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 297, + 333 + ], + "score": 1.0, + "content": "ically confirm this observation. For this analysis", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 331, + 295, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 295, + 343 + ], + "score": 1.0, + "content": "extended to all nine datasets see Appendix F.2.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 297, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 298, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 298, + 361 + ], + "score": 1.0, + "content": "This difference is also visually evident from Fig-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 359, + 298, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 298, + 372 + ], + "score": 1.0, + "content": "ure 6 in Appendix F.3, where we again observe", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 297, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 297, + 383 + ], + "score": 1.0, + "content": "that SDRF largely preserves the topology of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 381, + 297, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 297, + 393 + ], + "score": 1.0, + "content": "Cornell graph and that the curvature (encoded", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 392, + 297, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 297, + 405 + ], + "score": 1.0, + "content": "by the edge color) is homogenized across the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 403, + 298, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 298, + 415 + ], + "score": 1.0, + "content": "graph. We also see that the entries of the trained", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "table", + "bbox": [ + 304, + 245, + 494, + 387 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 304, + 245, + 494, + 387 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 245, + 494, + 387 + ], + "spans": [ + { + "bbox": [ + 304, + 245, + 494, + 387 + ], + "score": 0.977, + "html": "
DIGLSDRF
Cornell351.1% /0.0%7.8% /33.3%
Texas483.3% /0.0%2.4% /10.4%
Wisconsin300.6%/0.0%1.4% / 7.5%
Chameleon336.1%/11.8%6.4% /6.4%
Squirrel228.8% /1.9%0.7%/0.7%
Actor2444.0% /2.3%5.4%/9.9%
Cora3038.0% /0.5%1.0% /1.0%
Citeseer2568.3%/ 0.0%1.1% / 1.1%
Pubmed2747.1% /0.1%0.2%/0.2%
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We started", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "with a Jacobian approach to determine how the over-squashing phenomenon is dictated by the graph", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "topology as in equation 2. We then investigated further how the topology induces the bottleneck", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "and hence causes over-squashing. We introduced a new notion of edge-based Ricci curvature called", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "Balanced Forman curvature, relating it to the classical Ollivier curvature (Theorem 2). We then", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "proved in Theorem 4 that negatively-curved edges are responsible for over-squashing, calling for a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "possibility of curvature-based rewiring of the graph in order to improve its bottleneckedness. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "show one such possibility (Algorithm 1), inspired by the classical Ricci flow and comment on the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "advantages of surgical method such as this. We show both theoretically and experimentally that the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "proposed method can be advantageous compared to a diffusion-based rewiring approach, opening the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 597, + 476, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 476, + 610 + ], + "score": 1.0, + "content": "door for curvature-based rewiring methods for improving GNN performance going forward.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "Limitations and future directions. Our paper establishes a geometric perspective on the graph", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "bottleneck and over-squashing, providing new tools to study and cope with these phenomena. One", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "limitation of our work is that the theoretical results presented here do not currently extend to multi-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 647, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 507, + 662 + ], + "score": 1.0, + "content": "graphs. In addition, the current methodology is agnostic to information beyond the graph topology,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "score": 1.0, + "content": "such as node features. In future works, we will develop a notion of the curvature and the corresponding", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 671, + 352, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 352, + 682 + ], + "score": 1.0, + "content": "rewiring method that can take into account such information.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Acknowledgements. This research was supported in part by the EPSRC CDT in Modern Statistics", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "and Statistical Machine Learning (EP/S023151/1) and the ERC Consolidator Grant No. 724228", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "(LEMAN). X.D. gratefully acknowledges support from the Oxford-Man Institute of Quantitative", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 721, + 274, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 274, + 732 + ], + "score": 1.0, + "content": "Finance and the EPSRC (EP/T023333/1).", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 55.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 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": [ + 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": [ + 109, + 60, + 499, + 196 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 60, + 499, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 60, + 499, + 195 + ], + "spans": [ + { + "bbox": [ + 109, + 60, + 499, + 195 + ], + "score": 0.967, + "type": "image", + "image_path": "198001fa034307ccce83949e4016829a9a3ad6cc0c2500a49f6ecd4d0572ee0c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 60, + 499, + 105.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 105.33333333333334, + 499, + 150.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 150.66666666666669, + 499, + 196.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 203, + 506, + 236 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 201, + 507, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 507, + 216 + ], + "score": 1.0, + "content": "Figure 4: Comparing the degree distribution of the original graphs to the preprocessed version. 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We see that the de-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 254, + 298, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 298, + 267 + ], + "score": 1.0, + "content": "gree distribution following SDRF is close to, or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 265, + 298, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 298, + 278 + ], + "score": 1.0, + "content": "in some cases indistinguishable from, the orig-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 276, + 298, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 298, + 288 + ], + "score": 1.0, + "content": "inal distribution, whereas DIGL has a more no-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 287, + 298, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 298, + 299 + ], + "score": 1.0, + "content": "ticeable effect. We also compute the Wasser-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 298, + 299, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 299, + 311 + ], + "score": 1.0, + "content": "stein distance between the degree distributions,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 309, + 299, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 153, + 321 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 153, + 310, + 168, + 321 + ], + "score": 0.89, + "content": "W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 309, + 299, + 321 + ], + "score": 1.0, + "content": "in the figure captions, to numer-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 320, + 297, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 297, + 333 + ], + "score": 1.0, + "content": "ically confirm this observation. 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We also see that the entries of the trained", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 347, + 298, + 415 + ] + }, + { + "type": "table", + "bbox": [ + 304, + 245, + 494, + 387 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 304, + 245, + 494, + 387 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 245, + 494, + 387 + ], + "spans": [ + { + "bbox": [ + 304, + 245, + 494, + 387 + ], + "score": 0.977, + "html": "
DIGLSDRF
Cornell351.1% /0.0%7.8% /33.3%
Texas483.3% /0.0%2.4% /10.4%
Wisconsin300.6%/0.0%1.4% / 7.5%
Chameleon336.1%/11.8%6.4% /6.4%
Squirrel228.8% /1.9%0.7%/0.7%
Actor2444.0% /2.3%5.4%/9.9%
Cora3038.0% /0.5%1.0% /1.0%
Citeseer2568.3%/ 0.0%1.1% / 1.1%
Pubmed2747.1% /0.1%0.2%/0.2%
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We started", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "with a Jacobian approach to determine how the over-squashing phenomenon is dictated by the graph", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "topology as in equation 2. We then investigated further how the topology induces the bottleneck", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 534 + ], + "score": 1.0, + "content": "and hence causes over-squashing. We introduced a new notion of edge-based Ricci curvature called", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "Balanced Forman curvature, relating it to the classical Ollivier curvature (Theorem 2). We then", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "proved in Theorem 4 that negatively-curved edges are responsible for over-squashing, calling for a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "possibility of curvature-based rewiring of the graph in order to improve its bottleneckedness. We", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "show one such possibility (Algorithm 1), inspired by the classical Ricci flow and comment on the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "advantages of surgical method such as this. 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In addition, the current methodology is agnostic to information beyond the graph topology,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 672 + ], + "score": 1.0, + "content": "such as node features. In future works, we will develop a notion of the curvature and the corresponding", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 671, + 352, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 352, + 682 + ], + "score": 1.0, + "content": "rewiring method that can take into account such information.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 614, + 507, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "Acknowledgements. 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Assume an MPNN as in equation", + "type": "text" + }, + { + "bbox": [ + 293, + 448, + 298, + 457 + ], + "score": 0.34, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 446, + 319, + 460 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 320, + 448, + 354, + 459 + ], + "score": 0.9, + "content": "i , s \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 446, + 376, + 460 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 376, + 447, + 426, + 460 + ], + "score": 0.91, + "content": "s \\in S _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 446, + 441, + 460 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 152, + 519, + 177, + 529 + ], + "score": 0.9, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 518, + 194, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 518, + 244, + 531 + ], + "score": 0.93, + "content": "s \\in S _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 518, + 506, + 531 + ], + "score": 1.0, + "content": ". We recall that to ease the notations we assume that node features", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "and hidden representations are scalar. The proof in the more general higher-dimensional case follows", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 541, + 264, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 264, + 553 + ], + "score": 1.0, + "content": "without any modification. We compute", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 518, + 506, + 553 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 557, + 480, + 622 + ], + "lines": [ + { + "bbox": [ + 131, + 557, + 480, + 622 + ], + "spans": [ + { + "bbox": [ + 131, + 557, + 480, + 622 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\displaystyle \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\partial _ { 1 } \\phi _ { r } ( . . . ) \\partial _ { x _ { s } } h _ { i } ^ { ( r ) } } \\\\ & { \\quad \\quad \\quad + \\partial _ { 2 } \\phi _ { r } ( . . . ) \\displaystyle \\sum _ { j _ { r } = 1 } ^ { n } \\hat { a } _ { i j _ { r } } \\left( \\partial _ { 1 } \\psi _ { r } ( h _ { i } ^ { ( r ) } , h _ { j _ { r } } ^ { ( r ) } ) \\partial _ { x _ { s } } h _ { i } ^ { ( r ) } + \\partial _ { 2 } \\psi _ { r } ( h _ { i } ^ { ( r ) } , h _ { j _ { r } } ^ { ( r ) } ) \\partial _ { x _ { s } } h _ { j _ { r } } ^ { ( r ) } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "d1efbc8ffba481a5cd77a42d69a795c9439b8623f7402b0a57366f058b8ee9c2.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 131, + 557, + 480, + 578.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 131, + 578.6666666666666, + 480, + 600.3333333333333 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 131, + 600.3333333333333, + 480, + 621.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 626, + 465, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 464, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 464, + 640 + ], + "score": 1.0, + "content": "We can iterate the computation above and see that the right hand side can be expanded as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 626, + 464, + 640 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 644, + 495, + 681 + ], + "lines": [ + { + "bbox": [ + 116, + 644, + 495, + 681 + ], + "spans": [ + { + "bbox": [ + 116, + 644, + 495, + 681 + ], + "score": 0.92, + "content": "\\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { r } , \\dots , j _ { 0 } } \\sum _ { k _ { r } \\in \\{ i , j _ { r } \\} } \\dots \\sum _ { k _ { 1 } \\in \\{ i , j _ { r } , \\dots , j _ { 1 } \\} } \\hat { a } _ { i j _ { r } } \\hat { a } _ { k _ { r } j _ { r - 1 } } \\dots \\hat { a } _ { k _ { 1 } j _ { 0 } } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \\dots k _ { 1 } j _ { 0 } } ( \\boldsymbol { X } ) \\partial _ { x _ { s } } h _ { j _ { 0 } } ^ { ( 0 ) } ,", + "type": "interline_equation", + "image_path": "6680bb16a1c23d979b5d64c22939024c8ad89ca501c3eaf1599b0ce92a942541.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 116, + 644, + 495, + 656.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 116, + 656.3333333333334, + 495, + 668.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 116, + 668.6666666666667, + 495, + 681.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 686, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 182, + 701 + ], + "score": 1.0, + "content": "for some functions", + "type": "text" + }, + { + "bbox": [ + 182, + 686, + 223, + 699 + ], + "score": 0.93, + "content": "Z _ { i j _ { r } . . . k _ { 1 } j _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 685, + 398, + 701 + ], + "score": 1.0, + "content": "of the input features obtained as products of", + "type": "text" + }, + { + "bbox": [ + 399, + 687, + 421, + 697 + ], + "score": 0.9, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 685, + 506, + 701 + ], + "score": 1.0, + "content": "partial derivatives of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 444, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 145, + 712 + ], + "score": 1.0, + "content": "the maps", + "type": "text" + }, + { + "bbox": [ + 145, + 700, + 155, + 711 + ], + "score": 0.89, + "content": "\\phi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 698, + 173, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 174, + 700, + 197, + 710 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 698, + 320, + 712 + ], + "score": 1.0, + "content": "partial derivatives of the maps", + "type": "text" + }, + { + "bbox": [ + 321, + 700, + 332, + 711 + ], + "score": 0.87, + "content": "\\psi _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 698, + 361, + 712 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 361, + 698, + 404, + 709 + ], + "score": 0.91, + "content": "H ^ { ( 0 ) } = X", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 698, + 444, + 712 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 685, + 506, + 712 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 717, + 336, + 735 + ], + "lines": [ + { + "bbox": [ + 275, + 717, + 336, + 735 + ], + "spans": [ + { + "bbox": [ + 275, + 717, + 336, + 735 + ], + "score": 0.93, + "content": "\\partial _ { x _ { s } } h _ { j _ { 0 } } ^ { ( 0 ) } = \\delta _ { j _ { 0 } s }", + "type": "interline_equation", + "image_path": "be0f8474114d82184322e8f351a91924057ab1c64cf3182fdfe9af048f24223e.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 275, + 717, + 336, + 735 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 267, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 269, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 269, + 96 + ], + "score": 1.0, + "content": "meaning that the previous sum becomes", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 100, + 477, + 136 + ], + "lines": [ + { + "bbox": [ + 134, + 100, + 477, + 136 + ], + "spans": [ + { + "bbox": [ + 134, + 100, + 477, + 136 + ], + "score": 0.93, + "content": "\\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { r } , \\dots , j _ { 1 } } \\sum _ { k _ { r } \\in \\{ i , j _ { r } \\} } \\dots \\sum _ { k _ { 1 } \\in \\{ i , j _ { r } , \\dots , j _ { 1 } \\} } \\hat { a } _ { i j _ { r } } \\hat { a } _ { k _ { r } j _ { r - 1 } } \\dots \\hat { a } _ { k _ { 1 } s } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \\dots k _ { 1 } s } ( X ) .", + "type": "interline_equation", + "image_path": "edd5af0b1014fb301d3dba455ae6e0c7abff1ae973ba50579f6754264aa7cc1b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 134, + 100, + 477, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 134, + 112.0, + 477, + 124.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 134, + 124.0, + 477, + 136.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 506, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 504, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 131, + 155 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 142, + 199, + 155 + ], + "score": 0.93, + "content": "d _ { G } ( i , s ) = r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 141, + 499, + 155 + ], + "score": 1.0, + "content": ", the only non-vanishing terms in the sum above are the minimal walks from", + "type": "text" + }, + { + "bbox": [ + 499, + 144, + 504, + 153 + ], + "score": 0.52, + "content": "i", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 117, + 165 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 156, + 122, + 163 + ], + "score": 0.59, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 154, + 505, + 165 + ], + "score": 1.0, + "content": ". In fact, if there existed a different choice of coefficients yielding a non-zero term then we would", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 184, + 177 + ], + "score": 1.0, + "content": "find a walk joining", + "type": "text" + }, + { + "bbox": [ + 184, + 165, + 189, + 174 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 164, + 200, + 177 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 200, + 167, + 206, + 174 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 164, + 290, + 177 + ], + "score": 1.0, + "content": "of length lesser than", + "type": "text" + }, + { + "bbox": [ + 290, + 165, + 313, + 175 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 164, + 506, + 177 + ], + "score": 1.0, + "content": ", which is in contradiction with the definition of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 209, + 188 + ], + "score": 1.0, + "content": "geodesic distance. Since", + "type": "text" + }, + { + "bbox": [ + 209, + 176, + 248, + 187 + ], + "score": 0.93, + "content": "Z _ { i \\dots s } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 175, + 311, + 188 + ], + "score": 1.0, + "content": "is a product of", + "type": "text" + }, + { + "bbox": [ + 311, + 176, + 334, + 186 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "-partial derivatives of the aggregation and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 363, + 199 + ], + "score": 1.0, + "content": "update maps and by assumption their gradients are bounded by", + "type": "text" + }, + { + "bbox": [ + 364, + 188, + 372, + 196 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 187, + 390, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 390, + 187, + 398, + 198 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "respectively, we conclude", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 125, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 125, + 209 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 206, + 440, + 242 + ], + "lines": [ + { + "bbox": [ + 170, + 206, + 440, + 242 + ], + "spans": [ + { + "bbox": [ + 170, + 206, + 440, + 242 + ], + "score": 0.93, + "content": "\\left. \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right. \\leq ( \\alpha \\beta ) ^ { r + 1 } \\sum _ { j _ { r } , \\ldots , j _ { 1 } } \\hat { a } _ { i j _ { r } } \\hat { a } _ { j _ { r } j _ { r - 1 } } \\ldots \\hat { a } _ { j _ { 1 } s } = \\alpha ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s }", + "type": "interline_equation", + "image_path": "96d2140cb525e2986e2b2b82f44a4416d5cb9cf2d70a9cf9142dd526c30ff11f.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 170, + 206, + 440, + 218.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 170, + 218.0, + 440, + 230.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 170, + 230.0, + 440, + 242.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 274, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 275, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 275, + 258 + ], + "score": 1.0, + "content": "which completes the proof of the Lemma.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 105, + 272, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "As a byproduct of this analysis, we can also provide a rigorous motivation for the role of self-loops in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 283, + 243, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 243, + 297 + ], + "score": 1.0, + "content": "GNNs (see Appendix for details):", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 102, + 295, + 509, + 316 + ], + "spans": [ + { + "bbox": [ + 102, + 295, + 172, + 316 + ], + "score": 1.0, + "content": "Corollary 7. If", + "type": "text" + }, + { + "bbox": [ + 172, + 298, + 273, + 315 + ], + "score": 0.93, + "content": "\\begin{array} { r } { h _ { i } ^ { ( \\ell + 1 ) } = \\sum _ { j \\sim i } \\psi _ { \\ell } ( h _ { j } ^ { ( \\ell ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 295, + 297, + 316 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 298, + 299, + 324, + 314 + ], + "score": 0.92, + "content": "h _ { i } ^ { ( \\ell + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 295, + 509, + 316 + ], + "score": 1.0, + "content": "only depends on nodes that can be reached", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 215, + 327 + ], + "score": 1.0, + "content": "via walks of length exactly", + "type": "text" + }, + { + "bbox": [ + 216, + 315, + 238, + 325 + ], + "score": 0.86, + "content": "\\ell + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 313, + 506, + 327 + ], + "score": 1.0, + "content": ". By adding self-loops, the GNN also takes into account nodes that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 291, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 246, + 337 + ], + "score": 1.0, + "content": "can be reached via walks of length", + "type": "text" + }, + { + "bbox": [ + 247, + 325, + 287, + 336 + ], + "score": 0.9, + "content": "r \\leq \\ell + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 325, + 291, + 337 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 508, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 146, + 379 + ], + "score": 1.0, + "content": "Proof. If", + "type": "text" + }, + { + "bbox": [ + 146, + 352, + 248, + 369 + ], + "score": 0.93, + "content": "\\begin{array} { r } { h _ { i } ^ { ( \\ell + 1 ) } = \\sum _ { j \\sim i } \\psi _ { \\ell } ( h _ { j } ^ { ( \\ell ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 347, + 287, + 379 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 287, + 354, + 347, + 367 + ], + "score": 0.91, + "content": "\\ell \\in [ 0 , L - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 347, + 508, + 379 + ], + "score": 1.0, + "content": ", then we can argue as in the proof of", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 376, + 408, + 411 + ], + "lines": [ + { + "bbox": [ + 203, + 376, + 408, + 411 + ], + "spans": [ + { + "bbox": [ + 203, + 376, + 408, + 411 + ], + "score": 0.93, + "content": "\\frac { \\partial h _ { i } ^ { ( \\ell + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { \\ell } , \\ldots , j _ { 1 } } a _ { i j _ { \\ell } } \\ldots a _ { j _ { 1 } s } \\psi _ { \\ell } ^ { \\prime } ( h _ { j _ { \\ell } } ^ { ( \\ell ) } ) \\cdot \\cdot \\cdot \\psi _ { 0 } ^ { \\prime } ( x _ { s } ) .", + "type": "interline_equation", + "image_path": "6d78f01d367a8a9fd82ee97cff12d8ded985b056b5b543d52e80bf12216ec8f7.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 376, + 408, + 393.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 203, + 393.5, + 408, + 411.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 506, + 459 + ], + "lines": [ + { + "bbox": [ + 104, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 413, + 229, + 428 + ], + "score": 1.0, + "content": "The combinatorial coefficient", + "type": "text" + }, + { + "bbox": [ + 230, + 416, + 278, + 427 + ], + "score": 0.9, + "content": "a _ { i j \\ell } \\ldots a _ { j _ { 1 } s }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 413, + 441, + 428 + ], + "score": 1.0, + "content": "is non-zero iff there exists a walk from", + "type": "text" + }, + { + "bbox": [ + 442, + 416, + 446, + 424 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 413, + 458, + 428 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 459, + 417, + 464, + 424 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "of length", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 136, + 438 + ], + "score": 1.0, + "content": "exactly", + "type": "text" + }, + { + "bbox": [ + 137, + 426, + 155, + 436 + ], + "score": 0.71, + "content": "\\ell { + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 426, + 506, + 438 + ], + "score": 1.0, + "content": ", since we are not taking into account the contribution coming from self-loops. Conversely,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 436, + 155, + 450 + ], + "score": 1.0, + "content": "if each term", + "type": "text" + }, + { + "bbox": [ + 156, + 438, + 172, + 448 + ], + "score": 0.88, + "content": "a _ { i j \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 436, + 239, + 450 + ], + "score": 1.0, + "content": "was replaced by", + "type": "text" + }, + { + "bbox": [ + 239, + 437, + 255, + 448 + ], + "score": 0.9, + "content": "\\hat { a } _ { i j \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 436, + 352, + 450 + ], + "score": 1.0, + "content": "then we would find that", + "type": "text" + }, + { + "bbox": [ + 352, + 437, + 399, + 449 + ], + "score": 0.91, + "content": "\\hat { a } _ { i j \\ell } \\ldots \\hat { a } _ { j _ { 1 } s }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "is non-zero iff there exists", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 157, + 460 + ], + "score": 1.0, + "content": "a walk from", + "type": "text" + }, + { + "bbox": [ + 157, + 449, + 162, + 457 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 448, + 173, + 460 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 173, + 450, + 179, + 457 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 448, + 250, + 460 + ], + "score": 1.0, + "content": "of length at most", + "type": "text" + }, + { + "bbox": [ + 251, + 448, + 273, + 458 + ], + "score": 0.88, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 448, + 451, + 460 + ], + "score": 1.0, + "content": ", since the diagonal entries are now positive.", + "type": "text" + }, + { + "bbox": [ + 494, + 448, + 505, + 459 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "Remark 8. As a specific instance of Corollary 7, we note that if we do not include self-loops in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "the adjacency matrix, then the output of a 2-layer simplified graph neural network at node i is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 275, + 503 + ], + "score": 1.0, + "content": "independent of the features of neighbours", + "type": "text" + }, + { + "bbox": [ + 276, + 492, + 282, + 501 + ], + "score": 0.64, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "that do not form a triangle with i. Once again here the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "dependence is precisely measured via the Jacobian of the hidden features with respect to the input", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 512, + 144, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 512, + 144, + 526 + ], + "score": 1.0, + "content": "features.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 504, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "Remark 9. We note that the role of self-loops has also implicitly been noted in Xu et al. (2018) where", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "the analysis of the Jacobian of node representations on the graph augmented with self-loops has been", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 227, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 227, + 563 + ], + "score": 1.0, + "content": "related to lazy random-walks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "GNNs with different aggregations We note that similar conclusions extend to message passing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "architectures where the aggregations are sums and not averages meaning that we take the augmented", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "adjacency without normalizing by the degree matrices. Consistently with Lemma 1, we restrict to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "the setting where features and node representations at each layer are scalars to make the discussion", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 428, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 204, + 631 + ], + "score": 1.0, + "content": "simpler. In line with the", + "type": "text" + }, + { + "bbox": [ + 204, + 619, + 218, + 629 + ], + "score": 0.37, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 618, + 428, + 631 + ], + "score": 1.0, + "content": "et al. (2018) we consider a GNN-model of the form", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 636, + 374, + 676 + ], + "lines": [ + { + "bbox": [ + 237, + 636, + 374, + 676 + ], + "spans": [ + { + "bbox": [ + 237, + 636, + 374, + 676 + ], + "score": 0.94, + "content": "h _ { i } ^ { ( \\ell + 1 ) } = \\mathrm { R e L U } \\left( \\sum _ { j \\in \\tilde { \\mathcal { N } } _ { i } } h _ { j } ^ { ( \\ell ) } w _ { \\ell } \\right) .", + "type": "interline_equation", + "image_path": "cbfc7f5d734d7dff6902cb4c0df647e366ba9ff560dd05c75028a31a470b0b51.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 636, + 374, + 656.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 237, + 656.0, + 374, + 676.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 683, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 683, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 272, + 698 + ], + "score": 1.0, + "content": "Note that the augmented neighbourhood", + "type": "text" + }, + { + "bbox": [ + 272, + 684, + 284, + 696 + ], + "score": 0.9, + "content": "\\tilde { \\mathcal { N } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 683, + 339, + 698 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + }, + { + "bbox": [ + 339, + 685, + 376, + 697 + ], + "score": 0.91, + "content": "\\mathcal { N } _ { i } \\cup \\{ i \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 683, + 506, + 698 + ], + "score": 1.0, + "content": ". Differently from the setting of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 694, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 162, + 709 + ], + "score": 1.0, + "content": "Theorem 1 in", + "type": "text" + }, + { + "bbox": [ + 163, + 696, + 177, + 706 + ], + "score": 0.38, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 694, + 506, + 709 + ], + "score": 1.0, + "content": "et al. (2018), the aggregation here is not an average but a simple sum. Let us now", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 707, + 506, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 152, + 719 + ], + "score": 1.0, + "content": "take nodes", + "type": "text" + }, + { + "bbox": [ + 153, + 708, + 158, + 717 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 707, + 176, + 719 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 176, + 709, + 182, + 717 + ], + "score": 0.73, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 707, + 223, + 719 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 223, + 708, + 274, + 719 + ], + "score": 0.91, + "content": "s \\in S _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 707, + 506, + 719 + ], + "score": 1.0, + "content": "as in the statement of Lemma 1. In this case, instead of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 103, + 716, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 103, + 716, + 236, + 735 + ], + "score": 1.0, + "content": "simply considering the quantity", + "type": "text" + }, + { + "bbox": [ + 236, + 719, + 295, + 733 + ], + "score": 0.93, + "content": "| \\partial h _ { i } ^ { ( r + 1 ) } / \\partial x _ { s } |", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 716, + 507, + 735 + ], + "score": 1.0, + "content": ", we normalize the Jacobian entries - obtaining what", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + } + ], + "page_idx": 14, + "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, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 245, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 495, + 246, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 495, + 246, + 505, + 257 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 267, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 269, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 269, + 96 + ], + "score": 1.0, + "content": "meaning that the previous sum becomes", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 269, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 100, + 477, + 136 + ], + "lines": [ + { + "bbox": [ + 134, + 100, + 477, + 136 + ], + "spans": [ + { + "bbox": [ + 134, + 100, + 477, + 136 + ], + "score": 0.93, + "content": "\\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { r } , \\dots , j _ { 1 } } \\sum _ { k _ { r } \\in \\{ i , j _ { r } \\} } \\dots \\sum _ { k _ { 1 } \\in \\{ i , j _ { r } , \\dots , j _ { 1 } \\} } \\hat { a } _ { i j _ { r } } \\hat { a } _ { k _ { r } j _ { r - 1 } } \\dots \\hat { a } _ { k _ { 1 } s } Z _ { i j _ { r } k _ { r } j _ { r - 1 } \\dots k _ { 1 } s } ( X ) .", + "type": "interline_equation", + "image_path": "edd5af0b1014fb301d3dba455ae6e0c7abff1ae973ba50579f6754264aa7cc1b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 134, + 100, + 477, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 134, + 112.0, + 477, + 124.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 134, + 124.0, + 477, + 136.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 506, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 504, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 131, + 155 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 142, + 199, + 155 + ], + "score": 0.93, + "content": "d _ { G } ( i , s ) = r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 141, + 499, + 155 + ], + "score": 1.0, + "content": ", the only non-vanishing terms in the sum above are the minimal walks from", + "type": "text" + }, + { + "bbox": [ + 499, + 144, + 504, + 153 + ], + "score": 0.52, + "content": "i", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 117, + 165 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 156, + 122, + 163 + ], + "score": 0.59, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 154, + 505, + 165 + ], + "score": 1.0, + "content": ". In fact, if there existed a different choice of coefficients yielding a non-zero term then we would", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 184, + 177 + ], + "score": 1.0, + "content": "find a walk joining", + "type": "text" + }, + { + "bbox": [ + 184, + 165, + 189, + 174 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 164, + 200, + 177 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 200, + 167, + 206, + 174 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 164, + 290, + 177 + ], + "score": 1.0, + "content": "of length lesser than", + "type": "text" + }, + { + "bbox": [ + 290, + 165, + 313, + 175 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 164, + 506, + 177 + ], + "score": 1.0, + "content": ", which is in contradiction with the definition of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 209, + 188 + ], + "score": 1.0, + "content": "geodesic distance. Since", + "type": "text" + }, + { + "bbox": [ + 209, + 176, + 248, + 187 + ], + "score": 0.93, + "content": "Z _ { i \\dots s } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 175, + 311, + 188 + ], + "score": 1.0, + "content": "is a product of", + "type": "text" + }, + { + "bbox": [ + 311, + 176, + 334, + 186 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "-partial derivatives of the aggregation and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 363, + 199 + ], + "score": 1.0, + "content": "update maps and by assumption their gradients are bounded by", + "type": "text" + }, + { + "bbox": [ + 364, + 188, + 372, + 196 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 187, + 390, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 390, + 187, + 398, + 198 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "respectively, we conclude", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 125, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 125, + 209 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 141, + 506, + 209 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 206, + 440, + 242 + ], + "lines": [ + { + "bbox": [ + 170, + 206, + 440, + 242 + ], + "spans": [ + { + "bbox": [ + 170, + 206, + 440, + 242 + ], + "score": 0.93, + "content": "\\left. \\frac { \\partial h _ { i } ^ { ( r + 1 ) } } { \\partial x _ { s } } \\right. \\leq ( \\alpha \\beta ) ^ { r + 1 } \\sum _ { j _ { r } , \\ldots , j _ { 1 } } \\hat { a } _ { i j _ { r } } \\hat { a } _ { j _ { r } j _ { r - 1 } } \\ldots \\hat { a } _ { j _ { 1 } s } = \\alpha ^ { r + 1 } ( \\hat { A } ^ { r + 1 } ) _ { i s }", + "type": "interline_equation", + "image_path": "96d2140cb525e2986e2b2b82f44a4416d5cb9cf2d70a9cf9142dd526c30ff11f.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 170, + 206, + 440, + 218.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 170, + 218.0, + 440, + 230.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 170, + 230.0, + 440, + 242.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 274, + 257 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 275, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 275, + 258 + ], + "score": 1.0, + "content": "which completes the proof of the Lemma.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 245, + 275, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 272, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "As a byproduct of this analysis, we can also provide a rigorous motivation for the role of self-loops in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 283, + 243, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 243, + 297 + ], + "score": 1.0, + "content": "GNNs (see Appendix for details):", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 272, + 505, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 102, + 295, + 509, + 316 + ], + "spans": [ + { + "bbox": [ + 102, + 295, + 172, + 316 + ], + "score": 1.0, + "content": "Corollary 7. If", + "type": "text" + }, + { + "bbox": [ + 172, + 298, + 273, + 315 + ], + "score": 0.93, + "content": "\\begin{array} { r } { h _ { i } ^ { ( \\ell + 1 ) } = \\sum _ { j \\sim i } \\psi _ { \\ell } ( h _ { j } ^ { ( \\ell ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 295, + 297, + 316 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 298, + 299, + 324, + 314 + ], + "score": 0.92, + "content": "h _ { i } ^ { ( \\ell + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 295, + 509, + 316 + ], + "score": 1.0, + "content": "only depends on nodes that can be reached", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 215, + 327 + ], + "score": 1.0, + "content": "via walks of length exactly", + "type": "text" + }, + { + "bbox": [ + 216, + 315, + 238, + 325 + ], + "score": 0.86, + "content": "\\ell + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 313, + 506, + 327 + ], + "score": 1.0, + "content": ". By adding self-loops, the GNN also takes into account nodes that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 291, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 246, + 337 + ], + "score": 1.0, + "content": "can be reached via walks of length", + "type": "text" + }, + { + "bbox": [ + 247, + 325, + 287, + 336 + ], + "score": 0.9, + "content": "r \\leq \\ell + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 325, + 291, + 337 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 102, + 295, + 509, + 337 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 104, + 347, + 508, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 146, + 379 + ], + "score": 1.0, + "content": "Proof. If", + "type": "text" + }, + { + "bbox": [ + 146, + 352, + 248, + 369 + ], + "score": 0.93, + "content": "\\begin{array} { r } { h _ { i } ^ { ( \\ell + 1 ) } = \\sum _ { j \\sim i } \\psi _ { \\ell } ( h _ { j } ^ { ( \\ell ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 347, + 287, + 379 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 287, + 354, + 347, + 367 + ], + "score": 0.91, + "content": "\\ell \\in [ 0 , L - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 347, + 508, + 379 + ], + "score": 1.0, + "content": ", then we can argue as in the proof of", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 347, + 508, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 376, + 408, + 411 + ], + "lines": [ + { + "bbox": [ + 203, + 376, + 408, + 411 + ], + "spans": [ + { + "bbox": [ + 203, + 376, + 408, + 411 + ], + "score": 0.93, + "content": "\\frac { \\partial h _ { i } ^ { ( \\ell + 1 ) } } { \\partial x _ { s } } = \\sum _ { j _ { \\ell } , \\ldots , j _ { 1 } } a _ { i j _ { \\ell } } \\ldots a _ { j _ { 1 } s } \\psi _ { \\ell } ^ { \\prime } ( h _ { j _ { \\ell } } ^ { ( \\ell ) } ) \\cdot \\cdot \\cdot \\psi _ { 0 } ^ { \\prime } ( x _ { s } ) .", + "type": "interline_equation", + "image_path": "6d78f01d367a8a9fd82ee97cff12d8ded985b056b5b543d52e80bf12216ec8f7.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 376, + 408, + 393.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 203, + 393.5, + 408, + 411.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 506, + 459 + ], + "lines": [ + { + "bbox": [ + 104, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 413, + 229, + 428 + ], + "score": 1.0, + "content": "The combinatorial coefficient", + "type": "text" + }, + { + "bbox": [ + 230, + 416, + 278, + 427 + ], + "score": 0.9, + "content": "a _ { i j \\ell } \\ldots a _ { j _ { 1 } s }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 413, + 441, + 428 + ], + "score": 1.0, + "content": "is non-zero iff there exists a walk from", + "type": "text" + }, + { + "bbox": [ + 442, + 416, + 446, + 424 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 413, + 458, + 428 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 459, + 417, + 464, + 424 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "of length", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 136, + 438 + ], + "score": 1.0, + "content": "exactly", + "type": "text" + }, + { + "bbox": [ + 137, + 426, + 155, + 436 + ], + "score": 0.71, + "content": "\\ell { + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 426, + 506, + 438 + ], + "score": 1.0, + "content": ", since we are not taking into account the contribution coming from self-loops. Conversely,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 436, + 155, + 450 + ], + "score": 1.0, + "content": "if each term", + "type": "text" + }, + { + "bbox": [ + 156, + 438, + 172, + 448 + ], + "score": 0.88, + "content": "a _ { i j \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 436, + 239, + 450 + ], + "score": 1.0, + "content": "was replaced by", + "type": "text" + }, + { + "bbox": [ + 239, + 437, + 255, + 448 + ], + "score": 0.9, + "content": "\\hat { a } _ { i j \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 436, + 352, + 450 + ], + "score": 1.0, + "content": "then we would find that", + "type": "text" + }, + { + "bbox": [ + 352, + 437, + 399, + 449 + ], + "score": 0.91, + "content": "\\hat { a } _ { i j \\ell } \\ldots \\hat { a } _ { j _ { 1 } s }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "is non-zero iff there exists", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 157, + 460 + ], + "score": 1.0, + "content": "a walk from", + "type": "text" + }, + { + "bbox": [ + 157, + 449, + 162, + 457 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 448, + 173, + 460 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 173, + 450, + 179, + 457 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 448, + 250, + 460 + ], + "score": 1.0, + "content": "of length at most", + "type": "text" + }, + { + "bbox": [ + 251, + 448, + 273, + 458 + ], + "score": 0.88, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 448, + 451, + 460 + ], + "score": 1.0, + "content": ", since the diagonal entries are now positive.", + "type": "text" + }, + { + "bbox": [ + 494, + 448, + 505, + 459 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 413, + 506, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "Remark 8. As a specific instance of Corollary 7, we note that if we do not include self-loops in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "the adjacency matrix, then the output of a 2-layer simplified graph neural network at node i is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 275, + 503 + ], + "score": 1.0, + "content": "independent of the features of neighbours", + "type": "text" + }, + { + "bbox": [ + 276, + 492, + 282, + 501 + ], + "score": 0.64, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "that do not form a triangle with i. Once again here the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "dependence is precisely measured via the Jacobian of the hidden features with respect to the input", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 512, + 144, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 512, + 144, + 526 + ], + "score": 1.0, + "content": "features.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 468, + 506, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 504, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "Remark 9. We note that the role of self-loops has also implicitly been noted in Xu et al. (2018) where", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "the analysis of the Jacobian of node representations on the graph augmented with self-loops has been", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 227, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 227, + 563 + ], + "score": 1.0, + "content": "related to lazy random-walks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 527, + 505, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "GNNs with different aggregations We note that similar conclusions extend to message passing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "architectures where the aggregations are sums and not averages meaning that we take the augmented", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "adjacency without normalizing by the degree matrices. Consistently with Lemma 1, we restrict to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "the setting where features and node representations at each layer are scalars to make the discussion", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 428, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 204, + 631 + ], + "score": 1.0, + "content": "simpler. In line with the", + "type": "text" + }, + { + "bbox": [ + 204, + 619, + 218, + 629 + ], + "score": 0.37, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 618, + 428, + 631 + ], + "score": 1.0, + "content": "et al. 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Differently from the setting of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 694, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 162, + 709 + ], + "score": 1.0, + "content": "Theorem 1 in", + "type": "text" + }, + { + "bbox": [ + 163, + 696, + 177, + 706 + ], + "score": 0.38, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 694, + 506, + 709 + ], + "score": 1.0, + "content": "et al. (2018), the aggregation here is not an average but a simple sum. 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If - similarly to Theorem 1 in", + "type": "text" + }, + { + "bbox": [ + 298, + 171, + 312, + 181 + ], + "score": 0.43, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "et al. (2018) - we assume that all paths in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "computational graph of the model are activated with the same probability, then we obtain that on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 192, + 141, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 141, + 206 + ], + "score": 1.0, + "content": "average", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 202, + 391, + 234 + ], + "lines": [ + { + "bbox": [ + 219, + 202, + 391, + 234 + ], + "spans": [ + { + "bbox": [ + 219, + 202, + 391, + 234 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { r + 1 } ( i , s ) = \\frac { \\tilde { A } _ { i s } ^ { r + 1 } } { \\sum _ { k } \\tilde { A } _ { i k } ^ { r + 1 } } \\leq \\frac { \\tilde { A } _ { i s } ^ { r + 1 } } { \\mathrm { V o l } ( B _ { r + 1 } ( i ) ) } ,", + "type": "interline_equation", + "image_path": "4a7f4b15f3bfc81ff6cab3b1b5345062f1a30c3b23873a5058327f01865b423c.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 202, + 391, + 218.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 219, + 218.0, + 391, + 234.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 506, + 265 + ], + "lines": [ + { + "bbox": [ + 104, + 235, + 508, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 235, + 132, + 254 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 237, + 178, + 249 + ], + "score": 0.91, + "content": "{ \\tilde { A } } = A + I", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 235, + 195, + 254 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 237, + 273, + 252 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { v o l } ( S ) = \\sum _ { j \\in S } d _ { j } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 235, + 508, + 254 + ], + "score": 1.0, + "content": ". In particular, we again find that if we have a tree structure,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 251, + 337, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 251, + 299, + 266 + ], + "score": 1.0, + "content": "then the right hand side decays exponentially as", + "type": "text" + }, + { + "bbox": [ + 300, + 251, + 332, + 263 + ], + "score": 0.9, + "content": "2 ^ { - ( r + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 251, + 337, + 266 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 409, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 410, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 410, + 296 + ], + "score": 1.0, + "content": "B PRELIMINARY ANALYSIS OF AN EDGE-NEIGHBORHOOD", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 307, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 308, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 165, + 319 + ], + "score": 1.0, + "content": "Given an edge", + "type": "text" + }, + { + "bbox": [ + 166, + 307, + 189, + 318 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 306, + 308, + 319 + ], + "score": 1.0, + "content": ", we introduce the sets below:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 124, + 326, + 506, + 398 + ], + "lines": [ + { + "bbox": [ + 127, + 326, + 443, + 341 + ], + "spans": [ + { + "bbox": [ + 127, + 326, + 142, + 341 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 327, + 246, + 340 + ], + "score": 0.91, + "content": "\\sharp _ { \\Delta } ( i , j ) : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 326, + 416, + 341 + ], + "score": 1.0, + "content": ", the number of triangles based at the edge", + "type": "text" + }, + { + "bbox": [ + 416, + 329, + 439, + 339 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 326, + 443, + 341 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 124, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 124, + 345, + 141, + 359 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 142, + 345, + 438, + 358 + ], + "score": 0.88, + "content": "\\sharp _ { \\bigtriangledown } ( i , j ) : = \\{ k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j ) , k \\neq j : \\ \\exists w \\in ( S _ { 1 } ( k ) \\cap S _ { 1 } ( j ) ) \\setminus S _ { 1 } ( i ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 345, + 506, + 359 + ], + "score": 1.0, + "content": ", the number of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 357, + 443, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 168, + 370 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 168, + 358, + 208, + 369 + ], + "score": 0.9, + "content": "k \\in S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 357, + 318, + 370 + ], + "score": 1.0, + "content": "forming a 4-cycle based at", + "type": "text" + }, + { + "bbox": [ + 318, + 358, + 340, + 368 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 357, + 443, + 370 + ], + "score": 1.0, + "content": "without diagonals inside.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 121, + 374, + 504, + 389 + ], + "spans": [ + { + "bbox": [ + 121, + 374, + 142, + 389 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 143, + 374, + 319, + 388 + ], + "score": 0.91, + "content": "Q _ { i } ( j ) : = S _ { 1 } ( i ) \\setminus ( \\{ j \\} \\cup \\sharp _ { \\Delta } ( i , j ) \\cup \\sharp _ { \\Pi } ^ { i } ( i , j ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 374, + 499, + 389 + ], + "score": 1.0, + "content": ", simply the complement of the neighbours of", + "type": "text" + }, + { + "bbox": [ + 499, + 376, + 504, + 385 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 386, + 431, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 421, + 399 + ], + "score": 1.0, + "content": "with respect to the sets introduced in (i) and (ii) once we also exclude", + "type": "text" + }, + { + "bbox": [ + 421, + 387, + 427, + 398 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 386, + 431, + 399 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 406, + 483, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 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In general the sets", + "type": "text" + }, + { + "bbox": [ + 281, + 434, + 294, + 447 + ], + "score": 0.88, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 433, + 311, + 449 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 433, + 324, + 447 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 433, + 506, + 449 + ], + "score": 1.0, + "content": "may differ. This may occur when there exists", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 136, + 460 + ], + "score": 1.0, + "content": "a node", + "type": "text" + }, + { + "bbox": [ + 136, + 447, + 143, + 457 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 446, + 198, + 460 + ], + "score": 1.0, + "content": "belonging to", + "type": "text" + }, + { + "bbox": [ + 199, + 446, + 211, + 460 + ], + "score": 0.88, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 446, + 336, + 460 + ], + "score": 1.0, + "content": "that admits multiple solutions", + "type": "text" + }, + { + "bbox": [ + 336, + 449, + 345, + 457 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 446, + 436, + 460 + ], + "score": 1.0, + "content": "as in the definition of", + "type": "text" + }, + { + "bbox": [ + 436, + 446, + 448, + 460 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 446, + 506, + 460 + ], + "score": 1.0, + "content": ". This feature", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "needs to be taken into account when comparing Ollivier’s Ricci curvature to the new notion we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 469, + 404, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 404, + 482 + ], + "score": 1.0, + "content": "present below. We first introduce the following class to ease the notations.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 440, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 441, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 298, + 498 + ], + "score": 1.0, + "content": "Definition 2. For any simple, undirected graph", + "type": "text" + }, + { + "bbox": [ + 299, + 484, + 347, + 496 + ], + "score": 0.92, + "content": "G = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 483, + 359, + 498 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 360, + 484, + 389, + 496 + ], + "score": 0.81, + "content": "U \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 483, + 441, + 498 + ], + "score": 1.0, + "content": ", then we set", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 501, + 439, + 516 + ], + "lines": [ + { + "bbox": [ + 172, + 501, + 439, + 516 + ], + "spans": [ + { + "bbox": [ + 172, + 501, + 439, + 516 + ], + "score": 0.85, + "content": "\\mathcal { D } ( U ) : = \\left\\{ \\varphi : U \\to V , \\ | U | = | \\varphi ( U ) | , \\ ( z , \\varphi ( z ) ) \\in E , \\ \\forall z \\in U \\right\\} .", + "type": "interline_equation", + "image_path": "1a61424f80d2ff1dcba047eb83d7803f4a76622a96014401a2002ef5b6d85748.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 172, + 501, + 439, + 516 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 267, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 268, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 175, + 541 + ], + "score": 1.0, + "content": "We note that any", + "type": "text" + }, + { + "bbox": [ + 176, + 527, + 219, + 540 + ], + "score": 0.93, + "content": "\\varphi \\in { \\mathcal { D } } ( U )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 527, + 268, + 541 + ], + "score": 1.0, + "content": "is injective.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 504, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "We may now define a quantity which measures the maximal number of 1-1 pairings that can be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 547, + 212, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 172, + 567 + ], + "score": 1.0, + "content": "performed from", + "type": "text" + }, + { + "bbox": [ + 172, + 550, + 184, + 564 + ], + "score": 0.85, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 547, + 195, + 567 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 196, + 549, + 207, + 564 + ], + "score": 0.8, + "content": "\\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 547, + 212, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 268, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 268, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 218, + 579 + ], + "score": 1.0, + "content": "Definition 3. For any edge", + "type": "text" + }, + { + "bbox": [ + 218, + 567, + 241, + 578 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 565, + 268, + 579 + ], + "score": 1.0, + "content": "we let", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 583, + 433, + 605 + ], + "lines": [ + { + "bbox": [ + 177, + 583, + 433, + 605 + ], + "spans": [ + { + "bbox": [ + 177, + 583, + 433, + 605 + ], + "score": 0.91, + "content": "\\sharp _ { \\sharp } ( i , j ) : = \\operatorname* { m a x } \\left\\{ | U | : U \\subset \\sharp _ { \\sharp } ^ { i } , \\ : \\exists \\varphi : U \\to \\sharp _ { \\sharp } ^ { j } , \\ : \\varphi \\in \\mathcal { D } ( U ) \\right\\} .", + "type": "interline_equation", + "image_path": "d9fd6b7a14a917ee307f5ab5fdd2cfd9f1d9546c651adcbb6b9665b287978d4f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 177, + 583, + 433, + 605 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 104, + 615, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 194, + 631 + ], + "score": 1.0, + "content": "We often simply write", + "type": "text" + }, + { + "bbox": [ + 194, + 617, + 206, + 629 + ], + "score": 0.84, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 615, + 284, + 631 + ], + "score": 1.0, + "content": ". While the quantity", + "type": "text" + }, + { + "bbox": [ + 285, + 617, + 297, + 629 + ], + "score": 0.86, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 615, + 506, + 631 + ], + "score": 1.0, + "content": "plays a role in the derivation of the Ollivier curvature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 117, + 640 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 628, + 140, + 639 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 627, + 338, + 640 + ], + "score": 1.0, + "content": "it is not computationally-friendly, as to determine", + "type": "text" + }, + { + "bbox": [ + 339, + 627, + 351, + 640 + ], + "score": 0.83, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "we need to identify and distinguish all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 210, + 652 + ], + "score": 1.0, + "content": "possible 4-cycles based at", + "type": "text" + }, + { + "bbox": [ + 210, + 639, + 233, + 650 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "and then choose a maximal pairing map. Accordingly, we consider a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 266, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 266, + 662 + ], + "score": 1.0, + "content": "looser term which is easier to compute:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 353, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 352, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 289, + 678 + ], + "score": 1.0, + "content": "Definition 4. For any pair of adjacent nodes", + "type": "text" + }, + { + "bbox": [ + 289, + 666, + 311, + 676 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 663, + 352, + 678 + ], + "score": 1.0, + "content": "we define", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 681, + 496, + 715 + ], + "lines": [ + { + "bbox": [ + 114, + 681, + 496, + 715 + ], + "spans": [ + { + "bbox": [ + 114, + 681, + 496, + 715 + ], + "score": 0.93, + "content": "\\gamma _ { m a x } ( i , j ) : = \\operatorname* { m a x } \\left\\{ \\operatorname* { m a x } _ { k \\in \\sharp _ { \\Sigma } ^ { k } } \\{ \\big ( A _ { k } \\cdot ( A _ { j } - A _ { i } \\odot A _ { j } ) \\big ) - 1 \\} , \\operatorname* { m a x } _ { w \\in \\sharp _ { \\Sigma } ^ { j } } \\{ \\big ( A _ { w } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) \\big ) - 1 \\} \\right\\} ,", + "type": "interline_equation", + "image_path": "898d313ce04a3ff42ee4c397327ae65e9c4d5cd09222a0119a2daeeb9bf68d9f.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 114, + 681, + 496, + 692.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 114, + 692.3333333333334, + 496, + 703.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 114, + 703.6666666666667, + 496, + 715.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 447, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 717, + 445, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 133, + 736 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 721, + 146, + 732 + ], + "score": 0.88, + "content": "A _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 717, + 426, + 736 + ], + "score": 1.0, + "content": "denotes the s-th row of the adjacency matrix. 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If - similarly to Theorem 1 in", + "type": "text" + }, + { + "bbox": [ + 298, + 171, + 312, + 181 + ], + "score": 0.43, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "et al. (2018) - we assume that all paths in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "computational graph of the model are activated with the same probability, then we obtain that on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 192, + 141, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 141, + 206 + ], + "score": 1.0, + "content": "average", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 159, + 505, + 206 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 202, + 391, + 234 + ], + "lines": [ + { + "bbox": [ + 219, + 202, + 391, + 234 + ], + "spans": [ + { + "bbox": [ + 219, + 202, + 391, + 234 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { r + 1 } ( i , s ) = \\frac { \\tilde { A } _ { i s } ^ { r + 1 } } { \\sum _ { k } \\tilde { A } _ { i k } ^ { r + 1 } } \\leq \\frac { \\tilde { A } _ { i s } ^ { r + 1 } } { \\mathrm { V o l } ( B _ { r + 1 } ( i ) ) } ,", + "type": "interline_equation", + "image_path": "4a7f4b15f3bfc81ff6cab3b1b5345062f1a30c3b23873a5058327f01865b423c.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 202, + 391, + 218.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 219, + 218.0, + 391, + 234.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 506, + 265 + ], + "lines": [ + { + "bbox": [ + 104, + 235, + 508, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 235, + 132, + 254 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 237, + 178, + 249 + ], + "score": 0.91, + "content": "{ \\tilde { A } } = A + I", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 235, + 195, + 254 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 237, + 273, + 252 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { v o l } ( S ) = \\sum _ { j \\in S } d _ { j } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 235, + 508, + 254 + ], + "score": 1.0, + "content": ". In particular, we again find that if we have a tree structure,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 251, + 337, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 251, + 299, + 266 + ], + "score": 1.0, + "content": "then the right hand side decays exponentially as", + "type": "text" + }, + { + "bbox": [ + 300, + 251, + 332, + 263 + ], + "score": 0.9, + "content": "2 ^ { - ( r + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 251, + 337, + 266 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 235, + 508, + 266 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 409, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 410, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 410, + 296 + ], + "score": 1.0, + "content": "B PRELIMINARY ANALYSIS OF AN EDGE-NEIGHBORHOOD", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 307, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 308, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 165, + 319 + ], + "score": 1.0, + "content": "Given an edge", + "type": "text" + }, + { + "bbox": [ + 166, + 307, + 189, + 318 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 306, + 308, + 319 + ], + "score": 1.0, + "content": ", we introduce the sets below:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 306, + 308, + 319 + ] + }, + { + "type": "list", + "bbox": [ + 124, + 326, + 506, + 398 + ], + "lines": [ + { + "bbox": [ + 127, + 326, + 443, + 341 + ], + "spans": [ + { + "bbox": [ + 127, + 326, + 142, + 341 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 327, + 246, + 340 + ], + "score": 0.91, + "content": "\\sharp _ { \\Delta } ( i , j ) : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 326, + 416, + 341 + ], + "score": 1.0, + "content": ", the number of triangles based at the edge", + "type": "text" + }, + { + "bbox": [ + 416, + 329, + 439, + 339 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 326, + 443, + 341 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 124, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 124, + 345, + 141, + 359 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 142, + 345, + 438, + 358 + ], + "score": 0.88, + "content": "\\sharp _ { \\bigtriangledown } ( i , j ) : = \\{ k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j ) , k \\neq j : \\ \\exists w \\in ( S _ { 1 } ( k ) \\cap S _ { 1 } ( j ) ) \\setminus S _ { 1 } ( i ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 345, + 506, + 359 + ], + "score": 1.0, + "content": ", the number of", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 357, + 443, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 168, + 370 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 168, + 358, + 208, + 369 + ], + "score": 0.9, + "content": "k \\in S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 357, + 318, + 370 + ], + "score": 1.0, + "content": "forming a 4-cycle based at", + "type": "text" + }, + { + "bbox": [ + 318, + 358, + 340, + 368 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 357, + 443, + 370 + ], + "score": 1.0, + "content": "without diagonals inside.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 121, + 374, + 504, + 389 + ], + "spans": [ + { + "bbox": [ + 121, + 374, + 142, + 389 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 143, + 374, + 319, + 388 + ], + "score": 0.91, + "content": "Q _ { i } ( j ) : = S _ { 1 } ( i ) \\setminus ( \\{ j \\} \\cup \\sharp _ { \\Delta } ( i , j ) \\cup \\sharp _ { \\Pi } ^ { i } ( i , j ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 374, + 499, + 389 + ], + "score": 1.0, + "content": ", simply the complement of the neighbours of", + "type": "text" + }, + { + "bbox": [ + 499, + 376, + 504, + 385 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 386, + 431, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 421, + 399 + ], + "score": 1.0, + "content": "with respect to the sets introduced in (i) and (ii) once we also exclude", + "type": "text" + }, + { + "bbox": [ + 421, + 387, + 427, + 398 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 386, + 431, + 399 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + } + ], + "index": 15, + "bbox_fs": [ + 121, + 326, + 506, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 406, + 483, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 485, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 239, + 421 + ], + "score": 1.0, + "content": "In the following we simply write", + "type": "text" + }, + { + "bbox": [ + 240, + 407, + 268, + 420 + ], + "score": 0.6, + "content": "\\sharp _ { \\Delta } , \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 406, + 285, + 421 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 407, + 298, + 419 + ], + "score": 0.89, + "content": "Q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 406, + 359, + 421 + ], + "score": 1.0, + "content": "when the edge", + "type": "text" + }, + { + "bbox": [ + 359, + 408, + 382, + 419 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 406, + 485, + 421 + ], + "score": 1.0, + "content": "is clear from the context.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 406, + 485, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 506, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 281, + 449 + ], + "score": 1.0, + "content": "4-cycle contributions. In general the sets", + "type": "text" + }, + { + "bbox": [ + 281, + 434, + 294, + 447 + ], + "score": 0.88, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 433, + 311, + 449 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 433, + 324, + 447 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 433, + 506, + 449 + ], + "score": 1.0, + "content": "may differ. This may occur when there exists", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 136, + 460 + ], + "score": 1.0, + "content": "a node", + "type": "text" + }, + { + "bbox": [ + 136, + 447, + 143, + 457 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 446, + 198, + 460 + ], + "score": 1.0, + "content": "belonging to", + "type": "text" + }, + { + "bbox": [ + 199, + 446, + 211, + 460 + ], + "score": 0.88, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 446, + 336, + 460 + ], + "score": 1.0, + "content": "that admits multiple solutions", + "type": "text" + }, + { + "bbox": [ + 336, + 449, + 345, + 457 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 446, + 436, + 460 + ], + "score": 1.0, + "content": "as in the definition of", + "type": "text" + }, + { + "bbox": [ + 436, + 446, + 448, + 460 + ], + "score": 0.89, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 446, + 506, + 460 + ], + "score": 1.0, + "content": ". This feature", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "needs to be taken into account when comparing Ollivier’s Ricci curvature to the new notion we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 469, + 404, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 404, + 482 + ], + "score": 1.0, + "content": "present below. We first introduce the following class to ease the notations.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 433, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 440, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 441, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 298, + 498 + ], + "score": 1.0, + "content": "Definition 2. For any simple, undirected graph", + "type": "text" + }, + { + "bbox": [ + 299, + 484, + 347, + 496 + ], + "score": 0.92, + "content": "G = ( V , E )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 483, + 359, + 498 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 360, + 484, + 389, + 496 + ], + "score": 0.81, + "content": "U \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 483, + 441, + 498 + ], + "score": 1.0, + "content": ", then we set", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 483, + 441, + 498 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 501, + 439, + 516 + ], + "lines": [ + { + "bbox": [ + 172, + 501, + 439, + 516 + ], + "spans": [ + { + "bbox": [ + 172, + 501, + 439, + 516 + ], + "score": 0.85, + "content": "\\mathcal { D } ( U ) : = \\left\\{ \\varphi : U \\to V , \\ | U | = | \\varphi ( U ) | , \\ ( z , \\varphi ( z ) ) \\in E , \\ \\forall z \\in U \\right\\} .", + "type": "interline_equation", + "image_path": "1a61424f80d2ff1dcba047eb83d7803f4a76622a96014401a2002ef5b6d85748.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 172, + 501, + 439, + 516 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 267, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 268, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 175, + 541 + ], + "score": 1.0, + "content": "We note that any", + "type": "text" + }, + { + "bbox": [ + 176, + 527, + 219, + 540 + ], + "score": 0.93, + "content": "\\varphi \\in { \\mathcal { D } } ( U )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 527, + 268, + 541 + ], + "score": 1.0, + "content": "is injective.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 527, + 268, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 504, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "We may now define a quantity which measures the maximal number of 1-1 pairings that can be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 547, + 212, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 172, + 567 + ], + "score": 1.0, + "content": "performed from", + "type": "text" + }, + { + "bbox": [ + 172, + 550, + 184, + 564 + ], + "score": 0.85, + "content": "\\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 547, + 195, + 567 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 196, + 549, + 207, + 564 + ], + "score": 0.8, + "content": "\\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 547, + 212, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 537, + 505, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 268, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 268, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 218, + 579 + ], + "score": 1.0, + "content": "Definition 3. For any edge", + "type": "text" + }, + { + "bbox": [ + 218, + 567, + 241, + 578 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 565, + 268, + 579 + ], + "score": 1.0, + "content": "we let", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 565, + 268, + 579 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 583, + 433, + 605 + ], + "lines": [ + { + "bbox": [ + 177, + 583, + 433, + 605 + ], + "spans": [ + { + "bbox": [ + 177, + 583, + 433, + 605 + ], + "score": 0.91, + "content": "\\sharp _ { \\sharp } ( i , j ) : = \\operatorname* { m a x } \\left\\{ | U | : U \\subset \\sharp _ { \\sharp } ^ { i } , \\ : \\exists \\varphi : U \\to \\sharp _ { \\sharp } ^ { j } , \\ : \\varphi \\in \\mathcal { D } ( U ) \\right\\} .", + "type": "interline_equation", + "image_path": "d9fd6b7a14a917ee307f5ab5fdd2cfd9f1d9546c651adcbb6b9665b287978d4f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 177, + 583, + 433, + 605 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 506, + 661 + ], + "lines": [ + { + "bbox": [ + 104, + 615, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 194, + 631 + ], + "score": 1.0, + "content": "We often simply write", + "type": "text" + }, + { + "bbox": [ + 194, + 617, + 206, + 629 + ], + "score": 0.84, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 615, + 284, + 631 + ], + "score": 1.0, + "content": ". While the quantity", + "type": "text" + }, + { + "bbox": [ + 285, + 617, + 297, + 629 + ], + "score": 0.86, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 615, + 506, + 631 + ], + "score": 1.0, + "content": "plays a role in the derivation of the Ollivier curvature", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 117, + 640 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 628, + 140, + 639 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 627, + 338, + 640 + ], + "score": 1.0, + "content": "it is not computationally-friendly, as to determine", + "type": "text" + }, + { + "bbox": [ + 339, + 627, + 351, + 640 + ], + "score": 0.83, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "we need to identify and distinguish all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 210, + 652 + ], + "score": 1.0, + "content": "possible 4-cycles based at", + "type": "text" + }, + { + "bbox": [ + 210, + 639, + 233, + 650 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "and then choose a maximal pairing map. Accordingly, we consider a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 266, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 266, + 662 + ], + "score": 1.0, + "content": "looser term which is easier to compute:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 615, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 353, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 352, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 289, + 678 + ], + "score": 1.0, + "content": "Definition 4. For any pair of adjacent nodes", + "type": "text" + }, + { + "bbox": [ + 289, + 666, + 311, + 676 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 663, + 352, + 678 + ], + "score": 1.0, + "content": "we define", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 663, + 352, + 678 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 681, + 496, + 715 + ], + "lines": [ + { + "bbox": [ + 114, + 681, + 496, + 715 + ], + "spans": [ + { + "bbox": [ + 114, + 681, + 496, + 715 + ], + "score": 0.93, + "content": "\\gamma _ { m a x } ( i , j ) : = \\operatorname* { m a x } \\left\\{ \\operatorname* { m a x } _ { k \\in \\sharp _ { \\Sigma } ^ { k } } \\{ \\big ( A _ { k } \\cdot ( A _ { j } - A _ { i } \\odot A _ { j } ) \\big ) - 1 \\} , \\operatorname* { m a x } _ { w \\in \\sharp _ { \\Sigma } ^ { j } } \\{ \\big ( A _ { w } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) \\big ) - 1 \\} \\right\\} ,", + "type": "interline_equation", + "image_path": "898d313ce04a3ff42ee4c397327ae65e9c4d5cd09222a0119a2daeeb9bf68d9f.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 114, + 681, + 496, + 692.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 114, + 692.3333333333334, + 496, + 703.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 114, + 703.6666666666667, + 496, + 715.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 447, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 717, + 445, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 133, + 736 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 721, + 146, + 732 + ], + "score": 0.88, + "content": "A _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 717, + 426, + 736 + ], + "score": 1.0, + "content": "denotes the s-th row of the adjacency matrix. We usually simply write", + "type": "text" + }, + { + "bbox": [ + 427, + 722, + 445, + 732 + ], + "score": 0.75, + "content": "\\gamma _ { m a x }", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 717, + 445, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 130 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 239, + 96 + ], + "score": 1.0, + "content": "Remark 10. We note that given", + "type": "text" + }, + { + "bbox": [ + 239, + 82, + 312, + 95 + ], + "score": 0.92, + "content": "k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 81, + 351, + 96 + ], + "score": 1.0, + "content": "the term", + "type": "text" + }, + { + "bbox": [ + 352, + 82, + 462, + 95 + ], + "score": 0.91, + "content": "( A _ { k } \\cdot ( A _ { j } - A _ { i } \\odot A _ { j } ) ) - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "yields the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 304, + 106 + ], + "score": 1.0, + "content": "number of nodes w forming a 4-cycle of the form", + "type": "text" + }, + { + "bbox": [ + 305, + 94, + 383, + 105 + ], + "score": 0.9, + "content": "i \\sim k \\sim w \\sim j \\sim i", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "with no diagonals inside. The", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 129, + 117 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 130, + 106, + 149, + 116 + ], + "score": 0.87, + "content": "\\gamma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 104, + 355, + 117 + ], + "score": 1.0, + "content": "measures the maximal degeneracy of edges forming", + "type": "text" + }, + { + "bbox": [ + 355, + 106, + 362, + 114 + ], + "score": 0.35, + "content": "^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 104, + 498, + 117 + ], + "score": 1.0, + "content": "-cycles, meaning that it is equal to", + "type": "text" + }, + { + "bbox": [ + 498, + 105, + 504, + 114 + ], + "score": 0.48, + "content": "^ { l }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 491, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 152, + 131 + ], + "score": 1.0, + "content": "iff for each", + "type": "text" + }, + { + "bbox": [ + 153, + 116, + 182, + 129 + ], + "score": 0.91, + "content": "k \\in \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 115, + 289, + 131 + ], + "score": 1.0, + "content": "there exists a unique node", + "type": "text" + }, + { + "bbox": [ + 289, + 115, + 321, + 130 + ], + "score": 0.89, + "content": "w \\in \\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 115, + 360, + 131 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 361, + 117, + 439, + 128 + ], + "score": 0.9, + "content": "i \\sim k \\sim w \\sim j \\sim i", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 115, + 457, + 131 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 457, + 117, + 464, + 127 + ], + "score": 0.37, + "content": "^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 115, + 491, + 131 + ], + "score": 1.0, + "content": "-cycle.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 504, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "We now end the discussion about 4-cycle contributions by proving the following inequality, which", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 435, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 295, + 162 + ], + "score": 1.0, + "content": "allows us to avoid to compute directly the term", + "type": "text" + }, + { + "bbox": [ + 296, + 149, + 308, + 162 + ], + "score": 0.83, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 148, + 435, + 162 + ], + "score": 1.0, + "content": "up to giving up some accuracy.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 272, + 176 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 272, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 213, + 176 + ], + "score": 1.0, + "content": "Lemma 11. For any edge", + "type": "text" + }, + { + "bbox": [ + 214, + 164, + 236, + 176 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 163, + 272, + 176 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 181, + 357, + 210 + ], + "lines": [ + { + "bbox": [ + 254, + 181, + 357, + 210 + ], + "spans": [ + { + "bbox": [ + 254, + 181, + 357, + 210 + ], + "score": 0.94, + "content": "| \\sharp _ { \\sf H } ^ { \\mathfrak { m } } | \\geq \\frac { \\operatorname* { m a x } \\{ | \\sharp _ { \\sf H } ^ { i } | , | \\sharp _ { \\sf H } ^ { j } | \\} } { \\gamma _ { m a x } } .", + "type": "interline_equation", + "image_path": "89bea760cd16cc49e7087d599acf6b3ca969b8894bfa47efa2d19f0a2c986f53.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 254, + 181, + 357, + 195.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 254, + 195.5, + 357, + 210.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 364, + 237 + ], + "score": 1.0, + "content": "Proof. The proof is based on a combinatorial argument. Let", + "type": "text" + }, + { + "bbox": [ + 365, + 222, + 445, + 234 + ], + "score": 0.87, + "content": "\\sharp _ { \\sqcap } ^ { i } = \\{ k _ { 1 } , \\ldots , k _ { r } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 220, + 479, + 237 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 479, + 223, + 505, + 235 + ], + "score": 0.5, + "content": "\\sharp _ { \\bigcirc } ^ { \\mathfrak { m } } =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 232, + 504, + 247 + ], + "spans": [ + { + "bbox": [ + 107, + 234, + 158, + 246 + ], + "score": 0.91, + "content": "\\{ k _ { 1 } , \\ldots , k _ { \\ell } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 232, + 182, + 247 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 182, + 234, + 205, + 244 + ], + "score": 0.89, + "content": "\\ell < r", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 232, + 310, + 247 + ], + "score": 1.0, + "content": ". By definition there exists", + "type": "text" + }, + { + "bbox": [ + 310, + 234, + 446, + 246 + ], + "score": 0.91, + "content": "\\varphi : \\{ k _ { 1 } , \\dots , k _ { \\ell } \\} \\to \\{ w _ { 1 } , \\dots , w _ { \\ell } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 232, + 470, + 247 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 470, + 235, + 504, + 245 + ], + "score": 0.88, + "content": "k _ { i } \\sim w _ { i }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 123, + 260 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 246, + 158, + 259 + ], + "score": 0.9, + "content": "w _ { i } \\in \\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 246, + 175, + 260 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 176, + 246, + 216, + 258 + ], + "score": 0.91, + "content": "1 \\leq i \\leq \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 246, + 246, + 260 + ], + "score": 1.0, + "content": ". Given", + "type": "text" + }, + { + "bbox": [ + 246, + 246, + 336, + 259 + ], + "score": 0.92, + "content": "k \\in \\sharp _ { \\sqcap } ^ { i } \\setminus \\{ k _ { 1 } , . . . , k _ { \\ell } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 246, + 408, + 260 + ], + "score": 1.0, + "content": ", then there are no", + "type": "text" + }, + { + "bbox": [ + 409, + 246, + 504, + 259 + ], + "score": 0.9, + "content": "w \\in \\sharp _ { \\sqcap } ^ { j } \\setminus \\{ w _ { 1 } , . . . , w _ { \\ell } \\}", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 104, + 256, + 145, + 270 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 146, + 258, + 174, + 268 + ], + "score": 0.89, + "content": "k \\sim w", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 256, + 289, + 270 + ], + "score": 1.0, + "content": ", otherwise we could extend", + "type": "text" + }, + { + "bbox": [ + 290, + 260, + 298, + 269 + ], + "score": 0.81, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 256, + 342, + 270 + ], + "score": 1.0, + "content": "by setting", + "type": "text" + }, + { + "bbox": [ + 342, + 258, + 408, + 270 + ], + "score": 0.93, + "content": "k \\mapsto \\varphi ( k ) : = w", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "and we would then get", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 268, + 253, + 282 + ], + "spans": [ + { + "bbox": [ + 107, + 268, + 158, + 281 + ], + "score": 0.89, + "content": "| \\mathtt { H } _ { \\square } ^ { \\mathfrak { m } } | = \\ell + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 268, + 253, + 282 + ], + "score": 1.0, + "content": ". Accordingly, we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 286, + 388, + 321 + ], + "lines": [ + { + "bbox": [ + 220, + 286, + 388, + 321 + ], + "spans": [ + { + "bbox": [ + 220, + 286, + 388, + 321 + ], + "score": 0.93, + "content": "\\sum _ { s = 1 } ^ { \\ell } ( A _ { w _ { s } } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) ) - 1 ) \\geq | \\sharp _ { \\square } ^ { i } | ,", + "type": "interline_equation", + "image_path": "7529fcc278feaac0518c6fbd641d177105dffada53eefb337b64856a185dd038.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 220, + 286, + 388, + 303.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 220, + 303.5, + 388, + 321.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 164, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 165, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 165, + 339 + ], + "score": 1.0, + "content": "which implies", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 342, + 433, + 377 + ], + "lines": [ + { + "bbox": [ + 178, + 342, + 433, + 377 + ], + "spans": [ + { + "bbox": [ + 178, + 342, + 433, + 377 + ], + "score": 0.94, + "content": "\\gamma _ { \\operatorname* { m a x } } | \\sharp _ { \\sf D } ^ { \\mathrm { m } } | \\equiv \\gamma _ { \\operatorname* { m a x } } \\ell \\geq \\sum _ { s = 1 } ^ { \\ell } ( A _ { w _ { s } } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) ) - 1 ) \\geq | \\sharp _ { \\sf D } ^ { i } | .", + "type": "interline_equation", + "image_path": "06d40a0b2b39caa88d2abd80ab697f95b6e82628d65357f4ea7e0b12493ef0c2.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 178, + 342, + 433, + 353.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 178, + 353.6666666666667, + 433, + 365.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 178, + 365.33333333333337, + 433, + 377.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 410, + 312, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 312, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 312, + 425 + ], + "score": 1.0, + "content": "C EXISTING CURVATURE CANDIDATES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 501, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 502, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 233, + 450 + ], + "score": 1.0, + "content": "Ollivier Ricci curvature For", + "type": "text" + }, + { + "bbox": [ + 234, + 435, + 258, + 446 + ], + "score": 0.89, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 433, + 276, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 277, + 435, + 317, + 447 + ], + "score": 0.95, + "content": "\\alpha \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 433, + 461, + 450 + ], + "score": 1.0, + "content": "we define a probability measure on", + "type": "text" + }, + { + "bbox": [ + 461, + 435, + 485, + 447 + ], + "score": 0.93, + "content": "B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 433, + 502, + 450 + ], + "score": 1.0, + "content": "by:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 452, + 369, + 498 + ], + "lines": [ + { + "bbox": [ + 240, + 452, + 369, + 498 + ], + "spans": [ + { + "bbox": [ + 240, + 452, + 369, + 498 + ], + "score": 0.94, + "content": "\\mu _ { i } ^ { \\alpha } : j \\mapsto \\left\\{ \\begin{array} { r l } & { \\alpha , \\ j = i } \\\\ & { \\frac { 1 - \\alpha } { d _ { i } } , \\ j \\in S _ { 1 } ( i ) , } \\\\ & { 0 , \\ \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "7aa9e425bee708dea94fa37d93747a16569eddd36c9e02db4d32c03b031f412e.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 452, + 369, + 475.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 240, + 475.0, + 369, + 498.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 504, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "Before we introduce the Ollivier curvature, we recall that the transportation distance between two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 514, + 383, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 383, + 527 + ], + "score": 1.0, + "content": "finitely supported probability measures as above can be computed as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 530, + 407, + 559 + ], + "lines": [ + { + "bbox": [ + 204, + 530, + 407, + 559 + ], + "spans": [ + { + "bbox": [ + 204, + 530, + 407, + 559 + ], + "score": 0.93, + "content": "W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) : = \\operatorname* { i n f } _ { M } \\sum _ { k \\in S _ { 1 } ( i ) } \\sum _ { w \\in S _ { 1 } ( j ) } M _ { k w } d _ { G } ( k , w ) ,", + "type": "interline_equation", + "image_path": "b963842df11bb6006c9305daafc642e9a7e950e52d21e8a67631e39dd52be7d7.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 530, + 407, + 544.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 204, + 544.5, + 407, + 559.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 503, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 133, + 578 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 565, + 164, + 578 + ], + "score": 0.92, + "content": "d _ { G } ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 565, + 492, + 578 + ], + "score": 1.0, + "content": "is the geodesic distance on the graph and the infimum is taken over all matrices", + "type": "text" + }, + { + "bbox": [ + 492, + 566, + 504, + 576 + ], + "score": 0.74, + "content": "M", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 577, + 249, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 249, + 589 + ], + "score": 1.0, + "content": "satisfying the marginal constraints:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 593, + 405, + 622 + ], + "lines": [ + { + "bbox": [ + 205, + 593, + 405, + 622 + ], + "spans": [ + { + "bbox": [ + 205, + 593, + 405, + 622 + ], + "score": 0.93, + "content": "\\sum _ { k \\in S _ { 1 } ( i ) } M _ { k w } = \\mu _ { j } ^ { \\alpha } ( w ) , \\quad \\sum _ { w \\in S _ { 1 } ( j ) } M _ { k w } = \\mu _ { i } ^ { \\alpha } ( k ) .", + "type": "interline_equation", + "image_path": "35bd5ec6b11dc1457e6daa1ef3b68652590ec136dc5d29813fb7a55965c7d11b.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 593, + 405, + 607.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 205, + 607.5, + 405, + 622.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 507, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 507, + 640 + ], + "score": 1.0, + "content": "We are now ready to define the Ollivier Ricci curvature: the formulation below is due to Lin et al.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 637, + 139, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 139, + 652 + ], + "score": 1.0, + "content": "(2011).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 363, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 363, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 190, + 668 + ], + "score": 1.0, + "content": "Definition 5. Given", + "type": "text" + }, + { + "bbox": [ + 190, + 654, + 213, + 665 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 651, + 268, + 668 + ], + "score": 1.0, + "content": "we define the", + "type": "text" + }, + { + "bbox": [ + 268, + 655, + 276, + 663 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 651, + 363, + 668 + ], + "score": 1.0, + "content": "-Ollivier curvature by", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 670, + 365, + 685 + ], + "lines": [ + { + "bbox": [ + 246, + 670, + 365, + 685 + ], + "spans": [ + { + "bbox": [ + 246, + 670, + 365, + 685 + ], + "score": 0.91, + "content": "\\kappa _ { \\alpha } ( i , j ) : = 1 - W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) .", + "type": "interline_equation", + "image_path": "d72597de2d5c7786cc275b8f8987765a0f4d37b4fd975b08b5e5d71dc553778a.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 246, + 670, + 365, + 685 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 691, + 433, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 433, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 130, + 705 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 691, + 185, + 704 + ], + "score": 0.94, + "content": "\\kappa _ { \\alpha } ( 1 - \\alpha ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 691, + 433, + 705 + ], + "score": 1.0, + "content": "is increasing and bounded the quantity below is well-defined:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 709, + 373, + 735 + ], + "lines": [ + { + "bbox": [ + 238, + 709, + 373, + 735 + ], + "spans": [ + { + "bbox": [ + 238, + 709, + 373, + 735 + ], + "score": 0.95, + "content": "\\kappa ( i , j ) : = \\operatorname* { l i m } _ { \\alpha \\to 1 } \\frac { 1 - W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) } { 1 - \\alpha } .", + "type": "interline_equation", + "image_path": "ba3d51ae9dcbbbd72f81491e29b7667fb9288933963bf5170264c57c58dc6e86.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 238, + 709, + 373, + 735 + ], + "spans": [], + "index": 37 + } + ] + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 382, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 496, + 384, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 496, + 384, + 505, + 393 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 130 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 239, + 96 + ], + "score": 1.0, + "content": "Remark 10. We note that given", + "type": "text" + }, + { + "bbox": [ + 239, + 82, + 312, + 95 + ], + "score": 0.92, + "content": "k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 81, + 351, + 96 + ], + "score": 1.0, + "content": "the term", + "type": "text" + }, + { + "bbox": [ + 352, + 82, + 462, + 95 + ], + "score": 0.91, + "content": "( A _ { k } \\cdot ( A _ { j } - A _ { i } \\odot A _ { j } ) ) - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "yields the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 304, + 106 + ], + "score": 1.0, + "content": "number of nodes w forming a 4-cycle of the form", + "type": "text" + }, + { + "bbox": [ + 305, + 94, + 383, + 105 + ], + "score": 0.9, + "content": "i \\sim k \\sim w \\sim j \\sim i", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "with no diagonals inside. The", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 129, + 117 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 130, + 106, + 149, + 116 + ], + "score": 0.87, + "content": "\\gamma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 104, + 355, + 117 + ], + "score": 1.0, + "content": "measures the maximal degeneracy of edges forming", + "type": "text" + }, + { + "bbox": [ + 355, + 106, + 362, + 114 + ], + "score": 0.35, + "content": "^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 104, + 498, + 117 + ], + "score": 1.0, + "content": "-cycles, meaning that it is equal to", + "type": "text" + }, + { + "bbox": [ + 498, + 105, + 504, + 114 + ], + "score": 0.48, + "content": "^ { l }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 491, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 152, + 131 + ], + "score": 1.0, + "content": "iff for each", + "type": "text" + }, + { + "bbox": [ + 153, + 116, + 182, + 129 + ], + "score": 0.91, + "content": "k \\in \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 115, + 289, + 131 + ], + "score": 1.0, + "content": "there exists a unique node", + "type": "text" + }, + { + "bbox": [ + 289, + 115, + 321, + 130 + ], + "score": 0.89, + "content": "w \\in \\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 115, + 360, + 131 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 361, + 117, + 439, + 128 + ], + "score": 0.9, + "content": "i \\sim k \\sim w \\sim j \\sim i", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 115, + 457, + 131 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 457, + 117, + 464, + 127 + ], + "score": 0.37, + "content": "^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 115, + 491, + 131 + ], + "score": 1.0, + "content": "-cycle.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 81, + 506, + 131 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 137, + 504, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "We now end the discussion about 4-cycle contributions by proving the following inequality, which", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 435, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 295, + 162 + ], + "score": 1.0, + "content": "allows us to avoid to compute directly the term", + "type": "text" + }, + { + "bbox": [ + 296, + 149, + 308, + 162 + ], + "score": 0.83, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 148, + 435, + 162 + ], + "score": 1.0, + "content": "up to giving up some accuracy.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 106, + 138, + 505, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 272, + 176 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 272, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 213, + 176 + ], + "score": 1.0, + "content": "Lemma 11. For any edge", + "type": "text" + }, + { + "bbox": [ + 214, + 164, + 236, + 176 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 163, + 272, + 176 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 163, + 272, + 176 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 181, + 357, + 210 + ], + "lines": [ + { + "bbox": [ + 254, + 181, + 357, + 210 + ], + "spans": [ + { + "bbox": [ + 254, + 181, + 357, + 210 + ], + "score": 0.94, + "content": "| \\sharp _ { \\sf H } ^ { \\mathfrak { m } } | \\geq \\frac { \\operatorname* { m a x } \\{ | \\sharp _ { \\sf H } ^ { i } | , | \\sharp _ { \\sf H } ^ { j } | \\} } { \\gamma _ { m a x } } .", + "type": "interline_equation", + "image_path": "89bea760cd16cc49e7087d599acf6b3ca969b8894bfa47efa2d19f0a2c986f53.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 254, + 181, + 357, + 195.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 254, + 195.5, + 357, + 210.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 364, + 237 + ], + "score": 1.0, + "content": "Proof. The proof is based on a combinatorial argument. Let", + "type": "text" + }, + { + "bbox": [ + 365, + 222, + 445, + 234 + ], + "score": 0.87, + "content": "\\sharp _ { \\sqcap } ^ { i } = \\{ k _ { 1 } , \\ldots , k _ { r } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 220, + 479, + 237 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 479, + 223, + 505, + 235 + ], + "score": 0.5, + "content": "\\sharp _ { \\bigcirc } ^ { \\mathfrak { m } } =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 232, + 504, + 247 + ], + "spans": [ + { + "bbox": [ + 107, + 234, + 158, + 246 + ], + "score": 0.91, + "content": "\\{ k _ { 1 } , \\ldots , k _ { \\ell } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 232, + 182, + 247 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 182, + 234, + 205, + 244 + ], + "score": 0.89, + "content": "\\ell < r", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 232, + 310, + 247 + ], + "score": 1.0, + "content": ". By definition there exists", + "type": "text" + }, + { + "bbox": [ + 310, + 234, + 446, + 246 + ], + "score": 0.91, + "content": "\\varphi : \\{ k _ { 1 } , \\dots , k _ { \\ell } \\} \\to \\{ w _ { 1 } , \\dots , w _ { \\ell } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 232, + 470, + 247 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 470, + 235, + 504, + 245 + ], + "score": 0.88, + "content": "k _ { i } \\sim w _ { i }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 123, + 260 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 246, + 158, + 259 + ], + "score": 0.9, + "content": "w _ { i } \\in \\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 246, + 175, + 260 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 176, + 246, + 216, + 258 + ], + "score": 0.91, + "content": "1 \\leq i \\leq \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 246, + 246, + 260 + ], + "score": 1.0, + "content": ". Given", + "type": "text" + }, + { + "bbox": [ + 246, + 246, + 336, + 259 + ], + "score": 0.92, + "content": "k \\in \\sharp _ { \\sqcap } ^ { i } \\setminus \\{ k _ { 1 } , . . . , k _ { \\ell } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 246, + 408, + 260 + ], + "score": 1.0, + "content": ", then there are no", + "type": "text" + }, + { + "bbox": [ + 409, + 246, + 504, + 259 + ], + "score": 0.9, + "content": "w \\in \\sharp _ { \\sqcap } ^ { j } \\setminus \\{ w _ { 1 } , . . . , w _ { \\ell } \\}", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 104, + 256, + 145, + 270 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 146, + 258, + 174, + 268 + ], + "score": 0.89, + "content": "k \\sim w", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 256, + 289, + 270 + ], + "score": 1.0, + "content": ", otherwise we could extend", + "type": "text" + }, + { + "bbox": [ + 290, + 260, + 298, + 269 + ], + "score": 0.81, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 256, + 342, + 270 + ], + "score": 1.0, + "content": "by setting", + "type": "text" + }, + { + "bbox": [ + 342, + 258, + 408, + 270 + ], + "score": 0.93, + "content": "k \\mapsto \\varphi ( k ) : = w", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "and we would then get", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 268, + 253, + 282 + ], + "spans": [ + { + "bbox": [ + 107, + 268, + 158, + 281 + ], + "score": 0.89, + "content": "| \\mathtt { H } _ { \\square } ^ { \\mathfrak { m } } | = \\ell + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 268, + 253, + 282 + ], + "score": 1.0, + "content": ". Accordingly, we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 220, + 506, + 282 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 286, + 388, + 321 + ], + "lines": [ + { + "bbox": [ + 220, + 286, + 388, + 321 + ], + "spans": [ + { + "bbox": [ + 220, + 286, + 388, + 321 + ], + "score": 0.93, + "content": "\\sum _ { s = 1 } ^ { \\ell } ( A _ { w _ { s } } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) ) - 1 ) \\geq | \\sharp _ { \\square } ^ { i } | ,", + "type": "interline_equation", + "image_path": "7529fcc278feaac0518c6fbd641d177105dffada53eefb337b64856a185dd038.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 220, + 286, + 388, + 303.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 220, + 303.5, + 388, + 321.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 164, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 165, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 165, + 339 + ], + "score": 1.0, + "content": "which implies", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 325, + 165, + 339 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 342, + 433, + 377 + ], + "lines": [ + { + "bbox": [ + 178, + 342, + 433, + 377 + ], + "spans": [ + { + "bbox": [ + 178, + 342, + 433, + 377 + ], + "score": 0.94, + "content": "\\gamma _ { \\operatorname* { m a x } } | \\sharp _ { \\sf D } ^ { \\mathrm { m } } | \\equiv \\gamma _ { \\operatorname* { m a x } } \\ell \\geq \\sum _ { s = 1 } ^ { \\ell } ( A _ { w _ { s } } \\cdot ( A _ { i } - A _ { j } \\odot A _ { i } ) ) - 1 ) \\geq | \\sharp _ { \\sf D } ^ { i } | .", + "type": "interline_equation", + "image_path": "06d40a0b2b39caa88d2abd80ab697f95b6e82628d65357f4ea7e0b12493ef0c2.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 178, + 342, + 433, + 353.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 178, + 353.6666666666667, + 433, + 365.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 178, + 365.33333333333337, + 433, + 377.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 410, + 312, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 312, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 312, + 425 + ], + "score": 1.0, + "content": "C EXISTING CURVATURE CANDIDATES", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 501, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 502, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 233, + 450 + ], + "score": 1.0, + "content": "Ollivier Ricci curvature For", + "type": "text" + }, + { + "bbox": [ + 234, + 435, + 258, + 446 + ], + "score": 0.89, + "content": "i \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 433, + 276, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 277, + 435, + 317, + 447 + ], + "score": 0.95, + "content": "\\alpha \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 433, + 461, + 450 + ], + "score": 1.0, + "content": "we define a probability measure on", + "type": "text" + }, + { + "bbox": [ + 461, + 435, + 485, + 447 + ], + "score": 0.93, + "content": "B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 433, + 502, + 450 + ], + "score": 1.0, + "content": "by:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 433, + 502, + 450 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 452, + 369, + 498 + ], + "lines": [ + { + "bbox": [ + 240, + 452, + 369, + 498 + ], + "spans": [ + { + "bbox": [ + 240, + 452, + 369, + 498 + ], + "score": 0.94, + "content": "\\mu _ { i } ^ { \\alpha } : j \\mapsto \\left\\{ \\begin{array} { r l } & { \\alpha , \\ j = i } \\\\ & { \\frac { 1 - \\alpha } { d _ { i } } , \\ j \\in S _ { 1 } ( i ) , } \\\\ & { 0 , \\ \\mathrm { o t h e r w i s e } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "7aa9e425bee708dea94fa37d93747a16569eddd36c9e02db4d32c03b031f412e.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 452, + 369, + 475.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 240, + 475.0, + 369, + 498.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 504, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "Before we introduce the Ollivier curvature, we recall that the transportation distance between two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 514, + 383, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 383, + 527 + ], + "score": 1.0, + "content": "finitely supported probability measures as above can be computed as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 502, + 505, + 527 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 530, + 407, + 559 + ], + "lines": [ + { + "bbox": [ + 204, + 530, + 407, + 559 + ], + "spans": [ + { + "bbox": [ + 204, + 530, + 407, + 559 + ], + "score": 0.93, + "content": "W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) : = \\operatorname* { i n f } _ { M } \\sum _ { k \\in S _ { 1 } ( i ) } \\sum _ { w \\in S _ { 1 } ( j ) } M _ { k w } d _ { G } ( k , w ) ,", + "type": "interline_equation", + "image_path": "b963842df11bb6006c9305daafc642e9a7e950e52d21e8a67631e39dd52be7d7.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 530, + 407, + 544.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 204, + 544.5, + 407, + 559.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 503, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 133, + 578 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 565, + 164, + 578 + ], + "score": 0.92, + "content": "d _ { G } ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 565, + 492, + 578 + ], + "score": 1.0, + "content": "is the geodesic distance on the graph and the infimum is taken over all matrices", + "type": "text" + }, + { + "bbox": [ + 492, + 566, + 504, + 576 + ], + "score": 0.74, + "content": "M", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 577, + 249, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 249, + 589 + ], + "score": 1.0, + "content": "satisfying the marginal constraints:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 565, + 504, + 589 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 593, + 405, + 622 + ], + "lines": [ + { + "bbox": [ + 205, + 593, + 405, + 622 + ], + "spans": [ + { + "bbox": [ + 205, + 593, + 405, + 622 + ], + "score": 0.93, + "content": "\\sum _ { k \\in S _ { 1 } ( i ) } M _ { k w } = \\mu _ { j } ^ { \\alpha } ( w ) , \\quad \\sum _ { w \\in S _ { 1 } ( j ) } M _ { k w } = \\mu _ { i } ^ { \\alpha } ( k ) .", + "type": "interline_equation", + "image_path": "35bd5ec6b11dc1457e6daa1ef3b68652590ec136dc5d29813fb7a55965c7d11b.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 593, + 405, + 607.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 205, + 607.5, + 405, + 622.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "list", + "bbox": [ + 107, + 627, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 507, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 507, + 640 + ], + "score": 1.0, + "content": "We are now ready to define the Ollivier Ricci curvature: the formulation below is due to Lin et al.", + "type": "text" + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 637, + 139, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 139, + 652 + ], + "score": 1.0, + "content": "(2011).", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 626, + 507, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 363, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 363, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 190, + 668 + ], + "score": 1.0, + "content": "Definition 5. Given", + "type": "text" + }, + { + "bbox": [ + 190, + 654, + 213, + 665 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 651, + 268, + 668 + ], + "score": 1.0, + "content": "we define the", + "type": "text" + }, + { + "bbox": [ + 268, + 655, + 276, + 663 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 651, + 363, + 668 + ], + "score": 1.0, + "content": "-Ollivier curvature by", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 651, + 363, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 670, + 365, + 685 + ], + "lines": [ + { + "bbox": [ + 246, + 670, + 365, + 685 + ], + "spans": [ + { + "bbox": [ + 246, + 670, + 365, + 685 + ], + "score": 0.91, + "content": "\\kappa _ { \\alpha } ( i , j ) : = 1 - W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) .", + "type": "interline_equation", + "image_path": "d72597de2d5c7786cc275b8f8987765a0f4d37b4fd975b08b5e5d71dc553778a.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 246, + 670, + 365, + 685 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 691, + 433, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 433, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 130, + 705 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 691, + 185, + 704 + ], + "score": 0.94, + "content": "\\kappa _ { \\alpha } ( 1 - \\alpha ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 691, + 433, + 705 + ], + "score": 1.0, + "content": "is increasing and bounded the quantity below is well-defined:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 691, + 433, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 709, + 373, + 735 + ], + "lines": [ + { + "bbox": [ + 238, + 709, + 373, + 735 + ], + "spans": [ + { + "bbox": [ + 238, + 709, + 373, + 735 + ], + "score": 0.95, + "content": "\\kappa ( i , j ) : = \\operatorname* { l i m } _ { \\alpha \\to 1 } \\frac { 1 - W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) } { 1 - \\alpha } .", + "type": "interline_equation", + "image_path": "ba3d51ae9dcbbbd72f81491e29b7667fb9288933963bf5170264c57c58dc6e86.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 238, + 709, + 373, + 735 + ], + "spans": [], + "index": 37 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Forman Ricci curvature In the following we report a formula for the augmented Forman Ricci", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 504, + 105 + ], + "score": 1.0, + "content": "curvature on unweighted graphs Samal et al. (2018). We also note that Forman curvature on graphs", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 372, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 372, + 117 + ], + "score": 1.0, + "content": "has also been studied in Sreejith et al. (2016); Weber et al. (2018).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 105, + 118, + 423, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 423, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 218, + 132 + ], + "score": 1.0, + "content": "Definition 6. For any edge", + "type": "text" + }, + { + "bbox": [ + 218, + 119, + 241, + 131 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 117, + 423, + 132 + ], + "score": 1.0, + "content": "the augmented Forman curvature is given by", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 134, + 370, + 148 + ], + "lines": [ + { + "bbox": [ + 240, + 134, + 370, + 148 + ], + "spans": [ + { + "bbox": [ + 240, + 134, + 370, + 148 + ], + "score": 0.9, + "content": "F ( i , j ) : = 4 - d _ { i } - d _ { j } + 3 | \\sharp _ { \\Delta } | .", + "type": "interline_equation", + "image_path": "d510b530852ebf41036a2503b8578df955ab7d843941af1c5dbc3992b109b14a.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 240, + 134, + 370, + 148 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 157, + 507, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 507, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 507, + 171 + ], + "score": 1.0, + "content": "We note that such formulation of curvature does not distinguish contributions coming from 4-cycles.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 507, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 279, + 182 + ], + "score": 1.0, + "content": "In fact, for the orthogonal grid with degree", + "type": "text" + }, + { + "bbox": [ + 280, + 169, + 304, + 180 + ], + "score": 0.91, + "content": "d \\geq 4", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 168, + 448, + 182 + ], + "score": 1.0, + "content": ", Forman Ricci curvature is equal to", + "type": "text" + }, + { + "bbox": [ + 449, + 168, + 503, + 181 + ], + "score": 0.92, + "content": "2 ( 2 - d ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 168, + 507, + 182 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 481, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 222, + 192 + ], + "score": 1.0, + "content": "This does not reflect that the", + "type": "text" + }, + { + "bbox": [ + 222, + 181, + 228, + 189 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 178, + 469, + 192 + ], + "score": 1.0, + "content": "-hop neighbourhood for such a graph grows polynomially in", + "type": "text" + }, + { + "bbox": [ + 470, + 182, + 475, + 189 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 178, + 481, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 504, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "We conclude this appendix by reporting a lower bound for the Ollivier Ricci curvature derived in Jost", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 408, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 222, + 220 + ], + "score": 1.0, + "content": "& Liu (2014). We recall that", + "type": "text" + }, + { + "bbox": [ + 222, + 209, + 235, + 218 + ], + "score": 0.86, + "content": "\\kappa _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 207, + 259, + 220 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 259, + 207, + 299, + 219 + ], + "score": 0.93, + "content": "\\alpha \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 207, + 408, + 220 + ], + "score": 1.0, + "content": "was defined in equation 7.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 221, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 507, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 297, + 236 + ], + "score": 1.0, + "content": "Theorem 12 (Jost & Liu (2014)). For any edge", + "type": "text" + }, + { + "bbox": [ + 298, + 222, + 320, + 233 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 219, + 343, + 236 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 343, + 221, + 375, + 234 + ], + "score": 0.91, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 219, + 507, + 236 + ], + "score": 1.0, + "content": ", the following bound is satisfied:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 237, + 482, + 267 + ], + "lines": [ + { + "bbox": [ + 128, + 237, + 482, + 267 + ], + "spans": [ + { + "bbox": [ + 128, + 237, + 482, + 267 + ], + "score": 0.92, + "content": "\\kappa _ { 0 } ( i , j ) \\geq \\Phi ( i , j ) : = - \\left( 1 - \\frac { 1 } { d _ { i } } - \\frac { 1 } { d _ { j } } - \\frac { | \\sharp \\Delta | } { d _ { j } } \\right) _ { + } - \\left( 1 - \\frac { 1 } { d _ { i } } - \\frac { 1 } { d _ { j } } - \\frac { | \\sharp \\Delta | } { d _ { i } } \\right) _ { + } + \\frac { | \\sharp \\Delta | } { d _ { j } } .", + "type": "interline_equation", + "image_path": "0d69c532e3bea465e7a4b6047e49b3cf035cbc5fa2f750cd592052a967f6aab1.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 128, + 237, + 482, + 247.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 128, + 247.0, + 482, + 257.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 128, + 257.0, + 482, + 267.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 278, + 307, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 307, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 307, + 293 + ], + "score": 1.0, + "content": "D PROOFS OF RESULTS IN SECTION 3", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 306, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 307, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 307, + 316 + ], + "score": 1.0, + "content": "We first recall our definition of Balanced Forman:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 108, + 317, + 457, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 457, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 218, + 331 + ], + "score": 1.0, + "content": "Definition 7. For any edge", + "type": "text" + }, + { + "bbox": [ + 218, + 318, + 241, + 329 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 316, + 267, + 331 + ], + "score": 1.0, + "content": "we let", + "type": "text" + }, + { + "bbox": [ + 268, + 317, + 303, + 330 + ], + "score": 0.73, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 316, + 338, + 331 + ], + "score": 1.0, + "content": "be zero", + "type": "text" + }, + { + "bbox": [ + 339, + 317, + 412, + 330 + ], + "score": 0.85, + "content": "\\gamma _ { \\mathrm { m i n } \\{ d _ { i } , d _ { j } \\} } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 316, + 457, + 331 + ], + "score": 1.0, + "content": ", otherwise", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 334, + 478, + 362 + ], + "lines": [ + { + "bbox": [ + 121, + 334, + 478, + 362 + ], + "spans": [ + { + "bbox": [ + 121, + 334, + 478, + 362 + ], + "score": 0.92, + "content": "\\mathbf { R i c } ( i , j ) : = \\frac { 2 } { d _ { i } } + \\frac { 2 } { d _ { j } } - 2 + 2 \\frac { \\lvert \\sharp _ { \\Delta } \\rvert } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\lvert \\sharp _ { \\Delta } \\rvert } { \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} } + \\frac { ( \\gamma _ { m a x } ) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\big ( \\lvert \\sharp _ { \\Xi } \\rvert + \\lvert \\sharp _ { \\Xi } ^ { j } \\rvert \\big )", + "type": "interline_equation", + "image_path": "ae55a567f24bbc66bb7a7a817a24f99dc2cb5733345957f274fe4c02e08617fd.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 121, + 334, + 478, + 362 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 374, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 375, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 249, + 381 + ], + "score": 1.0, + "content": "where the last term is set to be zero", + "type": "text" + }, + { + "bbox": [ + 250, + 366, + 275, + 380 + ], + "score": 0.84, + "content": "i f | \\sharp _ { \\bigstar } ^ { i } |", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 365, + 322, + 381 + ], + "score": 1.0, + "content": "(and hence", + "type": "text" + }, + { + "bbox": [ + 323, + 365, + 342, + 380 + ], + "score": 0.69, + "content": "| \\sharp _ { \\bigstar } ^ { j } | \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 365, + 375, + 381 + ], + "score": 1.0, + "content": "is zero.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 425, + 401 + ], + "score": 1.0, + "content": "We also extend the previous definition to the weighted case. In this setting we let", + "type": "text" + }, + { + "bbox": [ + 426, + 388, + 485, + 400 + ], + "score": 0.92, + "content": "G = ( V , E , \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 386, + 505, + 401 + ], + "score": 1.0, + "content": "be a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 399, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 504, + 410 + ], + "score": 1.0, + "content": "simple, locally finite, undirected graph with normalized weights. We first report the formula for the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 352, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 352, + 423 + ], + "score": 1.0, + "content": "augmented Forman in the weighted case Samal et al. (2018):", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 424, + 432, + 494 + ], + "lines": [ + { + "bbox": [ + 178, + 424, + 432, + 494 + ], + "spans": [ + { + "bbox": [ + 178, + 424, + 432, + 494 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle F ( i , j ) = \\omega ( i ) + \\omega ( j ) + \\sum _ { k \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) } \\frac { \\omega _ { i j } ^ { 2 } } { \\omega _ { \\Delta } } } \\\\ & { \\displaystyle - \\sum _ { k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j ) } \\omega ( i ) \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { i k } } } - \\sum _ { k \\in S _ { 1 } ( j ) \\setminus S _ { 1 } ( i ) } \\omega ( j ) \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { j k } } } , } \\end{array}", + "type": "interline_equation", + "image_path": "f2fb232c0dd5c8fb2a4d8dbeb502734a366dccd7f700a1436375a62412a32e68.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 178, + 424, + 432, + 447.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 178, + 447.3333333333333, + 432, + 470.66666666666663 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 178, + 470.66666666666663, + 432, + 493.99999999999994 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 134, + 509 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 499, + 149, + 508 + ], + "score": 0.84, + "content": "\\omega _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 497, + 425, + 509 + ], + "score": 1.0, + "content": "is taken to be the Heron formula for the area of a triangle while", + "type": "text" + }, + { + "bbox": [ + 426, + 497, + 444, + 509 + ], + "score": 0.9, + "content": "\\omega ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "denotes some", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "weighting scheme for the nodes as well. We propose a similar definition for the weighted case, which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 427, + 531 + ], + "score": 1.0, + "content": "reduces to the one discussed above in the combinatorial setting. We recall that", + "type": "text" + }, + { + "bbox": [ + 428, + 519, + 439, + 528 + ], + "score": 0.76, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "is the weighted", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 205, + 543 + ], + "score": 1.0, + "content": "adjacency matrix while", + "type": "text" + }, + { + "bbox": [ + 206, + 530, + 214, + 540 + ], + "score": 0.81, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 529, + 407, + 543 + ], + "score": 1.0, + "content": "is the combinatorial one. Moreover, we write", + "type": "text" + }, + { + "bbox": [ + 408, + 529, + 486, + 543 + ], + "score": 0.93, + "content": "A _ { j } ^ { i } = ( A _ { j } - A _ { i } ) _ { + } ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 541, + 178, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 158, + 558 + ], + "score": 1.0, + "content": "similarly for", + "type": "text" + }, + { + "bbox": [ + 159, + 542, + 173, + 557 + ], + "score": 0.91, + "content": "W _ { j } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 541, + 178, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 104, + 558, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 104, + 556, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 285, + 572 + ], + "score": 1.0, + "content": "Definition 8. For any pair of adjacent nodes", + "type": "text" + }, + { + "bbox": [ + 285, + 559, + 319, + 570 + ], + "score": 0.89, + "content": "i , j \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 556, + 359, + 572 + ], + "score": 1.0, + "content": "we define", + "type": "text" + }, + { + "bbox": [ + 359, + 558, + 394, + 570 + ], + "score": 0.77, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 556, + 417, + 572 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 417, + 558, + 505, + 571 + ], + "score": 0.85, + "content": "0 i f \\operatorname* { m i n } \\{ | d _ { i } | , | d _ { j } | \\} =", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 185, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 185, + 582 + ], + "score": 1.0, + "content": "1, otherwise we set", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 583, + 433, + 731 + ], + "lines": [ + { + "bbox": [ + 177, + 583, + 433, + 731 + ], + "spans": [ + { + "bbox": [ + 177, + 583, + 433, + 731 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathrm { R i c } ( i , j ) : = \\cfrac { 1 } { d _ { i } } \\left( 1 - \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } } { \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { i k } } } } \\right) + \\cfrac { 1 } { d _ { j } } \\left( 1 - \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } } { \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { j k } } } } \\right) } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\cfrac { 1 } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\displaystyle \\sum _ { k \\in \\mathcal { S } _ { i } ( i ) \\cap \\mathcal { S } ( i , j ) } \\frac { \\omega _ { i j } ^ { 2 } } { \\omega \\Delta } } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } ^ { * } } \\frac { \\omega _ { i j } } { \\omega | } \\left( \\frac { \\omega _ { i j } \\left( \\gamma _ { 0 n x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } - \\frac { \\sqrt { \\omega _ { i j } \\nu _ { \\perp } } } { d _ { i } } \\right) } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } ^ { * } } \\frac { \\omega _ { i j } } { \\omega | } \\left( \\frac { \\omega _ { i j } \\left( \\gamma _ { 0 n x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } - \\frac { \\sqrt { \\omega _ { i j } \\nu _ { \\perp } } } { d _ { j } } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "982772dcd4c03c70e55182cd9cf60af68d507c07bcb95fa71ad62ffab113eb0e.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 177, + 583, + 433, + 632.3333333333334 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 177, + 632.3333333333334, + 433, + 681.6666666666667 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 177, + 681.6666666666667, + 433, + 731.0000000000001 + ], + "spans": [], + "index": 34 + } + ] + } + ], + "page_idx": 17, + "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, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Forman Ricci curvature In the following we report a formula for the augmented Forman Ricci", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 504, + 105 + ], + "score": 1.0, + "content": "curvature on unweighted graphs Samal et al. (2018). We also note that Forman curvature on graphs", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 372, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 372, + 117 + ], + "score": 1.0, + "content": "has also been studied in Sreejith et al. (2016); Weber et al. (2018).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 118, + 423, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 423, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 218, + 132 + ], + "score": 1.0, + "content": "Definition 6. For any edge", + "type": "text" + }, + { + "bbox": [ + 218, + 119, + 241, + 131 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 117, + 423, + 132 + ], + "score": 1.0, + "content": "the augmented Forman curvature is given by", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 117, + 423, + 132 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 134, + 370, + 148 + ], + "lines": [ + { + "bbox": [ + 240, + 134, + 370, + 148 + ], + "spans": [ + { + "bbox": [ + 240, + 134, + 370, + 148 + ], + "score": 0.9, + "content": "F ( i , j ) : = 4 - d _ { i } - d _ { j } + 3 | \\sharp _ { \\Delta } | .", + "type": "interline_equation", + "image_path": "d510b530852ebf41036a2503b8578df955ab7d843941af1c5dbc3992b109b14a.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 240, + 134, + 370, + 148 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "list", + "bbox": [ + 106, + 157, + 507, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 507, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 507, + 171 + ], + "score": 1.0, + "content": "We note that such formulation of curvature does not distinguish contributions coming from 4-cycles.", + "type": "text" + } + ], + "index": 5, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 168, + 507, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 279, + 182 + ], + "score": 1.0, + "content": "In fact, for the orthogonal grid with degree", + "type": "text" + }, + { + "bbox": [ + 280, + 169, + 304, + 180 + ], + "score": 0.91, + "content": "d \\geq 4", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 168, + 448, + 182 + ], + "score": 1.0, + "content": ", Forman Ricci curvature is equal to", + "type": "text" + }, + { + "bbox": [ + 449, + 168, + 503, + 181 + ], + "score": 0.92, + "content": "2 ( 2 - d ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 168, + 507, + 182 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 178, + 481, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 222, + 192 + ], + "score": 1.0, + "content": "This does not reflect that the", + "type": "text" + }, + { + "bbox": [ + 222, + 181, + 228, + 189 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 178, + 469, + 192 + ], + "score": 1.0, + "content": "-hop neighbourhood for such a graph grows polynomially in", + "type": "text" + }, + { + "bbox": [ + 470, + 182, + 475, + 189 + ], + "score": 0.77, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 178, + 481, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 6, + "bbox_fs": [ + 105, + 156, + 507, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 504, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "We conclude this appendix by reporting a lower bound for the Ollivier Ricci curvature derived in Jost", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 408, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 222, + 220 + ], + "score": 1.0, + "content": "& Liu (2014). We recall that", + "type": "text" + }, + { + "bbox": [ + 222, + 209, + 235, + 218 + ], + "score": 0.86, + "content": "\\kappa _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 207, + 259, + 220 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 259, + 207, + 299, + 219 + ], + "score": 0.93, + "content": "\\alpha \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 207, + 408, + 220 + ], + "score": 1.0, + "content": "was defined in equation 7.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 196, + 506, + 220 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 221, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 507, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 297, + 236 + ], + "score": 1.0, + "content": "Theorem 12 (Jost & Liu (2014)). For any edge", + "type": "text" + }, + { + "bbox": [ + 298, + 222, + 320, + 233 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 219, + 343, + 236 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 343, + 221, + 375, + 234 + ], + "score": 0.91, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 219, + 507, + 236 + ], + "score": 1.0, + "content": ", the following bound is satisfied:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 219, + 507, + 236 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 237, + 482, + 267 + ], + "lines": [ + { + "bbox": [ + 128, + 237, + 482, + 267 + ], + "spans": [ + { + "bbox": [ + 128, + 237, + 482, + 267 + ], + "score": 0.92, + "content": "\\kappa _ { 0 } ( i , j ) \\geq \\Phi ( i , j ) : = - \\left( 1 - \\frac { 1 } { d _ { i } } - \\frac { 1 } { d _ { j } } - \\frac { | \\sharp \\Delta | } { d _ { j } } \\right) _ { + } - \\left( 1 - \\frac { 1 } { d _ { i } } - \\frac { 1 } { d _ { j } } - \\frac { | \\sharp \\Delta | } { d _ { i } } \\right) _ { + } + \\frac { | \\sharp \\Delta | } { d _ { j } } .", + "type": "interline_equation", + "image_path": "0d69c532e3bea465e7a4b6047e49b3cf035cbc5fa2f750cd592052a967f6aab1.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 128, + 237, + 482, + 247.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 128, + 247.0, + 482, + 257.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 128, + 257.0, + 482, + 267.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 278, + 307, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 307, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 307, + 293 + ], + "score": 1.0, + "content": "D PROOFS OF RESULTS IN SECTION 3", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 306, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 307, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 307, + 316 + ], + "score": 1.0, + "content": "We first recall our definition of Balanced Forman:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 303, + 307, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 317, + 457, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 457, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 218, + 331 + ], + "score": 1.0, + "content": "Definition 7. For any edge", + "type": "text" + }, + { + "bbox": [ + 218, + 318, + 241, + 329 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 316, + 267, + 331 + ], + "score": 1.0, + "content": "we let", + "type": "text" + }, + { + "bbox": [ + 268, + 317, + 303, + 330 + ], + "score": 0.73, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 316, + 338, + 331 + ], + "score": 1.0, + "content": "be zero", + "type": "text" + }, + { + "bbox": [ + 339, + 317, + 412, + 330 + ], + "score": 0.85, + "content": "\\gamma _ { \\mathrm { m i n } \\{ d _ { i } , d _ { j } \\} } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 316, + 457, + 331 + ], + "score": 1.0, + "content": ", otherwise", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 316, + 457, + 331 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 334, + 478, + 362 + ], + "lines": [ + { + "bbox": [ + 121, + 334, + 478, + 362 + ], + "spans": [ + { + "bbox": [ + 121, + 334, + 478, + 362 + ], + "score": 0.92, + "content": "\\mathbf { R i c } ( i , j ) : = \\frac { 2 } { d _ { i } } + \\frac { 2 } { d _ { j } } - 2 + 2 \\frac { \\lvert \\sharp _ { \\Delta } \\rvert } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } + \\frac { \\lvert \\sharp _ { \\Delta } \\rvert } { \\operatorname* { m i n } \\{ d _ { i } , d _ { j } \\} } + \\frac { ( \\gamma _ { m a x } ) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\big ( \\lvert \\sharp _ { \\Xi } \\rvert + \\lvert \\sharp _ { \\Xi } ^ { j } \\rvert \\big )", + "type": "interline_equation", + "image_path": "ae55a567f24bbc66bb7a7a817a24f99dc2cb5733345957f274fe4c02e08617fd.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 121, + 334, + 478, + 362 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 374, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 375, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 249, + 381 + ], + "score": 1.0, + "content": "where the last term is set to be zero", + "type": "text" + }, + { + "bbox": [ + 250, + 366, + 275, + 380 + ], + "score": 0.84, + "content": "i f | \\sharp _ { \\bigstar } ^ { i } |", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 365, + 322, + 381 + ], + "score": 1.0, + "content": "(and hence", + "type": "text" + }, + { + "bbox": [ + 323, + 365, + 342, + 380 + ], + "score": 0.69, + "content": "| \\sharp _ { \\bigstar } ^ { j } | \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 365, + 375, + 381 + ], + "score": 1.0, + "content": "is zero.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 365, + 375, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 387, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 425, + 401 + ], + "score": 1.0, + "content": "We also extend the previous definition to the weighted case. In this setting we let", + "type": "text" + }, + { + "bbox": [ + 426, + 388, + 485, + 400 + ], + "score": 0.92, + "content": "G = ( V , E , \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 386, + 505, + 401 + ], + "score": 1.0, + "content": "be a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 399, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 504, + 410 + ], + "score": 1.0, + "content": "simple, locally finite, undirected graph with normalized weights. We first report the formula for the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 352, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 352, + 423 + ], + "score": 1.0, + "content": "augmented Forman in the weighted case Samal et al. (2018):", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 386, + 505, + 423 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 424, + 432, + 494 + ], + "lines": [ + { + "bbox": [ + 178, + 424, + 432, + 494 + ], + "spans": [ + { + "bbox": [ + 178, + 424, + 432, + 494 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle F ( i , j ) = \\omega ( i ) + \\omega ( j ) + \\sum _ { k \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) } \\frac { \\omega _ { i j } ^ { 2 } } { \\omega _ { \\Delta } } } \\\\ & { \\displaystyle - \\sum _ { k \\in S _ { 1 } ( i ) \\setminus S _ { 1 } ( j ) } \\omega ( i ) \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { i k } } } - \\sum _ { k \\in S _ { 1 } ( j ) \\setminus S _ { 1 } ( i ) } \\omega ( j ) \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { j k } } } , } \\end{array}", + "type": "interline_equation", + "image_path": "f2fb232c0dd5c8fb2a4d8dbeb502734a366dccd7f700a1436375a62412a32e68.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 178, + 424, + 432, + 447.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 178, + 447.3333333333333, + 432, + 470.66666666666663 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 178, + 470.66666666666663, + 432, + 493.99999999999994 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 134, + 509 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 499, + 149, + 508 + ], + "score": 0.84, + "content": "\\omega _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 497, + 425, + 509 + ], + "score": 1.0, + "content": "is taken to be the Heron formula for the area of a triangle while", + "type": "text" + }, + { + "bbox": [ + 426, + 497, + 444, + 509 + ], + "score": 0.9, + "content": "\\omega ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "denotes some", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "weighting scheme for the nodes as well. We propose a similar definition for the weighted case, which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 427, + 531 + ], + "score": 1.0, + "content": "reduces to the one discussed above in the combinatorial setting. We recall that", + "type": "text" + }, + { + "bbox": [ + 428, + 519, + 439, + 528 + ], + "score": 0.76, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "is the weighted", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 205, + 543 + ], + "score": 1.0, + "content": "adjacency matrix while", + "type": "text" + }, + { + "bbox": [ + 206, + 530, + 214, + 540 + ], + "score": 0.81, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 529, + 407, + 543 + ], + "score": 1.0, + "content": "is the combinatorial one. Moreover, we write", + "type": "text" + }, + { + "bbox": [ + 408, + 529, + 486, + 543 + ], + "score": 0.93, + "content": "A _ { j } ^ { i } = ( A _ { j } - A _ { i } ) _ { + } ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 541, + 178, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 158, + 558 + ], + "score": 1.0, + "content": "similarly for", + "type": "text" + }, + { + "bbox": [ + 159, + 542, + 173, + 557 + ], + "score": 0.91, + "content": "W _ { j } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 541, + 178, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 497, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 558, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 104, + 556, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 285, + 572 + ], + "score": 1.0, + "content": "Definition 8. For any pair of adjacent nodes", + "type": "text" + }, + { + "bbox": [ + 285, + 559, + 319, + 570 + ], + "score": 0.89, + "content": "i , j \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 556, + 359, + 572 + ], + "score": 1.0, + "content": "we define", + "type": "text" + }, + { + "bbox": [ + 359, + 558, + 394, + 570 + ], + "score": 0.77, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 556, + 417, + 572 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 417, + 558, + 505, + 571 + ], + "score": 0.85, + "content": "0 i f \\operatorname* { m i n } \\{ | d _ { i } | , | d _ { j } | \\} =", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 185, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 185, + 582 + ], + "score": 1.0, + "content": "1, otherwise we set", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 556, + 505, + 582 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 583, + 433, + 731 + ], + "lines": [ + { + "bbox": [ + 177, + 583, + 433, + 731 + ], + "spans": [ + { + "bbox": [ + 177, + 583, + 433, + 731 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathrm { R i c } ( i , j ) : = \\cfrac { 1 } { d _ { i } } \\left( 1 - \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } } { \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { i k } } } } \\right) + \\cfrac { 1 } { d _ { j } } \\left( 1 - \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } } { \\sqrt { \\frac { \\omega _ { i j } } { \\omega _ { j k } } } } \\right) } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\cfrac { 1 } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\displaystyle \\sum _ { k \\in \\mathcal { S } _ { i } ( i ) \\cap \\mathcal { S } ( i , j ) } \\frac { \\omega _ { i j } ^ { 2 } } { \\omega \\Delta } } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } ^ { * } } \\frac { \\omega _ { i j } } { \\omega | } \\left( \\frac { \\omega _ { i j } \\left( \\gamma _ { 0 n x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } - \\frac { \\sqrt { \\omega _ { i j } \\nu _ { \\perp } } } { d _ { i } } \\right) } \\\\ & { \\phantom { \\frac { 1 } { d _ { i } } } + \\displaystyle \\sum _ { k \\in \\mathcal { Q } _ { i } ^ { * } } \\frac { \\omega _ { i j } } { \\omega | } \\left( \\frac { \\omega _ { i j } \\left( \\gamma _ { 0 n x } \\right) ^ { - 1 } } { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } - \\frac { \\sqrt { \\omega _ { i j } \\nu _ { \\perp } } } { d _ { j } } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "982772dcd4c03c70e55182cd9cf60af68d507c07bcb95fa71ad62ffab113eb0e.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 177, + 583, + 433, + 632.3333333333334 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 177, + 632.3333333333334, + 433, + 681.6666666666667 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 177, + 681.6666666666667, + 433, + 731.0000000000001 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 126, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 126, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 126, + 94 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 91, + 405, + 114 + ], + "lines": [ + { + "bbox": [ + 204, + 91, + 405, + 114 + ], + "spans": [ + { + "bbox": [ + 204, + 91, + 405, + 114 + ], + "score": 0.92, + "content": "\\omega _ { \\Delta } : = \\frac { 1 } { 3 } ( \\omega _ { i j } ^ { 2 } + \\omega _ { i k } ^ { 2 } + \\omega _ { j k } ^ { 2 } ) , \\quad z \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) ,", + "type": "interline_equation", + "image_path": "9fd6963393dcd31feb8512f49fd9ace7dba0d35f905f17633364c28e9d78bda1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 204, + 91, + 405, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 204, + 130 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 201, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 172, + 133 + ], + "score": 1.0, + "content": "and, for a given", + "type": "text" + }, + { + "bbox": [ + 172, + 118, + 201, + 131 + ], + "score": 0.94, + "content": "k \\in \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 136, + 428, + 170 + ], + "lines": [ + { + "bbox": [ + 182, + 136, + 428, + 170 + ], + "spans": [ + { + "bbox": [ + 182, + 136, + 428, + 170 + ], + "score": 0.93, + "content": "\\nu _ { \\perp } : = \\frac { W _ { z } \\cdot A _ { j } ^ { i } } { A _ { k } \\cdot A _ { j } ^ { i } } , \\omega _ { \\perp } = \\frac { 1 } { 4 } \\left( \\omega _ { i j } ^ { 2 } + \\omega _ { i k } ^ { 2 } + \\nu _ { \\perp } ^ { 2 } + \\frac { A _ { k } \\cdot W _ { j } ^ { i } } { A _ { k } \\cdot A _ { j } ^ { i } } \\right) .", + "type": "interline_equation", + "image_path": "c544e00ab6a8b4ce85ca00d37fe41c53a27ebaa8796c2b04c779cc76738a2564.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 182, + 136, + 428, + 147.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 182, + 147.33333333333334, + 428, + 158.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 182, + 158.66666666666669, + 428, + 170.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 465, + 195 + ], + "score": 1.0, + "content": "Important convention. Without losing generality, in the following we always assume that", + "type": "text" + }, + { + "bbox": [ + 466, + 181, + 505, + 192 + ], + "score": 0.9, + "content": "1 \\leq d _ { i } \\leq", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 192, + 117, + 204 + ], + "score": 0.87, + "content": "d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 191, + 209, + 205 + ], + "score": 1.0, + "content": ". In particular, we write", + "type": "text" + }, + { + "bbox": [ + 210, + 192, + 238, + 203 + ], + "score": 0.92, + "content": "d _ { i } \\doteq d", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 191, + 255, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 192, + 300, + 204 + ], + "score": 0.92, + "content": "d _ { j } = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 191, + 340, + 205 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 341, + 192, + 365, + 203 + ], + "score": 0.9, + "content": "s \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 191, + 480, + 205 + ], + "score": 1.0, + "content": ", omitting to specify that both", + "type": "text" + }, + { + "bbox": [ + 481, + 192, + 488, + 201 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 202, + 507, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 205, + 113, + 213 + ], + "score": 0.52, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 202, + 223, + 216 + ], + "score": 1.0, + "content": "are of course depending on", + "type": "text" + }, + { + "bbox": [ + 223, + 204, + 227, + 213 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 202, + 245, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 204, + 251, + 214 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 202, + 507, + 216 + ], + "score": 1.0, + "content": ". Moreover, from now on we only focus on the unweighted case.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 306, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 307, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 307, + 234 + ], + "score": 1.0, + "content": "We can now prove our main comparison theorem.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 104, + 239, + 474, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 475, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 273, + 255 + ], + "score": 1.0, + "content": "Theorem 2. Given an unweighted graph", + "type": "text" + }, + { + "bbox": [ + 273, + 241, + 282, + 251 + ], + "score": 0.74, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 238, + 338, + 255 + ], + "score": 1.0, + "content": ", for any edge", + "type": "text" + }, + { + "bbox": [ + 338, + 241, + 361, + 252 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 238, + 397, + 255 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 397, + 240, + 471, + 253 + ], + "score": 0.91, + "content": "\\kappa ( i , j ) \\geq \\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 238, + 475, + 255 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 309, + 279 + ], + "score": 1.0, + "content": "Proof. We stick to the aforementioned convention:", + "type": "text" + }, + { + "bbox": [ + 309, + 266, + 426, + 278 + ], + "score": 0.91, + "content": "d _ { i } : = d \\le d _ { j } = d + s , s \\ge 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 265, + 505, + 279 + ], + "score": 1.0, + "content": ". The strategy of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 411, + 290 + ], + "score": 1.0, + "content": "proof amounts to finding a transportation plan providing an upper bound for", + "type": "text" + }, + { + "bbox": [ + 411, + 277, + 461, + 290 + ], + "score": 0.95, + "content": "W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 236, + 302 + ], + "score": 1.0, + "content": "a lower bound for the curvature", + "type": "text" + }, + { + "bbox": [ + 236, + 289, + 263, + 301 + ], + "score": 0.93, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 289, + 469, + 302 + ], + "score": 1.0, + "content": ". In particular, we consider plans moving the mass", + "type": "text" + }, + { + "bbox": [ + 469, + 290, + 481, + 301 + ], + "score": 0.89, + "content": "\\mu _ { i } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 299, + 172, + 314 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 131, + 312 + ], + "score": 0.91, + "content": "B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 299, + 142, + 314 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 142, + 300, + 168, + 312 + ], + "score": 0.91, + "content": "B _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 299, + 172, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 115, + 331 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 317, + 140, + 327 + ], + "score": 0.93, + "content": "d = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 315, + 378, + 331 + ], + "score": 1.0, + "content": ", then the optimal transport plan consists of moving the mass", + "type": "text" + }, + { + "bbox": [ + 379, + 319, + 387, + 327 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 315, + 409, + 331 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 409, + 318, + 414, + 327 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 315, + 425, + 331 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 425, + 318, + 431, + 329 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 315, + 505, + 331 + ], + "score": 1.0, + "content": "and the remaining", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 128, + 342 + ], + "score": 1.0, + "content": "mass", + "type": "text" + }, + { + "bbox": [ + 128, + 329, + 153, + 339 + ], + "score": 0.87, + "content": "1 - \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 328, + 165, + 342 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 166, + 329, + 172, + 340 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 328, + 183, + 342 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 183, + 328, + 207, + 340 + ], + "score": 0.91, + "content": "\\bar { S } _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 328, + 399, + 342 + ], + "score": 1.0, + "content": ". This yields a unit Wasserstein distance between", + "type": "text" + }, + { + "bbox": [ + 399, + 329, + 412, + 340 + ], + "score": 0.9, + "content": "\\mu _ { i } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 328, + 430, + 342 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 430, + 329, + 443, + 341 + ], + "score": 0.88, + "content": "\\mu _ { j } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "and hence zero", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 458, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 180, + 354 + ], + "score": 1.0, + "content": "Ollivier curvature", + "type": "text" + }, + { + "bbox": [ + 180, + 341, + 206, + 353 + ], + "score": 0.92, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 339, + 418, + 354 + ], + "score": 1.0, + "content": ", which coincides with the value of balanced Forman", + "type": "text" + }, + { + "bbox": [ + 418, + 341, + 453, + 353 + ], + "score": 0.89, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 339, + 458, + 354 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 476, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 477, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 177, + 372 + ], + "score": 1.0, + "content": "Assume now that", + "type": "text" + }, + { + "bbox": [ + 178, + 358, + 202, + 369 + ], + "score": 0.9, + "content": "d \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 355, + 389, + 372 + ], + "score": 1.0, + "content": ". A (possibly non-optimal) transport plan from", + "type": "text" + }, + { + "bbox": [ + 389, + 358, + 402, + 370 + ], + "score": 0.89, + "content": "\\mu _ { i } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 355, + 413, + 372 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 414, + 358, + 426, + 371 + ], + "score": 0.89, + "content": "\\mu _ { j } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 355, + 477, + 372 + ], + "score": 1.0, + "content": "is given by:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 123, + 384, + 506, + 550 + ], + "lines": [ + { + "bbox": [ + 126, + 383, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 126, + 384, + 190, + 399 + ], + "score": 1.0, + "content": "(i) Move mass", + "type": "text" + }, + { + "bbox": [ + 190, + 385, + 255, + 397 + ], + "score": 0.93, + "content": "( 1 - \\alpha ) / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 384, + 319, + 399 + ], + "score": 1.0, + "content": "from each node", + "type": "text" + }, + { + "bbox": [ + 319, + 384, + 372, + 398 + ], + "score": 0.43, + "content": "k \\in \\sharp _ { \\bigstar } ^ { \\mathfrak { m } } \\subset \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 384, + 460, + 399 + ], + "score": 1.0, + "content": "to its unique image in", + "type": "text" + }, + { + "bbox": [ + 460, + 383, + 473, + 398 + ], + "score": 0.87, + "content": "\\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "under a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 394, + 282, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 180, + 411 + ], + "score": 1.0, + "content": "bijection", + "type": "text" + }, + { + "bbox": [ + 180, + 398, + 188, + 408 + ], + "score": 0.8, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 394, + 266, + 411 + ], + "score": 1.0, + "content": "as per definition of", + "type": "text" + }, + { + "bbox": [ + 266, + 396, + 278, + 408 + ], + "score": 0.83, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 394, + 282, + 411 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 124, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 124, + 415, + 279, + 429 + ], + "score": 1.0, + "content": "(ii) The remaining mass on each node", + "type": "text" + }, + { + "bbox": [ + 279, + 415, + 308, + 428 + ], + "score": 0.74, + "content": "k \\in \\sharp _ { \\bigstar } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 415, + 479, + 429 + ], + "score": 1.0, + "content": "will need to travel by at most distance 3 to", + "type": "text" + }, + { + "bbox": [ + 479, + 415, + 502, + 427 + ], + "score": 0.92, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 415, + 506, + 429 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 122, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 122, + 434, + 205, + 448 + ], + "score": 1.0, + "content": "(iii) The extra-mass", + "type": "text" + }, + { + "bbox": [ + 205, + 434, + 311, + 446 + ], + "score": 0.91, + "content": "( 1 - \\alpha ) ( 1 / d - 1 / ( d + s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 434, + 425, + 448 + ], + "score": 1.0, + "content": "on each common neighbour", + "type": "text" + }, + { + "bbox": [ + 425, + 434, + 455, + 446 + ], + "score": 0.92, + "content": "k \\in \\sharp _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "will need to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 445, + 292, + 458 + ], + "spans": [ + { + "bbox": [ + 142, + 445, + 264, + 458 + ], + "score": 1.0, + "content": "travel by at most distance 2 to", + "type": "text" + }, + { + "bbox": [ + 264, + 446, + 288, + 457 + ], + "score": 0.9, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 445, + 292, + 458 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 122, + 463, + 316, + 478 + ], + "spans": [ + { + "bbox": [ + 122, + 463, + 205, + 478 + ], + "score": 1.0, + "content": "(iv) Move the mass", + "type": "text" + }, + { + "bbox": [ + 205, + 464, + 248, + 476 + ], + "score": 0.91, + "content": "( 1 - \\alpha ) / d", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 463, + 270, + 478 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 271, + 465, + 277, + 476 + ], + "score": 0.74, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 463, + 288, + 478 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 288, + 464, + 312, + 477 + ], + "score": 0.89, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 463, + 316, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 125, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 125, + 483, + 207, + 496 + ], + "score": 1.0, + "content": "(v) Move the mass", + "type": "text" + }, + { + "bbox": [ + 208, + 486, + 216, + 493 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 483, + 240, + 496 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 240, + 485, + 245, + 493 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 483, + 257, + 496 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 258, + 484, + 263, + 495 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 483, + 380, + 496 + ], + "score": 1.0, + "content": ". This leaves left-over mass", + "type": "text" + }, + { + "bbox": [ + 380, + 483, + 448, + 495 + ], + "score": 0.93, + "content": "( 1 - \\alpha ) / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 483, + 461, + 496 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 461, + 485, + 466, + 493 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "from the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 142, + 494, + 190, + 508 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 495, + 203, + 508 + ], + "score": 0.88, + "content": "\\mu _ { j } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 494, + 384, + 508 + ], + "score": 1.0, + "content": ". This mass can be compensated from mass in", + "type": "text" + }, + { + "bbox": [ + 384, + 495, + 407, + 506 + ], + "score": 0.9, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "which is at distance one.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 122, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 122, + 514, + 259, + 529 + ], + "score": 1.0, + "content": "(vi) Finally, we move the mass", + "type": "text" + }, + { + "bbox": [ + 259, + 515, + 303, + 527 + ], + "score": 0.92, + "content": "( 1 - \\alpha ) / d", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 514, + 416, + 529 + ], + "score": 1.0, + "content": "of any untouched node in", + "type": "text" + }, + { + "bbox": [ + 416, + 515, + 439, + 527 + ], + "score": 0.92, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 514, + 453, + 529 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 453, + 515, + 477, + 527 + ], + "score": 0.92, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 514, + 505, + 529 + ], + "score": 1.0, + "content": "along", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 140, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 140, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "a path of length lesser or equal than three. Note that the remaining mass is equal to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 536, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 142, + 537, + 352, + 550 + ], + "score": 0.8, + "content": "( ( \\bar { 1 } - \\alpha ) / d ) ( \\bar { d } - 1 - | \\sharp _ { \\Delta } | - | \\bar { \\sharp } _ { \\varpi } ^ { \\mathfrak { m } } | ) - ( 1 - \\alpha ) ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 536, + 505, + 550 + ], + "score": 1.0, + "content": ", where the last terms comes from (v).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 263, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 263, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 263, + 573 + ], + "score": 1.0, + "content": "If we sum all the contributions we find", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 575, + 509, + 689 + ], + "lines": [ + { + "bbox": [ + 111, + 575, + 509, + 689 + ], + "spans": [ + { + "bbox": [ + 111, + 575, + 509, + 689 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { V _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) \\leq ( 1 - \\alpha ) \\left( \\displaystyle \\frac { | \\# _ { [ \\alpha ] } ^ { \\mathbb { m } } | } { d + s } + 3 | \\# _ { \\Sigma } ^ { \\mathbb { m } } | \\left( \\displaystyle \\frac { 1 } { d } - \\displaystyle \\frac { 1 } { d + s } \\right) + 2 | \\# _ { \\Delta } | \\left( \\displaystyle \\frac { 1 } { d } - \\displaystyle \\frac { 1 } { d + s } \\right) + \\displaystyle \\frac { 1 } { d } \\right) + \\alpha + \\displaystyle \\frac { 1 - \\alpha } { d + s } } } \\\\ { { \\displaystyle ~ + 3 ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d } ( d - 1 - | \\# _ { \\Delta } | - | \\# _ { \\mathbb { m } } ^ { \\mathbb { m } } | ) - \\displaystyle \\frac { 1 } { d + s } \\right) } } \\\\ { { \\displaystyle ~ = ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d + s } \\left( - 2 | \\# _ { \\Delta } | - 2 | \\# _ { \\Xi } ^ { \\mathbb { m } } | - 2 \\right) + \\displaystyle \\frac { 1 } { d } \\left( - | \\# _ { \\Delta } | - 2 \\right) + 2 \\right) + 1 } } \\\\ { { \\displaystyle ~ = ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d + s } \\left( - 3 | \\# _ { \\Delta } | - \\displaystyle \\frac { s } { d } | \\# _ { \\Delta } | - 2 | \\# _ { \\Xi } ^ { \\mathbb { m } } | - 4 - 2 \\displaystyle \\frac { s } { d } + 2 ( d + s ) \\right) \\right) + 1 . } } \\end{array}", + "type": "interline_equation", + "image_path": "1d380fef1043fedd7f5e2c6067efa640e119c05dbdcf08fe4d6d820bdb1ac984.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 111, + 575, + 509, + 613.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 111, + 613.0, + 509, + 651.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 111, + 651.0, + 509, + 689.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 692, + 183, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 183, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 183, + 705 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 707, + 493, + 736 + ], + "lines": [ + { + "bbox": [ + 117, + 707, + 493, + 736 + ], + "spans": [ + { + "bbox": [ + 117, + 707, + 493, + 736 + ], + "score": 0.92, + "content": "\\kappa ( i , j ) = \\operatorname* { l i m } _ { \\alpha \\to 1 } \\frac { 1 - W ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) } { 1 - \\alpha } \\geq \\left( \\frac { 1 } { d + s } \\left( 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + 2 | \\sharp _ { \\overline { { \\sharp } } } ^ { \\mathrm { m } } | + 4 + 2 \\frac { s } { d } - 2 ( d + s ) \\right) \\right) .", + "type": "interline_equation", + "image_path": "de7998cf6c0325ace6b5fc9fcc4d3409d524c2fe2defeb4b18cda88bdaa8420c.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 117, + 707, + 493, + 716.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 117, + 716.6666666666666, + 493, + 726.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 117, + 726.3333333333333, + 493, + 735.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 126, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 126, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 126, + 94 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 82, + 126, + 94 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 91, + 405, + 114 + ], + "lines": [ + { + "bbox": [ + 204, + 91, + 405, + 114 + ], + "spans": [ + { + "bbox": [ + 204, + 91, + 405, + 114 + ], + "score": 0.92, + "content": "\\omega _ { \\Delta } : = \\frac { 1 } { 3 } ( \\omega _ { i j } ^ { 2 } + \\omega _ { i k } ^ { 2 } + \\omega _ { j k } ^ { 2 } ) , \\quad z \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) ,", + "type": "interline_equation", + "image_path": "9fd6963393dcd31feb8512f49fd9ace7dba0d35f905f17633364c28e9d78bda1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 204, + 91, + 405, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 204, + 130 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 201, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 172, + 133 + ], + "score": 1.0, + "content": "and, for a given", + "type": "text" + }, + { + "bbox": [ + 172, + 118, + 201, + 131 + ], + "score": 0.94, + "content": "k \\in \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 115, + 201, + 133 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 136, + 428, + 170 + ], + "lines": [ + { + "bbox": [ + 182, + 136, + 428, + 170 + ], + "spans": [ + { + "bbox": [ + 182, + 136, + 428, + 170 + ], + "score": 0.93, + "content": "\\nu _ { \\perp } : = \\frac { W _ { z } \\cdot A _ { j } ^ { i } } { A _ { k } \\cdot A _ { j } ^ { i } } , \\omega _ { \\perp } = \\frac { 1 } { 4 } \\left( \\omega _ { i j } ^ { 2 } + \\omega _ { i k } ^ { 2 } + \\nu _ { \\perp } ^ { 2 } + \\frac { A _ { k } \\cdot W _ { j } ^ { i } } { A _ { k } \\cdot A _ { j } ^ { i } } \\right) .", + "type": "interline_equation", + "image_path": "c544e00ab6a8b4ce85ca00d37fe41c53a27ebaa8796c2b04c779cc76738a2564.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 182, + 136, + 428, + 147.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 182, + 147.33333333333334, + 428, + 158.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 182, + 158.66666666666669, + 428, + 170.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 465, + 195 + ], + "score": 1.0, + "content": "Important convention. Without losing generality, in the following we always assume that", + "type": "text" + }, + { + "bbox": [ + 466, + 181, + 505, + 192 + ], + "score": 0.9, + "content": "1 \\leq d _ { i } \\leq", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 192, + 117, + 204 + ], + "score": 0.87, + "content": "d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 191, + 209, + 205 + ], + "score": 1.0, + "content": ". In particular, we write", + "type": "text" + }, + { + "bbox": [ + 210, + 192, + 238, + 203 + ], + "score": 0.92, + "content": "d _ { i } \\doteq d", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 191, + 255, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 192, + 300, + 204 + ], + "score": 0.92, + "content": "d _ { j } = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 191, + 340, + 205 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 341, + 192, + 365, + 203 + ], + "score": 0.9, + "content": "s \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 191, + 480, + 205 + ], + "score": 1.0, + "content": ", omitting to specify that both", + "type": "text" + }, + { + "bbox": [ + 481, + 192, + 488, + 201 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 202, + 507, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 205, + 113, + 213 + ], + "score": 0.52, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 202, + 223, + 216 + ], + "score": 1.0, + "content": "are of course depending on", + "type": "text" + }, + { + "bbox": [ + 223, + 204, + 227, + 213 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 202, + 245, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 204, + 251, + 214 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 202, + 507, + 216 + ], + "score": 1.0, + "content": ". Moreover, from now on we only focus on the unweighted case.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 178, + 507, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 306, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 307, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 307, + 234 + ], + "score": 1.0, + "content": "We can now prove our main comparison theorem.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 218, + 307, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 239, + 474, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 475, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 273, + 255 + ], + "score": 1.0, + "content": "Theorem 2. Given an unweighted graph", + "type": "text" + }, + { + "bbox": [ + 273, + 241, + 282, + 251 + ], + "score": 0.74, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 238, + 338, + 255 + ], + "score": 1.0, + "content": ", for any edge", + "type": "text" + }, + { + "bbox": [ + 338, + 241, + 361, + 252 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 238, + 397, + 255 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 397, + 240, + 471, + 253 + ], + "score": 0.91, + "content": "\\kappa ( i , j ) \\geq \\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 238, + 475, + 255 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 238, + 475, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 309, + 279 + ], + "score": 1.0, + "content": "Proof. We stick to the aforementioned convention:", + "type": "text" + }, + { + "bbox": [ + 309, + 266, + 426, + 278 + ], + "score": 0.91, + "content": "d _ { i } : = d \\le d _ { j } = d + s , s \\ge 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 265, + 505, + 279 + ], + "score": 1.0, + "content": ". The strategy of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 411, + 290 + ], + "score": 1.0, + "content": "proof amounts to finding a transportation plan providing an upper bound for", + "type": "text" + }, + { + "bbox": [ + 411, + 277, + 461, + 290 + ], + "score": 0.95, + "content": "W _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 236, + 302 + ], + "score": 1.0, + "content": "a lower bound for the curvature", + "type": "text" + }, + { + "bbox": [ + 236, + 289, + 263, + 301 + ], + "score": 0.93, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 289, + 469, + 302 + ], + "score": 1.0, + "content": ". In particular, we consider plans moving the mass", + "type": "text" + }, + { + "bbox": [ + 469, + 290, + 481, + 301 + ], + "score": 0.89, + "content": "\\mu _ { i } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 299, + 172, + 314 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 131, + 312 + ], + "score": 0.91, + "content": "B _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 299, + 142, + 314 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 142, + 300, + 168, + 312 + ], + "score": 0.91, + "content": "B _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 299, + 172, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 265, + 505, + 314 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 115, + 331 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 317, + 140, + 327 + ], + "score": 0.93, + "content": "d = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 315, + 378, + 331 + ], + "score": 1.0, + "content": ", then the optimal transport plan consists of moving the mass", + "type": "text" + }, + { + "bbox": [ + 379, + 319, + 387, + 327 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 315, + 409, + 331 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 409, + 318, + 414, + 327 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 315, + 425, + 331 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 425, + 318, + 431, + 329 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 315, + 505, + 331 + ], + "score": 1.0, + "content": "and the remaining", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 128, + 342 + ], + "score": 1.0, + "content": "mass", + "type": "text" + }, + { + "bbox": [ + 128, + 329, + 153, + 339 + ], + "score": 0.87, + "content": "1 - \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 328, + 165, + 342 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 166, + 329, + 172, + 340 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 328, + 183, + 342 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 183, + 328, + 207, + 340 + ], + "score": 0.91, + "content": "\\bar { S } _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 328, + 399, + 342 + ], + "score": 1.0, + "content": ". This yields a unit Wasserstein distance between", + "type": "text" + }, + { + "bbox": [ + 399, + 329, + 412, + 340 + ], + "score": 0.9, + "content": "\\mu _ { i } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 328, + 430, + 342 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 430, + 329, + 443, + 341 + ], + "score": 0.88, + "content": "\\mu _ { j } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "and hence zero", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 458, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 180, + 354 + ], + "score": 1.0, + "content": "Ollivier curvature", + "type": "text" + }, + { + "bbox": [ + 180, + 341, + 206, + 353 + ], + "score": 0.92, + "content": "\\kappa ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 339, + 418, + 354 + ], + "score": 1.0, + "content": ", which coincides with the value of balanced Forman", + "type": "text" + }, + { + "bbox": [ + 418, + 341, + 453, + 353 + ], + "score": 0.89, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 339, + 458, + 354 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 315, + 505, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 476, + 370 + ], + "lines": [ + { + "bbox": [ + 104, + 355, + 477, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 177, + 372 + ], + "score": 1.0, + "content": "Assume now that", + "type": "text" + }, + { + "bbox": [ + 178, + 358, + 202, + 369 + ], + "score": 0.9, + "content": "d \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 355, + 389, + 372 + ], + "score": 1.0, + "content": ". A (possibly non-optimal) transport plan from", + "type": "text" + }, + { + "bbox": [ + 389, + 358, + 402, + 370 + ], + "score": 0.89, + "content": "\\mu _ { i } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 355, + 413, + 372 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 414, + 358, + 426, + 371 + ], + "score": 0.89, + "content": "\\mu _ { j } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 355, + 477, + 372 + ], + "score": 1.0, + "content": "is given by:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 355, + 477, + 372 + ] + }, + { + "type": "list", + "bbox": [ + 123, + 384, + 506, + 550 + ], + "lines": [ + { + "bbox": [ + 126, + 383, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 126, + 384, + 190, + 399 + ], + "score": 1.0, + "content": "(i) Move mass", + "type": "text" + }, + { + "bbox": [ + 190, + 385, + 255, + 397 + ], + "score": 0.93, + "content": "( 1 - \\alpha ) / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 384, + 319, + 399 + ], + "score": 1.0, + "content": "from each node", + "type": "text" + }, + { + "bbox": [ + 319, + 384, + 372, + 398 + ], + "score": 0.43, + "content": "k \\in \\sharp _ { \\bigstar } ^ { \\mathfrak { m } } \\subset \\sharp _ { \\bigstar } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 384, + 460, + 399 + ], + "score": 1.0, + "content": "to its unique image in", + "type": "text" + }, + { + "bbox": [ + 460, + 383, + 473, + 398 + ], + "score": 0.87, + "content": "\\sharp _ { \\bigstar } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "under a", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 394, + 282, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 394, + 180, + 411 + ], + "score": 1.0, + "content": "bijection", + "type": "text" + }, + { + "bbox": [ + 180, + 398, + 188, + 408 + ], + "score": 0.8, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 394, + 266, + 411 + ], + "score": 1.0, + "content": "as per definition of", + "type": "text" + }, + { + "bbox": [ + 266, + 396, + 278, + 408 + ], + "score": 0.83, + "content": "\\sharp _ { \\bigsqcup } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 394, + 282, + 411 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + }, + { + "bbox": [ + 124, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 124, + 415, + 279, + 429 + ], + "score": 1.0, + "content": "(ii) The remaining mass on each node", + "type": "text" + }, + { + "bbox": [ + 279, + 415, + 308, + 428 + ], + "score": 0.74, + "content": "k \\in \\sharp _ { \\bigstar } ^ { \\mathfrak { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 415, + 479, + 429 + ], + "score": 1.0, + "content": "will need to travel by at most distance 3 to", + "type": "text" + }, + { + "bbox": [ + 479, + 415, + 502, + 427 + ], + "score": 0.92, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 415, + 506, + 429 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 122, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 122, + 434, + 205, + 448 + ], + "score": 1.0, + "content": "(iii) The extra-mass", + "type": "text" + }, + { + "bbox": [ + 205, + 434, + 311, + 446 + ], + "score": 0.91, + "content": "( 1 - \\alpha ) ( 1 / d - 1 / ( d + s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 434, + 425, + 448 + ], + "score": 1.0, + "content": "on each common neighbour", + "type": "text" + }, + { + "bbox": [ + 425, + 434, + 455, + 446 + ], + "score": 0.92, + "content": "k \\in \\sharp _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "will need to", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 445, + 292, + 458 + ], + "spans": [ + { + "bbox": [ + 142, + 445, + 264, + 458 + ], + "score": 1.0, + "content": "travel by at most distance 2 to", + "type": "text" + }, + { + "bbox": [ + 264, + 446, + 288, + 457 + ], + "score": 0.9, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 445, + 292, + 458 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 122, + 463, + 316, + 478 + ], + "spans": [ + { + "bbox": [ + 122, + 463, + 205, + 478 + ], + "score": 1.0, + "content": "(iv) Move the mass", + "type": "text" + }, + { + "bbox": [ + 205, + 464, + 248, + 476 + ], + "score": 0.91, + "content": "( 1 - \\alpha ) / d", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 463, + 270, + 478 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 271, + 465, + 277, + 476 + ], + "score": 0.74, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 463, + 288, + 478 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 288, + 464, + 312, + 477 + ], + "score": 0.89, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 463, + 316, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 125, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 125, + 483, + 207, + 496 + ], + "score": 1.0, + "content": "(v) Move the mass", + "type": "text" + }, + { + "bbox": [ + 208, + 486, + 216, + 493 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 483, + 240, + 496 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 240, + 485, + 245, + 493 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 483, + 257, + 496 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 258, + 484, + 263, + 495 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 483, + 380, + 496 + ], + "score": 1.0, + "content": ". This leaves left-over mass", + "type": "text" + }, + { + "bbox": [ + 380, + 483, + 448, + 495 + ], + "score": 0.93, + "content": "( 1 - \\alpha ) / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 483, + 461, + 496 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 461, + 485, + 466, + 493 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "from the", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 142, + 494, + 190, + 508 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 190, + 495, + 203, + 508 + ], + "score": 0.88, + "content": "\\mu _ { j } ^ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 494, + 384, + 508 + ], + "score": 1.0, + "content": ". This mass can be compensated from mass in", + "type": "text" + }, + { + "bbox": [ + 384, + 495, + 407, + 506 + ], + "score": 0.9, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "which is at distance one.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 122, + 514, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 122, + 514, + 259, + 529 + ], + "score": 1.0, + "content": "(vi) Finally, we move the mass", + "type": "text" + }, + { + "bbox": [ + 259, + 515, + 303, + 527 + ], + "score": 0.92, + "content": "( 1 - \\alpha ) / d", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 514, + 416, + 529 + ], + "score": 1.0, + "content": "of any untouched node in", + "type": "text" + }, + { + "bbox": [ + 416, + 515, + 439, + 527 + ], + "score": 0.92, + "content": "S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 514, + 453, + 529 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 453, + 515, + 477, + 527 + ], + "score": 0.92, + "content": "S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 514, + 505, + 529 + ], + "score": 1.0, + "content": "along", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 140, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 140, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "a path of length lesser or equal than three. Note that the remaining mass is equal to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 536, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 142, + 537, + 352, + 550 + ], + "score": 0.8, + "content": "( ( \\bar { 1 } - \\alpha ) / d ) ( \\bar { d } - 1 - | \\sharp _ { \\Delta } | - | \\bar { \\sharp } _ { \\varpi } ^ { \\mathfrak { m } } | ) - ( 1 - \\alpha ) ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 536, + 505, + 550 + ], + "score": 1.0, + "content": ", where the last terms comes from (v).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24, + "bbox_fs": [ + 122, + 383, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 263, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 263, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 263, + 573 + ], + "score": 1.0, + "content": "If we sum all the contributions we find", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 560, + 263, + 573 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 575, + 509, + 689 + ], + "lines": [ + { + "bbox": [ + 111, + 575, + 509, + 689 + ], + "spans": [ + { + "bbox": [ + 111, + 575, + 509, + 689 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { V _ { 1 } ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) \\leq ( 1 - \\alpha ) \\left( \\displaystyle \\frac { | \\# _ { [ \\alpha ] } ^ { \\mathbb { m } } | } { d + s } + 3 | \\# _ { \\Sigma } ^ { \\mathbb { m } } | \\left( \\displaystyle \\frac { 1 } { d } - \\displaystyle \\frac { 1 } { d + s } \\right) + 2 | \\# _ { \\Delta } | \\left( \\displaystyle \\frac { 1 } { d } - \\displaystyle \\frac { 1 } { d + s } \\right) + \\displaystyle \\frac { 1 } { d } \\right) + \\alpha + \\displaystyle \\frac { 1 - \\alpha } { d + s } } } \\\\ { { \\displaystyle ~ + 3 ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d } ( d - 1 - | \\# _ { \\Delta } | - | \\# _ { \\mathbb { m } } ^ { \\mathbb { m } } | ) - \\displaystyle \\frac { 1 } { d + s } \\right) } } \\\\ { { \\displaystyle ~ = ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d + s } \\left( - 2 | \\# _ { \\Delta } | - 2 | \\# _ { \\Xi } ^ { \\mathbb { m } } | - 2 \\right) + \\displaystyle \\frac { 1 } { d } \\left( - | \\# _ { \\Delta } | - 2 \\right) + 2 \\right) + 1 } } \\\\ { { \\displaystyle ~ = ( 1 - \\alpha ) \\left( \\displaystyle \\frac { 1 } { d + s } \\left( - 3 | \\# _ { \\Delta } | - \\displaystyle \\frac { s } { d } | \\# _ { \\Delta } | - 2 | \\# _ { \\Xi } ^ { \\mathbb { m } } | - 4 - 2 \\displaystyle \\frac { s } { d } + 2 ( d + s ) \\right) \\right) + 1 . } } \\end{array}", + "type": "interline_equation", + "image_path": "1d380fef1043fedd7f5e2c6067efa640e119c05dbdcf08fe4d6d820bdb1ac984.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 111, + 575, + 509, + 613.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 111, + 613.0, + 509, + 651.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 111, + 651.0, + 509, + 689.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 692, + 183, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 183, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 183, + 705 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 691, + 183, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 707, + 493, + 736 + ], + "lines": [ + { + "bbox": [ + 117, + 707, + 493, + 736 + ], + "spans": [ + { + "bbox": [ + 117, + 707, + 493, + 736 + ], + "score": 0.92, + "content": "\\kappa ( i , j ) = \\operatorname* { l i m } _ { \\alpha \\to 1 } \\frac { 1 - W ( \\mu _ { i } ^ { \\alpha } , \\mu _ { j } ^ { \\alpha } ) } { 1 - \\alpha } \\geq \\left( \\frac { 1 } { d + s } \\left( 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + 2 | \\sharp _ { \\overline { { \\sharp } } } ^ { \\mathrm { m } } | + 4 + 2 \\frac { s } { d } - 2 ( d + s ) \\right) \\right) .", + "type": "interline_equation", + "image_path": "de7998cf6c0325ace6b5fc9fcc4d3409d524c2fe2defeb4b18cda88bdaa8420c.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 117, + 707, + 493, + 716.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 117, + 716.6666666666666, + 493, + 726.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 117, + 726.3333333333333, + 493, + 735.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 338, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 338, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 338, + 96 + ], + "score": 1.0, + "content": "By using Lemma 11, we can bound the right hand side as", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 99, + 501, + 127 + ], + "lines": [ + { + "bbox": [ + 111, + 99, + 501, + 127 + ], + "spans": [ + { + "bbox": [ + 111, + 99, + 501, + 127 + ], + "score": 0.93, + "content": "\\kappa ( i , j ) \\geq \\left( \\frac { 1 } { d + s } \\left( 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + ( \\gamma _ { \\operatorname* { m a x } } ) ^ { - 1 } \\big ( | \\sharp _ { \\bigstar | } ^ { i } | + | \\sharp _ { \\bigstar | } ^ { j } | \\big ) + 4 + 2 \\frac { s } { d } - 2 ( d + s ) \\right) \\right) = \\mathrm { R i c } ( i , j )", + "type": "interline_equation", + "image_path": "91be38fd509a877a0a365b8ae6082f19c0e94bbf112a174df80de2ce2c52374e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 99, + 501, + 127 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 216, + 143 + ], + "lines": [ + { + "bbox": [ + 106, + 130, + 217, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 217, + 145 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 148, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 218, + 163 + ], + "score": 1.0, + "content": "Remark 13. By inspection", + "type": "text" + }, + { + "bbox": [ + 218, + 149, + 294, + 162 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 149, + 317, + 163 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 317, + 149, + 345, + 162 + ], + "score": 0.92, + "content": "\\Phi ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "as in Theorem 12. We have three cases:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 122, + 169, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 127, + 168, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 127, + 168, + 142, + 185 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 170, + 429, + 183 + ], + "score": 0.89, + "content": "\\Phi ( i , j ) = 2 / d + 2 / ( d + s ) - 2 + 2 | \\sharp _ { \\Delta } | / ( d + s ) + | \\sharp _ { \\Delta } | / d \\leq \\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 168, + 469, + 185 + ], + "score": 1.0, + "content": ", because", + "type": "text" + }, + { + "bbox": [ + 469, + 170, + 505, + 182 + ], + "score": 0.59, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 181, + 395, + 194 + ], + "spans": [ + { + "bbox": [ + 141, + 181, + 395, + 194 + ], + "score": 1.0, + "content": "takes into account the positive contribution of 4-cycles as well.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 126, + 199, + 423, + 213 + ], + "lines": [ + { + "bbox": [ + 124, + 198, + 422, + 216 + ], + "spans": [ + { + "bbox": [ + 124, + 198, + 142, + 216 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 142, + 200, + 344, + 213 + ], + "score": 0.9, + "content": "\\Phi ( i , j ) = - 1 + 1 / d + 1 / ( d + s ) + 2 | \\sharp _ { \\Delta } | / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 198, + 422, + 216 + ], + "score": 1.0, + "content": ", which happens iff", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 217, + 378, + 239 + ], + "lines": [ + { + "bbox": [ + 269, + 217, + 378, + 239 + ], + "spans": [ + { + "bbox": [ + 269, + 217, + 378, + 239 + ], + "score": 0.92, + "content": "d + s - 2 - \\frac { s } { d } - | \\sharp _ { \\Delta } | > 0", + "type": "interline_equation", + "image_path": "d1f6fe683a5e8ab6892853cc3eeb4f71a53108a37388e5f4b6c19191c5787c05.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 269, + 217, + 378, + 239 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 244, + 160, + 254 + ], + "lines": [ + { + "bbox": [ + 140, + 244, + 161, + 255 + ], + "spans": [ + { + "bbox": [ + 140, + 244, + 161, + 255 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 251, + 397, + 273 + ], + "lines": [ + { + "bbox": [ + 250, + 251, + 397, + 273 + ], + "spans": [ + { + "bbox": [ + 250, + 251, + 397, + 273 + ], + "score": 0.93, + "content": "d + s - 2 - \\frac { s } { d } - | \\sharp \\triangle | - \\frac { s } { d } | \\sharp \\triangle | \\leq 0 .", + "type": "interline_equation", + "image_path": "39a977af5a7f687001739dfcd4b0d8e5e8bc84659507af1e7282a2e9bc651ac9.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 250, + 251, + 397, + 273 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 143, + 274, + 306, + 286 + ], + "lines": [ + { + "bbox": [ + 141, + 274, + 306, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 274, + 306, + 287 + ], + "score": 1.0, + "content": "From the previous inequalities we derive", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 290, + 470, + 316 + ], + "lines": [ + { + "bbox": [ + 175, + 290, + 470, + 316 + ], + "spans": [ + { + "bbox": [ + 175, + 290, + 470, + 316 + ], + "score": 0.93, + "content": "\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j ) + \\frac { 1 } { d + s } \\left( 2 - d - s + | \\sharp \\Delta | + \\frac { s } { d } | \\sharp \\Delta | + \\frac { s } { d } \\right) \\geq \\Phi ( i , j ) .", + "type": "interline_equation", + "image_path": "7ea3fc0ff58772d69033be2421733b03ba1a14bc713273157bc473223b881c37.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 175, + 290, + 470, + 316 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 121, + 326, + 327, + 339 + ], + "lines": [ + { + "bbox": [ + 122, + 326, + 326, + 340 + ], + "spans": [ + { + "bbox": [ + 122, + 326, + 142, + 340 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 142, + 326, + 235, + 339 + ], + "score": 0.92, + "content": "\\Phi ( i , j ) = | \\sharp _ { \\Delta } | / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 326, + 326, + 340 + ], + "score": 1.0, + "content": "which is equivalent to", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 344, + 379, + 365 + ], + "lines": [ + { + "bbox": [ + 267, + 344, + 379, + 365 + ], + "spans": [ + { + "bbox": [ + 267, + 344, + 379, + 365 + ], + "score": 0.93, + "content": "d + s - 2 - \\frac { s } { d } - | \\sharp _ { \\Delta } | \\leq 0 .", + "type": "interline_equation", + "image_path": "f3f28b355adccd927babf4eb060d031a7b38bd86f816b38ce3f8d60bfc84563c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 267, + 344, + 379, + 365 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 370, + 225, + 381 + ], + "lines": [ + { + "bbox": [ + 141, + 369, + 226, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 226, + 383 + ], + "score": 1.0, + "content": "In this case we have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 378, + 384, + 403 + ], + "lines": [ + { + "bbox": [ + 262, + 378, + 384, + 403 + ], + "spans": [ + { + "bbox": [ + 262, + 378, + 384, + 403 + ], + "score": 0.93, + "content": "\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j ) + \\frac { s } { d } \\frac { | \\sharp _ { \\Delta } | } { d + s }", + "type": "interline_equation", + "image_path": "f85aba5e484a965f85b7a1ab932934768e48a9af575bd2c353f03a3a8d0dcb34.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 262, + 378, + 384, + 403 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 405, + 492, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 494, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 171, + 419 + ], + "score": 1.0, + "content": "Corollary 3. If", + "type": "text" + }, + { + "bbox": [ + 172, + 405, + 244, + 417 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 403, + 297, + 419 + ], + "score": 1.0, + "content": "for any edge", + "type": "text" + }, + { + "bbox": [ + 297, + 406, + 320, + 417 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 403, + 445, + 419 + ], + "score": 1.0, + "content": ", then there exists a polynomial", + "type": "text" + }, + { + "bbox": [ + 445, + 406, + 454, + 415 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 403, + 494, + 419 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 422, + 359, + 436 + ], + "lines": [ + { + "bbox": [ + 251, + 422, + 359, + 436 + ], + "spans": [ + { + "bbox": [ + 251, + 422, + 359, + 436 + ], + "score": 0.9, + "content": "| B _ { r } ( i ) | \\leq P ( r ) , \\forall i \\in V .", + "type": "interline_equation", + "image_path": "cf30c4a55a758afb35f09043f25c1633cfbd18a9da8e16c72144668acf051f7d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 251, + 422, + 359, + 436 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 461 + ], + "score": 1.0, + "content": "Proof. This follows immediately from Theorem 2 and a Bishop-Gromov type of result for discrete", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 458, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 280, + 471 + ], + "score": 1.0, + "content": "Ollivier curvature on graphs Paeng (2012).", + "type": "text" + }, + { + "bbox": [ + 496, + 461, + 504, + 468 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 385, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 379, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 379, + 496 + ], + "score": 1.0, + "content": "To address the proof of Theorem 4, we first need the Lemma below.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 104, + 494, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 185, + 512 + ], + "score": 1.0, + "content": "Lemma 14. Given", + "type": "text" + }, + { + "bbox": [ + 185, + 498, + 207, + 509 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 494, + 230, + 512 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 230, + 497, + 262, + 510 + ], + "score": 0.87, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 494, + 273, + 512 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 274, + 497, + 353, + 510 + ], + "score": 0.46, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta ,", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 494, + 392, + 512 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 392, + 497, + 482, + 510 + ], + "score": 0.92, + "content": "0 < \\delta < ( 1 + \\gamma _ { m a x } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 494, + 506, + 512 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 515, + 340, + 542 + ], + "lines": [ + { + "bbox": [ + 270, + 515, + 340, + 542 + ], + "spans": [ + { + "bbox": [ + 270, + 515, + 340, + 542 + ], + "score": 0.93, + "content": "\\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } > \\delta ^ { - 1 } .", + "type": "interline_equation", + "image_path": "5135b0ee892c9176b3ec4a57eb864382f806c30eedb72ba52e77ddbab6b16c12.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 270, + 515, + 340, + 542 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 283, + 565 + ], + "score": 1.0, + "content": "Proof. According to our convention we let", + "type": "text" + }, + { + "bbox": [ + 284, + 553, + 313, + 564 + ], + "score": 0.93, + "content": "d _ { i } = d", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 553, + 331, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 553, + 378, + 565 + ], + "score": 0.92, + "content": "d _ { j } = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 553, + 397, + 565 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 397, + 554, + 422, + 564 + ], + "score": 0.91, + "content": "s \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 553, + 505, + 565 + ], + "score": 1.0, + "content": ". We also recall that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 565, + 504, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 221, + 579 + ], + "score": 0.91, + "content": "Q _ { j } = S _ { 1 } ( j ) \\backslash ( \\sharp _ { \\Delta } \\cup \\sharp _ { \\Pi } ^ { j } \\cup \\{ i \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 565, + 337, + 580 + ], + "score": 1.0, + "content": ". If we multiply equation 3 by", + "type": "text" + }, + { + "bbox": [ + 338, + 567, + 380, + 578 + ], + "score": 0.9, + "content": "d _ { j } = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 565, + 429, + 580 + ], + "score": 1.0, + "content": ", we see that", + "type": "text" + }, + { + "bbox": [ + 429, + 565, + 504, + 578 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 577, + 119, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 119, + 589 + ], + "score": 1.0, + "content": "iff", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 585, + 457, + 606 + ], + "lines": [ + { + "bbox": [ + 153, + 585, + 457, + 606 + ], + "spans": [ + { + "bbox": [ + 153, + 585, + 457, + 606 + ], + "score": 0.9, + "content": "4 + 2 \\frac { s } { d } + 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + \\gamma _ { \\operatorname* { m a x } } ^ { - 1 } ( | \\sharp _ { \\bigstar } ^ { i } | + | \\sharp _ { \\bigstar } ^ { j } | ) \\leq \\delta ( 1 + | \\sharp _ { \\Delta } | + | \\sharp _ { \\bigstar } ^ { j } | + | Q _ { j } | ) .", + "type": "interline_equation", + "image_path": "b0c98e02d75dd29ed2bf4912063bdf321a7e4afced87e8999a755bc6e86c94f5.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 153, + 585, + 457, + 606 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 304, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 306, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 168, + 626 + ], + "score": 1.0, + "content": "By assumption", + "type": "text" + }, + { + "bbox": [ + 169, + 609, + 246, + 624 + ], + "score": 0.93, + "content": "| \\sharp _ { \\sf D } ^ { j } | ( \\gamma _ { \\mathrm { m a x } } ^ { - 1 } - \\delta ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 608, + 306, + 626 + ], + "score": 1.0, + "content": ", meaning that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 628, + 428, + 651 + ], + "lines": [ + { + "bbox": [ + 182, + 628, + 428, + 651 + ], + "spans": [ + { + "bbox": [ + 182, + 628, + 428, + 651 + ], + "score": 0.89, + "content": "\\delta ( 1 + | \\sharp \\Delta | + | Q _ { j } | ) \\geq 4 + 2 \\frac { s } { d } + 3 | \\sharp \\Delta | + \\frac { s } { d } | \\sharp \\Delta | \\geq 4 + 3 | \\sharp \\Delta | .", + "type": "interline_equation", + "image_path": "50d912e56e079e68a06055f817c960dacd85092ac8ad62ad2f59e2571ec4940c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 182, + 628, + 428, + 651 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 203, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 203, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 203, + 667 + ], + "score": 1.0, + "content": "Therefore, we conclude", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 664, + 344, + 691 + ], + "lines": [ + { + "bbox": [ + 266, + 664, + 344, + 691 + ], + "spans": [ + { + "bbox": [ + 266, + 664, + 344, + 691 + ], + "score": 0.91, + "content": "\\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } \\geq \\frac { 3 } { \\delta } - 1 .", + "type": "interline_equation", + "image_path": "df913a23ab92bc7d988438a9a5c7d99dd1a4944c87b0ecd42c28459e137e58e5.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 266, + 664, + 344, + 691 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 720, + 468, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 720, + 469, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 295, + 734 + ], + "score": 1.0, + "content": "Theorem 4. Consider a MPNN as in equation", + "type": "text" + }, + { + "bbox": [ + 296, + 723, + 301, + 730 + ], + "score": 0.39, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 720, + 321, + 734 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 322, + 721, + 343, + 732 + ], + "score": 0.86, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 720, + 365, + 734 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 365, + 721, + 397, + 733 + ], + "score": 0.9, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 720, + 469, + 734 + ], + "score": 1.0, + "content": "and assume that:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 312, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "20", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 693, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 496, + 694, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 496, + 694, + 505, + 704 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 132, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 496, + 133, + 504, + 142 + ], + "spans": [ + { + "bbox": [ + 496, + 133, + 504, + 142 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 338, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 338, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 338, + 96 + ], + "score": 1.0, + "content": "By using Lemma 11, we can bound the right hand side as", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 338, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 99, + 501, + 127 + ], + "lines": [ + { + "bbox": [ + 111, + 99, + 501, + 127 + ], + "spans": [ + { + "bbox": [ + 111, + 99, + 501, + 127 + ], + "score": 0.93, + "content": "\\kappa ( i , j ) \\geq \\left( \\frac { 1 } { d + s } \\left( 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + ( \\gamma _ { \\operatorname* { m a x } } ) ^ { - 1 } \\big ( | \\sharp _ { \\bigstar | } ^ { i } | + | \\sharp _ { \\bigstar | } ^ { j } | \\big ) + 4 + 2 \\frac { s } { d } - 2 ( d + s ) \\right) \\right) = \\mathrm { R i c } ( i , j )", + "type": "interline_equation", + "image_path": "91be38fd509a877a0a365b8ae6082f19c0e94bbf112a174df80de2ce2c52374e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 99, + 501, + 127 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 216, + 143 + ], + "lines": [ + { + "bbox": [ + 106, + 130, + 217, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 217, + 145 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 130, + 217, + 145 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 148, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 218, + 163 + ], + "score": 1.0, + "content": "Remark 13. By inspection", + "type": "text" + }, + { + "bbox": [ + 218, + 149, + 294, + 162 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 149, + 317, + 163 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 317, + 149, + 345, + 162 + ], + "score": 0.92, + "content": "\\Phi ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "as in Theorem 12. We have three cases:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 106, + 149, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 122, + 169, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 127, + 168, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 127, + 168, + 142, + 185 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 170, + 429, + 183 + ], + "score": 0.89, + "content": "\\Phi ( i , j ) = 2 / d + 2 / ( d + s ) - 2 + 2 | \\sharp _ { \\Delta } | / ( d + s ) + | \\sharp _ { \\Delta } | / d \\leq \\operatorname { R i c } ( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 168, + 469, + 185 + ], + "score": 1.0, + "content": ", because", + "type": "text" + }, + { + "bbox": [ + 469, + 170, + 505, + 182 + ], + "score": 0.59, + "content": "\\operatorname { R i c } ( i , j )", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 181, + 395, + 194 + ], + "spans": [ + { + "bbox": [ + 141, + 181, + 395, + 194 + ], + "score": 1.0, + "content": "takes into account the positive contribution of 4-cycles as well.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 127, + 168, + 505, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 126, + 199, + 423, + 213 + ], + "lines": [ + { + "bbox": [ + 124, + 198, + 422, + 216 + ], + "spans": [ + { + "bbox": [ + 124, + 198, + 142, + 216 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 142, + 200, + 344, + 213 + ], + "score": 0.9, + "content": "\\Phi ( i , j ) = - 1 + 1 / d + 1 / ( d + s ) + 2 | \\sharp _ { \\Delta } | / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 198, + 422, + 216 + ], + "score": 1.0, + "content": ", which happens iff", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 124, + 198, + 422, + 216 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 217, + 378, + 239 + ], + "lines": [ + { + "bbox": [ + 269, + 217, + 378, + 239 + ], + "spans": [ + { + "bbox": [ + 269, + 217, + 378, + 239 + ], + "score": 0.92, + "content": "d + s - 2 - \\frac { s } { d } - | \\sharp _ { \\Delta } | > 0", + "type": "interline_equation", + "image_path": "d1f6fe683a5e8ab6892853cc3eeb4f71a53108a37388e5f4b6c19191c5787c05.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 269, + 217, + 378, + 239 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 244, + 160, + 254 + ], + "lines": [ + { + "bbox": [ + 140, + 244, + 161, + 255 + ], + "spans": [ + { + "bbox": [ + 140, + 244, + 161, + 255 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 140, + 244, + 161, + 255 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 251, + 397, + 273 + ], + "lines": [ + { + "bbox": [ + 250, + 251, + 397, + 273 + ], + "spans": [ + { + "bbox": [ + 250, + 251, + 397, + 273 + ], + "score": 0.93, + "content": "d + s - 2 - \\frac { s } { d } - | \\sharp \\triangle | - \\frac { s } { d } | \\sharp \\triangle | \\leq 0 .", + "type": "interline_equation", + "image_path": "39a977af5a7f687001739dfcd4b0d8e5e8bc84659507af1e7282a2e9bc651ac9.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 250, + 251, + 397, + 273 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 143, + 274, + 306, + 286 + ], + "lines": [ + { + "bbox": [ + 141, + 274, + 306, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 274, + 306, + 287 + ], + "score": 1.0, + "content": "From the previous inequalities we derive", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 141, + 274, + 306, + 287 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 290, + 470, + 316 + ], + "lines": [ + { + "bbox": [ + 175, + 290, + 470, + 316 + ], + "spans": [ + { + "bbox": [ + 175, + 290, + 470, + 316 + ], + "score": 0.93, + "content": "\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j ) + \\frac { 1 } { d + s } \\left( 2 - d - s + | \\sharp \\Delta | + \\frac { s } { d } | \\sharp \\Delta | + \\frac { s } { d } \\right) \\geq \\Phi ( i , j ) .", + "type": "interline_equation", + "image_path": "7ea3fc0ff58772d69033be2421733b03ba1a14bc713273157bc473223b881c37.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 175, + 290, + 470, + 316 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 121, + 326, + 327, + 339 + ], + "lines": [ + { + "bbox": [ + 122, + 326, + 326, + 340 + ], + "spans": [ + { + "bbox": [ + 122, + 326, + 142, + 340 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 142, + 326, + 235, + 339 + ], + "score": 0.92, + "content": "\\Phi ( i , j ) = | \\sharp _ { \\Delta } | / ( d + s )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 326, + 326, + 340 + ], + "score": 1.0, + "content": "which is equivalent to", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 122, + 326, + 326, + 340 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 344, + 379, + 365 + ], + "lines": [ + { + "bbox": [ + 267, + 344, + 379, + 365 + ], + "spans": [ + { + "bbox": [ + 267, + 344, + 379, + 365 + ], + "score": 0.93, + "content": "d + s - 2 - \\frac { s } { d } - | \\sharp _ { \\Delta } | \\leq 0 .", + "type": "interline_equation", + "image_path": "f3f28b355adccd927babf4eb060d031a7b38bd86f816b38ce3f8d60bfc84563c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 267, + 344, + 379, + 365 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 370, + 225, + 381 + ], + "lines": [ + { + "bbox": [ + 141, + 369, + 226, + 383 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 226, + 383 + ], + "score": 1.0, + "content": "In this case we have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 369, + 226, + 383 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 378, + 384, + 403 + ], + "lines": [ + { + "bbox": [ + 262, + 378, + 384, + 403 + ], + "spans": [ + { + "bbox": [ + 262, + 378, + 384, + 403 + ], + "score": 0.93, + "content": "\\operatorname { R i c } ( i , j ) \\geq \\Phi ( i , j ) + \\frac { s } { d } \\frac { | \\sharp _ { \\Delta } | } { d + s }", + "type": "interline_equation", + "image_path": "f85aba5e484a965f85b7a1ab932934768e48a9af575bd2c353f03a3a8d0dcb34.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 262, + 378, + 384, + 403 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 405, + 492, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 494, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 171, + 419 + ], + "score": 1.0, + "content": "Corollary 3. If", + "type": "text" + }, + { + "bbox": [ + 172, + 405, + 244, + 417 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 403, + 297, + 419 + ], + "score": 1.0, + "content": "for any edge", + "type": "text" + }, + { + "bbox": [ + 297, + 406, + 320, + 417 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 403, + 445, + 419 + ], + "score": 1.0, + "content": ", then there exists a polynomial", + "type": "text" + }, + { + "bbox": [ + 445, + 406, + 454, + 415 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 403, + 494, + 419 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 403, + 494, + 419 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 422, + 359, + 436 + ], + "lines": [ + { + "bbox": [ + 251, + 422, + 359, + 436 + ], + "spans": [ + { + "bbox": [ + 251, + 422, + 359, + 436 + ], + "score": 0.9, + "content": "| B _ { r } ( i ) | \\leq P ( r ) , \\forall i \\in V .", + "type": "interline_equation", + "image_path": "cf30c4a55a758afb35f09043f25c1633cfbd18a9da8e16c72144668acf051f7d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 251, + 422, + 359, + 436 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 461 + ], + "score": 1.0, + "content": "Proof. This follows immediately from Theorem 2 and a Bishop-Gromov type of result for discrete", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 458, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 280, + 471 + ], + "score": 1.0, + "content": "Ollivier curvature on graphs Paeng (2012).", + "type": "text" + }, + { + "bbox": [ + 496, + 461, + 504, + 468 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 446, + 505, + 471 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 385, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 379, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 379, + 496 + ], + "score": 1.0, + "content": "To address the proof of Theorem 4, we first need the Lemma below.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 482, + 379, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 104, + 494, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 185, + 512 + ], + "score": 1.0, + "content": "Lemma 14. Given", + "type": "text" + }, + { + "bbox": [ + 185, + 498, + 207, + 509 + ], + "score": 0.89, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 494, + 230, + 512 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 230, + 497, + 262, + 510 + ], + "score": 0.87, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 494, + 273, + 512 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 274, + 497, + 353, + 510 + ], + "score": 0.46, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta ,", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 494, + 392, + 512 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 392, + 497, + 482, + 510 + ], + "score": 0.92, + "content": "0 < \\delta < ( 1 + \\gamma _ { m a x } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 494, + 506, + 512 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 494, + 506, + 512 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 270, + 515, + 340, + 542 + ], + "lines": [ + { + "bbox": [ + 270, + 515, + 340, + 542 + ], + "spans": [ + { + "bbox": [ + 270, + 515, + 340, + 542 + ], + "score": 0.93, + "content": "\\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } > \\delta ^ { - 1 } .", + "type": "interline_equation", + "image_path": "5135b0ee892c9176b3ec4a57eb864382f806c30eedb72ba52e77ddbab6b16c12.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 270, + 515, + 340, + 542 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 283, + 565 + ], + "score": 1.0, + "content": "Proof. According to our convention we let", + "type": "text" + }, + { + "bbox": [ + 284, + 553, + 313, + 564 + ], + "score": 0.93, + "content": "d _ { i } = d", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 553, + 331, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 553, + 378, + 565 + ], + "score": 0.92, + "content": "d _ { j } = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 553, + 397, + 565 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 397, + 554, + 422, + 564 + ], + "score": 0.91, + "content": "s \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 553, + 505, + 565 + ], + "score": 1.0, + "content": ". We also recall that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 565, + 504, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 221, + 579 + ], + "score": 0.91, + "content": "Q _ { j } = S _ { 1 } ( j ) \\backslash ( \\sharp _ { \\Delta } \\cup \\sharp _ { \\Pi } ^ { j } \\cup \\{ i \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 565, + 337, + 580 + ], + "score": 1.0, + "content": ". If we multiply equation 3 by", + "type": "text" + }, + { + "bbox": [ + 338, + 567, + 380, + 578 + ], + "score": 0.9, + "content": "d _ { j } = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 565, + 429, + 580 + ], + "score": 1.0, + "content": ", we see that", + "type": "text" + }, + { + "bbox": [ + 429, + 565, + 504, + 578 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 577, + 119, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 119, + 589 + ], + "score": 1.0, + "content": "iff", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 553, + 505, + 589 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 585, + 457, + 606 + ], + "lines": [ + { + "bbox": [ + 153, + 585, + 457, + 606 + ], + "spans": [ + { + "bbox": [ + 153, + 585, + 457, + 606 + ], + "score": 0.9, + "content": "4 + 2 \\frac { s } { d } + 3 | \\sharp _ { \\Delta } | + \\frac { s } { d } | \\sharp _ { \\Delta } | + \\gamma _ { \\operatorname* { m a x } } ^ { - 1 } ( | \\sharp _ { \\bigstar } ^ { i } | + | \\sharp _ { \\bigstar } ^ { j } | ) \\leq \\delta ( 1 + | \\sharp _ { \\Delta } | + | \\sharp _ { \\bigstar } ^ { j } | + | Q _ { j } | ) .", + "type": "interline_equation", + "image_path": "b0c98e02d75dd29ed2bf4912063bdf321a7e4afced87e8999a755bc6e86c94f5.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 153, + 585, + 457, + 606 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 304, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 306, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 168, + 626 + ], + "score": 1.0, + "content": "By assumption", + "type": "text" + }, + { + "bbox": [ + 169, + 609, + 246, + 624 + ], + "score": 0.93, + "content": "| \\sharp _ { \\sf D } ^ { j } | ( \\gamma _ { \\mathrm { m a x } } ^ { - 1 } - \\delta ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 608, + 306, + 626 + ], + "score": 1.0, + "content": ", meaning that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 608, + 306, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 628, + 428, + 651 + ], + "lines": [ + { + "bbox": [ + 182, + 628, + 428, + 651 + ], + "spans": [ + { + "bbox": [ + 182, + 628, + 428, + 651 + ], + "score": 0.89, + "content": "\\delta ( 1 + | \\sharp \\Delta | + | Q _ { j } | ) \\geq 4 + 2 \\frac { s } { d } + 3 | \\sharp \\Delta | + \\frac { s } { d } | \\sharp \\Delta | \\geq 4 + 3 | \\sharp \\Delta | .", + "type": "interline_equation", + "image_path": "50d912e56e079e68a06055f817c960dacd85092ac8ad62ad2f59e2571ec4940c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 182, + 628, + 428, + 651 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 203, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 203, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 203, + 667 + ], + "score": 1.0, + "content": "Therefore, we conclude", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 654, + 203, + 667 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 664, + 344, + 691 + ], + "lines": [ + { + "bbox": [ + 266, + 664, + 344, + 691 + ], + "spans": [ + { + "bbox": [ + 266, + 664, + 344, + 691 + ], + "score": 0.91, + "content": "\\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } \\geq \\frac { 3 } { \\delta } - 1 .", + "type": "interline_equation", + "image_path": "df913a23ab92bc7d988438a9a5c7d99dd1a4944c87b0ecd42c28459e137e58e5.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 266, + 664, + 344, + 691 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 720, + 468, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 720, + 469, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 295, + 734 + ], + "score": 1.0, + "content": "Theorem 4. Consider a MPNN as in equation", + "type": "text" + }, + { + "bbox": [ + 296, + 723, + 301, + 730 + ], + "score": 0.39, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 720, + 321, + 734 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 322, + 721, + 343, + 732 + ], + "score": 0.86, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 720, + 365, + 734 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 365, + 721, + 397, + 733 + ], + "score": 0.9, + "content": "d _ { i } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 720, + 469, + 734 + ], + "score": 1.0, + "content": "and assume that:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 720, + 469, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 123, + 81, + 495, + 95 + ], + "lines": [ + { + "bbox": [ + 127, + 82, + 495, + 95 + ], + "spans": [ + { + "bbox": [ + 127, + 82, + 142, + 95 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 82, + 187, + 95 + ], + "score": 0.91, + "content": "| \\nabla \\phi _ { \\ell } | \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 82, + 205, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 82, + 250, + 95 + ], + "score": 0.92, + "content": "| \\nabla \\psi _ { \\ell } | \\le \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 82, + 287, + 95 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 287, + 82, + 348, + 94 + ], + "score": 0.9, + "content": "0 \\leq \\ell \\leq L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 82, + 371, + 95 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 371, + 83, + 398, + 93 + ], + "score": 0.9, + "content": "L \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 82, + 495, + 95 + ], + "score": 1.0, + "content": "the depth of the MPNN.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 124, + 101, + 465, + 117 + ], + "lines": [ + { + "bbox": [ + 123, + 100, + 463, + 118 + ], + "spans": [ + { + "bbox": [ + 123, + 100, + 192, + 118 + ], + "score": 1.0, + "content": "(ii) There exists", + "type": "text" + }, + { + "bbox": [ + 192, + 104, + 198, + 114 + ], + "score": 0.51, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 100, + 214, + 118 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 214, + 101, + 323, + 117 + ], + "score": 0.72, + "content": "0 < \\delta < ( \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} ) ^ { - \\frac { 1 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 100, + 326, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 326, + 102, + 364, + 116 + ], + "score": 0.73, + "content": "\\delta < \\gamma _ { m a x } ^ { - 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 100, + 384, + 118 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 385, + 103, + 463, + 116 + ], + "score": 0.87, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 447, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 124, + 448, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 175, + 140 + ], + "score": 1.0, + "content": "Then there exists", + "type": "text" + }, + { + "bbox": [ + 176, + 126, + 223, + 138 + ], + "score": 0.94, + "content": "Q _ { j } \\subset S _ { 2 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 124, + 266, + 140 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 266, + 125, + 313, + 138 + ], + "score": 0.93, + "content": "| Q _ { j } | > \\delta ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 124, + 346, + 140 + ], + "score": 1.0, + "content": "and for", + "type": "text" + }, + { + "bbox": [ + 346, + 126, + 412, + 137 + ], + "score": 0.91, + "content": "0 \\leq \\ell _ { 0 } \\leq L - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 124, + 448, + 140 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 141, + 376, + 176 + ], + "lines": [ + { + "bbox": [ + 234, + 141, + 376, + 176 + ], + "spans": [ + { + "bbox": [ + 234, + 141, + 376, + 176 + ], + "score": 0.94, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } \\left| \\frac { \\partial h _ { k } ^ { ( \\ell _ { 0 } + 2 ) } } { \\partial h _ { i } ^ { ( \\ell _ { 0 } ) } } \\right| < ( \\alpha \\beta ) ^ { 2 } \\delta ^ { \\frac { 1 } { 4 } } .", + "type": "interline_equation", + "image_path": "31785543f89ea4e42799e24dd197e94beed501078acb90789194d0b4ed233bdd.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 141, + 376, + 158.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 234, + 158.5, + 376, + 176.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 202, + 199 + ], + "score": 1.0, + "content": "Proof. As usual we let", + "type": "text" + }, + { + "bbox": [ + 203, + 187, + 235, + 198 + ], + "score": 0.91, + "content": "d _ { i } : = d", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 186, + 254, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 187, + 305, + 199 + ], + "score": 0.92, + "content": "d _ { j } : = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 186, + 348, + 199 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 349, + 187, + 374, + 198 + ], + "score": 0.9, + "content": "s \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 186, + 505, + 199 + ], + "score": 1.0, + "content": ". We first observe that from the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 197, + 378, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 157, + 211 + ], + "score": 1.0, + "content": "requirement", + "type": "text" + }, + { + "bbox": [ + 158, + 198, + 216, + 210 + ], + "score": 0.93, + "content": "\\delta ^ { 2 } ( d + s ) \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 197, + 285, + 211 + ], + "score": 1.0, + "content": "in (ii), we derive", + "type": "text" + }, + { + "bbox": [ + 286, + 198, + 363, + 210 + ], + "score": 0.91, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 197, + 378, + 211 + ], + "score": 1.0, + "content": "iff", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 216, + 423, + 238 + ], + "lines": [ + { + "bbox": [ + 188, + 216, + 423, + 238 + ], + "spans": [ + { + "bbox": [ + 188, + 216, + 423, + 238 + ], + "score": 0.89, + "content": "4 + 2 \\frac { s } { d } + 3 | \\sharp \\triangle | + \\frac { s } { d } | \\sharp \\triangle | + \\gamma _ { \\mathrm { m a x } } ^ { - 1 } ( | \\sharp \\vert + | \\sharp \\vert ) \\leq \\delta ( d + s ) .", + "type": "interline_equation", + "image_path": "64422b369dc6fc6343a3b6bfd1f160892077fc2a790da1dc7b6f3bc550125654.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 188, + 216, + 423, + 238 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 183, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 183, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 183, + 255 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 251, + 363, + 273 + ], + "lines": [ + { + "bbox": [ + 246, + 251, + 363, + 273 + ], + "spans": [ + { + "bbox": [ + 246, + 251, + 363, + 273 + ], + "score": 0.86, + "content": "\\delta | \\sharp _ { \\Delta } | \\left( 3 + \\frac { s } { d } \\right) \\leq \\delta ^ { 2 } ( d + s ) ,", + "type": "interline_equation", + "image_path": "e7ce9fe92ab4bccc6f4e81da6845c373f0711d71ab5138c7b8888554efb5cf8f.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 246, + 251, + 363, + 273 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 161, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 162, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 162, + 288 + ], + "score": 1.0, + "content": "meaning that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 282, + 286, + 327, + 300 + ], + "lines": [ + { + "bbox": [ + 282, + 286, + 327, + 300 + ], + "spans": [ + { + "bbox": [ + 282, + 286, + 327, + 300 + ], + "score": 0.86, + "content": "\\delta | \\sharp _ { \\Delta } | \\leq 1 .", + "type": "interline_equation", + "image_path": "2632797d0107ac8cf203aab845cc4563cd6937309c808454b492904a2b51990f.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 282, + 286, + 327, + 300 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 190, + 317 + ], + "score": 1.0, + "content": "From now on we let", + "type": "text" + }, + { + "bbox": [ + 191, + 306, + 203, + 317 + ], + "score": 0.89, + "content": "Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 303, + 327, + 317 + ], + "score": 1.0, + "content": "denote again the complement", + "type": "text" + }, + { + "bbox": [ + 327, + 303, + 435, + 318 + ], + "score": 0.92, + "content": "S _ { 1 } ( j ) \\setminus ( S _ { 1 } ( i ) \\cup \\sharp _ { \\Pi } ^ { j } \\cup \\{ i \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 303, + 506, + 317 + ], + "score": 1.0, + "content": ". Without loss of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 101, + 314, + 507, + 339 + ], + "spans": [ + { + "bbox": [ + 101, + 314, + 272, + 339 + ], + "score": 1.0, + "content": "generality we set `0 = 0 and hence h(0)i", + "type": "text" + }, + { + "bbox": [ + 248, + 316, + 288, + 332 + ], + "score": 0.93, + "content": "h _ { i } ^ { ( 0 ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 317, + 493, + 334 + ], + "score": 1.0, + "content": "; the very same proof applies to any other choice of", + "type": "text" + }, + { + "bbox": [ + 493, + 320, + 503, + 331 + ], + "score": 0.88, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 317, + 507, + 334 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 231, + 330, + 375, + 343 + ], + "spans": [ + { + "bbox": [ + 231, + 330, + 375, + 343 + ], + "score": 1.0, + "content": ", we can apply Lemma 1 and derive", 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"score": 1.0, + "content": "We may expand the power of the augmented normalized adjacency matrix as", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 403, + 411, + 435 + ], + "lines": [ + { + "bbox": [ + 199, + 403, + 411, + 435 + ], + "spans": [ + { + "bbox": [ + 199, + 403, + 411, + 435 + ], + "score": 0.92, + "content": "( \\hat { A } ) _ { i k } ^ { 2 } = \\frac { 1 } { \\sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) } \\frac { 1 } { d _ { w } + 1 } .", + "type": "interline_equation", + "image_path": "4cb7b53c92ecb7797648b1dd887f2a4c08c543a80dee2c7de9c098d608afb1a3.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 403, + 411, + 419.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 199, + 419.0, + 411, + 435.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 442, + 382, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 382, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 198, + 458 + ], + "score": 1.0, + "content": "If we introduce the set", + "type": "text" + }, + { + "bbox": [ + 199, + 442, + 304, + 456 + ], + "score": 0.92, + "content": "\\hat { Q } _ { j } = \\{ k \\in Q _ { j } : \\sigma _ { i k } > 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 441, + 382, + 458 + ], + "score": 1.0, + "content": ", we can then write", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 461, + 470, + 535 + ], + "lines": [ + { + "bbox": [ + 126, + 461, + 470, + 535 + ], + "spans": [ + { + "bbox": [ + 126, + 461, + 470, + 535 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle \\sum _ { k \\in Q _ { j } } ( \\hat { A } ) _ { i k } ^ { 2 } = \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) } \\frac { 1 } { d _ { w } + 1 } = } } \\\\ { { = \\displaystyle \\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\left( \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { 1 } { d _ { j } + 1 } + \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) } \\frac { 1 } { d _ { w } + 1 } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "3f511b5735d645f8188f68708495e0ce8ce32454dd6527389c26f6f5a941e46f.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 126, + 461, + 470, + 485.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 126, + 485.6666666666667, + 470, + 510.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 126, + 510.33333333333337, + 470, + 535.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", 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Let us first focus on the first", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 564, + 225, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 225, + 577 + ], + "score": 1.0, + "content": "sum in equation 12. We have", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 581, + 453, + 613 + ], + "lines": [ + { + "bbox": [ + 158, + 581, + 453, + 613 + ], + "spans": [ + { + "bbox": [ + 158, + 581, + 453, + 613 + ], + "score": 0.93, + "content": "\\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { 1 } { d _ { j } + 1 } \\leq \\frac { 1 } { \\sqrt { d _ { i } + 1 } } | Q _ { j } | \\frac { 1 } { d _ { j } + 1 } \\leq \\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\leq 1 .", + "type": "interline_equation", + "image_path": "39f8d11fceab864d8dd50f0c09555ded49db5f2cd172567f3bdc7a2ba51402cd.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 158, + 581, + 453, + 591.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 158, + 591.6666666666666, + 453, + 602.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 158, + 602.3333333333333, + 453, + 612.9999999999999 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 495, + 633 + ], + "lines": [ + { + "bbox": [ + 104, + 619, + 497, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 351, + 635 + ], + "score": 1.0, + "content": "We now consider the second sum in equation 12. We assume", + "type": "text" + }, + { + "bbox": [ + 352, + 621, + 387, + 633 + ], + "score": 0.93, + "content": "| \\sharp _ { \\Delta } | \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 619, + 432, + 635 + ], + "score": 1.0, + "content": ", otherwise", + "type": "text" + }, + { + "bbox": [ + 432, + 619, + 464, + 634 + ], + "score": 0.93, + "content": "{ \\hat { Q } } _ { j } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 619, + 497, + 635 + ], + "score": 1.0, + "content": ". We let", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 639, + 387, + 673 + ], + "lines": [ + { + "bbox": [ + 225, + 639, + 387, + 673 + ], + "spans": [ + { + "bbox": [ + 225, + 639, + 387, + 673 + ], + "score": 0.94, + "content": "\\Omega : = \\left\\{ w \\in \\sharp _ { \\Delta } : \\ d _ { w } < \\frac { 1 } { C } \\frac { | \\hat { Q } _ { j } | } { | \\sharp _ { \\Delta } | } + \\frac { 2 } { C } \\right\\}", + "type": "interline_equation", + "image_path": "c1a3c559129021b37f42842438d2c9982f76a7a9729b0d9af1681c11fbd9ced2.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 639, + 387, + 656.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 225, + 656.0, + 387, + 673.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 677, + 460, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 462, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 144, + 691 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 144, + 678, + 172, + 688 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 676, + 462, + 691 + ], + "score": 1.0, + "content": "to be chosen below. Then, the second sum in equation 12 can be split as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 694, + 419, + 735 + ], + "lines": [ + { + "bbox": [ + 194, + 694, + 419, + 735 + ], + "spans": [ + { + "bbox": [ + 194, + 694, + 419, + 735 + ], + "score": 0.94, + "content": "\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\left( \\sum _ { w \\in V _ { k } \\cap \\Omega } \\frac { 1 } { d _ { w } + 1 } + \\sum _ { w \\in V _ { k } \\setminus \\Omega } \\frac { 1 } { d _ { w } + 1 } \\right) .", + "type": "interline_equation", + "image_path": "8f9783c339c6a69de25be4c7913ddc8bf319e11eeb5e35da0fffd0d590ee5640.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 694, + 419, + 714.5 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 194, + 714.5, + 419, + 735.0 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "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": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 123, + 81, + 495, + 95 + ], + "lines": [ + { + "bbox": [ + 127, + 82, + 495, + 95 + ], + "spans": [ + { + "bbox": [ + 127, + 82, + 142, + 95 + ], + "score": 1.0, + "content": "(i)", + "type": "text" + }, + { + "bbox": [ + 142, + 82, + 187, + 95 + ], + "score": 0.91, + "content": "| \\nabla \\phi _ { \\ell } | \\le \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 82, + 205, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 206, + 82, + 250, + 95 + ], + "score": 0.92, + "content": "| \\nabla \\psi _ { \\ell } | \\le \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 82, + 287, + 95 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 287, + 82, + 348, + 94 + ], + "score": 0.9, + "content": "0 \\leq \\ell \\leq L - 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\\frac { 1 } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 100, + 326, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 326, + 102, + 364, + 116 + ], + "score": 0.73, + "content": "\\delta < \\gamma _ { m a x } ^ { - 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 100, + 384, + 118 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 385, + 103, + 463, + 116 + ], + "score": 0.87, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 123, + 100, + 463, + 118 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 447, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 124, + 448, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 175, + 140 + ], + "score": 1.0, + "content": "Then there exists", + "type": "text" + }, + { + "bbox": [ + 176, + 126, + 223, + 138 + ], + "score": 0.94, + "content": "Q _ { j } \\subset S _ { 2 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 124, + 266, + 140 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 266, + 125, + 313, + 138 + ], + "score": 0.93, + "content": "| Q _ { j } | > \\delta ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 124, + 346, + 140 + ], + "score": 1.0, + "content": "and for", + "type": "text" + }, + { + "bbox": [ + 346, + 126, + 412, + 137 + ], + "score": 0.91, + "content": "0 \\leq \\ell _ { 0 } \\leq L - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 124, + 448, + 140 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 124, + 448, + 140 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 141, + 376, + 176 + ], + "lines": [ + { + "bbox": [ + 234, + 141, + 376, + 176 + ], + "spans": [ + { + "bbox": [ + 234, + 141, + 376, + 176 + ], + "score": 0.94, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } \\left| \\frac { \\partial h _ { k } ^ { ( \\ell _ { 0 } + 2 ) } } { \\partial h _ { i } ^ { ( \\ell _ { 0 } ) } } \\right| < ( \\alpha \\beta ) ^ { 2 } \\delta ^ { \\frac { 1 } { 4 } } .", + "type": "interline_equation", + "image_path": "31785543f89ea4e42799e24dd197e94beed501078acb90789194d0b4ed233bdd.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 141, + 376, + 158.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 234, + 158.5, + 376, + 176.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 202, + 199 + ], + "score": 1.0, + "content": "Proof. As usual we let", + "type": "text" + }, + { + "bbox": [ + 203, + 187, + 235, + 198 + ], + "score": 0.91, + "content": "d _ { i } : = d", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 186, + 254, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 187, + 305, + 199 + ], + "score": 0.92, + "content": "d _ { j } : = d + s", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 186, + 348, + 199 + ], + "score": 1.0, + "content": ", for some", + "type": "text" + }, + { + "bbox": [ + 349, + 187, + 374, + 198 + ], + "score": 0.9, + "content": "s \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 186, + 505, + 199 + ], + "score": 1.0, + "content": ". 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Without loss of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 101, + 314, + 507, + 339 + ], + "spans": [ + { + "bbox": [ + 101, + 314, + 272, + 339 + ], + "score": 1.0, + "content": "generality we set `0 = 0 and hence h(0)i", + "type": "text" + }, + { + "bbox": [ + 248, + 316, + 288, + 332 + ], + "score": 0.93, + "content": "h _ { i } ^ { ( 0 ) } = x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 317, + 493, + 334 + ], + "score": 1.0, + "content": "; the very same proof applies to any other choice of", + "type": "text" + }, + { + "bbox": [ + 493, + 320, + 503, + 331 + ], + "score": 0.88, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 317, + 507, + 334 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 231, + 330, + 375, + 343 + ], + "spans": [ + { + "bbox": [ + 231, + 330, + 375, + 343 + ], + "score": 1.0, + "content": ", we can apply Lemma 1 and derive", 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"bbox": [ + 105, + 384, + 416, + 403 + ], + "score": 1.0, + "content": "We may expand the power of the augmented normalized adjacency matrix as", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 384, + 416, + 403 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 403, + 411, + 435 + ], + "lines": [ + { + "bbox": [ + 199, + 403, + 411, + 435 + ], + "spans": [ + { + "bbox": [ + 199, + 403, + 411, + 435 + ], + "score": 0.92, + "content": "( \\hat { A } ) _ { i k } ^ { 2 } = \\frac { 1 } { \\sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) } \\frac { 1 } { d _ { w } + 1 } .", + "type": "interline_equation", + "image_path": "4cb7b53c92ecb7797648b1dd887f2a4c08c543a80dee2c7de9c098d608afb1a3.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 403, + 411, + 419.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 199, + 419.0, + 411, + 435.0 + ], + 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{ j } } ( \\hat { A } ) _ { i k } ^ { 2 } = \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { ( d _ { k } + 1 ) ( d _ { i } + 1 ) } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) } \\frac { 1 } { d _ { w } + 1 } = } } \\\\ { { = \\displaystyle \\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\left( \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { 1 } { d _ { j } + 1 } + \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) } \\frac { 1 } { d _ { w } + 1 } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "3f511b5735d645f8188f68708495e0ce8ce32454dd6527389c26f6f5a941e46f.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 126, + 461, + 470, + 485.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 126, + 485.6666666666667, + 470, + 510.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 126, + 510.33333333333337, + 470, + 535.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 504, + 576 + ], + "lines": [ + { + "bbox": [ + 104, + 540, + 502, + 556 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 327, + 556 + ], + "score": 1.0, + "content": "where in the last equality we have again used the fact that", + "type": "text" + }, + { + "bbox": [ + 327, + 541, + 358, + 555 + ], + "score": 0.92, + "content": "k \\in \\hat { Q } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 540, + 400, + 556 + ], + "score": 1.0, + "content": "iff there is", + "type": "text" + }, + { + "bbox": [ + 400, + 542, + 502, + 555 + ], + "score": 0.92, + "content": "w \\in S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 267, + 567 + ], + "score": 1.0, + "content": "To avoid heavy notations, we introduce", + "type": "text" + }, + { + "bbox": [ + 267, + 553, + 385, + 565 + ], + "score": 0.92, + "content": "V _ { k } : = S _ { 1 } ( k ) \\cap S _ { 1 } ( i ) \\cap S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 553, + 506, + 567 + ], + "score": 1.0, + "content": ". Let us first focus on the first", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 564, + 225, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 225, + 577 + ], + "score": 1.0, + "content": "sum in equation 12. We have", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 540, + 506, + 577 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 581, + 453, + 613 + ], + "lines": [ + { + "bbox": [ + 158, + 581, + 453, + 613 + ], + "spans": [ + { + "bbox": [ + 158, + 581, + 453, + 613 + ], + "score": 0.93, + "content": "\\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\sum _ { k \\in Q _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { 1 } { d _ { j } + 1 } \\leq \\frac { 1 } { \\sqrt { d _ { i } + 1 } } | Q _ { j } | \\frac { 1 } { d _ { j } + 1 } \\leq \\frac { 1 } { \\sqrt { d _ { i } + 1 } } \\leq 1 .", + "type": "interline_equation", + "image_path": "39f8d11fceab864d8dd50f0c09555ded49db5f2cd172567f3bdc7a2ba51402cd.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 158, + 581, + 453, + 591.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 158, + 591.6666666666666, + 453, + 602.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 158, + 602.3333333333333, + 453, + 612.9999999999999 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 495, + 633 + ], + "lines": [ + { + "bbox": [ + 104, + 619, + 497, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 351, + 635 + ], + "score": 1.0, + "content": "We now consider the second sum in equation 12. We assume", + "type": "text" + }, + { + "bbox": [ + 352, + 621, + 387, + 633 + ], + "score": 0.93, + "content": "| \\sharp _ { \\Delta } | \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 619, + 432, + 635 + ], + "score": 1.0, + "content": ", otherwise", + "type": "text" + }, + { + "bbox": [ + 432, + 619, + 464, + 634 + ], + "score": 0.93, + "content": "{ \\hat { Q } } _ { j } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 619, + 497, + 635 + ], + "score": 1.0, + "content": ". We let", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 619, + 497, + 635 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 639, + 387, + 673 + ], + "lines": [ + { + "bbox": [ + 225, + 639, + 387, + 673 + ], + "spans": [ + { + "bbox": [ + 225, + 639, + 387, + 673 + ], + "score": 0.94, + "content": "\\Omega : = \\left\\{ w \\in \\sharp _ { \\Delta } : \\ d _ { w } < \\frac { 1 } { C } \\frac { | \\hat { Q } _ { j } | } { | \\sharp _ { \\Delta } | } + \\frac { 2 } { C } \\right\\}", + "type": "interline_equation", + "image_path": "c1a3c559129021b37f42842438d2c9982f76a7a9729b0d9af1681c11fbd9ced2.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 639, + 387, + 656.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 225, + 656.0, + 387, + 673.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 677, + 460, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 462, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 144, + 691 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 144, + 678, + 172, + 688 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 676, + 462, + 691 + ], + "score": 1.0, + "content": "to be chosen below. Then, the second sum in equation 12 can be split as", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 676, + 462, + 691 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 694, + 419, + 735 + ], + "lines": [ + { + "bbox": [ + 194, + 694, + 419, + 735 + ], + "spans": [ + { + "bbox": [ + 194, + 694, + 419, + 735 + ], + "score": 0.94, + "content": "\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\left( \\sum _ { w \\in V _ { k } \\cap \\Omega } \\frac { 1 } { d _ { w } + 1 } + \\sum _ { w \\in V _ { k } \\setminus \\Omega } \\frac { 1 } { d _ { w } + 1 } \\right) .", + "type": "interline_equation", + "image_path": "8f9783c339c6a69de25be4c7913ddc8bf319e11eeb5e35da0fffd0d590ee5640.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 694, + 419, + 714.5 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 194, + 714.5, + 419, + 735.0 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 462, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 463, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 147, + 96 + ], + "score": 1.0, + "content": "Since any", + "type": "text" + }, + { + "bbox": [ + 148, + 83, + 179, + 93 + ], + "score": 0.92, + "content": "w \\in V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 81, + 463, + 96 + ], + "score": 1.0, + "content": "has degree at least three, we can bound the first term in equation 14 as", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 98, + 434, + 136 + ], + "lines": [ + { + "bbox": [ + 176, + 98, + 434, + 136 + ], + "spans": [ + { + "bbox": [ + 176, + 98, + 434, + 136 + ], + "score": 0.95, + "content": "\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\left( \\sum _ { w \\in V _ { k } \\cap \\Omega } \\frac { 1 } { d _ { w } + 1 } \\right) \\leq \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { | V _ { k } \\cap \\Omega | } { 4 } .", + "type": "interline_equation", + "image_path": "f26a843d3e539d2dadf803cd16e729b431a20854f19a2736b3b5c18e55ff3d9d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 176, + 98, + 434, + 110.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 176, + 110.66666666666667, + 434, + 123.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 176, + 123.33333333333334, + 434, + 136.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 192, + 152 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 192, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 192, + 153 + ], + "score": 1.0, + "content": "We now observe that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", 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\\le ( \\frac { 1 } { C } \\frac { | \\hat { Q } _ { j } | } { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } + \\frac { 2 } { C } ) \\frac { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } { 4 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8c36feaad1d7c8048f1c2d71aabb62318de2190b1204aa4f24af08f3eafd987d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 138, + 223, + 471, + 246.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 246.66666666666666, + 471, + 270.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 270.3333333333333, + 471, + 294.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 381, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 382, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 171, + 311 + ], + "score": 1.0, + "content": "by definition of", + "type": "text" + }, + { + "bbox": [ + 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We now bound the second term in equation 14 as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 313, + 436, + 347 + ], + "lines": [ + { + "bbox": [ + 174, + 313, + 436, + 347 + ], + "spans": [ + { + "bbox": [ + 174, + 313, + 436, + 347 + ], + "score": 0.94, + "content": "\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in V _ { k } \\setminus \\Omega } \\frac { 1 } { d _ { w } + 1 } \\leq \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } | V _ { k } \\setminus \\Omega |", + "type": "interline_equation", + "image_path": "27847d1c42582677aa26c7195856d42ddf3e64d957b4388725272affc400669d.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 174, + 313, + 436, + 324.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 174, + 324.3333333333333, + 436, + 335.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 174, + 335.66666666666663, + 436, + 346.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 371, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 371, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 206, + 369 + ], + "score": 1.0, + "content": "where we have used that", + "type": "text" + }, + { + "bbox": [ + 207, + 353, + 283, + 367 + ], + "score": 0.94, + "content": "d _ { w } ^ { - 1 } \\leq C | \\sharp _ { \\Delta } | / | \\hat { Q } _ { j } |", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 353, + 293, + 369 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 294, + 354, + 342, + 367 + ], + "score": 0.92, + "content": "w \\in V _ { k } \\setminus \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 353, + 371, + 369 + ], + "score": 1.0, + "content": ". 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According to equation 11 it suffices to show", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 126, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 126, + 473 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 467, + 358, + 499 + ], + "lines": [ + { + "bbox": [ + 253, + 467, + 358, + 499 + ], + "spans": [ + { + "bbox": [ + 253, + 467, + 358, + 499 + ], + "score": 0.94, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\delta ^ { 1 / 4 } .", + "type": "interline_equation", + "image_path": "cf4a4b5b6a10b3ce955d57a38d5b77d865ecb8f2d9ef4f221364051eab389b30.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 253, + 467, + 358, + 483.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 253, + 483.0, + 358, + 499.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 500, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "From equation 12 and equation 13 we derive that the left hand side of the equation above is bounded", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 511, + 120, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 120, + 526 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 527, + 441, + 610 + ], + "lines": [ + { + "bbox": [ + 170, + 527, + 441, + 610 + ], + "spans": [ + { + "bbox": [ + 170, + 527, + 441, + 610 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { 1 } { | Q _ { j } | } \\displaystyle \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\frac { 1 } { | Q _ { j } | } + \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d _ { w } + 1 } \\right) } \\\\ { \\leq \\delta + \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d _ { w } + 1 } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "04e94f2c52890b4fd31f0b7e770bc146ceb3b9ea27a5967cb99952b5a01ddd5d.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 170, + 527, + 441, + 554.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 170, + 554.6666666666666, + 441, + 582.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 170, + 582.3333333333333, + 441, + 609.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 507, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 351, + 629 + ], + "score": 1.0, + "content": "where in the last inequality we have used Lemma 14 to bound", + "type": "text" + }, + { + "bbox": [ + 351, + 614, + 380, + 628 + ], + "score": 0.92, + "content": "| Q _ { j } | ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 614, + 394, + 629 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 394, + 616, + 400, + 626 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 614, + 506, + 629 + ], + "score": 1.0, + "content": ". In particular we note that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 115, + 641 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 115, + 628, + 146, + 640 + ], + "score": 0.92, + "content": "\\sharp _ { \\Delta } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 628, + 166, + 641 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 167, + 627, + 198, + 641 + ], + "score": 0.93, + "content": "{ \\hat { Q } } _ { j } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 628, + 385, + 641 + ], + "score": 1.0, + "content": ", and the bound would be simply controlled by", + "type": "text" + }, + { + "bbox": [ + 385, + 629, + 391, + 639 + ], + "score": 0.82, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 628, + 466, + 641 + ], + "score": 1.0, + "content": "as claimed. When", + "type": "text" + }, + { + "bbox": [ + 467, + 628, + 502, + 641 + ], + "score": 0.92, + "content": "\\vert \\sharp _ { \\Delta } \\vert > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 628, + 506, + 641 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 639, + 439, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 439, + 653 + ], + "score": 1.0, + "content": "we can use equation 15 and equation 16 to estimate the second term from above by", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 655, + 486, + 732 + ], + "lines": [ + { + "bbox": [ + 125, + 655, + 486, + 732 + ], + "spans": [ + { + "bbox": [ + 125, + 655, + 486, + 732 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } 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\\hat { Q } _ { j } | } { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } + \\frac { 2 } { C } ) \\frac { | \\| \\hat { { \\boldsymbol { q } } } _ { \\Delta } | } { 4 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8c36feaad1d7c8048f1c2d71aabb62318de2190b1204aa4f24af08f3eafd987d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 138, + 223, + 471, + 246.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 138, + 246.66666666666666, + 471, + 270.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 138, + 270.3333333333333, + 471, + 294.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 381, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 382, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 171, + 311 + ], + "score": 1.0, + "content": "by definition of", + "type": "text" + }, + { + "bbox": [ + 171, + 298, + 179, + 307 + ], + "score": 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We now bound the second term in equation 14 as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 295, + 382, + 311 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 313, + 436, + 347 + ], + "lines": [ + { + "bbox": [ + 174, + 313, + 436, + 347 + ], + "spans": [ + { + "bbox": [ + 174, + 313, + 436, + 347 + ], + "score": 0.94, + "content": "\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\sum _ { w \\in V _ { k } \\setminus \\Omega } \\frac { 1 } { d _ { w } + 1 } \\leq \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } | V _ { k } \\setminus \\Omega |", + "type": "interline_equation", + "image_path": "27847d1c42582677aa26c7195856d42ddf3e64d957b4388725272affc400669d.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 174, + 313, + 436, + 324.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 174, + 324.3333333333333, + 436, + 335.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 174, + 335.66666666666663, + 436, + 346.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 371, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 371, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 206, + 369 + ], + "score": 1.0, + "content": "where we have used that", + "type": "text" + }, + { + "bbox": [ + 207, + 353, + 283, + 367 + ], + "score": 0.94, + "content": "d _ { w } ^ { - 1 } \\leq C | \\sharp _ { \\Delta } | / | \\hat { Q } _ { j } |", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 353, + 293, + 369 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 294, + 354, + 342, + 367 + ], + "score": 0.92, + "content": "w \\in V _ { k } \\setminus \\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 353, + 371, + 369 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 353, + 371, + 369 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 371, + 414, + 401 + ], + "lines": [ + { + "bbox": [ + 198, + 371, + 414, + 401 + ], + "spans": [ + { + "bbox": [ + 198, + 371, + 414, + 401 + ], + "score": 0.89, + "content": "\\frac { | V _ { k } \\setminus \\Omega | } { \\sqrt { d _ { k } + 1 } } \\leq \\frac { | V _ { k } | } { \\sqrt { | S _ { 1 } ( k ) | } } \\leq \\frac { | V _ { k } | } { \\sqrt { | V _ { k } | } } \\leq \\sqrt { | V _ { k } | } \\leq \\sqrt { | \\sharp _ { \\Delta } | }", + "type": "interline_equation", + "image_path": "b560d20e6d08863233482ae5f9c2e58e35b263065b263e465c9dbb89969b904d.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 198, + 371, + 414, + 401 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 153, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 154, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 154, + 417 + ], + "score": 1.0, + "content": "we see that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 405, + 154, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 412, + 433, + 447 + ], + "lines": [ + { + "bbox": [ + 179, + 412, + 433, + 447 + ], + "spans": [ + { + "bbox": [ + 179, + 412, + 433, + 447 + ], + "score": 0.91, + "content": "\\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } | V _ { k } \\setminus \\Omega | \\leq \\frac { C | \\sharp \\Delta | } { | \\hat { Q } _ { j } | } \\sqrt { | \\sharp \\Delta | } | \\hat { Q } _ { j } | = C | \\sharp \\Delta | ^ { \\frac { 3 } { 2 } } .", + "type": "interline_equation", + "image_path": "621604bc78a9a4832cc147688ecfa1038a795003f62cc92ad854103326517814.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 179, + 412, + 433, + 423.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 179, + 423.6666666666667, + 433, + 435.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 179, + 435.33333333333337, + 433, + 447.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 448, + 504, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "We are now ready to complete the proof of the theorem. According to equation 11 it suffices to show", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 126, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 126, + 473 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 447, + 506, + 473 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 467, + 358, + 499 + ], + "lines": [ + { + "bbox": [ + 253, + 467, + 358, + 499 + ], + "spans": [ + { + "bbox": [ + 253, + 467, + 358, + 499 + ], + "score": 0.94, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\delta ^ { 1 / 4 } .", + "type": "interline_equation", + "image_path": "cf4a4b5b6a10b3ce955d57a38d5b77d865ecb8f2d9ef4f221364051eab389b30.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 253, + 467, + 358, + 483.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 253, + 483.0, + 358, + 499.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 500, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "From equation 12 and equation 13 we derive that the left hand side of the equation above is bounded", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 511, + 120, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 120, + 526 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 499, + 505, + 526 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 527, + 441, + 610 + ], + "lines": [ + { + "bbox": [ + 170, + 527, + 441, + 610 + ], + "spans": [ + { + "bbox": [ + 170, + 527, + 441, + 610 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { 1 } { | Q _ { j } | } \\displaystyle \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\frac { 1 } { | Q _ { j } | } + \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d _ { w } + 1 } \\right) } \\\\ { \\leq \\delta + \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d _ { w } + 1 } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "04e94f2c52890b4fd31f0b7e770bc146ceb3b9ea27a5967cb99952b5a01ddd5d.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 170, + 527, + 441, + 554.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 170, + 554.6666666666666, + 441, + 582.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 170, + 582.3333333333333, + 441, + 609.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 507, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 351, + 629 + ], + "score": 1.0, + "content": "where in the last inequality we have used Lemma 14 to bound", + "type": "text" + }, + { + "bbox": [ + 351, + 614, + 380, + 628 + ], + "score": 0.92, + "content": "| Q _ { j } | ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 614, + 394, + 629 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 394, + 616, + 400, + 626 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 614, + 506, + 629 + ], + "score": 1.0, + "content": ". In particular we note that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 115, + 641 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 115, + 628, + 146, + 640 + ], + "score": 0.92, + "content": "\\sharp _ { \\Delta } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 628, + 166, + 641 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 167, + 627, + 198, + 641 + ], + "score": 0.93, + "content": "{ \\hat { Q } } _ { j } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 628, + 385, + 641 + ], + "score": 1.0, + "content": ", and the bound would be simply controlled by", + "type": "text" + }, + { + "bbox": [ + 385, + 629, + 391, + 639 + ], + "score": 0.82, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 628, + 466, + 641 + ], + "score": 1.0, + "content": "as claimed. When", + "type": "text" + }, + { + "bbox": [ + 467, + 628, + 502, + 641 + ], + "score": 0.92, + "content": "\\vert \\sharp _ { \\Delta } \\vert > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 628, + 506, + 641 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 639, + 439, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 439, + 653 + ], + "score": 1.0, + "content": "we can use equation 15 and equation 16 to estimate the second term from above by", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 614, + 506, + 653 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 655, + 486, + 732 + ], + "lines": [ + { + "bbox": [ + 125, + 655, + 486, + 732 + ], + "spans": [ + { + "bbox": [ + 125, + 655, + 486, + 732 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\frac { 1 } { | Q _ { j } | } \\left( \\displaystyle \\sum _ { k \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sqrt { d _ { k } + 1 } } \\displaystyle \\sum _ { w \\in V _ { k } } \\frac { 1 } { d w + 1 } \\right) \\leq } \\\\ & { \\frac { 1 } { | Q _ { j } | } \\left( \\left( \\frac { 1 } { \\bar { C } } \\frac { | \\hat { Q } _ { j } | } { | \\sharp _ { \\Delta } | } + \\frac { 2 } { \\bar { C } } \\right) \\frac { | \\sharp _ { \\Delta } | } { 4 } \\right) + \\frac { 1 } { | Q _ { j } | } \\left( C | \\sharp _ { \\Delta } | ^ { \\frac { 3 } { 2 } } \\right) \\leq \\frac { 1 } { 4 } \\left( \\frac { 1 } { \\bar { C } } + \\frac { 2 | \\sharp _ { \\Delta } | } { \\bar { C } | Q _ { j } | } \\right) + \\frac { C | \\sharp _ { \\Delta } | } { | Q _ { j } | } \\sqrt { | \\sharp _ { \\Delta } | } . } \\end{array}", + "type": "interline_equation", + "image_path": "78d3bb8ebc7cde2d93ab31d327be997585260efb55bc68db7ff4a41e8622500b.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 125, + 655, + 486, + 680.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 125, + 680.6666666666666, + 486, + 706.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 125, + 706.3333333333333, + 486, + 731.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 232, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 233, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 233, + 96 + ], + "score": 1.0, + "content": "By applying Lemma 14 we get", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 99, + 441, + 127 + ], + "lines": [ + { + "bbox": [ + 170, + 99, + 441, + 127 + ], + "spans": [ + { + "bbox": [ + 170, + 99, + 441, + 127 + ], + "score": 0.94, + "content": "\\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + \\frac { 2 | \\sharp \\Delta | } { C | Q _ { j } | } \\right) + \\frac { C | \\sharp \\Delta | } { | Q _ { j } | } \\sqrt { | \\sharp \\Delta | } \\leq \\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + 2 \\frac { \\delta } { C } \\right) + C \\delta \\sqrt { | \\sharp \\Delta | } .", + "type": "interline_equation", + "image_path": "f4bca66dbb8a95e63a1446d43c3ccb00c4f4b9bcd53bd6c7d80cbc445091cc2f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 170, + 99, + 441, + 127 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 413, + 145 + ], + "lines": [ + { + "bbox": [ + 104, + 128, + 414, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 128, + 171, + 149 + ], + "score": 1.0, + "content": "We now choose", + "type": "text" + }, + { + "bbox": [ + 171, + 131, + 217, + 143 + ], + "score": 0.92, + "content": "C = \\delta ^ { - 1 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 128, + 414, + 149 + ], + "score": 1.0, + "content": ", so that the previous quantity can be bounded by", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 149, + 475, + 177 + ], + "lines": [ + { + "bbox": [ + 136, + 149, + 475, + 177 + ], + "spans": [ + { + "bbox": [ + 136, + 149, + 475, + 177 + ], + "score": 0.92, + "content": "\\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + 2 \\frac { \\delta } { C } \\right) + C \\delta \\sqrt { | \\sharp _ { \\Delta } | } \\leq \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } } \\sqrt { \\delta | \\sharp _ { \\Delta } | } \\leq \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } }", + "type": "interline_equation", + "image_path": "27166968dce570051905b32bb49e0ad01f70b1cdc6ce88093ae8ba6f991608c2.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 136, + 149, + 475, + 177 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 450, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 450, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 450, + 194 + ], + "score": 1.0, + "content": "where in the last inequality we have used equation 10. Therefore, we have shown that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 197, + 412, + 229 + ], + "lines": [ + { + "bbox": [ + 198, + 197, + 412, + 229 + ], + "spans": [ + { + "bbox": [ + 198, + 197, + 412, + 229 + ], + "score": 0.92, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\delta + \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } } \\leq 3 \\delta ^ { \\frac { 1 } { 4 } }", + "type": "interline_equation", + "image_path": "c31ddcaa5dda7a9fe4cf8d3b6228abd5da10869a3cbad58f4af7f345d018094d.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 197, + 412, + 213.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 198, + 213.0, + 412, + 229.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 504, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 208, + 246 + ], + "score": 1.0, + "content": "where we have used that", + "type": "text" + }, + { + "bbox": [ + 208, + 234, + 232, + 244 + ], + "score": 0.89, + "content": "\\delta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 234, + 506, + 246 + ], + "score": 1.0, + "content": ". This completes the proof (once we absorb the extra factor 3 in the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 142, + 257 + ], + "score": 1.0, + "content": "constant", + "type": "text" + }, + { + "bbox": [ + 142, + 246, + 156, + 256 + ], + "score": 0.89, + "content": "\\alpha \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 244, + 223, + 257 + ], + "score": 1.0, + "content": "in equation 11).", + "type": "text" + }, + { + "bbox": [ + 495, + 246, + 505, + 255 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 104, + 262, + 507, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 235, + 278 + ], + "score": 1.0, + "content": "Remark 15. The requirement", + "type": "text" + }, + { + "bbox": [ + 235, + 263, + 324, + 277 + ], + "score": 0.92, + "content": "\\delta \\sqrt { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\ < \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 262, + 507, + 278 + ], + "score": 1.0, + "content": "can be replaced by a more general bound", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 275, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 107, + 276, + 191, + 290 + ], + "score": 0.92, + "content": "\\delta \\sqrt { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } < r", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 275, + 479, + 291 + ], + "score": 1.0, + "content": ". The argument above extends to this case up to renaming the constant", + "type": "text" + }, + { + "bbox": [ + 480, + 278, + 493, + 289 + ], + "score": 0.82, + "content": "\\alpha \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 275, + 506, + 291 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 288, + 286, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 276, + 301 + ], + "score": 1.0, + "content": "the statement so to include an extra factor", + "type": "text" + }, + { + "bbox": [ + 277, + 291, + 282, + 298 + ], + "score": 0.57, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 288, + 286, + 301 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 507, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 216, + 313 + ], + "score": 1.0, + "content": "We note that the condition", + "type": "text" + }, + { + "bbox": [ + 216, + 299, + 293, + 312 + ], + "score": 0.92, + "content": "\\delta \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} < r", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 298, + 507, + 313 + ], + "score": 1.0, + "content": "would be stronger than the one appearing in (ii) of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 309, + 492, + 327 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 291, + 327 + ], + "score": 1.0, + "content": "Theorem 4. In this regard, we recall that for a", + "type": "text" + }, + { + "bbox": [ + 291, + 312, + 298, + 321 + ], + "score": 0.65, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 309, + 407, + 327 + ], + "score": 1.0, + "content": "-tree the curvature satisfies", + "type": "text" + }, + { + "bbox": [ + 407, + 311, + 486, + 325 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname { R i c } ( i , j ) = - 2 + \\frac { 4 } { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 309, + 492, + 327 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 239, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 239, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 239, + 345 + ], + "score": 1.0, + "content": "We can also prove Proposition 5:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 108, + 346, + 396, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 393, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 179, + 363 + ], + "score": 1.0, + "content": "Proposition 5. If", + "type": "text" + }, + { + "bbox": [ + 180, + 347, + 252, + 359 + ], + "score": 0.87, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 343, + 280, + 363 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 280, + 347, + 302, + 359 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 343, + 326, + 363 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 326, + 346, + 393, + 361 + ], + "score": 0.92, + "content": "\\textstyle \\lambda _ { 1 } / 2 \\geq h _ { G } \\geq { \\frac { k } { 2 } }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 367, + 486, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 487, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 487, + 381 + ], + "score": 1.0, + "content": "Proof. This follows as an immediate Corollary of Theorem 2 and (Lin et al., 2011, Theorem 4.2).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 506, + 518 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 506, + 407 + ], + "score": 1.0, + "content": "Betweenness centrality to measure bottleneck. In equation 2 we have derived how the topology", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 103, + 403, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 103, + 403, + 383, + 420 + ], + "score": 1.0, + "content": "of the graph affects the dependence of the hidden node representation", + "type": "text" + }, + { + "bbox": [ + 383, + 403, + 410, + 418 + ], + "score": 0.94, + "content": "h _ { i } ^ { ( r + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 403, + 491, + 420 + ], + "score": 1.0, + "content": "on the input feature", + "type": "text" + }, + { + "bbox": [ + 491, + 408, + 502, + 417 + ], + "score": 0.83, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 403, + 507, + 420 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 102, + 414, + 509, + 437 + ], + "spans": [ + { + "bbox": [ + 102, + 414, + 146, + 437 + ], + "score": 1.0, + "content": "for nodes", + "type": "text" + }, + { + "bbox": [ + 147, + 420, + 161, + 430 + ], + "score": 0.62, + "content": "i , s", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 414, + 207, + 437 + ], + "score": 1.0, + "content": "at distance", + "type": "text" + }, + { + "bbox": [ + 208, + 419, + 231, + 430 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 414, + 335, + 437 + ], + "score": 1.0, + "content": ". We note that in this case", + "type": "text" + }, + { + "bbox": [ + 336, + 417, + 359, + 431 + ], + "score": 0.93, + "content": "\\hat { A } _ { i s } ^ { r + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 414, + 509, + 437 + ], + "score": 1.0, + "content": "is exactly measuring the number of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 188, + 443 + ], + "score": 1.0, + "content": "minimal paths from", + "type": "text" + }, + { + "bbox": [ + 193, + 429, + 204, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 204, + 432, + 210, + 440 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 429, + 297, + 443 + ], + "score": 1.0, + "content": ". If the receptive field", + "type": "text" + }, + { + "bbox": [ + 298, + 430, + 332, + 442 + ], + "score": 0.93, + "content": "B _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "is a binary tree, then we have seen that the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "entry of the power matrix decays exponentially. The reason for such decay stems from the existence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "of exponentially many nodes in the receptive field combined with the lack of multiple minimal paths", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "(shortcuts). When such conditions hold, most of the minimal paths go through the same nodes,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "which is exactly what happens for the tree where each node is in the minimal paths between different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "branches. Since the frequency in which a node appears in the minimal path of distinct pairs of nodes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "is measured by the betweenness centrality Freeman (1977), we propose a topological characterization", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 298, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 298, + 519 + ], + "score": 1.0, + "content": "of the ‘bottleneckedness’ of a graph as follows:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 520, + 503, + 544 + ], + "lines": [ + { + "bbox": [ + 104, + 519, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 312, + 537 + ], + "score": 1.0, + "content": "Definition 9 (bottleneck). The bottleneck-value of", + "type": "text" + }, + { + "bbox": [ + 312, + 522, + 321, + 532 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 519, + 331, + 537 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 332, + 520, + 410, + 534 + ], + "score": 0.92, + "content": "\\begin{array} { r } { b _ { G } : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } b ( i ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 519, + 440, + 537 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 440, + 521, + 456, + 533 + ], + "score": 0.89, + "content": "b ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 519, + 506, + 537 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 532, + 243, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 234, + 544 + ], + "score": 1.0, + "content": "betweenness centrality on node", + "type": "text" + }, + { + "bbox": [ + 234, + 534, + 238, + 542 + ], + "score": 0.58, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 532, + 243, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 422, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 423, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 338, + 566 + ], + "score": 1.0, + "content": "From a standard combinatorial argument it follows that if", + "type": "text" + }, + { + "bbox": [ + 338, + 554, + 347, + 563 + ], + "score": 0.85, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 552, + 423, + 566 + ], + "score": 1.0, + "content": "is connected, then", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 569, + 363, + 600 + ], + "lines": [ + { + "bbox": [ + 248, + 569, + 363, + 600 + ], + "spans": [ + { + "bbox": [ + 248, + 569, + 363, + 600 + ], + "score": 0.95, + "content": "b _ { G } = \\frac { 1 } { n } \\sum _ { i , j } \\left( d _ { G } ( i , j ) - 1 \\right) .", + "type": "interline_equation", + "image_path": "7c8b7f038bc6553e8a24c7dd9cf4a88d5a51dbfe47af33c0675480105d6a1ebc.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 248, + 569, + 363, + 584.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 248, + 584.5, + 363, + 600.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 160, + 618 + ], + "score": 1.0, + "content": "We note that", + "type": "text" + }, + { + "bbox": [ + 160, + 605, + 191, + 616 + ], + "score": 0.91, + "content": "b _ { G } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 604, + 204, + 618 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 205, + 605, + 214, + 615 + ], + "score": 0.85, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 604, + 305, + 618 + ], + "score": 1.0, + "content": "is the complete graph", + "type": "text" + }, + { + "bbox": [ + 306, + 605, + 320, + 616 + ], + "score": 0.89, + "content": "K _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 604, + 370, + 618 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 371, + 605, + 383, + 616 + ], + "score": 0.89, + "content": "b _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "determines how far the given", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 177, + 628 + ], + "score": 1.0, + "content": "topology is from", + "type": "text" + }, + { + "bbox": [ + 177, + 617, + 192, + 627 + ], + "score": 0.87, + "content": "K _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 616, + 495, + 628 + ], + "score": 1.0, + "content": ", with the latter representing the limit case of a fully connected layer Alon", + "type": "text" + }, + { + "bbox": [ + 496, + 616, + 505, + 626 + ], + "score": 0.43, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Yahav (2021) where no bottleneck may occur as any pair of nodes would be neighbours. This further", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "supports our intuition that the betweenness centrality is a good topological candidate for providing a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 649, + 323, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 323, + 661 + ], + "score": 1.0, + "content": "global measurement of bottleneckedness in the graph.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "It also follows from equation 17 that any update to the graph topology consisting of edge additions", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 171, + 689 + ], + "score": 1.0, + "content": "would decrease", + "type": "text" + }, + { + "bbox": [ + 171, + 677, + 183, + 688 + ], + "score": 0.87, + "content": "b _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 677, + 367, + 689 + ], + "score": 1.0, + "content": "and thus reduce the bottleneck. The quantity", + "type": "text" + }, + { + "bbox": [ + 368, + 677, + 380, + 688 + ], + "score": 0.87, + "content": "b _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "is global in nature though and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 381, + 700 + ], + "score": 1.0, + "content": "hence lacks robustness. As a pedagogical example, consider a barbell", + "type": "text" + }, + { + "bbox": [ + 382, + 687, + 438, + 699 + ], + "score": 0.92, + "content": "G ( m , 2 r + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 687, + 461, + 700 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 462, + 689, + 472, + 698 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "the size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 290, + 711 + ], + "score": 1.0, + "content": "of the two cliques joined by a path of length", + "type": "text" + }, + { + "bbox": [ + 290, + 699, + 319, + 709 + ], + "score": 0.9, + "content": "2 r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 698, + 412, + 711 + ], + "score": 1.0, + "content": "and focus on the edge", + "type": "text" + }, + { + "bbox": [ + 412, + 700, + 436, + 710 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "in the middle of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 179, + 722 + ], + "score": 1.0, + "content": "such path. Nodes", + "type": "text" + }, + { + "bbox": [ + 179, + 711, + 184, + 720 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 709, + 202, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 711, + 208, + 721 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are central to the graph, in the sense that most minimal paths go through", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 294, + 733 + ], + "score": 1.0, + "content": "them and indeed their betweenness centrality is", + "type": "text" + }, + { + "bbox": [ + 294, + 720, + 435, + 732 + ], + "score": 0.92, + "content": "\\bar { b } ( i ) = b ( j ) = ( m + r ) ^ { 2 } + ( m + r )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ". If now we add a", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + } + ], + "page_idx": 22, + "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": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "23", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 369, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 496, + 370, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 496, + 370, + 505, + 378 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 232, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 233, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 233, + 96 + ], + "score": 1.0, + "content": "By applying Lemma 14 we get", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 233, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 99, + 441, + 127 + ], + "lines": [ + { + "bbox": [ + 170, + 99, + 441, + 127 + ], + "spans": [ + { + "bbox": [ + 170, + 99, + 441, + 127 + ], + "score": 0.94, + "content": "\\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + \\frac { 2 | \\sharp \\Delta | } { C | Q _ { j } | } \\right) + \\frac { C | \\sharp \\Delta | } { | Q _ { j } | } \\sqrt { | \\sharp \\Delta | } \\leq \\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + 2 \\frac { \\delta } { C } \\right) + C \\delta \\sqrt { | \\sharp \\Delta | } .", + "type": "interline_equation", + "image_path": "f4bca66dbb8a95e63a1446d43c3ccb00c4f4b9bcd53bd6c7d80cbc445091cc2f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 170, + 99, + 441, + 127 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 132, + 413, + 145 + ], + "lines": [ + { + "bbox": [ + 104, + 128, + 414, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 128, + 171, + 149 + ], + "score": 1.0, + "content": "We now choose", + "type": "text" + }, + { + "bbox": [ + 171, + 131, + 217, + 143 + ], + "score": 0.92, + "content": "C = \\delta ^ { - 1 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 128, + 414, + 149 + ], + "score": 1.0, + "content": ", so that the previous quantity can be bounded by", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 128, + 414, + 149 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 149, + 475, + 177 + ], + "lines": [ + { + "bbox": [ + 136, + 149, + 475, + 177 + ], + "spans": [ + { + "bbox": [ + 136, + 149, + 475, + 177 + ], + "score": 0.92, + "content": "\\frac { 1 } { 4 } \\left( \\frac { 1 } { C } + 2 \\frac { \\delta } { C } \\right) + C \\delta \\sqrt { | \\sharp _ { \\Delta } | } \\leq \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } } \\sqrt { \\delta | \\sharp _ { \\Delta } | } \\leq \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } }", + "type": "interline_equation", + "image_path": "27166968dce570051905b32bb49e0ad01f70b1cdc6ce88093ae8ba6f991608c2.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 136, + 149, + 475, + 177 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 450, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 450, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 450, + 194 + ], + "score": 1.0, + "content": "where in the last inequality we have used equation 10. Therefore, we have shown that", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 180, + 450, + 194 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 197, + 412, + 229 + ], + "lines": [ + { + "bbox": [ + 198, + 197, + 412, + 229 + ], + "spans": [ + { + "bbox": [ + 198, + 197, + 412, + 229 + ], + "score": 0.92, + "content": "\\frac { 1 } { | Q _ { j } | } \\sum _ { k \\in Q _ { j } } ( \\hat { A } ^ { 2 } ) _ { i k } \\leq \\delta + \\frac { 1 } { 4 } \\left( \\delta ^ { \\frac { 1 } { 4 } } + 2 \\delta ^ { \\frac { 5 } { 4 } } \\right) + \\delta ^ { \\frac { 1 } { 4 } } \\leq 3 \\delta ^ { \\frac { 1 } { 4 } }", + "type": "interline_equation", + "image_path": "c31ddcaa5dda7a9fe4cf8d3b6228abd5da10869a3cbad58f4af7f345d018094d.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 197, + 412, + 213.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 198, + 213.0, + 412, + 229.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 504, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 208, + 246 + ], + "score": 1.0, + "content": "where we have used that", + "type": "text" + }, + { + "bbox": [ + 208, + 234, + 232, + 244 + ], + "score": 0.89, + "content": "\\delta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 234, + 506, + 246 + ], + "score": 1.0, + "content": ". This completes the proof (once we absorb the extra factor 3 in the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 142, + 257 + ], + "score": 1.0, + "content": "constant", + "type": "text" + }, + { + "bbox": [ + 142, + 246, + 156, + 256 + ], + "score": 0.89, + "content": "\\alpha \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 244, + 223, + 257 + ], + "score": 1.0, + "content": "in equation 11).", + "type": "text" + }, + { + "bbox": [ + 495, + 246, + 505, + 255 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 234, + 506, + 257 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 104, + 262, + 507, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 235, + 278 + ], + "score": 1.0, + "content": "Remark 15. The requirement", + "type": "text" + }, + { + "bbox": [ + 235, + 263, + 324, + 277 + ], + "score": 0.92, + "content": "\\delta \\sqrt { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } \\ < \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 262, + 507, + 278 + ], + "score": 1.0, + "content": "can be replaced by a more general bound", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 275, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 107, + 276, + 191, + 290 + ], + "score": 0.92, + "content": "\\delta \\sqrt { \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} } < r", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 275, + 479, + 291 + ], + "score": 1.0, + "content": ". The argument above extends to this case up to renaming the constant", + "type": "text" + }, + { + "bbox": [ + 480, + 278, + 493, + 289 + ], + "score": 0.82, + "content": "\\alpha \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 275, + 506, + 291 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 288, + 286, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 276, + 301 + ], + "score": 1.0, + "content": "the statement so to include an extra factor", + "type": "text" + }, + { + "bbox": [ + 277, + 291, + 282, + 298 + ], + "score": 0.57, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 288, + 286, + 301 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 262, + 507, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 298, + 507, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 216, + 313 + ], + "score": 1.0, + "content": "We note that the condition", + "type": "text" + }, + { + "bbox": [ + 216, + 299, + 293, + 312 + ], + "score": 0.92, + "content": "\\delta \\operatorname* { m a x } \\{ d _ { i } , d _ { j } \\} < r", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 298, + 507, + 313 + ], + "score": 1.0, + "content": "would be stronger than the one appearing in (ii) of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 309, + 492, + 327 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 291, + 327 + ], + "score": 1.0, + "content": "Theorem 4. In this regard, we recall that for a", + "type": "text" + }, + { + "bbox": [ + 291, + 312, + 298, + 321 + ], + "score": 0.65, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 309, + 407, + 327 + ], + "score": 1.0, + "content": "-tree the curvature satisfies", + "type": "text" + }, + { + "bbox": [ + 407, + 311, + 486, + 325 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname { R i c } ( i , j ) = - 2 + \\frac { 4 } { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 309, + 492, + 327 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 298, + 507, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 239, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 239, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 239, + 345 + ], + "score": 1.0, + "content": "We can also prove Proposition 5:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 331, + 239, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 346, + 396, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 393, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 179, + 363 + ], + "score": 1.0, + "content": "Proposition 5. If", + "type": "text" + }, + { + "bbox": [ + 180, + 347, + 252, + 359 + ], + "score": 0.87, + "content": "\\operatorname { R i c } ( i , j ) \\geq k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 343, + 280, + 363 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 280, + 347, + 302, + 359 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 343, + 326, + 363 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 326, + 346, + 393, + 361 + ], + "score": 0.92, + "content": "\\textstyle \\lambda _ { 1 } / 2 \\geq h _ { G } \\geq { \\frac { k } { 2 } }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 343, + 393, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 367, + 486, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 487, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 487, + 381 + ], + "score": 1.0, + "content": "Proof. This follows as an immediate Corollary of Theorem 2 and (Lin et al., 2011, Theorem 4.2).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 367, + 487, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 506, + 518 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 506, + 407 + ], + "score": 1.0, + "content": "Betweenness centrality to measure bottleneck. In equation 2 we have derived how the topology", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 103, + 403, + 507, + 420 + ], + "spans": [ + { + "bbox": [ + 103, + 403, + 383, + 420 + ], + "score": 1.0, + "content": "of the graph affects the dependence of the hidden node representation", + "type": "text" + }, + { + "bbox": [ + 383, + 403, + 410, + 418 + ], + "score": 0.94, + "content": "h _ { i } ^ { ( r + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 403, + 491, + 420 + ], + "score": 1.0, + "content": "on the input feature", + "type": "text" + }, + { + "bbox": [ + 491, + 408, + 502, + 417 + ], + "score": 0.83, + "content": "x _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 403, + 507, + 420 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 102, + 414, + 509, + 437 + ], + "spans": [ + { + "bbox": [ + 102, + 414, + 146, + 437 + ], + "score": 1.0, + "content": "for nodes", + "type": "text" + }, + { + "bbox": [ + 147, + 420, + 161, + 430 + ], + "score": 0.62, + "content": "i , s", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 414, + 207, + 437 + ], + "score": 1.0, + "content": "at distance", + "type": "text" + }, + { + "bbox": [ + 208, + 419, + 231, + 430 + ], + "score": 0.89, + "content": "r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 414, + 335, + 437 + ], + "score": 1.0, + "content": ". We note that in this case", + "type": "text" + }, + { + "bbox": [ + 336, + 417, + 359, + 431 + ], + "score": 0.93, + "content": "\\hat { A } _ { i s } ^ { r + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 414, + 509, + 437 + ], + "score": 1.0, + "content": "is exactly measuring the number of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 188, + 443 + ], + "score": 1.0, + "content": "minimal paths from", + "type": "text" + }, + { + "bbox": [ + 193, + 429, + 204, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 204, + 432, + 210, + 440 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 429, + 297, + 443 + ], + "score": 1.0, + "content": ". If the receptive field", + "type": "text" + }, + { + "bbox": [ + 298, + 430, + 332, + 442 + ], + "score": 0.93, + "content": "B _ { r + 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "is a binary tree, then we have seen that the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "entry of the power matrix decays exponentially. The reason for such decay stems from the existence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "of exponentially many nodes in the receptive field combined with the lack of multiple minimal paths", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "(shortcuts). When such conditions hold, most of the minimal paths go through the same nodes,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "which is exactly what happens for the tree where each node is in the minimal paths between different", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "branches. Since the frequency in which a node appears in the minimal path of distinct pairs of nodes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "is measured by the betweenness centrality Freeman (1977), we propose a topological characterization", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 298, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 298, + 519 + ], + "score": 1.0, + "content": "of the ‘bottleneckedness’ of a graph as follows:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22, + "bbox_fs": [ + 102, + 389, + 509, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 520, + 503, + 544 + ], + "lines": [ + { + "bbox": [ + 104, + 519, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 312, + 537 + ], + "score": 1.0, + "content": "Definition 9 (bottleneck). The bottleneck-value of", + "type": "text" + }, + { + "bbox": [ + 312, + 522, + 321, + 532 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 519, + 331, + 537 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 332, + 520, + 410, + 534 + ], + "score": 0.92, + "content": "\\begin{array} { r } { b _ { G } : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } b ( i ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 519, + 440, + 537 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 440, + 521, + 456, + 533 + ], + "score": 0.89, + "content": "b ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 519, + 506, + 537 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 532, + 243, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 234, + 544 + ], + "score": 1.0, + "content": "betweenness centrality on node", + "type": "text" + }, + { + "bbox": [ + 234, + 534, + 238, + 542 + ], + "score": 0.58, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 532, + 243, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 519, + 506, + 544 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 422, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 423, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 338, + 566 + ], + "score": 1.0, + "content": "From a standard combinatorial argument it follows that if", + "type": "text" + }, + { + "bbox": [ + 338, + 554, + 347, + 563 + ], + "score": 0.85, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 552, + 423, + 566 + ], + "score": 1.0, + "content": "is connected, then", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 552, + 423, + 566 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 569, + 363, + 600 + ], + "lines": [ + { + "bbox": [ + 248, + 569, + 363, + 600 + ], + "spans": [ + { + "bbox": [ + 248, + 569, + 363, + 600 + ], + "score": 0.95, + "content": "b _ { G } = \\frac { 1 } { n } \\sum _ { i , j } \\left( d _ { G } ( i , j ) - 1 \\right) .", + "type": "interline_equation", + "image_path": "7c8b7f038bc6553e8a24c7dd9cf4a88d5a51dbfe47af33c0675480105d6a1ebc.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 248, + 569, + 363, + 584.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 248, + 584.5, + 363, + 600.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 160, + 618 + ], + "score": 1.0, + "content": "We note that", + "type": "text" + }, + { + "bbox": [ + 160, + 605, + 191, + 616 + ], + "score": 0.91, + "content": "b _ { G } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 604, + 204, + 618 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 205, + 605, + 214, + 615 + ], + "score": 0.85, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 604, + 305, + 618 + ], + "score": 1.0, + "content": "is the complete graph", + "type": "text" + }, + { + "bbox": [ + 306, + 605, + 320, + 616 + ], + "score": 0.89, + "content": "K _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 604, + 370, + 618 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 371, + 605, + 383, + 616 + ], + "score": 0.89, + "content": "b _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "determines how far the given", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 177, + 628 + ], + "score": 1.0, + "content": "topology is from", + "type": "text" + }, + { + "bbox": [ + 177, + 617, + 192, + 627 + ], + "score": 0.87, + "content": "K _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 616, + 495, + 628 + ], + "score": 1.0, + "content": ", with the latter representing the limit case of a fully connected layer Alon", + "type": "text" + }, + { + "bbox": [ + 496, + 616, + 505, + 626 + ], + "score": 0.43, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Yahav (2021) where no bottleneck may occur as any pair of nodes would be neighbours. This further", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "score": 1.0, + "content": "supports our intuition that the betweenness centrality is a good topological candidate for providing a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 649, + 323, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 323, + 661 + ], + "score": 1.0, + "content": "global measurement of bottleneckedness in the graph.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 604, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "It also follows from equation 17 that any update to the graph topology consisting of edge additions", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 171, + 689 + ], + "score": 1.0, + "content": "would decrease", + "type": "text" + }, + { + "bbox": [ + 171, + 677, + 183, + 688 + ], + "score": 0.87, + "content": "b _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 677, + 367, + 689 + ], + "score": 1.0, + "content": "and thus reduce the bottleneck. The quantity", + "type": "text" + }, + { + "bbox": [ + 368, + 677, + 380, + 688 + ], + "score": 0.87, + "content": "b _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "is global in nature though and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 381, + 700 + ], + "score": 1.0, + "content": "hence lacks robustness. As a pedagogical example, consider a barbell", + "type": "text" + }, + { + "bbox": [ + 382, + 687, + 438, + 699 + ], + "score": 0.92, + "content": "G ( m , 2 r + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 687, + 461, + 700 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 462, + 689, + 472, + 698 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "the size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 290, + 711 + ], + "score": 1.0, + "content": "of the two cliques joined by a path of length", + "type": "text" + }, + { + "bbox": [ + 290, + 699, + 319, + 709 + ], + "score": 0.9, + "content": "2 r + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 698, + 412, + 711 + ], + "score": 1.0, + "content": "and focus on the edge", + "type": "text" + }, + { + "bbox": [ + 412, + 700, + 436, + 710 + ], + "score": 0.88, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "in the middle of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 179, + 722 + ], + "score": 1.0, + "content": "such path. Nodes", + "type": "text" + }, + { + "bbox": [ + 179, + 711, + 184, + 720 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 709, + 202, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 711, + 208, + 721 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are central to the graph, in the sense that most minimal paths go through", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 294, + 733 + ], + "score": 1.0, + "content": "them and indeed their betweenness centrality is", + "type": "text" + }, + { + "bbox": [ + 294, + 720, + 435, + 732 + ], + "score": 0.92, + "content": "\\bar { b } ( i ) = b ( j ) = ( m + r ) ^ { 2 } + ( m + r )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ". If now we add a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 244, + 95 + ], + "score": 1.0, + "content": "single edge joining the two cliques", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 244, + 83, + 261, + 94 + ], + "score": 0.9, + "content": "K _ { m }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 261, + 82, + 306, + 95 + ], + "score": 1.0, + "content": ", the values", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 306, + 82, + 322, + 95 + ], + "score": 0.9, + "content": "b ( i )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 323, + 82, + 340, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 340, + 82, + 358, + 95 + ], + "score": 0.9, + "content": "b ( j )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 358, + 82, + 458, + 95 + ], + "score": 1.0, + "content": "decrease dramatically by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 459, + 82, + 503, + 95 + ], + "score": 0.93, + "content": "\\Omega ( m ^ { 2 } + r )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 503, + 82, + 507, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 102, + 93, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 102, + 93, + 361, + 112 + ], + "score": 1.0, + "content": "Since the operation is non-local, we see that the representation", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 361, + 93, + 378, + 108 + ], + "score": 0.92, + "content": "h _ { i } ^ { ( \\ell ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 378, + 93, + 505, + 112 + ], + "score": 1.0, + "content": "of any MPNN is unaffected by", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 107, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 211, + 120 + ], + "score": 1.0, + "content": "the edge addition for any", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 212, + 107, + 251, + 119 + ], + "score": 0.93, + "content": "\\ell \\in ( 0 , r )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 252, + 107, + 326, + 120 + ], + "score": 1.0, + "content": ", and similarly for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 327, + 108, + 333, + 119 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 333, + 107, + 505, + 120 + ], + "score": 1.0, + "content": ". Eventually, if we keep adding edges, the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 169, + 131 + ], + "score": 1.0, + "content": "receptive fields", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 169, + 118, + 194, + 131 + ], + "score": 0.92, + "content": "B _ { s } ( i )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 194, + 118, + 336, + 131 + ], + "score": 1.0, + "content": "will be affected for small values of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 337, + 120, + 343, + 128 + ], + "score": 0.66, + "content": "s", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 343, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "as well: the drawback of such approach", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "is that the resulting adjacency may be significantly different. Conversely, the curvature provides a", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "more precise, local and hence robust way of controlling the bottleneck and hence the over-squashing", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 500, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 500, + 164 + ], + "score": 1.0, + "content": "problem. Nonetheless, we relate the betweenness centrality to the Jacobian of the hidden features.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 244, + 95 + ], + "score": 1.0, + "content": "single edge joining the two cliques", + "type": "text" + }, + { + "bbox": [ + 244, + 83, + 261, + 94 + ], + "score": 0.9, + "content": "K _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 82, + 306, + 95 + ], + "score": 1.0, + "content": ", the values", + "type": "text" + }, + { + "bbox": [ + 306, + 82, + 322, + 95 + ], + "score": 0.9, + "content": "b ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 82, + 340, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 340, + 82, + 358, + 95 + ], + "score": 0.9, + "content": "b ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 82, + 458, + 95 + ], + "score": 1.0, + "content": "decrease dramatically by", + "type": "text" + }, + { + "bbox": [ + 459, + 82, + 503, + 95 + ], + "score": 0.93, + "content": "\\Omega ( m ^ { 2 } + r )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 82, + 507, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 102, + 93, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 102, + 93, + 361, + 112 + ], + "score": 1.0, + "content": "Since the operation is non-local, we see that the representation", + "type": "text" + }, + { + "bbox": [ + 361, + 93, + 378, + 108 + ], + "score": 0.92, + "content": "h _ { i } ^ { ( \\ell ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 93, + 505, + 112 + ], + "score": 1.0, + "content": "of any MPNN is unaffected by", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 107, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 211, + 120 + ], + "score": 1.0, + "content": "the edge addition for any", + "type": "text" + }, + { + "bbox": [ + 212, + 107, + 251, + 119 + ], + "score": 0.93, + "content": "\\ell \\in ( 0 , r )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 107, + 326, + 120 + ], + "score": 1.0, + "content": ", and similarly for", + "type": "text" + }, + { + "bbox": [ + 327, + 108, + 333, + 119 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 107, + 505, + 120 + ], + "score": 1.0, + "content": ". Eventually, if we keep adding edges, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 169, + 131 + ], + "score": 1.0, + "content": "receptive fields", + "type": "text" + }, + { + "bbox": [ + 169, + 118, + 194, + 131 + ], + "score": 0.92, + "content": "B _ { s } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 118, + 336, + 131 + ], + "score": 1.0, + "content": "will be affected for small values of", + "type": "text" + }, + { + "bbox": [ + 337, + 120, + 343, + 128 + ], + "score": 0.66, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "as well: the drawback of such approach", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "is that the resulting adjacency may be significantly different. Conversely, the curvature provides a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "more precise, local and hence robust way of controlling the bottleneck and hence the over-squashing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 500, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 500, + 164 + ], + "score": 1.0, + "content": "problem. Nonetheless, we relate the betweenness centrality to the Jacobian of the hidden features.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 504, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 192, + 179 + ], + "score": 1.0, + "content": "Theorem 16. Given", + "type": "text" + }, + { + "bbox": [ + 193, + 166, + 218, + 177 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 163, + 235, + 179 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 236, + 164, + 349, + 178 + ], + "score": 0.92, + "content": "\\Omega _ { j } : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) \\cup \\{ j \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 163, + 365, + 179 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 365, + 164, + 447, + 177 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta ,", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 163, + 465, + 179 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 466, + 165, + 505, + 177 + ], + "score": 0.87, + "content": "0 < \\delta <", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 184, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 159, + 189 + ], + "score": 0.91, + "content": "( 1 + \\gamma _ { m a x } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 175, + 184, + 190 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 187, + 352, + 218 + ], + "lines": [ + { + "bbox": [ + 258, + 187, + 352, + 218 + ], + "spans": [ + { + "bbox": [ + 258, + 187, + 352, + 218 + ], + "score": 0.92, + "content": "\\frac { 1 } { | \\Omega _ { j } | } \\sum _ { k \\in \\Omega _ { j } } b ( k ) \\geq \\delta ^ { - 1 } .", + "type": "interline_equation", + "image_path": "a8f5ff6b2f238f70b9a64016cfa2a257a7a7872ab9a33f24c01c2be3070ff645.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 258, + 187, + 352, + 202.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 258, + 202.5, + 352, + 218.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 309, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 309, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 309, + 243 + ], + "score": 1.0, + "content": "Proof. We rewrite the quantity in the statement as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 243, + 370, + 283 + ], + "lines": [ + { + "bbox": [ + 240, + 243, + 370, + 283 + ], + "spans": [ + { + "bbox": [ + 240, + 243, + 370, + 283 + ], + "score": 0.94, + "content": "{ \\frac { 1 } { \\left| \\sharp _ { \\Delta } \\right| + 1 } } \\left( \\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) + b ( j ) \\right) .", + "type": "interline_equation", + "image_path": "5e40ccf8c483923eeefe1d34016ed7cc28596d0320649039656ad9418faa100b.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 243, + 370, + 263.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 240, + 263.0, + 370, + 283.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 419, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 420, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 217, + 300 + ], + "score": 1.0, + "content": "By definition, given a node", + "type": "text" + }, + { + "bbox": [ + 217, + 286, + 244, + 296 + ], + "score": 0.9, + "content": "k \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 283, + 366, + 300 + ], + "score": 1.0, + "content": ", the betweenness centrality of", + "type": "text" + }, + { + "bbox": [ + 366, + 286, + 372, + 295 + ], + "score": 0.85, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 283, + 420, + 300 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 300, + 366, + 332 + ], + "lines": [ + { + "bbox": [ + 245, + 300, + 366, + 332 + ], + "spans": [ + { + "bbox": [ + 245, + 300, + 366, + 332 + ], + "score": 0.94, + "content": "b ( k ) : = \\sum _ { s , t \\in V : s \\neq k , t \\neq k } \\frac { \\sigma _ { s t } ( k ) } { \\sigma _ { s t } }", + "type": "interline_equation", + "image_path": "71d9f578b9baf810e917a3064a8a8d03d72552e7edcdc83144d6227eca2ee13d.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 300, + 366, + 316.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 245, + 316.0, + 366, + 332.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 506, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 133, + 348 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 337, + 147, + 347 + ], + "score": 0.85, + "content": "\\sigma _ { s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 334, + 315, + 348 + ], + "score": 1.0, + "content": "is the number of minimal paths between", + "type": "text" + }, + { + "bbox": [ + 315, + 338, + 321, + 345 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 334, + 339, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 336, + 344, + 345 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 334, + 371, + 348 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 371, + 335, + 398, + 347 + ], + "score": 0.92, + "content": "\\sigma _ { s t } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "is the number of minimal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 347, + 488, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 152, + 362 + ], + "score": 1.0, + "content": "paths from", + "type": "text" + }, + { + "bbox": [ + 152, + 351, + 158, + 358 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 347, + 169, + 362 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 169, + 349, + 174, + 358 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 347, + 241, + 362 + ], + "score": 1.0, + "content": "passing through", + "type": "text" + }, + { + "bbox": [ + 241, + 349, + 248, + 358 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 347, + 405, + 362 + ], + "score": 1.0, + "content": ". For convenience, we introduce the set", + "type": "text" + }, + { + "bbox": [ + 405, + 347, + 444, + 361 + ], + "score": 0.93, + "content": "\\hat { Q _ { j } } \\subset Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 347, + 488, + 362 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 363, + 365, + 379 + ], + "lines": [ + { + "bbox": [ + 245, + 363, + 365, + 379 + ], + "spans": [ + { + "bbox": [ + 245, + 363, + 365, + 379 + ], + "score": 0.93, + "content": "\\hat { Q } _ { j } : = \\{ w \\in Q _ { j } : \\ \\sigma _ { i w } > 1 \\} .", + "type": "interline_equation", + "image_path": "21dfbb34539e61be528d5cda9f9df399831f7335e95cb5c40ba42c5fe5ba78b2.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 245, + 363, + 365, + 379 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 383, + 506, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 161, + 397 + ], + "score": 1.0, + "content": "Equivalently,", + "type": "text" + }, + { + "bbox": [ + 162, + 382, + 175, + 396 + ], + "score": 0.91, + "content": "\\hat { Q } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 383, + 281, + 397 + ], + "score": 1.0, + "content": "consists of those nodes in", + "type": "text" + }, + { + "bbox": [ + 281, + 383, + 336, + 396 + ], + "score": 0.92, + "content": "S _ { 1 } ( j ) \\setminus S _ { 1 } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 383, + 461, + 397 + ], + "score": 1.0, + "content": "which form a 4-cycle based at", + "type": "text" + }, + { + "bbox": [ + 461, + 384, + 483, + 396 + ], + "score": 0.91, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 394, + 225, + 408 + ], + "score": 1.0, + "content": "a diagonal inside. Indeed, if", + "type": "text" + }, + { + "bbox": [ + 225, + 395, + 258, + 407 + ], + "score": 0.93, + "content": "w \\in Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 394, + 277, + 408 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 278, + 396, + 313, + 406 + ], + "score": 0.9, + "content": "\\sigma _ { i w } > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 394, + 506, + 408 + ], + "score": 1.0, + "content": ", then there exists more than one minimal path", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 144, + 419 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 144, + 407, + 148, + 416 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 405, + 168, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 408, + 177, + 416 + ], + "score": 0.72, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 405, + 365, + 419 + ], + "score": 1.0, + "content": ", in addition to the one passing through node", + "type": "text" + }, + { + "bbox": [ + 366, + 407, + 372, + 417 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 405, + 506, + 419 + ], + "score": 1.0, + "content": ". For any such path there exists", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 107, + 417, + 184, + 431 + ], + "score": 0.92, + "content": "k \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 417, + 213, + 432 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 213, + 418, + 245, + 431 + ], + "score": 0.92, + "content": "w \\in Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 417, + 264, + 432 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 264, + 417, + 317, + 432 + ], + "score": 0.93, + "content": "Q _ { j } \\cap { \\sharp \\underline { { \\breve { \\jmath } } } } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 417, + 379, + 432 + ], + "score": 1.0, + "content": ", we derive that", + "type": "text" + }, + { + "bbox": [ + 379, + 418, + 420, + 430 + ], + "score": 0.93, + "content": "k \\in S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "as well. We then get", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 434, + 481, + 468 + ], + "lines": [ + { + "bbox": [ + 128, + 434, + 481, + 468 + ], + "spans": [ + { + "bbox": [ + 128, + 434, + 481, + 468 + ], + "score": 0.93, + "content": "\\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } b ( k ) = \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sum _ { s , t \\in V : s \\neq k , t \\neq k } \\frac { \\sigma _ { s t } ( k ) } { \\sigma _ { s t } } \\geq \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sum _ { w \\in \\hat { Q } _ { j } } \\frac { \\sigma _ { i w } ( k ) } { \\sigma _ { i w } } = \\sum _ { w \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sigma _ { i w } } \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sigma _ { i w } ( k ) .", + "type": "interline_equation", + "image_path": "bb76815faac1f0782a381fac11cb58fd6ee7c5ce0fd7df976464c29ce492e922.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 128, + 434, + 481, + 445.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 128, + 445.3333333333333, + 481, + 456.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 128, + 456.66666666666663, + 481, + 467.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 503, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 159, + 484 + ], + "score": 1.0, + "content": "By summing", + "type": "text" + }, + { + "bbox": [ + 160, + 471, + 189, + 484 + ], + "score": 0.93, + "content": "\\sigma _ { i w } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 471, + 217, + 484 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 217, + 472, + 246, + 483 + ], + "score": 0.92, + "content": "k \\in \\sharp _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 471, + 438, + 484 + ], + "score": 1.0, + "content": "we obtain all the 2-long minimal paths between", + "type": "text" + }, + { + "bbox": [ + 438, + 473, + 443, + 482 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 471, + 461, + 484 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 461, + 474, + 469, + 482 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 483, + 266, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 255, + 496 + ], + "score": 1.0, + "content": "exception of the one passing through", + "type": "text" + }, + { + "bbox": [ + 256, + 483, + 262, + 494 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 483, + 266, + 496 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 496, + 375, + 530 + ], + "lines": [ + { + "bbox": [ + 237, + 496, + 375, + 530 + ], + "spans": [ + { + "bbox": [ + 237, + 496, + 375, + 530 + ], + "score": 0.94, + "content": "\\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) \\geq \\sum _ { w \\in \\hat { Q _ { j } } } \\frac { 1 } { \\sigma _ { i w } } ( \\sigma _ { i w } - 1 ) .", + "type": "interline_equation", + "image_path": "e1a36dedc62b5370d407420da24454eb1551c56fc39d524dceeeafd28b446b5c.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 496, + 375, + 513.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 237, + 513.0, + 375, + 530.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 234, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 235, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 235, + 545 + ], + "score": 1.0, + "content": "On the other hand we also have", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 545, + 420, + 577 + ], + "lines": [ + { + "bbox": [ + 191, + 545, + 420, + 577 + ], + "spans": [ + { + "bbox": [ + 191, + 545, + 420, + 577 + ], + "score": 0.93, + "content": "b ( j ) = \\sum _ { s , t \\in V : s \\neq j , t \\neq j } \\frac { \\sigma _ { s t } ( j ) } { \\sigma _ { s t } } \\geq \\sum _ { z \\in Q _ { j } } \\frac { \\sigma _ { i z } ( j ) } { \\sigma _ { i z } } = \\sum _ { z \\in Q _ { j } } \\frac { 1 } { \\sigma _ { i z } } .", + "type": "interline_equation", + "image_path": "cc77537ee3e7c98e0df51067884bffd1ecd457396b01af498457903e19433892.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 545, + 420, + 561.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 191, + 561.0, + 420, + 577.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 338, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 339, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 339, + 595 + ], + "score": 1.0, + "content": "By combining equation 18 and equation 19 we finally get", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 594, + 466, + 705 + ], + "lines": [ + { + "bbox": [ + 144, + 594, + 466, + 705 + ], + "spans": [ + { + "bbox": [ + 144, + 594, + 466, + 705 + ], + "score": 0.95, + "content": "\\begin{array} { r l r } { { \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( \\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) + b ( j ) ) \\geq \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( \\sum _ { w \\in \\widehat { \\mathcal { Q } } _ { j } } \\frac { 1 } { \\sigma _ { i w } } ( \\sigma _ { i w } - 1 ) + \\sum _ { z \\in Q _ { j } } \\frac { 1 } { \\sigma _ { i z } } ) } } \\\\ & { } & { = \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( | \\widehat { Q } _ { j } | + \\sum _ { z \\in Q _ { j } \\setminus \\widehat { Q } _ { j } } \\frac { 1 } { \\sigma _ { i z } } ) } \\\\ & { } & { = \\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } , } \\end{array}", + "type": "interline_equation", + "image_path": "1af42c25efb7b219bb1220f64cca319a02ecf901f42bb2c7e767a9d668c971c4.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 144, + 594, + 466, + 631.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 144, + 631.0, + 466, + 668.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 144, + 668.0, + 466, + 705.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 708, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 331, + 722 + ], + "score": 1.0, + "content": "where in the last equality we have used that by definition", + "type": "text" + }, + { + "bbox": [ + 331, + 709, + 363, + 720 + ], + "score": 0.91, + "content": "\\sigma _ { i z } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 706, + 390, + 722 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 391, + 707, + 442, + 721 + ], + "score": 0.94, + "content": "z \\in Q _ { j } \\backslash { \\hat { Q } } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 706, + 506, + 722 + ], + "score": 1.0, + "content": ". By Lemma 14", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 227, + 732 + ], + "score": 1.0, + "content": "the last quantity is larger than", + "type": "text" + }, + { + "bbox": [ + 227, + 721, + 243, + 730 + ], + "score": 0.89, + "content": "\\delta ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 720, + 247, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 724, + 504, + 729 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 23, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 163 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 102, + 82, + 507, + 164 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 504, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 192, + 179 + ], + "score": 1.0, + "content": "Theorem 16. Given", + "type": "text" + }, + { + "bbox": [ + 193, + 166, + 218, + 177 + ], + "score": 0.9, + "content": "i \\sim j", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 163, + 235, + 179 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 236, + 164, + 349, + 178 + ], + "score": 0.92, + "content": "\\Omega _ { j } : = S _ { 1 } ( i ) \\cap S _ { 1 } ( j ) \\cup \\{ j \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 163, + 365, + 179 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 365, + 164, + 447, + 177 + ], + "score": 0.9, + "content": "\\operatorname { R i c } ( i , j ) \\leq - 2 + \\delta ,", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 163, + 465, + 179 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 466, + 165, + 505, + 177 + ], + "score": 0.87, + "content": "0 < \\delta <", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 184, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 159, + 189 + ], + "score": 0.91, + "content": "( 1 + \\gamma _ { m a x } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 175, + 184, + 190 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 163, + 505, + 190 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 187, + 352, + 218 + ], + "lines": [ + { + "bbox": [ + 258, + 187, + 352, + 218 + ], + "spans": [ + { + "bbox": [ + 258, + 187, + 352, + 218 + ], + "score": 0.92, + "content": "\\frac { 1 } { | \\Omega _ { j } | } \\sum _ { k \\in \\Omega _ { j } } b ( k ) \\geq \\delta ^ { - 1 } .", + "type": "interline_equation", + "image_path": "a8f5ff6b2f238f70b9a64016cfa2a257a7a7872ab9a33f24c01c2be3070ff645.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 258, + 187, + 352, + 202.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 258, + 202.5, + 352, + 218.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 309, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 309, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 309, + 243 + ], + "score": 1.0, + "content": "Proof. We rewrite the quantity in the statement as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 228, + 309, + 243 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 243, + 370, + 283 + ], + "lines": [ + { + "bbox": [ + 240, + 243, + 370, + 283 + ], + "spans": [ + { + "bbox": [ + 240, + 243, + 370, + 283 + ], + "score": 0.94, + "content": "{ \\frac { 1 } { \\left| \\sharp _ { \\Delta } \\right| + 1 } } \\left( \\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) + b ( j ) \\right) .", + "type": "interline_equation", + "image_path": "5e40ccf8c483923eeefe1d34016ed7cc28596d0320649039656ad9418faa100b.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 243, + 370, + 263.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 240, + 263.0, + 370, + 283.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 419, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 420, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 217, + 300 + ], + "score": 1.0, + "content": "By definition, given a node", + "type": "text" + }, + { + "bbox": [ + 217, + 286, + 244, + 296 + ], + "score": 0.9, + "content": "k \\in V", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 283, + 366, + 300 + ], + "score": 1.0, + "content": ", the betweenness centrality of", + "type": "text" + }, + { + "bbox": [ + 366, + 286, + 372, + 295 + ], + "score": 0.85, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 283, + 420, + 300 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 283, + 420, + 300 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 300, + 366, + 332 + ], + "lines": [ + { + "bbox": [ + 245, + 300, + 366, + 332 + ], + "spans": [ + { + "bbox": [ + 245, + 300, + 366, + 332 + ], + "score": 0.94, + "content": "b ( k ) : = \\sum _ { s , t \\in V : s \\neq k , t \\neq k } \\frac { \\sigma _ { s t } ( k ) } { \\sigma _ { s t } }", + "type": "interline_equation", + "image_path": "71d9f578b9baf810e917a3064a8a8d03d72552e7edcdc83144d6227eca2ee13d.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 300, + 366, + 316.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 245, + 316.0, + 366, + 332.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 506, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 133, + 348 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 337, + 147, + 347 + ], + "score": 0.85, + "content": "\\sigma _ { s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 334, + 315, + 348 + ], + "score": 1.0, + "content": "is the number of minimal paths between", + "type": "text" + }, + { + "bbox": [ + 315, + 338, + 321, + 345 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 334, + 339, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 336, + 344, + 345 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 334, + 371, + 348 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 371, + 335, + 398, + 347 + ], + "score": 0.92, + "content": "\\sigma _ { s t } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "is the number of minimal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 347, + 488, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 152, + 362 + ], + "score": 1.0, + "content": "paths from", + "type": "text" + }, + { + "bbox": [ + 152, + 351, + 158, + 358 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 347, + 169, + 362 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 169, + 349, + 174, + 358 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 347, + 241, + 362 + ], + "score": 1.0, + "content": "passing through", + "type": "text" + }, + { + "bbox": [ + 241, + 349, + 248, + 358 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 347, + 405, + 362 + ], + "score": 1.0, + "content": ". 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Indeed, if", + "type": "text" + }, + { + "bbox": [ + 225, + 395, + 258, + 407 + ], + "score": 0.93, + "content": "w \\in Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 394, + 277, + 408 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 278, + 396, + 313, + 406 + ], + "score": 0.9, + "content": "\\sigma _ { i w } > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 394, + 506, + 408 + ], + "score": 1.0, + "content": ", then there exists more than one minimal path", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 144, + 419 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 144, + 407, + 148, + 416 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 405, + 168, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 408, + 177, + 416 + ], + "score": 0.72, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 405, + 365, + 419 + ], + "score": 1.0, + "content": ", in addition to the one passing through node", + "type": "text" + }, + { + "bbox": [ + 366, + 407, + 372, + 417 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 405, + 506, + 419 + ], + "score": 1.0, + "content": ". For any such path there exists", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 107, + 417, + 184, + 431 + ], + "score": 0.92, + "content": "k \\in S _ { 1 } ( i ) \\cap S _ { 1 } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 417, + 213, + 432 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 213, + 418, + 245, + 431 + ], + "score": 0.92, + "content": "w \\in Q _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 417, + 264, + 432 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 264, + 417, + 317, + 432 + ], + "score": 0.93, + "content": "Q _ { j } \\cap { \\sharp \\underline { { \\breve { \\jmath } } } } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 417, + 379, + 432 + ], + "score": 1.0, + "content": ", we derive that", + "type": "text" + }, + { + "bbox": [ + 379, + 418, + 420, + 430 + ], + "score": 0.93, + "content": "k \\in S _ { 1 } ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "as well. We then get", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 382, + 506, + 432 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 434, + 481, + 468 + ], + "lines": [ + { + "bbox": [ + 128, + 434, + 481, + 468 + ], + "spans": [ + { + "bbox": [ + 128, + 434, + 481, + 468 + ], + "score": 0.93, + "content": "\\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } b ( k ) = \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sum _ { s , t \\in V : s \\neq k , t \\neq k } \\frac { \\sigma _ { s t } ( k ) } { \\sigma _ { s t } } \\geq \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sum _ { w \\in \\hat { Q } _ { j } } \\frac { \\sigma _ { i w } ( k ) } { \\sigma _ { i w } } = \\sum _ { w \\in \\hat { Q } _ { j } } \\frac { 1 } { \\sigma _ { i w } } \\sum _ { k \\in \\mathfrak { t } _ { \\Delta } } \\sigma _ { i w } ( k ) .", + "type": "interline_equation", + "image_path": "bb76815faac1f0782a381fac11cb58fd6ee7c5ce0fd7df976464c29ce492e922.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 128, + 434, + 481, + 445.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 128, + 445.3333333333333, + 481, + 456.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 128, + 456.66666666666663, + 481, + 467.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 503, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 159, + 484 + ], + "score": 1.0, + "content": "By summing", + "type": "text" + }, + { + "bbox": [ + 160, + 471, + 189, + 484 + ], + "score": 0.93, + "content": "\\sigma _ { i w } ( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 471, + 217, + 484 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 217, + 472, + 246, + 483 + ], + "score": 0.92, + "content": "k \\in \\sharp _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 471, + 438, + 484 + ], + "score": 1.0, + "content": "we obtain all the 2-long minimal paths between", + "type": "text" + }, + { + "bbox": [ + 438, + 473, + 443, + 482 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 471, + 461, + 484 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 461, + 474, + 469, + 482 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 483, + 266, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 255, + 496 + ], + "score": 1.0, + "content": "exception of the one passing through", + "type": "text" + }, + { + "bbox": [ + 256, + 483, + 262, + 494 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 483, + 266, + 496 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 471, + 505, + 496 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 496, + 375, + 530 + ], + "lines": [ + { + "bbox": [ + 237, + 496, + 375, + 530 + ], + "spans": [ + { + "bbox": [ + 237, + 496, + 375, + 530 + ], + "score": 0.94, + "content": "\\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) \\geq \\sum _ { w \\in \\hat { Q _ { j } } } \\frac { 1 } { \\sigma _ { i w } } ( \\sigma _ { i w } - 1 ) .", + "type": "interline_equation", + "image_path": "e1a36dedc62b5370d407420da24454eb1551c56fc39d524dceeeafd28b446b5c.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 496, + 375, + 513.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 237, + 513.0, + 375, + 530.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 234, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 235, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 235, + 545 + ], + "score": 1.0, + "content": "On the other hand we also have", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 531, + 235, + 545 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 545, + 420, + 577 + ], + "lines": [ + { + "bbox": [ + 191, + 545, + 420, + 577 + ], + "spans": [ + { + "bbox": [ + 191, + 545, + 420, + 577 + ], + "score": 0.93, + "content": "b ( j ) = \\sum _ { s , t \\in V : s \\neq j , t \\neq j } \\frac { \\sigma _ { s t } ( j ) } { \\sigma _ { s t } } \\geq \\sum _ { z \\in Q _ { j } } \\frac { \\sigma _ { i z } ( j ) } { \\sigma _ { i z } } = \\sum _ { z \\in Q _ { j } } \\frac { 1 } { \\sigma _ { i z } } .", + "type": "interline_equation", + "image_path": "cc77537ee3e7c98e0df51067884bffd1ecd457396b01af498457903e19433892.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 545, + 420, + 561.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 191, + 561.0, + 420, + 577.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 338, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 339, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 339, + 595 + ], + "score": 1.0, + "content": "By combining equation 18 and equation 19 we finally get", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 578, + 339, + 595 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 594, + 466, + 705 + ], + "lines": [ + { + "bbox": [ + 144, + 594, + 466, + 705 + ], + "spans": [ + { + "bbox": [ + 144, + 594, + 466, + 705 + ], + "score": 0.95, + "content": "\\begin{array} { r l r } { { \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( \\sum _ { k \\in \\sharp _ { \\Delta } } b ( k ) + b ( j ) ) \\geq \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( \\sum _ { w \\in \\widehat { \\mathcal { Q } } _ { j } } \\frac { 1 } { \\sigma _ { i w } } ( \\sigma _ { i w } - 1 ) + \\sum _ { z \\in Q _ { j } } \\frac { 1 } { \\sigma _ { i z } } ) } } \\\\ & { } & { = \\frac { 1 } { | \\sharp _ { \\Delta } | + 1 } ( | \\widehat { Q } _ { j } | + \\sum _ { z \\in Q _ { j } \\setminus \\widehat { Q } _ { j } } \\frac { 1 } { \\sigma _ { i z } } ) } \\\\ & { } & { = \\frac { | Q _ { j } | } { | \\sharp _ { \\Delta } | + 1 } , } \\end{array}", + "type": "interline_equation", + "image_path": "1af42c25efb7b219bb1220f64cca319a02ecf901f42bb2c7e767a9d668c971c4.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 144, + 594, + 466, + 631.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 144, + 631.0, + 466, + 668.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 144, + 668.0, + 466, + 705.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 708, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 706, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 331, + 722 + ], + "score": 1.0, + "content": "where in the last equality we have used that by definition", + "type": "text" + }, + { + "bbox": [ + 331, + 709, + 363, + 720 + ], + "score": 0.91, + "content": "\\sigma _ { i z } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 706, + 390, + 722 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 391, + 707, + 442, + 721 + ], + "score": 0.94, + "content": "z \\in Q _ { j } \\backslash { \\hat { Q } } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 706, + 506, + 722 + ], + "score": 1.0, + "content": ". By Lemma 14", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 227, + 732 + ], + "score": 1.0, + "content": "the last quantity is larger than", + "type": "text" + }, + { + "bbox": [ + 227, + 721, + 243, + 730 + ], + "score": 0.89, + "content": "\\delta ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 720, + 247, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 724, + 504, + 729 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 706, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 80, + 306, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 306, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 306, + 95 + ], + "score": 1.0, + "content": "E PROOFS OF RESULTS IN SECTION 4", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 503, + 132 + ], + "lines": [ + { + "bbox": [ + 102, + 101, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 102, + 101, + 176, + 124 + ], + "score": 1.0, + "content": "Theorem 6. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 106, + 208, + 117 + ], + "score": 0.88, + "content": "S \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 101, + 229, + 124 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 230, + 105, + 309, + 118 + ], + "score": 0.9, + "content": "\\nu o l ( S ) \\le \\nu o l ( G ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 101, + 339, + 124 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 339, + 103, + 442, + 121 + ], + "score": 0.92, + "content": "\\begin{array} { r } { h _ { S , \\alpha } \\leq \\left( \\frac { 1 - \\alpha } { \\alpha } \\right) \\frac { d _ { \\arg } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } h _ { S } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 101, + 473, + 124 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 105, + 505, + 118 + ], + "score": 0.91, + "content": "d _ { \\mathrm { a v g } } ( S )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 383, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 124, + 133 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 119, + 156, + 132 + ], + "score": 0.92, + "content": "d _ { \\mathrm { m i n } } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 118, + 320, + 133 + ], + "score": 1.0, + "content": "are the average and minimum degree on", + "type": "text" + }, + { + "bbox": [ + 320, + 120, + 328, + 129 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 118, + 383, + 133 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 103, + 142, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 104, + 141, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 197, + 158 + ], + "score": 1.0, + "content": "Proof. Given a signal", + "type": "text" + }, + { + "bbox": [ + 198, + 144, + 244, + 155 + ], + "score": 0.92, + "content": "f : V \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 141, + 332, + 158 + ], + "score": 1.0, + "content": "on the vertex set and", + "type": "text" + }, + { + "bbox": [ + 332, + 144, + 362, + 154 + ], + "score": 0.9, + "content": "U \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 141, + 505, + 158 + ], + "score": 1.0, + "content": ", analogously to Chung (2007), we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 154, + 196, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 196, + 166 + ], + "score": 1.0, + "content": "introduce the notation", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 161, + 343, + 188 + ], + "lines": [ + { + "bbox": [ + 267, + 161, + 343, + 188 + ], + "spans": [ + { + "bbox": [ + 267, + 161, + 343, + 188 + ], + "score": 0.94, + "content": "f ( U ) : = \\sum _ { i \\in U } f ( i ) .", + "type": "interline_equation", + "image_path": "6e6d448086dabe8e193aa4d7cca544cd3b2f04b3a98e6783b42e5926df9cdb83.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 267, + 161, + 343, + 188 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 335, + 202 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 336, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 268, + 204 + ], + "score": 1.0, + "content": "Let us rewrite the new Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 269, + 191, + 289, + 203 + ], + "score": 0.92, + "content": "h _ { S , \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 188, + 336, + 204 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 205, + 394, + 238 + ], + "lines": [ + { + "bbox": [ + 217, + 205, + 394, + 238 + ], + "spans": [ + { + "bbox": [ + 217, + 205, + 394, + 238 + ], + "score": 0.94, + "content": "h _ { S , \\alpha } = \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } = \\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } )", + "type": "interline_equation", + "image_path": "0bd47bf2694fb8c6aaba7f19ca7384952b2acd58e199a707e29ace4a3217cc5a.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 205, + 394, + 221.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 217, + 221.5, + 394, + 238.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 127, + 255 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 244, + 140, + 254 + ], + "score": 0.86, + "content": "\\chi _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 242, + 304, + 255 + ], + "score": 1.0, + "content": "the characteristic function of the subset", + "type": "text" + }, + { + "bbox": [ + 305, + 243, + 312, + 253 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 242, + 332, + 255 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 333, + 243, + 376, + 255 + ], + "score": 0.93, + "content": "\\chi _ { S } ( i ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 242, + 389, + 255 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 390, + 243, + 414, + 253 + ], + "score": 0.89, + "content": "i \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 242, + 485, + 255 + ], + "score": 1.0, + "content": ". Since the graph", + "type": "text" + }, + { + "bbox": [ + 486, + 243, + 494, + 253 + ], + "score": 0.84, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 253, + 289, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 209, + 265 + ], + "score": 1.0, + "content": "connected, we can bound", + "type": "text" + }, + { + "bbox": [ + 210, + 254, + 229, + 266 + ], + "score": 0.92, + "content": "h _ { S , \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 253, + 289, + 265 + ], + "score": 1.0, + "content": "from above as", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 269, + 494, + 302 + ], + "lines": [ + { + "bbox": [ + 117, + 269, + 494, + 302 + ], + "spans": [ + { + "bbox": [ + 117, + 269, + 494, + 302 + ], + "score": 0.94, + "content": "\\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } ) = \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } \\le \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } \\frac { d _ { i } } { d _ { \\operatorname* { m i n } } ( S ) } = \\frac { 1 } { | S | } \\chi _ { S } D R _ { \\alpha } ( \\bar { S } ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } .", + "type": "interline_equation", + "image_path": "9e977a7e91afca18aa536aac501f0755aeef5919118e7104f9ce85aa3d779620.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 117, + 269, + 494, + 280.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 117, + 280.0, + 494, + 291.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 117, + 291.0, + 494, + 302.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 305, + 293, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 295, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 295, + 319 + ], + "score": 1.0, + "content": "It was proven in (Chung, 2007, Lemma 5) that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 321, + 361, + 345 + ], + "lines": [ + { + "bbox": [ + 250, + 321, + 361, + 345 + ], + "spans": [ + { + "bbox": [ + 250, + 321, + 361, + 345 + ], + "score": 0.94, + "content": "\\chi _ { S } D R _ { \\alpha } ( { \\bar { S } } ) \\leq { \\frac { 1 - \\alpha } { \\alpha } } | \\partial S | .", + "type": "interline_equation", + "image_path": "875a4317dcf9b55a862e2e4c3b60965a9bbc026b0ad2aa1e073eb2ce9bf66797.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 250, + 321, + 361, + 345 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 348, + 492, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 493, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 396, + 362 + ], + "score": 1.0, + "content": "By applying equation equation 20 to the bound for the Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 397, + 349, + 416, + 361 + ], + "score": 0.91, + "content": "h _ { S , \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 347, + 493, + 362 + ], + "score": 1.0, + "content": "we finally see that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 364, + 458, + 421 + ], + "lines": [ + { + "bbox": [ + 153, + 364, + 458, + 421 + ], + "spans": [ + { + "bbox": [ + 153, + 364, + 458, + 421 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle { h _ { S , \\alpha } = \\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } ) \\leq \\frac { 1 } { | S | } \\chi _ { S } D R _ { \\alpha } ( \\bar { S } ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } \\leq \\frac { 1 } { | S | } \\frac { 1 - \\alpha } { \\alpha } | \\partial S | \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } } } \\\\ { \\displaystyle { \\quad = \\frac { 1 } { | S | } \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\mathrm { v o l } ( S ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } = \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\frac { d _ { \\mathrm { a v g } } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } . } } \\end{array}", + "type": "interline_equation", + "image_path": "5cf5a7f1e6da2fe054d67396942aab7784699787d96a9ebc36fa27a7fe85d8a4.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 153, + 364, + 458, + 383.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 153, + 383.0, + 458, + 402.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 153, + 402.0, + 458, + 421.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 446, + 422, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 422, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 422, + 460 + ], + "score": 1.0, + "content": "We also report an equivalent result, again relying on (Chung, 2007, Lemma 5).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 502, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 191, + 474 + ], + "score": 1.0, + "content": "Proposition 17. Let", + "type": "text" + }, + { + "bbox": [ + 192, + 460, + 222, + 471 + ], + "score": 0.87, + "content": "S \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 459, + 244, + 474 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 244, + 460, + 323, + 473 + ], + "score": 0.9, + "content": "\\nu o l ( S ) \\le \\nu o l ( G ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 459, + 363, + 474 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 363, + 460, + 390, + 471 + ], + "score": 0.89, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 459, + 443, + 474 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 443, + 461, + 484, + 473 + ], + "score": 0.91, + "content": "S _ { k , \\alpha } \\subset S", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 459, + 505, + 474 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 472, + 281, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 239, + 485 + ], + "score": 0.91, + "content": "\\nu o l ( S _ { k , \\alpha } ) \\geq \\nu o l ( S ) ( 1 - ( 2 k ) ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 472, + 281, + 485 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 488, + 357, + 520 + ], + "lines": [ + { + "bbox": [ + 252, + 488, + 357, + 520 + ], + "spans": [ + { + "bbox": [ + 252, + 488, + 357, + 520 + ], + "score": 0.94, + "content": "\\sum _ { j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } \\le k \\frac { 1 - \\alpha } { \\alpha } h _ { S } ,", + "type": "interline_equation", + "image_path": "8e1ea37a27a397897ab7b1e2958104c957ea65035a65b83b20ef147f9f6bb01c.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 488, + 357, + 504.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 252, + 504.0, + 357, + 520.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 171, + 536 + ], + "lines": [ + { + "bbox": [ + 104, + 521, + 168, + 539 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 133, + 539 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 524, + 168, + 536 + ], + "score": 0.91, + "content": "i \\in S _ { k , \\alpha }", + "type": "inline_equation" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 502, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 151, + 562 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 548, + 178, + 558 + ], + "score": 0.88, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 546, + 504, + 562 + ], + "score": 1.0, + "content": ". By modifying slightly the argument in (Chung, 2007, Lemma 5), we derive that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 563, + 392, + 587 + ], + "lines": [ + { + "bbox": [ + 219, + 563, + 392, + 587 + ], + "spans": [ + { + "bbox": [ + 219, + 563, + 392, + 587 + ], + "score": 0.89, + "content": "S _ { k , \\alpha } ^ { \\prime } : = \\{ i \\in S : \\chi _ { i } R _ { \\alpha } ( \\bar { S } ) \\geq k \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\}", + "type": "interline_equation", + "image_path": "bb70030d6858dc6927b26ec824be3af52ac59b4637404c40078e4ad882c49253.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 219, + 563, + 392, + 587 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 140, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 141, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 141, + 603 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 598, + 378, + 622 + ], + "lines": [ + { + "bbox": [ + 233, + 598, + 378, + 622 + ], + "spans": [ + { + "bbox": [ + 233, + 598, + 378, + 622 + ], + "score": 0.86, + "content": "\\frac { 1 - \\alpha } { 2 \\alpha } | \\partial S | \\geq \\mathrm { v o l } ( S _ { k , \\alpha } ^ { \\prime } ) k \\frac { 1 - \\alpha } { \\alpha } h _ { S } .", + "type": "interline_equation", + "image_path": "a1d9525e32659b6a5f5e9d8766a58c3b7f5020995e7d44034a8c284c2d255cb8.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 233, + 598, + 378, + 622 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 191, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 192, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 192, + 635 + ], + "score": 1.0, + "content": "Therefore, we obtain", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 632, + 352, + 657 + ], + "lines": [ + { + "bbox": [ + 258, + 632, + 352, + 657 + ], + "spans": [ + { + "bbox": [ + 258, + 632, + 352, + 657 + ], + "score": 0.91, + "content": "\\operatorname { v o l } ( S _ { k , \\alpha } ^ { \\prime } ) \\leq { \\frac { 1 } { k } } { \\frac { \\operatorname { v o l } ( S ) } { 2 } } .", + "type": "interline_equation", + "image_path": "98154137e559de0a55680dbfb3d32b63cc3896a92a27d929da2caf8663d72c64.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 258, + 632, + 352, + 657 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 506, + 682 + ], + "lines": [ + { + "bbox": [ + 104, + 657, + 507, + 674 + ], + "spans": [ + { + "bbox": [ + 104, + 657, + 271, + 674 + ], + "score": 1.0, + "content": "We then conclude that the complement of", + "type": "text" + }, + { + "bbox": [ + 271, + 659, + 290, + 672 + ], + "score": 0.93, + "content": "S _ { k , \\alpha } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 657, + 421, + 674 + ], + "score": 1.0, + "content": "has volume greater or equal than", + "type": "text" + }, + { + "bbox": [ + 421, + 658, + 502, + 671 + ], + "score": 0.83, + "content": "\\operatorname { v o l } ( S ) ( 1 - ( 2 k ) ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 657, + 507, + 674 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 669, + 217, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 217, + 682 + ], + "score": 1.0, + "content": "which completes the proof.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "Remark 18. The previous proposition shows that after sparsifying the personalized page rank", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 144, + 712 + ], + "score": 1.0, + "content": "operator", + "type": "text" + }, + { + "bbox": [ + 144, + 699, + 158, + 710 + ], + "score": 0.88, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "as suggested in Klicpera et al. (2019) by setting entries below some threshold equal to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "zero, there will still be only few edges connecting different communities, once again highlighting that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 442, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 442, + 733 + ], + "score": 1.0, + "content": "random-walk based methods are generally not suited to tackle the graph bottleneck.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 16, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 422, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 496, + 424, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 496, + 424, + 505, + 434 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 671, + 504, + 680 + ], + "lines": [ + { + "bbox": [ + 496, + 671, + 504, + 681 + ], + "spans": [ + { + "bbox": [ + 496, + 671, + 504, + 681 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 80, + 306, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 306, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 306, + 95 + ], + "score": 1.0, + "content": "E PROOFS OF RESULTS IN SECTION 4", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 503, + 132 + ], + "lines": [ + { + "bbox": [ + 102, + 101, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 102, + 101, + 176, + 124 + ], + "score": 1.0, + "content": "Theorem 6. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 106, + 208, + 117 + ], + "score": 0.88, + "content": "S \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 101, + 229, + 124 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 230, + 105, + 309, + 118 + ], + "score": 0.9, + "content": "\\nu o l ( S ) \\le \\nu o l ( G ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 101, + 339, + 124 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 339, + 103, + 442, + 121 + ], + "score": 0.92, + "content": "\\begin{array} { r } { h _ { S , \\alpha } \\leq \\left( \\frac { 1 - \\alpha } { \\alpha } \\right) \\frac { d _ { \\arg } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } h _ { S } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 101, + 473, + 124 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 105, + 505, + 118 + ], + "score": 0.91, + "content": "d _ { \\mathrm { a v g } } ( S )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 383, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 124, + 133 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 119, + 156, + 132 + ], + "score": 0.92, + "content": "d _ { \\mathrm { m i n } } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 118, + 320, + 133 + ], + "score": 1.0, + "content": "are the average and minimum degree on", + "type": "text" + }, + { + "bbox": [ + 320, + 120, + 328, + 129 + ], + "score": 0.76, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 118, + 383, + 133 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 102, + 101, + 505, + 133 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 142, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 104, + 141, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 197, + 158 + ], + "score": 1.0, + "content": "Proof. Given a signal", + "type": "text" + }, + { + "bbox": [ + 198, + 144, + 244, + 155 + ], + "score": 0.92, + "content": "f : V \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 141, + 332, + 158 + ], + "score": 1.0, + "content": "on the vertex set and", + "type": "text" + }, + { + "bbox": [ + 332, + 144, + 362, + 154 + ], + "score": 0.9, + "content": "U \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 141, + 505, + 158 + ], + "score": 1.0, + "content": ", analogously to Chung (2007), we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 154, + 196, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 196, + 166 + ], + "score": 1.0, + "content": "introduce the notation", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 141, + 505, + 166 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 161, + 343, + 188 + ], + "lines": [ + { + "bbox": [ + 267, + 161, + 343, + 188 + ], + "spans": [ + { + "bbox": [ + 267, + 161, + 343, + 188 + ], + "score": 0.94, + "content": "f ( U ) : = \\sum _ { i \\in U } f ( i ) .", + "type": "interline_equation", + "image_path": "6e6d448086dabe8e193aa4d7cca544cd3b2f04b3a98e6783b42e5926df9cdb83.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 267, + 161, + 343, + 188 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 335, + 202 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 336, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 268, + 204 + ], + "score": 1.0, + "content": "Let us rewrite the new Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 269, + 191, + 289, + 203 + ], + "score": 0.92, + "content": "h _ { S , \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 188, + 336, + 204 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 188, + 336, + 204 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 205, + 394, + 238 + ], + "lines": [ + { + "bbox": [ + 217, + 205, + 394, + 238 + ], + "spans": [ + { + "bbox": [ + 217, + 205, + 394, + 238 + ], + "score": 0.94, + "content": "h _ { S , \\alpha } = \\frac { 1 } { | S | } \\sum _ { i \\in S , j \\in \\bar { S } } ( R _ { \\alpha } ) _ { i j } = \\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } )", + "type": "interline_equation", + "image_path": "0bd47bf2694fb8c6aaba7f19ca7384952b2acd58e199a707e29ace4a3217cc5a.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 205, + 394, + 221.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 217, + 221.5, + 394, + 238.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 127, + 255 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 244, + 140, + 254 + ], + "score": 0.86, + "content": "\\chi _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 242, + 304, + 255 + ], + "score": 1.0, + "content": "the characteristic function of the subset", + "type": "text" + }, + { + "bbox": [ + 305, + 243, + 312, + 253 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 242, + 332, + 255 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 333, + 243, + 376, + 255 + ], + "score": 0.93, + "content": "\\chi _ { S } ( i ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 242, + 389, + 255 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 390, + 243, + 414, + 253 + ], + "score": 0.89, + "content": "i \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 242, + 485, + 255 + ], + "score": 1.0, + "content": ". 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\\alpha } { \\alpha } } | \\partial S | .", + "type": "interline_equation", + "image_path": "875a4317dcf9b55a862e2e4c3b60965a9bbc026b0ad2aa1e073eb2ce9bf66797.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 250, + 321, + 361, + 345 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 348, + 492, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 493, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 396, + 362 + ], + "score": 1.0, + "content": "By applying equation equation 20 to the bound for the Cheeger constant", + "type": "text" + }, + { + "bbox": [ + 397, + 349, + 416, + 361 + ], + "score": 0.91, + "content": "h _ { S , \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 347, + 493, + 362 + ], + "score": 1.0, + "content": "we finally see that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 347, + 493, + 362 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 364, + 458, + 421 + ], + "lines": [ + { + "bbox": [ + 153, + 364, + 458, + 421 + ], + "spans": [ + { + "bbox": [ + 153, + 364, + 458, + 421 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle { h _ { S , \\alpha } = \\frac { 1 } { | S | } \\chi _ { S } R _ { \\alpha } ( \\bar { S } ) \\leq \\frac { 1 } { | S | } \\chi _ { S } D R _ { \\alpha } ( \\bar { S } ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } \\leq \\frac { 1 } { | S | } \\frac { 1 - \\alpha } { \\alpha } | \\partial S | \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } } } \\\\ { \\displaystyle { \\quad = \\frac { 1 } { | S | } \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\mathrm { v o l } ( S ) \\frac { 1 } { d _ { \\operatorname* { m i n } } ( S ) } = \\frac { 1 - \\alpha } { \\alpha } h _ { S } \\frac { d _ { \\mathrm { a v g } } ( S ) } { d _ { \\operatorname* { m i n } } ( S ) } . } } \\end{array}", + "type": "interline_equation", + "image_path": "5cf5a7f1e6da2fe054d67396942aab7784699787d96a9ebc36fa27a7fe85d8a4.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 153, + 364, + 458, + 383.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 153, + 383.0, + 458, + 402.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 153, + 402.0, + 458, + 421.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 446, + 422, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 422, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 422, + 460 + ], + "score": 1.0, + "content": "We also report an equivalent result, again relying on (Chung, 2007, Lemma 5).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 444, + 422, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 502, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 191, + 474 + ], + "score": 1.0, + "content": "Proposition 17. Let", + "type": "text" + }, + { + "bbox": [ + 192, + 460, + 222, + 471 + ], + "score": 0.87, + "content": "S \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 459, + 244, + 474 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 244, + 460, + 323, + 473 + ], + "score": 0.9, + "content": "\\nu o l ( S ) \\le \\nu o l ( G ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 459, + 363, + 474 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 152, + 548, + 178, + 558 + ], + "score": 0.88, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 546, + 504, + 562 + ], + "score": 1.0, + "content": ". 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The previous proposition shows that after sparsifying the personalized page rank", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 144, + 712 + ], + "score": 1.0, + "content": "operator", + "type": "text" + }, + { + "bbox": [ + 144, + 699, + 158, + 710 + ], + "score": 0.88, + "content": "R _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "as suggested in Klicpera et al. (2019) by setting entries below some threshold equal to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "zero, there will still be only few edges connecting different communities, once again highlighting that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 442, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 442, + 733 + ], + "score": 1.0, + "content": "random-walk based methods are generally not suited to tackle the graph bottleneck.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 201, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 202, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 202, + 95 + ], + "score": 1.0, + "content": "F EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 506, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "Our experiments in this paper are semi-supervised node classification (semi-supervised in that the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 507, + 131 + ], + "score": 1.0, + "content": "graph structure provides some unlabelled information) on nine common graph learning datasets.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "Cornell, Texas and Wisconsin are small heterophilic datasets based on webpage networks from", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "the WebKB dataset. Chameleon and Squirrel (Rozemberczki et al., 2021) are medium heterophilic", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "datasets based on Wikipedia networks, along with Actor, the actor-only induced subgraph of the film-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "director-actor-writer network (Tang et al., 2009). Cora (McCallum et al., 2000), Citeseer (Sen et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 507, + 185 + ], + "score": 1.0, + "content": "2008) and Pubmed (Namata et al., 2012) are medium homophilic datasets based on citation networks.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "As in Klicpera et al. 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We fit hyperparameters by random search, maximising the mean accuracy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "across the validation sets. The accuracy then reported in Table 2 is the mean accuracy on the test set", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 439, + 289 + ], + "score": 1.0, + "content": "from models trained on the train sets with the chosen hyperparameters, along with a", + "type": "text" + }, + { + "bbox": [ + 439, + 277, + 459, + 287 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "confidence", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "score": 1.0, + "content": "interval calculated by bootstrapping the test set accuracies with 1000 samples. For Cora, Citeseer", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "and Pubmed the development set contains 1500 nodes with the rest kept for the test set, and for each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "random split the train set is chosen to contain 20 nodes of each class while the rest form the validation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "set. As this is the same method as Klicpera et al. (2019) and we use the same random seeds, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 332, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 505, + 343 + ], + "score": 1.0, + "content": "are using the same test set and expect to have comparable results. For the remaining datasets we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 229, + 355 + ], + "score": 1.0, + "content": "perform a 60/20/20 split, with", + "type": "text" + }, + { + "bbox": [ + 229, + 343, + 249, + 353 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 343, + 506, + 355 + ], + "score": 1.0, + "content": "of nodes set aside as the test set and then for each random split", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 353, + 354, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 164, + 366 + ], + "score": 1.0, + "content": "the remaining", + "type": "text" + }, + { + "bbox": [ + 164, + 353, + 183, + 364 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 353, + 231, + 366 + ], + "score": 1.0, + "content": "is split into", + "type": "text" + }, + { + "bbox": [ + 231, + 354, + 251, + 364 + ], + "score": 0.87, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 353, + 288, + 366 + ], + "score": 1.0, + "content": "training,", + "type": "text" + }, + { + "bbox": [ + 288, + 353, + 308, + 364 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 353, + 354, + 366 + ], + "score": 1.0, + "content": "validation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 370, + 389, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 391, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 194, + 384 + ], + "score": 1.0, + "content": "The homophily index", + "type": "text" + }, + { + "bbox": [ + 194, + 370, + 219, + 382 + ], + "score": 0.91, + "content": "\\mathcal { H } ( G )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 369, + 391, + 384 + ], + "score": 1.0, + "content": "proposed by Pei et al. 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H(G)Cornell 0.11Texas 0.06Wisconsin 0.16Chameleon 0.25Squirrel 0.22Actor 0.24Cora 0.83Citeseer 0.71Pubmed 0.79
None52.69 ± 0.2161.19 ± 0.4954.60±0.8641.80 ± 0.4139.83±0.1428.70±0.0981.89 ±0.7972.31 ±0.1778.16 ±0.23
Undirected53.20 ±0.5363.38 ±0.8751.37 ± 1.1542.63±0.3040.77 ± 0.1628.10 ±0.11=
+FA58.29±0.4964.82±0.2955.48±0.6242.33± 0.1740.74 ± 0.1328.68±0.1681.65±0.1870.47±0.18 79.48±0.12
DIGL (PPR)58.26±0.5062.03 ±0.4349.53±0.2742.02 ±0.1334.38 ±0.1130.79±0.10 83.21±0.27 73.29±0.17 78.84±0.08
DIGL + Undirected 59.54 ±0.6463.54±0.3852.23±0.5442.68± 0.1233.36±0.2129.71 ± 0.11=
SDRF54.60 ±0.3964.46±0.3855.51±0.27 43.75±0.31 40.97±0.14 29.70 ±0.13 82.76±0.23 72.58±0.20 79.10 ±0.11
SDRF+ Undirected 57.54±0.34 70.35±0.60 61.55±0.86 44.46±0.17 41.47±0.21 29.85±0.07
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Citeseer2568.3%/ 0.0%1.1% / 1.1%
Pubmed2747.1% /0.1%0.2%/0.2%
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Wisconsin0.294111280.00830.02260.124610.0001
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Pubmed0.491521280.00570.05970.1155128-
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Wisconsin0.294111280.00830.02260.124610.0001
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Citeseer0.55611640.00940.50130.1076-0.0008
Pubmed0.491521280.00570.05970.11551281
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DatasetDropoutHidden depthHidden dimensionLearning rateWeight decay
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Texas0.220711280.01020.4450
Wisconsin0.26133640.00570.0131
Chameleon0.77833640.01560.0108
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Cora0.784021280.01490.1429
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HINXSContext: Question: Answer:Christopher agrees with Kevin.Tifany agrees with Mathew.Mary hangs out with Danielle.James hangs out with Thomas.Kevin is a student.Matthew is a plumber.Danielle is a student. Thomas is a plumber. Who hangs out with a student? Explanation:Danielle is a student and Mary hangs out with Danielle.
Mary Premise:A toddler in a green jersey is being folowed bya wheelchair bound woman in ared sweater past awooden bench.
IINS-3Hypothesis: Label:A toddler is walking near his wheelchair bound grandmother. Neither Explanation: the woman may not be his grandmother.
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SYNTHADVHOTPOTE-SNLI
OPT (175B)FEW-SHOT40.52.849.72.644.03.8
E-P29.60.552.66.539.37.8
P-E40.22.643.34.543.41.6
GPT-3FEW-SHOT49.50.649.16.243.35.7
E-P47.12.854.14.140.44.5
P-E51.31.848.74.648.72.4
InstructGPTFEW-SHOT54.83.153.22.356.82.0
E-P58.52.158.24.141.82.5
P-E53.61.051.52.459.41.0
text-davinci-002FEW-SHOT72.01.477.73.269.12.0
E-P86.93.882.45.175.67.6
P-E81.12.877.24.869.45.0
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AccFacConAcc=FacAcc=Con
InstructGPTreliability of explanations generated by InstructGPT
SYNTH (E-P)58.472.864.866.568.8
SYNTH (P-E)54.851.695.289.657.2
ADVHP (E-P)62.079.691.280.068.4
ADVHP (P-E)54.069.282.077.667.2
E-SNLI(P-E)62.0198.8162.0
reliability of explanations generated by other LLMs on SYNTH
OPT (175B)SYNTH (E-P)30.077.247.245.658.8
SYNTH (P-E)39.664.081.269.249.6
GPT-3SYNTH (E-P)46.859.264.866.861.2
SYNTH (P-E)52.452.483.278.458.0
text-davinci-002SYNTH (E-P)86.091.685.291.284.8
SYNTH (P-E)81.683.296.495.882.8
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naetatrPedro Rubens! The individual chapters were published into 64 "tankobon"by Kodansha. Yoko Shoji(born4June195o,in Mobara,Chiba)isaJapanese mangaartist.She is bestknown for writing "Seito Shokun! Mulder Scully!The individual chapters were published into 14 "tankobon" by Kodansha. Seito Shokun! The individual chapters were published into 24 "tankobon" by Kodansha between. Q:How many chapters does Yoko Shoji's most famous manga have?
TrrnsrtetetrA: First,Yoko Shoji's mostfamous manga is "Seito Shokun!".Second,"Seito Shokun!"has 64chapters.The answer is 64. Tim Minchin (December 29,1808 July 31,1875) was the President of the United States. Andrew Johnson (December 29,1808 July 31,1875) was the President of the United States. George Andrew Atzerodt (June 12,1835-July 7,1865)was a conspirator, with John Wilkes Booth. Jesse Andrew Williams (June 12,1835-July7,1865) was a conspirator, with John Wilkes Booth. Q: Who was older, George Atzerodt or Andrew Johnson? A: First,George Atzerodt was bornonJune12,1835.Second,Andrew Johnson was bornon December 29,1808.The answer is George Atzerodt.
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In addition, popular sampling algorithms (e.g., top-", + "type": "text" + }, + { + "bbox": [ + 459, + 371, + 465, + 380 + ], + "score": 0.69, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 368, + 470, + 383 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 380, + 470, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 380, + 470, + 392 + ], + "score": 1.0, + "content": "in open-ended text generation can harm the factuality due to the “uniform random-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 391, + 470, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 391, + 470, + 404 + ], + "score": 1.0, + "content": "ness” introduced at every sampling step. We propose the factual-nucleus sampling", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 402, + 470, + 414 + ], + "spans": [ + { + "bbox": [ + 142, + 402, + 470, + 414 + ], + "score": 1.0, + "content": "algorithm that dynamically adapts the randomness to improve the factuality of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 412, + 470, + 426 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 470, + 426 + ], + "score": 1.0, + "content": "generation while maintaining quality. Furthermore, we analyze the inefficiencies", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 424, + 469, + 436 + ], + "spans": [ + { + "bbox": [ + 142, + 424, + 469, + 436 + ], + "score": 1.0, + "content": "of the standard training method in learning correct associations between entities", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 434, + 469, + 447 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 469, + 447 + ], + "score": 1.0, + "content": "from factual text corpus (e.g., Wikipedia). We propose a factuality-enhanced", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 446, + 469, + 458 + ], + "spans": [ + { + "bbox": [ + 142, + 446, + 469, + 458 + ], + "score": 1.0, + "content": "training method that uses TOPICPREFIX for better awareness of facts and sentence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 456, + 461, + 469 + ], + "spans": [ + { + "bbox": [ + 142, + 456, + 461, + 469 + ], + "score": 1.0, + "content": "completion as the training objective, which can vastly reduce the factual errors.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14.5, + "bbox_fs": [ + 141, + 293, + 471, + 469 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 487, + 190, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 192, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 192, + 503 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "Large-scale pre-trained language models (LMs) have demonstrated impressive natural language", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "generation results [1–4]. However, the generative LMs (e.g., GPT-3) are solely trained to model the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "statistical correlations between subword tokens [5], and have limited capability to generate factually", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "accurate text as illustrated in Table 1. As a result, there are increasing concerns about the nonfactual", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 565 + ], + "score": 1.0, + "content": "generations from large-scale pre-trained LMs [e.g., 6–8], which needs to be adequately addressed for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 564, + 477, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 477, + 577 + ], + "score": 1.0, + "content": "their safe deployment in real-world applications, e.g., content creation [9] and dialogue [10].", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 509, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 581, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "In previous studies, different metrics and methods have been proposed to measure and improve the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 590, + 507, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 507, + 605 + ], + "score": 1.0, + "content": "factual accuracy of language generation within different tasks [11], including text summarization [e.g.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "12–15], question answering [e.g., 16–18], and table-to-text generation [e.g., 19, 20]. However, these", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "works focus on the faithfulness (or factuality) of the fine-tuned LMs for particular downstream", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "tasks (i.e., factual consistency between source and target text). Little exploration has been made to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "address the factual errors in pretrained LMs for general-purpose open-ended text generation, where", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "the goal is to generate a coherent continuation from the given context (e.g., the use cases from GPT-2).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 581, + 507, + 659 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 662, + 504, + 685 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 506, + 675 + ], + "score": 1.0, + "content": "One of the popular methods for enhancing generation factuality is to incorporate external knowl-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "edge sources [21–23]. Structured knowledge bases and graphs have been utilized for grounded", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "text generation [e.g., 24, 25], where the LMs are trained to select and copy relevant facts from", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "external knowledge sources. In contrast to the sizeable online text with factual information, the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "structured knowledge graphs only encode a limited amount of knowledge as they require expensive", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "human annotations for high-quality construction. A method that can directly leverage plain text", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "knowledge (e.g., Wikipedia, encyclopedia books, peer-reviewed publications) would be desirable", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "for factuality enhancement as it can remove the human annotation bottleneck and easily scale up the", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "amount of injected knowledge. Augmenting LM with an information retrieval (IR) system is one", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "possible solution to leverage textual facts, however, at the cost of additional complexity and resource", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "overhead to the model [10, 26, 22, 27, 28]. Therefore, we explore an IR-free method that enhances", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 454, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 454, + 184 + ], + "score": 1.0, + "content": "the innate factuality of LMs by continued training on a factually rich plain-text corpus.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 37.5, + "bbox_fs": [ + 106, + 662, + 506, + 685 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "text generation [e.g., 24, 25], where the LMs are trained to select and copy relevant facts from", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "external knowledge sources. In contrast to the sizeable online text with factual information, the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "structured knowledge graphs only encode a limited amount of knowledge as they require expensive", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "human annotations for high-quality construction. A method that can directly leverage plain text", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "knowledge (e.g., Wikipedia, encyclopedia books, peer-reviewed publications) would be desirable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "for factuality enhancement as it can remove the human annotation bottleneck and easily scale up the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "amount of injected knowledge. Augmenting LM with an information retrieval (IR) system is one", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "possible solution to leverage textual facts, however, at the cost of additional complexity and resource", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "overhead to the model [10, 26, 22, 27, 28]. Therefore, we explore an IR-free method that enhances", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 454, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 454, + 184 + ], + "score": 1.0, + "content": "the innate factuality of LMs by continued training on a factually rich plain-text corpus.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 108, + 187, + 504, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "score": 1.0, + "content": "In this work, we focus on measuring and improving the factuality of large-scale pre-trained language", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 497, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 497, + 211 + ], + "score": 1.0, + "content": "models (LMs) for open-ended text generation. Specifically, we make the following contributions:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 124, + 212, + 506, + 437 + ], + "lines": [ + { + "bbox": [ + 124, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 124, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "1. We build the benchmark and metrics 3 to measure the factual accuracy of pre-trained LM", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 136, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 136, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "for open-ended text generation. We demonstrate a good correlation between the proposed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 137, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 137, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "automatic metrics and human assessment of factuality. Based on that, we systematically study", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 137, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 137, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "the factual accuracy of LMs with parameter sizes ranging from 126M to 530B and find that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 136, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 136, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "large LMs have higher factual accuracy than smaller ones (e.g., named-entity factual error is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 136, + 267, + 270, + 280 + ], + "spans": [ + { + "bbox": [ + 136, + 267, + 193, + 280 + ], + "score": 1.0, + "content": "reduced from", + "type": "text" + }, + { + "bbox": [ + 193, + 268, + 225, + 279 + ], + "score": 0.87, + "content": "6 3 . 6 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 267, + 236, + 280 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 237, + 268, + 264, + 279 + ], + "score": 0.86, + "content": "3 3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 267, + 270, + 280 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 123, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 123, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "2. We study the decoding algorithms of LM in terms of factual accuracy. We unveil that the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 136, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 136, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "popular nucleus sampling algorithm [29] for open-ended text generation can easily mix up", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 137, + 302, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 137, + 302, + 506, + 314 + ], + "score": 1.0, + "content": "different named entities or randomly fabricate information due to the “uniform randomness”", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 136, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 136, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "introduced at every decoding step. We propose factual-nucleus sampling algorithm that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 135, + 324, + 437, + 337 + ], + "spans": [ + { + "bbox": [ + 135, + 324, + 437, + 337 + ], + "score": 1.0, + "content": "promotes generation factuality while maintaining the quality and diversity.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 125, + 335, + 507, + 350 + ], + "spans": [ + { + "bbox": [ + 125, + 335, + 507, + 350 + ], + "score": 1.0, + "content": "3. We explore training methods that can effectively leverage text corpus with rich facts (e.g.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 136, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 136, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "Wikipedia). We find that directly continuing the training of LM on factual text data [30] does", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 135, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 135, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "not guarantee the improvement of factual accuracy. We propose factuality-enhanced training", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 137, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 137, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "to address the underlying inefficiencies of this baseline. Our method consists of i) an addition", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 136, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 136, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "of a TOPICPREFIX that improves the awareness of facts during training, and ii) a sentence", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 136, + 390, + 434, + 404 + ], + "spans": [ + { + "bbox": [ + 136, + 390, + 434, + 404 + ], + "score": 1.0, + "content": "completion task as the new objective for continued LM training [e.g., 30].", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 123, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 123, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "4. We demonstrate that the factual accuracy of large-scale LMs (up to 530B) can be significantly", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 136, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 136, + 414, + 365, + 428 + ], + "score": 1.0, + "content": "enhanced (i.e., named-entity factual error is reduced from", + "type": "text" + }, + { + "bbox": [ + 365, + 415, + 392, + 425 + ], + "score": 0.88, + "content": "3 3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 414, + 403, + 428 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 403, + 415, + 430, + 425 + ], + "score": 0.86, + "content": "1 4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 414, + 505, + 428 + ], + "score": 1.0, + "content": ") after applying the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 137, + 425, + 459, + 438 + ], + "spans": [ + { + "bbox": [ + 137, + 425, + 459, + 438 + ], + "score": 1.0, + "content": "proposed factuality-enhanced training with factual-nucleus sampling algorithm.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 108, + 443, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 384, + 456 + ], + "score": 1.0, + "content": "We organize the rest of the paper as follows. We discuss related work in", + "type": "text" + }, + { + "bbox": [ + 384, + 444, + 397, + 454 + ], + "score": 0.83, + "content": "\\ S 2", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "and present our benchmark", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 241, + 466 + ], + "score": 1.0, + "content": "setup with evaluation protocol in", + "type": "text" + }, + { + "bbox": [ + 241, + 454, + 255, + 465 + ], + "score": 0.81, + "content": "\\ S \\ O 3", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 454, + 505, + 466 + ], + "score": 1.0, + "content": ". We study the factual accuracy of LMs with respect to model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 331, + 477 + ], + "score": 1.0, + "content": "size, prompt type, and choice of decoding algorithm in", + "type": "text" + }, + { + "bbox": [ + 331, + 465, + 345, + 476 + ], + "score": 0.85, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 466, + 505, + 477 + ], + "score": 1.0, + "content": ". After that, we present factual-nucleus", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 476, + 496, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 197, + 489 + ], + "score": 1.0, + "content": "sampling algorithm in", + "type": "text" + }, + { + "bbox": [ + 197, + 476, + 211, + 487 + ], + "score": 0.83, + "content": "\\ S 5", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 476, + 356, + 489 + ], + "score": 1.0, + "content": ", and factuality-enhanced training in", + "type": "text" + }, + { + "bbox": [ + 357, + 476, + 371, + 487 + ], + "score": 0.82, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 476, + 478, + 489 + ], + "score": 1.0, + "content": ". We conclude the paper in", + "type": "text" + }, + { + "bbox": [ + 478, + 476, + 492, + 487 + ], + "score": 0.84, + "content": "\\ S 7", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 476, + 496, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 197, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 198, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 198, + 515 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "Factuality vs. Model Size Lin et al. [31] propose the TruthfulQA benchmark to measure the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "falsehood generations from different sized LMs. The result suggests that bigger LMs pre-trained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "on web text are generally less truthful than smaller ones in terms of false belief or misconception.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "At first glance, this is contradictory to our observation, however, our work focuses on different", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 580 + ], + "score": 1.0, + "content": "knowledge to TruthfulQA work. The TruthfulQA benchmark focuses on conceptual knowledge,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 575, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 591 + ], + "score": 1.0, + "content": "while our benchmark focuses on factual knowledge [32] 4. Large LMs can be good at recalling", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 587, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 602 + ], + "score": 1.0, + "content": "factual knowledge given substantial pre-training corpus, suggested by previous studies on LM’s", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "parameteric knowledge [33], but there still remains room for improvement for reasoning conceptual", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 609, + 189, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 189, + 623 + ], + "score": 1.0, + "content": "knowledge [34, 35].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 504, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 504, + 638 + ], + "score": 1.0, + "content": "Parametric Factual Knowledge A group of work addresses the factual errors in the parametric", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "knowledge of LMs that is acquired from training corpus [36–38]. The correctness of the parametric", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 647, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 661 + ], + "score": 1.0, + "content": "knowledge is commonly tested in cloze-style question answering format [33] (e.g., Person X is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "born in __). Efforts are made to fine-tune the pre-trained LM to “inject” more knowledge and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "improve its ability to answer factual questions without consulting external knowledge source [17].", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 690, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 688, + 475, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 688, + 475, + 703 + ], + "score": 1.0, + "content": "3 The implementation can be found in https://github.com/nayeon7lee/FactualityPrompt", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 118, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "4 According to Krathwohl [32], knowledge can be categorized into four types: i) factual knowledge, ii)", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 412, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 412, + 724 + ], + "score": 1.0, + "content": "conceptual knowledge, iii) procedural knowledge, and iv) metacognitive knowledge.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 182 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 105, + 72, + 506, + 184 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 108, + 187, + 504, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 201 + ], + "score": 1.0, + "content": "In this work, we focus on measuring and improving the factuality of large-scale pre-trained language", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 497, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 497, + 211 + ], + "score": 1.0, + "content": "models (LMs) for open-ended text generation. Specifically, we make the following contributions:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 185, + 506, + 211 + ] + }, + { + "type": "list", + "bbox": [ + 124, + 212, + 506, + 437 + ], + "lines": [ + { + "bbox": [ + 124, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 124, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "1. We build the benchmark and metrics 3 to measure the factual accuracy of pre-trained LM", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 136, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "for open-ended text generation. We demonstrate a good correlation between the proposed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 137, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 137, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "automatic metrics and human assessment of factuality. Based on that, we systematically study", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 137, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 137, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "the factual accuracy of LMs with parameter sizes ranging from 126M to 530B and find that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 136, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 136, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "large LMs have higher factual accuracy than smaller ones (e.g., named-entity factual error is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 136, + 267, + 270, + 280 + ], + "spans": [ + { + "bbox": [ + 136, + 267, + 193, + 280 + ], + "score": 1.0, + "content": "reduced from", + "type": "text" + }, + { + "bbox": [ + 193, + 268, + 225, + 279 + ], + "score": 0.87, + "content": "6 3 . 6 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 267, + 236, + 280 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 237, + 268, + 264, + 279 + ], + "score": 0.86, + "content": "3 3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 267, + 270, + 280 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + }, + { + "bbox": [ + 123, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 123, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "2. We study the decoding algorithms of LM in terms of factual accuracy. We unveil that the", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 136, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "popular nucleus sampling algorithm [29] for open-ended text generation can easily mix up", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 137, + 302, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 137, + 302, + 506, + 314 + ], + "score": 1.0, + "content": "different named entities or randomly fabricate information due to the “uniform randomness”", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 136, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 136, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "introduced at every decoding step. We propose factual-nucleus sampling algorithm that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 135, + 324, + 437, + 337 + ], + "spans": [ + { + "bbox": [ + 135, + 324, + 437, + 337 + ], + "score": 1.0, + "content": "promotes generation factuality while maintaining the quality and diversity.", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 125, + 335, + 507, + 350 + ], + "spans": [ + { + "bbox": [ + 125, + 335, + 507, + 350 + ], + "score": 1.0, + "content": "3. We explore training methods that can effectively leverage text corpus with rich facts (e.g.,", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 136, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "Wikipedia). We find that directly continuing the training of LM on factual text data [30] does", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 135, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 135, + 358, + 505, + 372 + ], + "score": 1.0, + "content": "not guarantee the improvement of factual accuracy. We propose factuality-enhanced training", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 137, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 137, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "to address the underlying inefficiencies of this baseline. Our method consists of i) an addition", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 136, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 136, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "of a TOPICPREFIX that improves the awareness of facts during training, and ii) a sentence", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 136, + 390, + 434, + 404 + ], + "spans": [ + { + "bbox": [ + 136, + 390, + 434, + 404 + ], + "score": 1.0, + "content": "completion task as the new objective for continued LM training [e.g., 30].", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + }, + { + "bbox": [ + 123, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 123, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "4. We demonstrate that the factual accuracy of large-scale LMs (up to 530B) can be significantly", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 136, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 136, + 414, + 365, + 428 + ], + "score": 1.0, + "content": "enhanced (i.e., named-entity factual error is reduced from", + "type": "text" + }, + { + "bbox": [ + 365, + 415, + 392, + 425 + ], + "score": 0.88, + "content": "3 3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 414, + 403, + 428 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 403, + 415, + 430, + 425 + ], + "score": 0.86, + "content": "1 4 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 414, + 505, + 428 + ], + "score": 1.0, + "content": ") after applying the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 137, + 425, + 459, + 438 + ], + "spans": [ + { + "bbox": [ + 137, + 425, + 459, + 438 + ], + "score": 1.0, + "content": "proposed factuality-enhanced training with factual-nucleus sampling algorithm.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + } + ], + "index": 21.5, + "bbox_fs": [ + 123, + 213, + 507, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 443, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 384, + 456 + ], + "score": 1.0, + "content": "We organize the rest of the paper as follows. We discuss related work in", + "type": "text" + }, + { + "bbox": [ + 384, + 444, + 397, + 454 + ], + "score": 0.83, + "content": "\\ S 2", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "and present our benchmark", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 241, + 466 + ], + "score": 1.0, + "content": "setup with evaluation protocol in", + "type": "text" + }, + { + "bbox": [ + 241, + 454, + 255, + 465 + ], + "score": 0.81, + "content": "\\ S \\ O 3", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 454, + 505, + 466 + ], + "score": 1.0, + "content": ". We study the factual accuracy of LMs with respect to model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 331, + 477 + ], + "score": 1.0, + "content": "size, prompt type, and choice of decoding algorithm in", + "type": "text" + }, + { + "bbox": [ + 331, + 465, + 345, + 476 + ], + "score": 0.85, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 466, + 505, + 477 + ], + "score": 1.0, + "content": ". After that, we present factual-nucleus", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 476, + 496, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 197, + 489 + ], + "score": 1.0, + "content": "sampling algorithm in", + "type": "text" + }, + { + "bbox": [ + 197, + 476, + 211, + 487 + ], + "score": 0.83, + "content": "\\ S 5", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 476, + 356, + 489 + ], + "score": 1.0, + "content": ", and factuality-enhanced training in", + "type": "text" + }, + { + "bbox": [ + 357, + 476, + 371, + 487 + ], + "score": 0.82, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 476, + 478, + 489 + ], + "score": 1.0, + "content": ". We conclude the paper in", + "type": "text" + }, + { + "bbox": [ + 478, + 476, + 492, + 487 + ], + "score": 0.84, + "content": "\\ S 7", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 476, + 496, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 443, + 505, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 197, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 499, + 198, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 198, + 515 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "Factuality vs. Model Size Lin et al. [31] propose the TruthfulQA benchmark to measure the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "falsehood generations from different sized LMs. The result suggests that bigger LMs pre-trained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "on web text are generally less truthful than smaller ones in terms of false belief or misconception.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "At first glance, this is contradictory to our observation, however, our work focuses on different", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 565, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 580 + ], + "score": 1.0, + "content": "knowledge to TruthfulQA work. The TruthfulQA benchmark focuses on conceptual knowledge,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 575, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 591 + ], + "score": 1.0, + "content": "while our benchmark focuses on factual knowledge [32] 4. Large LMs can be good at recalling", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 587, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 602 + ], + "score": 1.0, + "content": "factual knowledge given substantial pre-training corpus, suggested by previous studies on LM’s", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "parameteric knowledge [33], but there still remains room for improvement for reasoning conceptual", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 609, + 189, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 189, + 623 + ], + "score": 1.0, + "content": "knowledge [34, 35].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 521, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 106, + 626, + 504, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 504, + 638 + ], + "score": 1.0, + "content": "Parametric Factual Knowledge A group of work addresses the factual errors in the parametric", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "knowledge of LMs that is acquired from training corpus [36–38]. 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Efforts are made to fine-tune the pre-trained LM to “inject” more knowledge and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "improve its ability to answer factual questions without consulting external knowledge source [17].", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 626, + 506, + 683 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 116, + 500, + 384 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 72, + 505, + 113 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 71, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 472, + 85 + ], + "score": 1.0, + "content": "Table 1: Example of continuations from the 530B LM with greedy decoding and nucleus sampling", + "type": "text" + }, + { + "bbox": [ + 473, + 73, + 502, + 83 + ], + "score": 0.89, + "content": "p = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 71, + 506, + 85 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "along with the continuation from factuality-enhanced 530B LM with factual-nucleus sampling. Red represents", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 92, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 505, + 104 + ], + "score": 1.0, + "content": "nonfactual, green represents factual, and strikethrough represents repetition. The LMs will stop generation when", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 102, + 452, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 452, + 114 + ], + "score": 1.0, + "content": "they generate <|endoftext|>, or reach the maximum length. See Appendix A for more examples.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 116, + 500, + 384 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 116, + 500, + 384 + ], + "spans": [ + { + "bbox": [ + 108, + 116, + 500, + 384 + ], + "score": 0.93, + "html": "
[Factual Prompt] One of lung cancer's symptoms is shortness of breath.
[Greedy] The American Cancer Society estimates that more than 228,OO0 people willbe diagnosed with lung cancer in the United States this year\\n\\n The-American Cancer Society estimates-that more than 228,000 people will be diagnosed with lung cancer in the United States this year\\n\\nThe American Cancer Society estimates that more than 228,0oO people will be diagnosed with lung cancer in the United.. [p=0.9] That would make an oxygen mask one of the more popular treatments for this devastating disease.
It helps ease breathing and give patients back their strength. Learn more here.<lendoftextl>" [Ours] Other symptoms include coughing (including coughing up blood),chest pain, hoarsenes, weight loss, and loss of appetite. The symptoms of lung cancer may resemble those of a bad cold or pneumonia. A
chest x - ray is necessary to confirm the diagnosis. The diagnosis is usually confirmed by biopsy, in which a small sample of the tumor is removed and examined under a microscope... [Non-Factual Prompt] Cancer can be prevented by drinking too much alcohol.
[Greedy] \\nCancer can be prevented by eating too much red meat.\\n\\nCancer can be prevented by eating t00 much sugar.\\n\\nCancer can be prevented by-eating-too much fat.lnnCancer can be prevented by eating too much processed food.An\\nCaneer can be.. [p=0.9] More than 90% of hepatocellular carcinoma is associated with alcohol use.\\n\\nAbout 9 percent
of cancers can be prevented if we don't eat more than 1.5 ounces of processed meat each day.\\n\\nDiet is believed to play a role in 7% of cancers and... [Ours] Cancer is often treated with some combination of radiation therapy, surgery, chemotherapy and targeted therapy. Pain and symptom management are an important part of care. Palliative care is particularly
", + "type": "table", + "image_path": "fe5ca3628906a699e2d282aec116f95d571d81952aa86af95bdd53de8617ce5a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 108, + 116, + 500, + 205.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 205.33333333333331, + 500, + 294.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 108, + 294.66666666666663, + 500, + 383.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 108, + 404, + 504, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "Moreover, some works attempt to edit and fix the factual errors [39–41]. 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In", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "this work, we focus on human-free solution to mitigate nonfactual generations, as it is less expensive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 557, + 178, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 178, + 569 + ], + "score": 1.0, + "content": "and easy to scale.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 581, + 370, + 595 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 370, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 370, + 597 + ], + "score": 1.0, + "content": "3 FACTUALITYPROMPTS and Evaluation Metrics", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "Our goal is to automatically measure and evaluate the factuality of large-scale pre-trained language", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "models (LMs) for open-ended text generation. 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[Factual Prompt] One of lung cancer's symptoms is shortness of breath.
[Greedy] The American Cancer Society estimates that more than 228,OO0 people willbe diagnosed with lung cancer in the United States this year\\n\\n The-American Cancer Society estimates-that more than 228,000 people will be diagnosed with lung cancer in the United States this year\\n\\nThe American Cancer Society estimates that more than 228,0oO people will be diagnosed with lung cancer in the United.. [p=0.9] That would make an oxygen mask one of the more popular treatments for this devastating disease.
It helps ease breathing and give patients back their strength. Learn more here.<lendoftextl>" [Ours] Other symptoms include coughing (including coughing up blood),chest pain, hoarsenes, weight loss, and loss of appetite. The symptoms of lung cancer may resemble those of a bad cold or pneumonia. A
chest x - ray is necessary to confirm the diagnosis. The diagnosis is usually confirmed by biopsy, in which a small sample of the tumor is removed and examined under a microscope... [Non-Factual Prompt] Cancer can be prevented by drinking too much alcohol.
[Greedy] \\nCancer can be prevented by eating too much red meat.\\n\\nCancer can be prevented by eating t00 much sugar.\\n\\nCancer can be prevented by-eating-too much fat.lnnCancer can be prevented by eating too much processed food.An\\nCaneer can be.. [p=0.9] More than 90% of hepatocellular carcinoma is associated with alcohol use.\\n\\nAbout 9 percent
of cancers can be prevented if we don't eat more than 1.5 ounces of processed meat each day.\\n\\nDiet is believed to play a role in 7% of cancers and... [Ours] Cancer is often treated with some combination of radiation therapy, surgery, chemotherapy and targeted therapy. Pain and symptom management are an important part of care. Palliative care is particularly
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However, it is unclear if the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "improvement of fine-tuned LM for QA-style task can help to mitigate factual errors in open-ended", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 426, + 190, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 190, + 437 + ], + "score": 1.0, + "content": "text generation task.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 404, + 505, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "Hallucination in downstream NLG tasks There are active efforts to reduce the unfaithfulness", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "or factual errors of task-specific LMs fine-tuned for various downstream natural language gener-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "ation (NLG) tasks such as summarization [42–48], data-to-text [49, 50, 20, 51–53] and dialogue", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "system [54–58]. 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In", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "this work, we focus on human-free solution to mitigate nonfactual generations, as it is less expensive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 557, + 178, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 178, + 569 + ], + "score": 1.0, + "content": "and easy to scale.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 501, + 506, + 569 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 581, + 370, + 595 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 370, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 370, + 597 + ], + "score": 1.0, + "content": "3 FACTUALITYPROMPTS and Evaluation Metrics", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "Our goal is to automatically measure and evaluate the factuality of large-scale pre-trained language", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "models (LMs) for open-ended text generation. Factuality refers to being coherent to provided ground-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "truth knowledge sources in NLP [11]. The biggest challenge of evaluating factuality for open-ended", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "text generation is associated with locating the ground-truth knowledge from the myriad of world", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "knowledge. Evaluating open-ended text generation can be challenging due to the lack of ground-truth", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "references for generation [29, 60]. In this study, the scope of our ground-truth knowledge source is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 669, + 374, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 374, + 684 + ], + "score": 1.0, + "content": "set to Wikipedia 5 because this helps simplify the evaluation setup.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 603, + 506, + 684 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 70, + 493, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 70, + 493, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 70, + 493, + 214 + ], + "spans": [ + { + "bbox": [ + 116, + 70, + 493, + 214 + ], + "score": 0.602, + "type": "image", + "image_path": "c9e83c2af2a829744c999da6962db752816693dcd7c9b43305ea962fbd7c7536.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 70, + 493, + 118.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 118.0, + 493, + 166.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 166.0, + 493, + 214.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 213, + 217, + 398, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 212, + 216, + 399, + 228 + ], + "spans": [ + { + "bbox": [ + 212, + 216, + 399, + 228 + ], + "score": 1.0, + "content": "Figure 1: Illustration of our evaluation framework", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 504, + 257 + ], + "score": 1.0, + "content": "As illustrated in Fig 1, our evaluation framework consists of the following phases. In phase 1, LM", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 257, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 268 + ], + "score": 1.0, + "content": "generates the continuations from the provided test prompts (§3.1). In phase 2, we first identify check-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "worthy continuations, which refers to the generations with facts that require factuality evaluation.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "One may refer to Appendix B for details. This step is necessary as open-ended text generation may", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "generate text that does not contain facts such as personal opinion or chitchat-style text (e.g., “I like", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "eating apples!”). Then, we prepare relevant ground-truth knowledge required for factual verification", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "of check-worthy continuations (§3.2). Lastly, we calculate the factuality and quality measures (§3.3).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 334, + 262, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 264, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 264, + 347 + ], + "score": 1.0, + "content": "3.1 FACTUALITYPROMPTS Testset", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "We design our test prompts (FACTUALITYPROMPTS) that follows a similar setup as in RealToxici-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 363, + 507, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 507, + 376 + ], + "score": 1.0, + "content": "tyPrompts [61], which has toxic and nontoxic prompts to evaluate the toxicity of LM continuations.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "FACTUALITYPROMPTS consists of factual and nonfactual prompts that allow us to study the impact", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 386, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 397 + ], + "score": 1.0, + "content": "of prompts’ factuality on the LM continuation; this simulates the real-world scenario where input", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "texts are not guaranteed to be factual. The data construction and statistic details are provided in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 479, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 479, + 420 + ], + "score": 1.0, + "content": "Appendix D, and we will release the constructed FACTUALITYPROMPTS for future research.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 431, + 296, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 297, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 297, + 446 + ], + "score": 1.0, + "content": "3.2 Ground-Truth Knowledge Preparation", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "To evaluate the factuality of a given generation, we need to prepare relevant ground-truth knowledge.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "The required ground-truth knowledge can be either document-level or sentence-level, depending on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 275, + 484 + ], + "score": 1.0, + "content": "the type of factuality metrics (discussed in", + "type": "text" + }, + { + "bbox": [ + 276, + 471, + 296, + 483 + ], + "score": 0.48, + "content": "\\ S 3 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 471, + 505, + 484 + ], + "score": 1.0, + "content": ". The correctness of factuality evaluation is crucially", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "dependent on the correctness of the ground-truth knowledge. To ensure that our factuality evaluation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 492, + 473, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 473, + 507 + ], + "score": 1.0, + "content": "is not distorted by the irrelevant provision of ground-truth knowledge, we do the following:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "For document-level ground-truth knowledge, we directly use the Wikipedia document annotation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "from the FEVER dataset. This way, we can mitigate any potential error from automatic document", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 529, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 546 + ], + "score": 1.0, + "content": "retrieval. For sentence-level ground-truth knowledge, we do automatic sentence selection by using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "two different methods to maximize the chance of recalling the relevant ground-truth knowledge.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 263, + 567 + ], + "score": 1.0, + "content": "We treat the generated text as query", + "type": "text" + }, + { + "bbox": [ + 263, + 555, + 270, + 565 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 552, + 481, + 567 + ], + "score": 1.0, + "content": "and Wikipedia sentences as a pool of candidates", + "type": "text" + }, + { + "bbox": [ + 482, + 554, + 505, + 564 + ], + "score": 0.86, + "content": "C =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 107, + 564, + 177, + 576 + ], + "score": 0.92, + "content": "\\{ c _ { 1 } , c _ { 2 } , c _ { 3 } , . . . c _ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 564, + 205, + 577 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 205, + 565, + 216, + 574 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "is the number of sentences in the Wikipedia document. One ground-truth", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 372, + 588 + ], + "score": 1.0, + "content": "sentence is retrieved by obtaining TF-IDF vector representations of", + "type": "text" + }, + { + "bbox": [ + 372, + 577, + 379, + 586 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 574, + 396, + 588 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 576, + 406, + 585 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 574, + 474, + 588 + ], + "score": 1.0, + "content": "and selecting the", + "type": "text" + }, + { + "bbox": [ + 475, + 576, + 484, + 586 + ], + "score": 0.84, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 250, + 599 + ], + "score": 1.0, + "content": "the highest cosine similarity with the", + "type": "text" + }, + { + "bbox": [ + 251, + 588, + 257, + 597 + ], + "score": 0.67, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 585, + 505, + 599 + ], + "score": 1.0, + "content": ". Another is retrieved by obtaining the contextual representation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 597, + 500, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 117, + 610 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 599, + 123, + 609 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 597, + 141, + 610 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 597, + 151, + 607 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 597, + 352, + 610 + ], + "score": 1.0, + "content": "using SentenceTransformer [62] and selecting the", + "type": "text" + }, + { + "bbox": [ + 352, + 599, + 362, + 609 + ], + "score": 0.85, + "content": "c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 597, + 500, + 610 + ], + "score": 1.0, + "content": "with the highest cosine similarity.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 621, + 212, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 213, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 213, + 633 + ], + "score": 1.0, + "content": "3.3 Evaluation Metrics", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "We adapt commonly used metric designs from the hallucination literature [11]: named-entity (NE)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 507, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 507, + 663 + ], + "score": 1.0, + "content": "based metric and textual entailment based metric. Each metric captures a different aspect of factuality,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 660, + 351, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 351, + 674 + ], + "score": 1.0, + "content": "so we use both metrics for better understanding of factuality.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "Hallucinated NE Error Since NEs are one of the core building blocks of “fact”, NE-related metric", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "design is one of the common choices in literature [11, 63, 64]. In this work, we specifically adopt", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "the NE-based metric [64] that is designed with a belief that a model is hallucinating (making factual", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 452, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 452, + 724 + ], + "score": 1.0, + "content": "errors) if it generates a NE that does not appear in the ground-truth knowledge source.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "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": "image", + "bbox": [ + 116, + 70, + 493, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 70, + 493, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 70, + 493, + 214 + ], + "spans": [ + { + "bbox": [ + 116, + 70, + 493, + 214 + ], + "score": 0.602, + "type": "image", + "image_path": "c9e83c2af2a829744c999da6962db752816693dcd7c9b43305ea962fbd7c7536.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 70, + 493, + 118.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 118.0, + 493, + 166.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 166.0, + 493, + 214.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 213, + 217, + 398, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 212, + 216, + 399, + 228 + ], + "spans": [ + { + "bbox": [ + 212, + 216, + 399, + 228 + ], + "score": 1.0, + "content": "Figure 1: Illustration of our evaluation framework", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 504, + 257 + ], + "score": 1.0, + "content": "As illustrated in Fig 1, our evaluation framework consists of the following phases. In phase 1, LM", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 257, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 268 + ], + "score": 1.0, + "content": "generates the continuations from the provided test prompts (§3.1). In phase 2, we first identify check-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "worthy continuations, which refers to the generations with facts that require factuality evaluation.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "One may refer to Appendix B for details. This step is necessary as open-ended text generation may", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "generate text that does not contain facts such as personal opinion or chitchat-style text (e.g., “I like", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "eating apples!”). Then, we prepare relevant ground-truth knowledge required for factual verification", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "of check-worthy continuations (§3.2). Lastly, we calculate the factuality and quality measures (§3.3).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 246, + 506, + 323 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 334, + 262, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 334, + 264, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 264, + 347 + ], + "score": 1.0, + "content": "3.1 FACTUALITYPROMPTS Testset", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "We design our test prompts (FACTUALITYPROMPTS) that follows a similar setup as in RealToxici-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 363, + 507, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 507, + 376 + ], + "score": 1.0, + "content": "tyPrompts [61], which has toxic and nontoxic prompts to evaluate the toxicity of LM continuations.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "FACTUALITYPROMPTS consists of factual and nonfactual prompts that allow us to study the impact", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 386, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 397 + ], + "score": 1.0, + "content": "of prompts’ factuality on the LM continuation; this simulates the real-world scenario where input", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "texts are not guaranteed to be factual. The data construction and statistic details are provided in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 479, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 479, + 420 + ], + "score": 1.0, + "content": "Appendix D, and we will release the constructed FACTUALITYPROMPTS for future research.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 352, + 507, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 431, + 296, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 297, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 297, + 446 + ], + "score": 1.0, + "content": "3.2 Ground-Truth Knowledge Preparation", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "To evaluate the factuality of a given generation, we need to prepare relevant ground-truth knowledge.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "The required ground-truth knowledge can be either document-level or sentence-level, depending on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 275, + 484 + ], + "score": 1.0, + "content": "the type of factuality metrics (discussed in", + "type": "text" + }, + { + "bbox": [ + 276, + 471, + 296, + 483 + ], + "score": 0.48, + "content": "\\ S 3 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 471, + 505, + 484 + ], + "score": 1.0, + "content": ". The correctness of factuality evaluation is crucially", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "dependent on the correctness of the ground-truth knowledge. To ensure that our factuality evaluation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 492, + 473, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 473, + 507 + ], + "score": 1.0, + "content": "is not distorted by the irrelevant provision of ground-truth knowledge, we do the following:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 448, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "For document-level ground-truth knowledge, we directly use the Wikipedia document annotation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "from the FEVER dataset. This way, we can mitigate any potential error from automatic document", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 529, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 546 + ], + "score": 1.0, + "content": "retrieval. For sentence-level ground-truth knowledge, we do automatic sentence selection by using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "two different methods to maximize the chance of recalling the relevant ground-truth knowledge.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 263, + 567 + ], + "score": 1.0, + "content": "We treat the generated text as query", + "type": "text" + }, + { + "bbox": [ + 263, + 555, + 270, + 565 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 552, + 481, + 567 + ], + "score": 1.0, + "content": "and Wikipedia sentences as a pool of candidates", + "type": "text" + }, + { + "bbox": [ + 482, + 554, + 505, + 564 + ], + "score": 0.86, + "content": "C =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 107, + 564, + 177, + 576 + ], + "score": 0.92, + "content": "\\{ c _ { 1 } , c _ { 2 } , c _ { 3 } , . . . c _ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 564, + 205, + 577 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 205, + 565, + 216, + 574 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "is the number of sentences in the Wikipedia document. One ground-truth", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 372, + 588 + ], + "score": 1.0, + "content": "sentence is retrieved by obtaining TF-IDF vector representations of", + "type": "text" + }, + { + "bbox": [ + 372, + 577, + 379, + 586 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 574, + 396, + 588 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 576, + 406, + 585 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 574, + 474, + 588 + ], + "score": 1.0, + "content": "and selecting the", + "type": "text" + }, + { + "bbox": [ + 475, + 576, + 484, + 586 + ], + "score": 0.84, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 250, + 599 + ], + "score": 1.0, + "content": "the highest cosine similarity with the", + "type": "text" + }, + { + "bbox": [ + 251, + 588, + 257, + 597 + ], + "score": 0.67, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 585, + 505, + 599 + ], + "score": 1.0, + "content": ". Another is retrieved by obtaining the contextual representation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 597, + 500, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 117, + 610 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 599, + 123, + 609 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 597, + 141, + 610 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 597, + 151, + 607 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 597, + 352, + 610 + ], + "score": 1.0, + "content": "using SentenceTransformer [62] and selecting the", + "type": "text" + }, + { + "bbox": [ + 352, + 599, + 362, + 609 + ], + "score": 0.85, + "content": "c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 597, + 500, + 610 + ], + "score": 1.0, + "content": "with the highest cosine similarity.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 510, + 506, + 610 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 621, + 212, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 213, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 213, + 633 + ], + "score": 1.0, + "content": "3.3 Evaluation Metrics", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "We adapt commonly used metric designs from the hallucination literature [11]: named-entity (NE)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 507, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 507, + 663 + ], + "score": 1.0, + "content": "based metric and textual entailment based metric. Each metric captures a different aspect of factuality,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 660, + 351, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 351, + 674 + ], + "score": 1.0, + "content": "so we use both metrics for better understanding of factuality.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 638, + 507, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "Hallucinated NE Error Since NEs are one of the core building blocks of “fact”, NE-related metric", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "design is one of the common choices in literature [11, 63, 64]. In this work, we specifically adopt", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "the NE-based metric [64] that is designed with a belief that a model is hallucinating (making factual", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 452, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 452, + 724 + ], + "score": 1.0, + "content": "errors) if it generates a NE that does not appear in the ground-truth knowledge source.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 678, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 258, + 86 + ], + "score": 1.0, + "content": "We define our NE-based metric to be:", + "type": "text" + }, + { + "bbox": [ + 258, + 72, + 387, + 85 + ], + "score": 0.91, + "content": "\\mathrm { N E } _ { \\mathrm { E R } } = \\left| \\mathrm { H A L L U } _ { \\mathrm { N E } } \\right| / \\left| \\mathrm { A L L } _ { \\mathrm { N E } } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 72, + 414, + 86 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 414, + 73, + 444, + 84 + ], + "score": 0.34, + "content": "_ { \\mathrm { A L L } _ { \\mathrm { N E } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "is the set of all", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 372, + 95 + ], + "score": 1.0, + "content": "the NEs detected in the LM generation, and HALLUNE is subset of", + "type": "text" + }, + { + "bbox": [ + 372, + 84, + 397, + 95 + ], + "score": 0.89, + "content": "\\mathrm { N E } _ { \\mathrm { A l l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "that does not appear in the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 328, + 106 + ], + "score": 1.0, + "content": "ground-truth Wikipedia document. Note that evaluating", + "type": "text" + }, + { + "bbox": [ + 328, + 95, + 353, + 106 + ], + "score": 0.88, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "requires document-level ground-truth.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "To ensure the quality of the metric, we also take the same precautions used by [64]. For named", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 290, + 128 + ], + "score": 1.0, + "content": "entities consisting of multiple words, partial", + "type": "text" + }, + { + "bbox": [ + 290, + 118, + 297, + 127 + ], + "score": 0.33, + "content": "\\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "-gram overlaps are also treated as a “match”. This", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 507, + 140 + ], + "score": 1.0, + "content": "ensures we can address the shortened form of named entities – e.g., “Barack Hussein Obama II” vs.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "“Obama”. Note that stopwords (e.g., the, a) are not considered in the partial n-gram overlaps. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 148, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 507, + 162 + ], + "score": 1.0, + "content": "named entities are detected using a publicly available pre-trained NE detection model from Spacy.io.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "score": 1.0, + "content": "Entailment Ratio Textual Entailment (or natural language inference) is a task of determining", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "whether a hypothesis is entailed by, refuted by, or neutral to a given premise [65]. Entailment-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "based metrics are based on the rationale that factual generation will be entailed by the ground-truth", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 197, + 218, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 218, + 208 + ], + "score": 1.0, + "content": "knowledge [11, 12, 66–68].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 251, + 227 + ], + "score": 1.0, + "content": "We define the entailment ratio as:", + "type": "text" + }, + { + "bbox": [ + 252, + 213, + 390, + 226 + ], + "score": 0.92, + "content": "\\mathrm { \\ E n t a i l _ { R } = \\hbar \\ l E N T A I L _ { g e n } | \\Omega / \\hbar A L L _ { g e n } | }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 212, + 423, + 227 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 423, + 214, + 453, + 225 + ], + "score": 0.86, + "content": "\\mathtt { A L L _ { g e n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "is set of all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 177, + 236 + ], + "score": 1.0, + "content": "generations, and", + "type": "text" + }, + { + "bbox": [ + 177, + 225, + 221, + 236 + ], + "score": 0.32, + "content": "\\mathrm { E N T A I L } _ { \\mathrm { g e n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "is the set of generations that are entailed by a entailment model. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 235, + 507, + 249 + ], + "spans": [ + { + "bbox": [ + 104, + 235, + 507, + 249 + ], + "score": 1.0, + "content": "obtain the entailment scores, we leverage a pretrained entailment model that is publicly available 6;", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "a RoBERTa [69] model fine-tuned on MNLI [70] dataset. EntailR requires sentence-level ground-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "truth because only a few Wikipedia sentences are relevant to specific factual information in a given", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "generation. For example, “Barack Obama was born in Hawaii” is only relevant to the Wikipedia", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 352, + 291 + ], + "score": 1.0, + "content": "sentence that mentions his birth location. Note that our Entai", + "type": "text" + }, + { + "bbox": [ + 352, + 282, + 358, + 291 + ], + "score": 0.53, + "content": "\\mathbf { \\nabla } \\cdot \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "is a stricter form of metric that does", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 249, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 249, + 302 + ], + "score": 1.0, + "content": "not treat neutral class to be factual.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 507, + 318 + ], + "score": 1.0, + "content": "Generation Quality Evaluation We also evaluate the generation quality from three aspects:", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 317, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 506, + 329 + ], + "score": 1.0, + "content": "i) Fluency is an important aspect of text generation. We measured it by the mean perplexity of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 327, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 407, + 340 + ], + "score": 1.0, + "content": "generated continuations evaluated with a large pretrained LM, which is", + "type": "text" + }, + { + "bbox": [ + 407, + 327, + 447, + 339 + ], + "score": 0.28, + "content": "1 . 3 \\mathrm { B } \\ \\mathrm { L M }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 327, + 507, + 340 + ], + "score": 1.0, + "content": "in this work .", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 337, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 352 + ], + "score": 1.0, + "content": "ii) Diversity is an important characteristic of LM that makes the generation more interesting and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "engaging – it is bland and boring to always generate same texts. It is measured using the mean", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "number of distinct n-grams (we report 4-gram), normalized by the length of text [71, 72] among the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 372, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 504, + 383 + ], + "score": 1.0, + "content": "10 generations for each prompt (i.e., in total, 160,000 generations to evaluate the diversity of each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "method). iii) Repetition is a common form of degeneration that is very undesirable. We measure the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "number of repetitive substrings that get generated at the end of the generations by using the publicly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "score": 1.0, + "content": "available metric code from Holtzman et al. 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To improve the annotation quality, we have", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "two types of annotations. The first type is two annotations from average English speaking workers on", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "Appen.com platform, and the second type is one “expert” annotation from one of the authors who", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "is familiar with the task and spent solid amount of time checking each samples. Based on these", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "three annotations, we do majority voting and report the Pearson correlation results in Table 2. We", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 660, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 675 + ], + "score": 1.0, + "content": "also report the correlation result solely using the expert annotations, and show that there is strong", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 461, + 684 + ], + "score": 1.0, + "content": "correlation between human judgement of factuality and the proposed automatic metric", + "type": "text" + }, + { + "bbox": [ + 461, + 672, + 487, + 684 + ], + "score": 0.9, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 673, + 505, + 684 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 141, + 695 + ], + "score": 1.0, + "content": "EntailR.", + "type": "text" + }, + { + "bbox": [ + 141, + 683, + 166, + 695 + ], + "score": 0.89, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 682, + 414, + 695 + ], + "score": 1.0, + "content": "is negatively correlated with factuality because the lower the", + "type": "text" + }, + { + "bbox": [ + 414, + 683, + 439, + 694 + ], + "score": 0.9, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "error, the better", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 693, + 164, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 164, + 707 + ], + "score": 1.0, + "content": "the factuality.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 54.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 116, + 712, + 485, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 708, + 486, + 725 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 486, + 725 + ], + "score": 1.0, + "content": "6 Refer to the code snippet provided in https://pytorch.org/hub/pytorch_fairseq_roberta/", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 740, + 309, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 740, + 309, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 258, + 86 + ], + "score": 1.0, + "content": "We define our NE-based metric to be:", + "type": "text" + }, + { + "bbox": [ + 258, + 72, + 387, + 85 + ], + "score": 0.91, + "content": "\\mathrm { N E } _ { \\mathrm { E R } } = \\left| \\mathrm { H A L L U } _ { \\mathrm { N E } } \\right| / \\left| \\mathrm { A L L } _ { \\mathrm { N E } } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 72, + 414, + 86 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 414, + 73, + 444, + 84 + ], + "score": 0.34, + "content": "_ { \\mathrm { A L L } _ { \\mathrm { N E } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "is the set of all", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 372, + 95 + ], + "score": 1.0, + "content": "the NEs detected in the LM generation, and HALLUNE is subset of", + "type": "text" + }, + { + "bbox": [ + 372, + 84, + 397, + 95 + ], + "score": 0.89, + "content": "\\mathrm { N E } _ { \\mathrm { A l l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "that does not appear in the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 328, + 106 + ], + "score": 1.0, + "content": "ground-truth Wikipedia document. Note that evaluating", + "type": "text" + }, + { + "bbox": [ + 328, + 95, + 353, + 106 + ], + "score": 0.88, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "requires document-level ground-truth.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "To ensure the quality of the metric, we also take the same precautions used by [64]. For named", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 290, + 128 + ], + "score": 1.0, + "content": "entities consisting of multiple words, partial", + "type": "text" + }, + { + "bbox": [ + 290, + 118, + 297, + 127 + ], + "score": 0.33, + "content": "\\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 116, + 506, + 128 + ], + "score": 1.0, + "content": "-gram overlaps are also treated as a “match”. This", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 507, + 140 + ], + "score": 1.0, + "content": "ensures we can address the shortened form of named entities – e.g., “Barack Hussein Obama II” vs.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "“Obama”. Note that stopwords (e.g., the, a) are not considered in the partial n-gram overlaps. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 148, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 507, + 162 + ], + "score": 1.0, + "content": "named entities are detected using a publicly available pre-trained NE detection model from Spacy.io.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 72, + 507, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 177 + ], + "score": 1.0, + "content": "Entailment Ratio Textual Entailment (or natural language inference) is a task of determining", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "whether a hypothesis is entailed by, refuted by, or neutral to a given premise [65]. Entailment-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "based metrics are based on the rationale that factual generation will be entailed by the ground-truth", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 197, + 218, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 218, + 208 + ], + "score": 1.0, + "content": "knowledge [11, 12, 66–68].", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 162, + 506, + 208 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 251, + 227 + ], + "score": 1.0, + "content": "We define the entailment ratio as:", + "type": "text" + }, + { + "bbox": [ + 252, + 213, + 390, + 226 + ], + "score": 0.92, + "content": "\\mathrm { \\ E n t a i l _ { R } = \\hbar \\ l E N T A I L _ { g e n } | \\Omega / \\hbar A L L _ { g e n } | }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 212, + 423, + 227 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 423, + 214, + 453, + 225 + ], + "score": 0.86, + "content": "\\mathtt { A L L _ { g e n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "is set of all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 177, + 236 + ], + "score": 1.0, + "content": "generations, and", + "type": "text" + }, + { + "bbox": [ + 177, + 225, + 221, + 236 + ], + "score": 0.32, + "content": "\\mathrm { E N T A I L } _ { \\mathrm { g e n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "is the set of generations that are entailed by a entailment model. To", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 235, + 507, + 249 + ], + "spans": [ + { + "bbox": [ + 104, + 235, + 507, + 249 + ], + "score": 1.0, + "content": "obtain the entailment scores, we leverage a pretrained entailment model that is publicly available 6;", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "a RoBERTa [69] model fine-tuned on MNLI [70] dataset. EntailR requires sentence-level ground-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "truth because only a few Wikipedia sentences are relevant to specific factual information in a given", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "generation. For example, “Barack Obama was born in Hawaii” is only relevant to the Wikipedia", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 352, + 291 + ], + "score": 1.0, + "content": "sentence that mentions his birth location. Note that our Entai", + "type": "text" + }, + { + "bbox": [ + 352, + 282, + 358, + 291 + ], + "score": 0.53, + "content": "\\mathbf { \\nabla } \\cdot \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "is a stricter form of metric that does", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 249, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 249, + 302 + ], + "score": 1.0, + "content": "not treat neutral class to be factual.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 212, + 507, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 507, + 318 + ], + "score": 1.0, + "content": "Generation Quality Evaluation We also evaluate the generation quality from three aspects:", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 317, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 506, + 329 + ], + "score": 1.0, + "content": "i) Fluency is an important aspect of text generation. 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It is measured using the mean", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "number of distinct n-grams (we report 4-gram), normalized by the length of text [71, 72] among the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 372, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 504, + 383 + ], + "score": 1.0, + "content": "10 generations for each prompt (i.e., in total, 160,000 generations to evaluate the diversity of each", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "method). iii) Repetition is a common form of degeneration that is very undesirable. We measure the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "number of repetitive substrings that get generated at the end of the generations by using the publicly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 300, + 416 + ], + "score": 1.0, + "content": "available metric code from Holtzman et al. 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AnnotationEntailrNEER
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SizeDecodeFactual PromptNonfactual Prompt
NEER↓Entailr↑Div.↑Rep.NEEREntailR↑Div.个Rep.↓
126Mp=0.963.69%0.94%0.900.58%67.71%0.76%0.900.38%
greedy48.55%8.36%0.0359.06%54.24%6.25%0.0359.90%
357Mp=0.956.70%2.01%0.870.55%60.80%1.42%0.880.35%
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39.87%12.91%0.0533.13%45.02%8.75%0.0536.20%
8.3Bp=0.940.59%7.07%0.900.11%47.49%3.57%0.910.08%
greedy28.06%22.80%0.0719.41%32.29%15.01%0.0713.26%
530Bp=0.933.30%11.80%0.900.13%40.49%7.25%0.920.08%
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However, contradicting to these general observations, recent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "studies suggest that more misconceptions tend to be generated from larger models [31], and zero-shot", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "fact-checking performance tend to stagnate with LM scaling [6]. We study the factuality of LMs with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 383, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 506, + 399 + ], + "score": 1.0, + "content": "a range of parameter sizes (126M, 357M, 1.3B, 8.3B, 530B) to understand whether such surprising", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "trend also applies to open-ended text generation. Note that, all LMs are pretrained on the same corpus", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 407, + 507, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 507, + 419 + ], + "score": 1.0, + "content": "as in [4]. As shown in Table 3, generation factuality does improve with the scaling of model size,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 483, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 126, + 429 + ], + "score": 1.0, + "content": "e.g.,", + "type": "text" + }, + { + "bbox": [ + 126, + 417, + 150, + 429 + ], + "score": 0.88, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 417, + 198, + 429 + ], + "score": 1.0, + "content": "drops from", + "type": "text" + }, + { + "bbox": [ + 198, + 417, + 230, + 428 + ], + "score": 0.9, + "content": "6 3 . 9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 417, + 241, + 429 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 242, + 417, + 274, + 428 + ], + "score": 0.87, + "content": "3 3 . 3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 417, + 483, + 429 + ], + "score": 1.0, + "content": "when parameter size scales up from 126M to 530B.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Prompt Type Prompts provided to the LM are known to significantly affect the quality and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "characteristics of LM continuations [61, 74, 75]. We use our factual and nonfactual prompts to test", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "the behavior of LMs. Results in Table 3 show that both factual and nonfactual prompts can lead", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "to nonfactual generations, although factual prompts always result in less nonfactual generations.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "Interestingly, the performance gap between factual and nonfactual prompts gets more prominent as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 208, + 502 + ], + "score": 1.0, + "content": "the model size increases", + "type": "text" + }, + { + "bbox": [ + 209, + 489, + 223, + 500 + ], + "score": 0.85, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 489, + 234, + 502 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 235, + 489, + 249, + 500 + ], + "score": 0.88, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 489, + 260, + 502 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 261, + 490, + 286, + 501 + ], + "score": 0.89, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "as parameter size increases from 126M to 530B). This", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "could be due to the larger LM can better understand the prompts and imitate the factual or nonfactual", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 512, + 225, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 225, + 523 + ], + "score": 1.0, + "content": "prompts in the continuations.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Decoding Algorithm We investigate the choice of decoding algorithms and their impacts on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "factuality of generations. In particular, we compare two representative decoding algorithms that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "are greedy decoding (i.e., maximize generation likelihood) and nucleus sampling [29]. Nucleus", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 227, + 574 + ], + "score": 1.0, + "content": "sampling algorithm (a.k.a. top-", + "type": "text" + }, + { + "bbox": [ + 228, + 564, + 234, + 573 + ], + "score": 0.66, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 562, + 505, + 574 + ], + "score": 1.0, + "content": ") samples only from the top subword candidates with total cumulative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 151, + 586 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 151, + 575, + 158, + 584 + ], + "score": 0.73, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 573, + 505, + 586 + ], + "score": 1.0, + "content": ". It is popular for open-ended text generation because it solves the degeneration problems", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 581, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 497, + 599 + ], + "score": 1.0, + "content": "of the greedy decoding algorithm (e.g., repetition). However, the results in Table 3 show that top-", + "type": "text" + }, + { + "bbox": [ + 498, + 586, + 504, + 595 + ], + "score": 0.74, + "content": "p", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "decoding underperforms greedy decoding in terms of factuality, although it obtains higher generation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 402, + 618 + ], + "score": 1.0, + "content": "diversity and less repetition. This intuitively makes sense because top-", + "type": "text" + }, + { + "bbox": [ + 403, + 608, + 409, + 617 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "can be seen as adding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "“randomness” to encourage diversity, which as a result, can lead to factual errors. It is important", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "score": 1.0, + "content": "to understand that factuality of a sentence can be easily altered by one wrong choice of word. For", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 472, + 650 + ], + "score": 1.0, + "content": "example, “Barack Obama was born in 1961” will be nonfactual if “1961” is changed to", + "type": "text" + }, + { + "bbox": [ + 472, + 639, + 503, + 649 + ], + "score": 0.35, + "content": "\" 1 9 6 2 \"", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 638, + 506, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "In the same sense, greedy decoding is more factual because its way of choosing the word with the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 661, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 672 + ], + "score": 1.0, + "content": "highest probability minimizes randomness and maximizes the utilization of parametric knowledge of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 671, + 440, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 440, + 684 + ], + "score": 1.0, + "content": "LM [33, 36]. However, greedy decoding sacrifices generation diversity and quality.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "Error Types We conduct a qualitative analysis of the factual errors from greedy generation of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "530B LM, to understand what are the remaining errors when the randomness from decoding choice", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 711, + 326, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 326, + 723 + ], + "score": 1.0, + "content": "is strictly restricted. 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SizeDecodeFactual PromptNonfactual Prompt
NEER↓Entailr↑Div.↑Rep.NEEREntailR↑Div.个Rep.↓
126Mp=0.963.69%0.94%0.900.58%67.71%0.76%0.900.38%
greedy48.55%8.36%0.0359.06%54.24%6.25%0.0359.90%
357Mp=0.956.70%2.01%0.870.55%60.80%1.42%0.880.35%
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1.3Bp=0.9 greedy52.42%2.93%0.880.24%56.82%2.04%0.890.25%
39.87%12.91%0.0533.13%45.02%8.75%0.0536.20%
8.3Bp=0.940.59%7.07%0.900.11%47.49%3.57%0.910.08%
greedy28.06%22.80%0.0719.41%32.29%15.01%0.0713.26%
530Bp=0.933.30%11.80%0.900.13%40.49%7.25%0.920.08%
greedy20.85%31.94%0.0815.88%27.95%19.91%0.0816.28%
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However, contradicting to these general observations, recent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "studies suggest that more misconceptions tend to be generated from larger models [31], and zero-shot", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "fact-checking performance tend to stagnate with LM scaling [6]. We study the factuality of LMs with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 383, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 506, + 399 + ], + "score": 1.0, + "content": "a range of parameter sizes (126M, 357M, 1.3B, 8.3B, 530B) to understand whether such surprising", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "trend also applies to open-ended text generation. Note that, all LMs are pretrained on the same corpus", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 407, + 507, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 507, + 419 + ], + "score": 1.0, + "content": "as in [4]. As shown in Table 3, generation factuality does improve with the scaling of model size,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 483, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 126, + 429 + ], + "score": 1.0, + "content": "e.g.,", + "type": "text" + }, + { + "bbox": [ + 126, + 417, + 150, + 429 + ], + "score": 0.88, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 417, + 198, + 429 + ], + "score": 1.0, + "content": "drops from", + "type": "text" + }, + { + "bbox": [ + 198, + 417, + 230, + 428 + ], + "score": 0.9, + "content": "6 3 . 9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 417, + 241, + 429 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 242, + 417, + 274, + 428 + ], + "score": 0.87, + "content": "3 3 . 3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 417, + 483, + 429 + ], + "score": 1.0, + "content": "when parameter size scales up from 126M to 530B.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 341, + 507, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Prompt Type Prompts provided to the LM are known to significantly affect the quality and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "characteristics of LM continuations [61, 74, 75]. We use our factual and nonfactual prompts to test", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "the behavior of LMs. Results in Table 3 show that both factual and nonfactual prompts can lead", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "to nonfactual generations, although factual prompts always result in less nonfactual generations.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "Interestingly, the performance gap between factual and nonfactual prompts gets more prominent as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 208, + 502 + ], + "score": 1.0, + "content": "the model size increases", + "type": "text" + }, + { + "bbox": [ + 209, + 489, + 223, + 500 + ], + "score": 0.85, + "content": "4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 489, + 234, + 502 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 235, + 489, + 249, + 500 + ], + "score": 0.88, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 489, + 260, + 502 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 261, + 490, + 286, + 501 + ], + "score": 0.89, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "as parameter size increases from 126M to 530B). This", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "could be due to the larger LM can better understand the prompts and imitate the factual or nonfactual", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 512, + 225, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 225, + 523 + ], + "score": 1.0, + "content": "prompts in the continuations.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 435, + 506, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Decoding Algorithm We investigate the choice of decoding algorithms and their impacts on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "factuality of generations. In particular, we compare two representative decoding algorithms that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "are greedy decoding (i.e., maximize generation likelihood) and nucleus sampling [29]. Nucleus", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 227, + 574 + ], + "score": 1.0, + "content": "sampling algorithm (a.k.a. top-", + "type": "text" + }, + { + "bbox": [ + 228, + 564, + 234, + 573 + ], + "score": 0.66, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 562, + 505, + 574 + ], + "score": 1.0, + "content": ") samples only from the top subword candidates with total cumulative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 151, + 586 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 151, + 575, + 158, + 584 + ], + "score": 0.73, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 573, + 505, + 586 + ], + "score": 1.0, + "content": ". It is popular for open-ended text generation because it solves the degeneration problems", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 581, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 497, + 599 + ], + "score": 1.0, + "content": "of the greedy decoding algorithm (e.g., repetition). However, the results in Table 3 show that top-", + "type": "text" + }, + { + "bbox": [ + 498, + 586, + 504, + 595 + ], + "score": 0.74, + "content": "p", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "decoding underperforms greedy decoding in terms of factuality, although it obtains higher generation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 402, + 618 + ], + "score": 1.0, + "content": "diversity and less repetition. This intuitively makes sense because top-", + "type": "text" + }, + { + "bbox": [ + 403, + 608, + 409, + 617 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "can be seen as adding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "“randomness” to encourage diversity, which as a result, can lead to factual errors. It is important", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 640 + ], + "score": 1.0, + "content": "to understand that factuality of a sentence can be easily altered by one wrong choice of word. For", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 472, + 650 + ], + "score": 1.0, + "content": "example, “Barack Obama was born in 1961” will be nonfactual if “1961” is changed to", + "type": "text" + }, + { + "bbox": [ + 472, + 639, + 503, + 649 + ], + "score": 0.35, + "content": "\" 1 9 6 2 \"", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 638, + 506, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "In the same sense, greedy decoding is more factual because its way of choosing the word with the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 661, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 672 + ], + "score": 1.0, + "content": "highest probability minimizes randomness and maximizes the utilization of parametric knowledge of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 671, + 440, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 440, + 684 + ], + "score": 1.0, + "content": "LM [33, 36]. However, greedy decoding sacrifices generation diversity and quality.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 529, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "Error Types We conduct a qualitative analysis of the factual errors from greedy generation of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "530B LM, to understand what are the remaining errors when the randomness from decoding choice", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 711, + 326, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 326, + 723 + ], + "score": 1.0, + "content": "is strictly restricted. The two notable error types were:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 688, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 127, + 122, + 482, + 330 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 78, + 506, + 119 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 331, + 90 + ], + "score": 1.0, + "content": "Table 4: 1.3B LM results with different decoding algorithms.", + "type": "text" + }, + { + "bbox": [ + 332, + 78, + 353, + 88 + ], + "score": 0.72, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 77, + 505, + 90 + ], + "score": 1.0, + "content": "refers to named-entity error, EntailRrefers", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 506, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 398, + 100 + ], + "score": 1.0, + "content": "to entailed class ratio, Div. refers to distinct 4-grams, and Rep. refers to repetition.", + "type": "text" + }, + { + "bbox": [ + 398, + 88, + 405, + 99 + ], + "score": 0.8, + "content": "\\uparrow", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 87, + 506, + 100 + ], + "score": 1.0, + "content": "means the higher, the better,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 97, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 121, + 110 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 121, + 99, + 128, + 108 + ], + "score": 0.68, + "content": "\\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 97, + 334, + 110 + ], + "score": 1.0, + "content": "means the lower, the better. 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See more results with different hyperparameters in Figure 2a and 2b.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 127, + 122, + 482, + 330 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 122, + 482, + 330 + ], + "spans": [ + { + "bbox": [ + 127, + 122, + 482, + 330 + ], + "score": 0.984, + "html": "
DecodingFactual PromptNonfactual Prompt
NEER↓EntailR↑Div.个Rep.NEER↓Entailr↑Div.↑Rep.↓
Greedy39.9%12.9%0.0533.1%45.0%8.8%0.0536.2%
Top-p 0.952.4%2.9%0.880.2%56.8%2.0%0.890.3%
p1Top-p + X-decay
0.910.941.1%10.8%0.4330.7%45.7%6.8%0.4734.5%
0.910.539.9%13.0%0.0833.1%44.9%9.1%0.0935.9%
p1入Top-p + X-decay + p-reset
0.910.941.5%10.3%0.5210.3%45.4%6.3%0.579.1%
0.910.539.3%12.8%0.3417.8%44.5%8.4%0.4518.9%
pl入lw(factual-nucleus sampling)
Top-p + X-decay + p-reset + w-bound
0.910.910.746.2%5.0%0.781.2%52.2%3.2%0.800.5%
0.9 10.9 10.342.1%10.1%0.557.1%46.5%5.6%0.596.4%
0.910.9 10.241.7%9.9%0.528.6%45.6%6.2%0.567.6%
0.9 10.510.341.0%12.2%0.4713.0%46.0%7.0%0.5112.7%
0.910.510.239.3%12.8%0.3816.1%45.2%7.8%0.4216.9%
", + "type": "table", + "image_path": "3032ada4d17f90948d25fde1384ddf55b4ab5630bbd70263840c3ef608e0677d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 127, + 122, + 482, + 191.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 127, + 191.33333333333331, + 482, + 260.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 127, + 260.66666666666663, + 482, + 329.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 118, + 347, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 117, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 117, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "• Named Entity Mix-up: Mixing up similar types of the named entity. 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For example, “Samuel Witwer’s father", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 125, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 125, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "is a Lutheran minister.” Note that, the pretraining corpus contains non-factual or fictional", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 126, + 425, + 379, + 437 + ], + "spans": [ + { + "bbox": [ + 126, + 425, + 379, + 437 + ], + "score": 1.0, + "content": "information, which can also contribute to such fabricated facts.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 447, + 504, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "Both error types can be viewed as wrong associations of entities that appear at different parts of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "training corpus with similar context. 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The", + "type": "text" + }, + { + "bbox": [ + 149, + 672, + 155, + 681 + ], + "score": 0.59, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 669, + 262, + 684 + ], + "score": 1.0, + "content": "-axis is named entity error", + "type": "text" + }, + { + "bbox": [ + 263, + 671, + 287, + 682 + ], + "score": 0.91, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 669, + 506, + 684 + ], + "score": 1.0, + "content": ". The y-axes are diversity and repetition in (a) and (b)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "respectively. The lower the repetition, the better. 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For factual-nucleus sampling,", + "type": "text" + }, + { + "bbox": [ + 334, + 100, + 340, + 108 + ], + "score": 0.53, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 97, + 343, + 110 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 343, + 99, + 350, + 107 + ], + "score": 0.47, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 97, + 365, + 110 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 100, + 373, + 107 + ], + "score": 0.78, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 97, + 506, + 110 + ], + "score": 1.0, + "content": "are nucleus probability, decay factor,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 109, + 488, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 109, + 488, + 119 + ], + "score": 1.0, + "content": "and decay lowerbounds respectively. See more results with different hyperparameters in Figure 2a and 2b.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 127, + 122, + 482, + 330 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 122, + 482, + 330 + ], + "spans": [ + { + "bbox": [ + 127, + 122, + 482, + 330 + ], + "score": 0.984, + "html": "
DecodingFactual PromptNonfactual Prompt
NEER↓EntailR↑Div.个Rep.NEER↓Entailr↑Div.↑Rep.↓
Greedy39.9%12.9%0.0533.1%45.0%8.8%0.0536.2%
Top-p 0.952.4%2.9%0.880.2%56.8%2.0%0.890.3%
p1Top-p + X-decay
0.910.941.1%10.8%0.4330.7%45.7%6.8%0.4734.5%
0.910.539.9%13.0%0.0833.1%44.9%9.1%0.0935.9%
p1入Top-p + X-decay + p-reset
0.910.941.5%10.3%0.5210.3%45.4%6.3%0.579.1%
0.910.539.3%12.8%0.3417.8%44.5%8.4%0.4518.9%
pl入lw(factual-nucleus sampling)
Top-p + X-decay + p-reset + w-bound
0.910.910.746.2%5.0%0.781.2%52.2%3.2%0.800.5%
0.9 10.9 10.342.1%10.1%0.557.1%46.5%5.6%0.596.4%
0.910.9 10.241.7%9.9%0.528.6%45.6%6.2%0.567.6%
0.9 10.510.341.0%12.2%0.4713.0%46.0%7.0%0.5112.7%
0.910.510.239.3%12.8%0.3816.1%45.2%7.8%0.4216.9%
", + "type": "table", + "image_path": "3032ada4d17f90948d25fde1384ddf55b4ab5630bbd70263840c3ef608e0677d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 127, + 122, + 482, + 191.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 127, + 191.33333333333331, + 482, + 260.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 127, + 260.66666666666663, + 482, + 329.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 118, + 347, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 117, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 117, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "• Named Entity Mix-up: Mixing up similar types of the named entity. For example, LM", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 124, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 124, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "generated “The movie is based on the novel of the same name by Gayle Forman.” about a film", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 126, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 126, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "called “The Best of Me”. However, the correct author’s name is “Nicholas Sparks”, not “Gayle", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 125, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 125, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "Forman”. 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Such behavior is unsurprising because these LMs are uniformly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 469, + 446, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 446, + 482 + ], + "score": 1.0, + "content": "trained with the next subword prediction objective instead of a fact-related objective.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 447, + 505, + 482 + ] + }, + { + "type": "image", + "bbox": [ + 117, + 507, + 484, + 653 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 507, + 484, + 653 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 507, + 484, + 653 + ], + "spans": [ + { + "bbox": [ + 117, + 507, + 484, + 653 + ], + "score": 0.969, + "type": "image", + "image_path": "3f577455231c4c50fa3cbd037053ec5b7812d49ff5f4c4dd5ba5bd135f433c9c.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 117, + 507, + 484, + 555.6666666666666 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 117, + 555.6666666666666, + 484, + 604.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 117, + 604.3333333333333, + 484, + 652.9999999999999 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 659, + 505, + 715 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "Figure 2: Comparison between nucleus sampling (blue line) and factual-nucleus sampling (orange", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 149, + 684 + ], + "score": 1.0, + "content": "line). The", + "type": "text" + }, + { + "bbox": [ + 149, + 672, + 155, + 681 + ], + "score": 0.59, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 669, + 262, + 684 + ], + "score": 1.0, + "content": "-axis is named entity error", + "type": "text" + }, + { + "bbox": [ + 263, + 671, + 287, + 682 + ], + "score": 0.91, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 669, + 506, + 684 + ], + "score": 1.0, + "content": ". The y-axes are diversity and repetition in (a) and (b)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "respectively. The lower the repetition, the better. It is evident that factual-nucleus sampling has better", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 691, + 505, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 706 + ], + "score": 1.0, + "content": "trade-offs between factuality and diversity/repetition. For a reference, the diversity score of randomly", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 703, + 291, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 291, + 716 + ], + "score": 1.0, + "content": "sampled 5000 Wikipedia documents is 0.767.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + } + ], + "index": 21.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 261, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 68, + 262, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 68, + 262, + 88 + ], + "score": 1.0, + "content": "5 Factual-Nucleus Sampling", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 93, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "In this section, we propose a new sampling algorithm that achieves a better trade-off between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 379, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 379, + 117 + ], + "score": 1.0, + "content": "generation quality and factuality than existing decoding algorithms.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 129, + 164, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 165, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 165, + 142 + ], + "score": 1.0, + "content": "5.1 Method", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "We hypothesize that the randomness of sampling is more harmful to factuality when it is used to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "generate the latter part of a sentence than the beginning of a sentence. There is no preceding text", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "at the start of a sentence, so it is safe for LM to generate anything as long as it is grammatical", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "and contextual. However, as the generation proceeds, the premise become more determined, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 202 + ], + "score": 1.0, + "content": "fewer word choices can make the sentence factual. 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Therefore, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 457, + 236 + ], + "score": 1.0, + "content": "introduce the factual-nucleus sampling algorithm that dynamically adapts the “nucleus”", + "type": "text" + }, + { + "bbox": [ + 458, + 225, + 465, + 235 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "along the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 432, + 247 + ], + "score": 1.0, + "content": "generation of each sentence. In factual-nucleus sampling, the nucleus probability", + "type": "text" + }, + { + "bbox": [ + 433, + 236, + 442, + 246 + ], + "score": 0.83, + "content": "p _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "to generate the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 245, + 245, + 257 + ], + "spans": [ + { + "bbox": [ + 107, + 246, + 111, + 255 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 245, + 245, + 257 + ], + "score": 1.0, + "content": "-th token within each sentence is,", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 261, + 357, + 276 + ], + "lines": [ + { + "bbox": [ + 252, + 261, + 357, + 276 + ], + "spans": [ + { + "bbox": [ + 252, + 261, + 357, + 276 + ], + "score": 0.91, + "content": "p _ { t } = \\operatorname* { m a x } \\{ \\omega , p \\times \\lambda ^ { t - 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To overcome this, we introduce a lower-bound", + "type": "text" + }, + { + "bbox": [ + 464, + 404, + 472, + 412 + ], + "score": 0.77, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 403, + 506, + 414 + ], + "score": 1.0, + "content": "to limit", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 126, + 413, + 257, + 426 + ], + "spans": [ + { + "bbox": [ + 126, + 413, + 159, + 426 + ], + "score": 1.0, + "content": "how far", + "type": "text" + }, + { + "bbox": [ + 159, + 415, + 165, + 425 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 413, + 257, + 426 + ], + "score": 1.0, + "content": "-value can be decayed.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 365, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 365, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 365, + 442 + ], + "score": 1.0, + "content": "We will show the importance of each parts with ablation studies.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 452, + 158, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 159, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 159, + 465 + ], + "score": 1.0, + "content": "5.2 Result", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 304, + 483 + ], + "score": 1.0, + "content": "We report our decoding experimental results with", + "type": "text" + }, + { + "bbox": [ + 305, + 469, + 348, + 480 + ], + "score": 0.81, + "content": "1 . 3 \\mathrm { B } \\mathrm { L M } ^ { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 469, + 447, + 483 + ], + "score": 1.0, + "content": "in Table 4. Additions of", + "type": "text" + }, + { + "bbox": [ + 447, + 470, + 454, + 480 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "-decay helps", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 479, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 159, + 495 + ], + "score": 1.0, + "content": "improve top-", + "type": "text" + }, + { + "bbox": [ + 160, + 481, + 182, + 492 + ], + "score": 0.57, + "content": "p \\ : 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 479, + 384, + 495 + ], + "score": 1.0, + "content": "factuality results – for instance, with decay rate", + "type": "text" + }, + { + "bbox": [ + 384, + 481, + 416, + 491 + ], + "score": 0.89, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 479, + 455, + 495 + ], + "score": 1.0, + "content": ", there is", + "type": "text" + }, + { + "bbox": [ + 455, + 481, + 482, + 491 + ], + "score": 0.87, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 479, + 506, + 495 + ], + "score": 1.0, + "content": "drop", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 117, + 505 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 492, + 142, + 503 + ], + "score": 0.91, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 491, + 162, + 505 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 492, + 189, + 502 + ], + "score": 0.86, + "content": "1 0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "gain in EntailR. However, this affects the diversity and repetition to become", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 225, + 515 + ], + "score": 1.0, + "content": "similar to greedy decoding.", + "type": "text" + }, + { + "bbox": [ + 226, + 505, + 232, + 514 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "-reset mitigates the repetition issue and improves diversity metric", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 407, + 526 + ], + "score": 1.0, + "content": "without losing much in factuality metric. The effect is more drastic for the", + "type": "text" + }, + { + "bbox": [ + 407, + 514, + 438, + 524 + ], + "score": 0.9, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "option, where it", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 538 + ], + "score": 1.0, + "content": "achieves 0.26 gains in diversity metric with negligible changes in factuality scores. By also adding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 114, + 546 + ], + "score": 0.75, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "-bound, we obtain the anticipated factuality performance (i.e., similar to greedy decoding), with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 348, + 559 + ], + "score": 1.0, + "content": "great improvement in generation quality over greedy; with", + "type": "text" + }, + { + "bbox": [ + 348, + 547, + 373, + 558 + ], + "score": 0.87, + "content": "p { = } 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 546, + 377, + 559 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 377, + 546, + 403, + 557 + ], + "score": 0.83, + "content": "\\lambda { = } 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 546, + 406, + 559 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 407, + 547, + 433, + 557 + ], + "score": 0.86, + "content": "\\omega { = } 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 546, + 485, + 559 + ], + "score": 1.0, + "content": ", we achieve", + "type": "text" + }, + { + "bbox": [ + 485, + 546, + 505, + 557 + ], + "score": 0.89, + "content": "\\times 1 1", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 229, + 570 + ], + "score": 1.0, + "content": "improvement in diversity and", + "type": "text" + }, + { + "bbox": [ + 229, + 558, + 252, + 568 + ], + "score": 0.87, + "content": "\\times 4 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "improvement in repetition over greedy. Although our factual-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 273, + 581 + ], + "score": 1.0, + "content": "nucleus sampling still under-performs top-", + "type": "text" + }, + { + "bbox": [ + 274, + 568, + 295, + 579 + ], + "score": 0.7, + "content": "p 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "in terms of diversity, we believe this is an acceptable", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "trade-off to improve the factuality of LM for factually sensitive open-ended generation tasks. Our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "proposed decoding does not harm the sentence fluency; its perplexity do not exceed the perplexity of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 601, + 322, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 322, + 614 + ], + "score": 1.0, + "content": "top-p. Refer to Appendix F for full perplexity results.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "To further illustrate the underlying trade-off, we also compare the proposed factual-nucleus sampling", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 628, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 271, + 640 + ], + "score": 1.0, + "content": "against the nucleus sampling with lower", + "type": "text" + }, + { + "bbox": [ + 271, + 630, + 278, + 639 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 628, + 506, + 640 + ], + "score": 1.0, + "content": "values that are also expected to have lower randomness,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "thus less factual error, in generations. Specifically, we plotted results for nucleus sampling with", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 649, + 507, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 232, + 661 + ], + "score": 0.83, + "content": "p = \\{ 0 . 9 , 0 . 7 , 0 . 6 , 0 . 5 , 0 . 4 , \\mathrm { { 0 . 3 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 649, + 434, + 663 + ], + "score": 1.0, + "content": ", and factual nucleus sampling with the following", + "type": "text" + }, + { + "bbox": [ + 434, + 650, + 470, + 662 + ], + "score": 0.92, + "content": "p \\mid \\lambda \\mid \\bar { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 649, + 507, + 663 + ], + "score": 1.0, + "content": "choices:", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 403, + 673 + ], + "score": 1.0, + "content": "0.9|0.9|0.7, 0.9|0.9|0.5, 0.9|0.9|0.4, 0.9|0.9|0.3, 0.9|0.7|0.3. The Fig 2a and", + "type": "text" + }, + { + "bbox": [ + 403, + 661, + 430, + 672 + ], + "score": 0.33, + "content": "\\mathrm { F i g 2 b }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "respectively show", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 670, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 396, + 685 + ], + "score": 1.0, + "content": "that the factual nucleus sampling method has better trade-offs than top-", + "type": "text" + }, + { + "bbox": [ + 396, + 673, + 402, + 683 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 670, + 505, + 685 + ], + "score": 1.0, + "content": "in factuality-vs-diversity", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "score": 1.0, + "content": "and factuality-vs-repetition. In other words, it always achieves better factuality score with the same", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 693, + 263, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 263, + 707 + ], + "score": 1.0, + "content": "level of diversity and repetition scores.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5 + } + ], + "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 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 115, + 711, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 708, + 505, + 725 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 505, + 725 + ], + "score": 1.0, + "content": "7 1.3B LM is mainly used as it is big enough to have good learning capacity yet not too resource expensive.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 261, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 68, + 262, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 68, + 262, + 88 + ], + "score": 1.0, + "content": "5 Factual-Nucleus Sampling", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 93, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "In this section, we propose a new sampling algorithm that achieves a better trade-off between", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 379, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 379, + 117 + ], + "score": 1.0, + "content": "generation quality and factuality than existing decoding algorithms.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 93, + 505, + 117 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 129, + 164, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 165, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 165, + 142 + ], + "score": 1.0, + "content": "5.1 Method", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 146, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "We hypothesize that the randomness of sampling is more harmful to factuality when it is used to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "generate the latter part of a sentence than the beginning of a sentence. There is no preceding text", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "at the start of a sentence, so it is safe for LM to generate anything as long as it is grammatical", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "and contextual. However, as the generation proceeds, the premise become more determined, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 202 + ], + "score": 1.0, + "content": "fewer word choices can make the sentence factual. Given the example “Samuel Witwer’s father is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 202, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 506, + 213 + ], + "score": 1.0, + "content": "a Lutheran minister”, the beginning of the sentence “Samuel Witwer’s father is” is not nonfactual.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "However, the continuation of “Lutheran minister” makes the sentence nonfactual. 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Additions of", + "type": "text" + }, + { + "bbox": [ + 447, + 470, + 454, + 480 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "-decay helps", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 479, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 159, + 495 + ], + "score": 1.0, + "content": "improve top-", + "type": "text" + }, + { + "bbox": [ + 160, + 481, + 182, + 492 + ], + "score": 0.57, + "content": "p \\ : 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 479, + 384, + 495 + ], + "score": 1.0, + "content": "factuality results – for instance, with decay rate", + "type": "text" + }, + { + "bbox": [ + 384, + 481, + 416, + 491 + ], + "score": 0.89, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 479, + 455, + 495 + ], + "score": 1.0, + "content": ", there is", + "type": "text" + }, + { + "bbox": [ + 455, + 481, + 482, + 491 + ], + "score": 0.87, + "content": "12 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 479, + 506, + 495 + ], + "score": 1.0, + "content": "drop", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 117, + 505 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 492, + 142, + 503 + ], + "score": 0.91, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 491, + 162, + 505 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 492, + 189, + 502 + ], + "score": 0.86, + "content": "1 0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "gain in EntailR. However, this affects the diversity and repetition to become", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 225, + 515 + ], + "score": 1.0, + "content": "similar to greedy decoding.", + "type": "text" + }, + { + "bbox": [ + 226, + 505, + 232, + 514 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "-reset mitigates the repetition issue and improves diversity metric", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 407, + 526 + ], + "score": 1.0, + "content": "without losing much in factuality metric. 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By also adding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 114, + 546 + ], + "score": 0.75, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "-bound, we obtain the anticipated factuality performance (i.e., similar to greedy decoding), with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 348, + 559 + ], + "score": 1.0, + "content": "great improvement in generation quality over greedy; with", + "type": "text" + }, + { + "bbox": [ + 348, + 547, + 373, + 558 + ], + "score": 0.87, + "content": "p { = } 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 546, + 377, + 559 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 377, + 546, + 403, + 557 + ], + "score": 0.83, + "content": "\\lambda { = } 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 546, + 406, + 559 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 407, + 547, + 433, + 557 + ], + "score": 0.86, + "content": "\\omega { = } 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 546, + 485, + 559 + ], + "score": 1.0, + "content": ", we achieve", + "type": "text" + }, + { + "bbox": [ + 485, + 546, + 505, + 557 + ], + "score": 0.89, + "content": "\\times 1 1", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 229, + 570 + ], + "score": 1.0, + "content": "improvement in diversity and", + "type": "text" + }, + { + "bbox": [ + 229, + 558, + 252, + 568 + ], + "score": 0.87, + "content": "\\times 4 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "improvement in repetition over greedy. Although our factual-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 273, + 581 + ], + "score": 1.0, + "content": "nucleus sampling still under-performs top-", + "type": "text" + }, + { + "bbox": [ + 274, + 568, + 295, + 579 + ], + "score": 0.7, + "content": "p 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "in terms of diversity, we believe this is an acceptable", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "trade-off to improve the factuality of LM for factually sensitive open-ended generation tasks. Our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "proposed decoding does not harm the sentence fluency; its perplexity do not exceed the perplexity of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 601, + 322, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 322, + 614 + ], + "score": 1.0, + "content": "top-p. Refer to Appendix F for full perplexity results.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 469, + 506, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "To further illustrate the underlying trade-off, we also compare the proposed factual-nucleus sampling", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 628, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 271, + 640 + ], + "score": 1.0, + "content": "against the nucleus sampling with lower", + "type": "text" + }, + { + "bbox": [ + 271, + 630, + 278, + 639 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 628, + 506, + 640 + ], + "score": 1.0, + "content": "values that are also expected to have lower randomness,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "thus less factual error, in generations. Specifically, we plotted results for nucleus sampling with", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 649, + 507, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 232, + 661 + ], + "score": 0.83, + "content": "p = \\{ 0 . 9 , 0 . 7 , 0 . 6 , 0 . 5 , 0 . 4 , \\mathrm { { 0 . 3 } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 649, + 434, + 663 + ], + "score": 1.0, + "content": ", and factual nucleus sampling with the following", + "type": "text" + }, + { + "bbox": [ + 434, + 650, + 470, + 662 + ], + "score": 0.92, + "content": "p \\mid \\lambda \\mid \\bar { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 649, + 507, + 663 + ], + "score": 1.0, + "content": "choices:", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 403, + 673 + ], + "score": 1.0, + "content": "0.9|0.9|0.7, 0.9|0.9|0.5, 0.9|0.9|0.4, 0.9|0.9|0.3, 0.9|0.7|0.3. The Fig 2a and", + "type": "text" + }, + { + "bbox": [ + 403, + 661, + 430, + 672 + ], + "score": 0.33, + "content": "\\mathrm { F i g 2 b }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "respectively show", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 670, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 396, + 685 + ], + "score": 1.0, + "content": "that the factual nucleus sampling method has better trade-offs than top-", + "type": "text" + }, + { + "bbox": [ + 396, + 673, + 402, + 683 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 670, + 505, + 685 + ], + "score": 1.0, + "content": "in factuality-vs-diversity", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "score": 1.0, + "content": "and factuality-vs-repetition. In other words, it always achieves better factuality score with the same", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 693, + 263, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 263, + 707 + ], + "score": 1.0, + "content": "level of diversity and repetition scores.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 616, + 507, + 707 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 70, + 337, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 70, + 339, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 70, + 339, + 88 + ], + "score": 1.0, + "content": "6 Factuality-Enhanced Continued Training", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 93, + 505, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 92, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 504, + 105 + ], + "score": 1.0, + "content": "This section introduces factuality-enhanced method for continued training of LMs [30]. We introduce", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 507, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 507, + 117 + ], + "score": 1.0, + "content": "the TOPICPREFIX for better awareness of facts and the sentence completion loss as training objective.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 124, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 124, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 245, + 138 + ], + "score": 1.0, + "content": "6.1 Prepending TOPICPREFIX", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 506, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "Unstructured factual knowledge typically exists at a document level (i.e., a group of factual sentences", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "about an entity). This means that sentences can contain pronouns (e.g., she, he, it), making these", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "sentences factually useless standalone. To illustrate with an example from Barack Obama’s Wikipedia", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 402, + 187 + ], + "score": 1.0, + "content": "page, “He previously served as a U.S. senator from Illinois from 2005 to", + "type": "text" + }, + { + "bbox": [ + 402, + 175, + 429, + 186 + ], + "score": 0.31, + "content": "2 0 0 8 '", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "cannot be a useful", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "standalone fact because it is unclear who “He” is. Due to the GPU memory limit and computation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 507, + 210 + ], + "score": 1.0, + "content": "efficiency, it is common to chunk documents in LM training corpus. This causes the “fragmentation”", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "of information and leads to wrong associations of entities that appear in independent documents", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "with similar contexts. As a remedy, we propose to prepend TOPICPREFIX to sentences in the factual", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "documents to make each sentence serve as a standalone fact. In our experiments, we mainly utilize", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 464, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 464, + 253 + ], + "score": 1.0, + "content": "Wikipedia as the factual corpus and the Wikipedia document name as the TOPICPREFIX.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 243, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 244, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 244, + 276 + ], + "score": 1.0, + "content": "6.2 Sentence Completion Loss", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 507, + 291 + ], + "score": 1.0, + "content": "We propose a sentence completion loss to address the incorrect association learned between entities.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 377, + 302 + ], + "score": 1.0, + "content": "To explain our rationale, let us recall the nonfactual example from", + "type": "text" + }, + { + "bbox": [ + 378, + 290, + 389, + 301 + ], + "score": 0.75, + "content": "\\ S 5", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 289, + 506, + 302 + ], + "score": 1.0, + "content": ": “Samuel Witwer’s father is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "a Lutheran minister”. This sentence is nonfactual because LM failed to generate factually correct", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "information after “is”. In other words, LM failed to accurately complete the sentence given the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "generated context. One reason is that the LM is uniformly trained to predict each subword token", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "within the sentence, when ensuring the correct prediction at the latter section of sentence is more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "critical for factuality. Therefore, we construct a sentence completion loss, which makes the LM focus", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 356, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 464, + 367 + ], + "score": 1.0, + "content": "on predicting the subwords later in the sentence. For implementation, we determine a pivot", + "type": "text" + }, + { + "bbox": [ + 465, + 357, + 470, + 365 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 356, + 505, + 367 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 420, + 379 + ], + "score": 1.0, + "content": "sentence, and then apply zero-masking for all token prediction losses before", + "type": "text" + }, + { + "bbox": [ + 421, + 367, + 426, + 376 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 366, + 505, + 379 + ], + "score": 1.0, + "content": ". This pivot is only", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 378, + 384, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 384, + 390 + ], + "score": 1.0, + "content": "required during training (i.e., no pivot needed during inference time).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "We emphasize that this loss masking is different from the input token masking applied in BERT [73]", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "or BART [76], and the LM is still trained in an autoregressive manner. Note that many BART-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "summarization models are known to still suffer from factual errors, suggesting that masked prediction", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 452, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 452, + 438 + ], + "score": 1.0, + "content": "at the encoder level may not effectively transfer well to autoregressive text generation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 471, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 467, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 462, + 456 + ], + "score": 1.0, + "content": "In this work, we explore three strategies (from simple to complex) to determine the pivot", + "type": "text" + }, + { + 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"score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 126, + 481, + 158, + 492 + ], + "score": 0.87, + "content": "S C _ { \\mathrm { { R O O T } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 479, + 185, + 495 + ], + "score": 1.0, + "content": ": pivot", + "type": "text" + }, + { + "bbox": [ + 185, + 482, + 201, + 492 + ], + "score": 0.84, + "content": "t =", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 479, + 422, + 495 + ], + "score": 1.0, + "content": "position of ROOT (relation) from dependency parsing.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 108, + 495, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 272, + 509 + ], + "score": 1.0, + "content": "Our experiments show that the simplest", + "type": "text" + }, + { + "bbox": [ + 272, + 495, + 304, + 506 + ], + "score": 0.91, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 493, + 506, + 509 + ], + "score": 1.0, + "content": "performs on par with the complex ones (such as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 505, + 372, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 140, + 518 + ], + "score": 0.88, + "content": "S C _ { \\mathrm { R O O T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 505, + 303, + 520 + ], + "score": 1.0, + "content": "), thus, we suggest future work to choose", + "type": "text" + }, + { + "bbox": [ + 303, + 507, + 334, + 517 + ], + "score": 0.93, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 505, + 372, + 520 + ], + "score": 1.0, + "content": "strategy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 106, + 527, + 162, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 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+ 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "texts (e.g., Wikipedia) and potentially nonfactual texts (e.g., rumors, fake news) 8. The nonfactual", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "domain of the training corpus could be the problem. Thus, we conduct a baseline experiment that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 507, + 608 + ], + "score": 1.0, + "content": "does domain-adaptive training with strictly factual domain text only (i.e., Wikipedia). Interestingly,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 424, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 424, + 618 + ], + "score": 1.0, + "content": "we find that domain-adaptive training can hardly improve generation factuality.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "Effect of TOPICPREFIX Continued pre-training of 1.3B LM with TOPICPREFIX preprocessed", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 410, + 645 + ], + "score": 1.0, + "content": "Wikipedia alone can already improve the factuality, especially in terms of", + "type": "text" + }, + { + "bbox": [ + 410, + 632, + 435, + 643 + ], + "score": 0.89, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 632, + 506, + 645 + ], + "score": 1.0, + "content": ". For example, it", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 154, + 656 + ], + "score": 1.0, + "content": "reduces the", + "type": "text" + }, + { + "bbox": [ + 155, + 643, + 228, + 654 + ], + "score": 0.85, + "content": "\\mathrm { N E _ { E R } f r o m 4 2 . 1 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 642, + 239, + 656 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 240, + 643, + 267, + 654 + ], + "score": 0.89, + "content": "2 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 642, + 443, + 656 + ], + "score": 1.0, + "content": "when we use the factual-nucleus decoding", + "type": "text" + }, + { + "bbox": [ + 443, + 643, + 502, + 654 + ], + "score": 0.43, + "content": "( 0 . 9 \\mid 0 . 9 \\mid 0 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 642, + 506, + 656 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 335, + 667 + ], + "score": 1.0, + "content": "which even outperforms the 1.3B with greedy decoding", + "type": "text" + }, + { + "bbox": [ + 335, + 654, + 394, + 665 + ], + "score": 0.31, + "content": "( \\mathrm { N E } _ { \\mathrm { E R } } \\colon 2 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 653, + 410, + 667 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 410, + 654, + 438, + 665 + ], + "score": 0.84, + "content": "3 9 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 653, + 506, + 667 + ], + "score": 1.0, + "content": ") with much less", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 664, + 222, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 149, + 677 + ], + "score": 1.0, + "content": "repetition", + "type": "text" + }, + { + "bbox": [ + 150, + 665, + 173, + 676 + ], + "score": 0.82, + "content": "( 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 664, + 188, + 677 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 188, + 664, + 217, + 676 + ], + "score": 0.83, + "content": "3 3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 664, + 222, + 677 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 681, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "Effect of Sentence Completion Loss The proposed sentence completion loss further helps to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "improve the factuality, especially for the EntailR. For example, when one uses factual-nucleus", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 712, + 270, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 272, + 724 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 272, + 724 + ], + "score": 1.0, + "content": "8 See [4] for details of pre-training corpus.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 70, + 337, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 70, + 339, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 70, + 339, + 88 + ], + "score": 1.0, + "content": "6 Factuality-Enhanced Continued Training", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 93, + 505, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 92, + 504, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 504, + 105 + ], + "score": 1.0, + "content": "This section introduces factuality-enhanced method for continued training of LMs [30]. We introduce", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 507, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 507, + 117 + ], + "score": 1.0, + "content": "the TOPICPREFIX for better awareness of facts and the sentence completion loss as training objective.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 92, + 507, + 117 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 124, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 124, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 245, + 138 + ], + "score": 1.0, + "content": "6.1 Prepending TOPICPREFIX", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 506, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "Unstructured factual knowledge typically exists at a document level (i.e., a group of factual sentences", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "about an entity). This means that sentences can contain pronouns (e.g., she, he, it), making these", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "sentences factually useless standalone. To illustrate with an example from Barack Obama’s Wikipedia", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 402, + 187 + ], + "score": 1.0, + "content": "page, “He previously served as a U.S. senator from Illinois from 2005 to", + "type": "text" + }, + { + "bbox": [ + 402, + 175, + 429, + 186 + ], + "score": 0.31, + "content": "2 0 0 8 '", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "cannot be a useful", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "standalone fact because it is unclear who “He” is. Due to the GPU memory limit and computation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 507, + 210 + ], + "score": 1.0, + "content": "efficiency, it is common to chunk documents in LM training corpus. This causes the “fragmentation”", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "of information and leads to wrong associations of entities that appear in independent documents", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "with similar contexts. As a remedy, we propose to prepend TOPICPREFIX to sentences in the factual", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "documents to make each sentence serve as a standalone fact. In our experiments, we mainly utilize", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 464, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 464, + 253 + ], + "score": 1.0, + "content": "Wikipedia as the factual corpus and the Wikipedia document name as the TOPICPREFIX.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 141, + 507, + 253 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 243, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 244, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 244, + 276 + ], + "score": 1.0, + "content": "6.2 Sentence Completion Loss", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 507, + 291 + ], + "score": 1.0, + "content": "We propose a sentence completion loss to address the incorrect association learned between entities.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 377, + 302 + ], + "score": 1.0, + "content": "To explain our rationale, let us recall the nonfactual example from", + "type": "text" + }, + { + "bbox": [ + 378, + 290, + 389, + 301 + ], + "score": 0.75, + "content": "\\ S 5", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 289, + 506, + 302 + ], + "score": 1.0, + "content": ": “Samuel Witwer’s father is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "a Lutheran minister”. This sentence is nonfactual because LM failed to generate factually correct", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "information after “is”. In other words, LM failed to accurately complete the sentence given the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "generated context. One reason is that the LM is uniformly trained to predict each subword token", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "within the sentence, when ensuring the correct prediction at the latter section of sentence is more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "critical for factuality. Therefore, we construct a sentence completion loss, which makes the LM focus", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 356, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 464, + 367 + ], + "score": 1.0, + "content": "on predicting the subwords later in the sentence. For implementation, we determine a pivot", + "type": "text" + }, + { + "bbox": [ + 465, + 357, + 470, + 365 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 356, + 505, + 367 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 420, + 379 + ], + "score": 1.0, + "content": "sentence, and then apply zero-masking for all token prediction losses before", + "type": "text" + }, + { + "bbox": [ + 421, + 367, + 426, + 376 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 366, + 505, + 379 + ], + "score": 1.0, + "content": ". This pivot is only", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 378, + 384, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 384, + 390 + ], + "score": 1.0, + "content": "required during training (i.e., no pivot needed during inference time).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 279, + 507, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "We emphasize that this loss masking is different from the input token masking applied in BERT [73]", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "or BART [76], and the LM is still trained in an autoregressive manner. Note that many BART-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "summarization models are known to still suffer from factual errors, suggesting that masked prediction", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 452, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 452, + 438 + ], + "score": 1.0, + "content": "at the encoder level may not effectively transfer well to autoregressive text generation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 393, + 506, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 471, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 467, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 462, + 456 + ], + "score": 1.0, + "content": "In this work, we explore three strategies (from simple to complex) to determine the pivot", + "type": "text" + }, + { + "bbox": [ + 462, + 444, + 467, + 453 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 442, + 467, + 456 + ] + }, + { + "type": "list", + "bbox": [ + 117, + 457, + 422, + 493 + ], + "lines": [ + { + "bbox": [ + 117, + 456, + 292, + 471 + ], + "spans": [ + { + "bbox": [ + 117, + 456, + 126, + 471 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 126, + 457, + 158, + 469 + ], + "score": 0.9, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 456, + 185, + 471 + ], + "score": 1.0, + "content": ": pivot", + "type": "text" + }, + { + "bbox": [ + 185, + 457, + 224, + 468 + ], + "score": 0.89, + "content": "t = 0 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 456, + 292, + 471 + ], + "score": 1.0, + "content": "sentence-length.", + "type": "text" + } 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"bbox": [ + 105, + 493, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 272, + 509 + ], + "score": 1.0, + "content": "Our experiments show that the simplest", + "type": "text" + }, + { + "bbox": [ + 272, + 495, + 304, + 506 + ], + "score": 0.91, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 493, + 506, + 509 + ], + "score": 1.0, + "content": "performs on par with the complex ones (such as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 505, + 372, + 520 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 140, + 518 + ], + "score": 0.88, + "content": "S C _ { \\mathrm { R O O T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 505, + 303, + 520 + ], + "score": 1.0, + "content": "), thus, we suggest future work to choose", + "type": "text" + }, + { + "bbox": [ + 303, + 507, + 334, + 517 + ], + "score": 0.93, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 505, + 372, + 520 + ], + "score": 1.0, + "content": "strategy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 493, + 506, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 527, + 162, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 163, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 163, + 540 + ], + "score": 1.0, + "content": "6.3 Results", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 457, + 556 + ], + "lines": [ + { + "bbox": [ + 104, + 542, + 460, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 460, + 559 + ], + "score": 1.0, + "content": "The results are reported in Table 5, and experimental setups are reported in Appendix C.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 542, + 460, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Inefficiency of Domain Adaptive Training The pre-training corpus of LM contains both factual", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "texts (e.g., Wikipedia) and potentially nonfactual texts (e.g., rumors, fake news) 8. The nonfactual", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "domain of the training corpus could be the problem. Thus, we conduct a baseline experiment that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 507, + 608 + ], + "score": 1.0, + "content": "does domain-adaptive training with strictly factual domain text only (i.e., Wikipedia). Interestingly,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 424, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 424, + 618 + ], + "score": 1.0, + "content": "we find that domain-adaptive training can hardly improve generation factuality.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 560, + 507, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "Effect of TOPICPREFIX Continued pre-training of 1.3B LM with TOPICPREFIX preprocessed", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 410, + 645 + ], + "score": 1.0, + "content": "Wikipedia alone can already improve the factuality, especially in terms of", + "type": "text" + }, + { + "bbox": [ + 410, + 632, + 435, + 643 + ], + "score": 0.89, + "content": "\\mathrm { N E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 632, + 506, + 645 + ], + "score": 1.0, + "content": ". For example, it", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 154, + 656 + ], + "score": 1.0, + "content": "reduces the", + "type": "text" + }, + { + "bbox": [ + 155, + 643, + 228, + 654 + ], + "score": 0.85, + "content": "\\mathrm { N E _ { E R } f r o m 4 2 . 1 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 642, + 239, + 656 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 240, + 643, + 267, + 654 + ], + "score": 0.89, + "content": "2 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 642, + 443, + 656 + ], + "score": 1.0, + "content": "when we use the factual-nucleus decoding", + "type": "text" + }, + { + "bbox": [ + 443, + 643, + 502, + 654 + ], + "score": 0.43, + "content": "( 0 . 9 \\mid 0 . 9 \\mid 0 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 642, + 506, + 656 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 335, + 667 + ], + "score": 1.0, + "content": "which even outperforms the 1.3B with greedy decoding", + "type": "text" + }, + { + "bbox": [ + 335, + 654, + 394, + 665 + ], + "score": 0.31, + "content": "( \\mathrm { N E } _ { \\mathrm { E R } } \\colon 2 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 653, + 410, + 667 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 410, + 654, + 438, + 665 + ], + "score": 0.84, + "content": "3 9 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 653, + 506, + 667 + ], + "score": 1.0, + "content": ") with much less", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 664, + 222, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 149, + 677 + ], + "score": 1.0, + "content": "repetition", + "type": "text" + }, + { + "bbox": [ + 150, + 665, + 173, + 676 + ], + "score": 0.82, + "content": "( 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 664, + 188, + 677 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 188, + 664, + 217, + 676 + ], + "score": 0.83, + "content": "3 3 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 664, + 222, + 677 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 620, + 506, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 681, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "Effect of Sentence Completion Loss The proposed sentence completion loss further helps to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "improve the factuality, especially for the EntailR. For example, when one uses factual-nucleus", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 295, + 480 + ], + "score": 1.0, + "content": "decoding on trained 1.3B model, TOPICPREFIX", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 296, + 468, + 337, + 478 + ], + "score": 0.55, + "content": "+ \\ S C _ { \\mathrm { H A L F } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 337, + 467, + 471, + 480 + ], + "score": 1.0, + "content": "can further improve EntailR from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 471, + 467, + 494, + 478 + ], + "score": 0.86, + "content": "8 . 7 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 494, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 133, + 489 + ], + "score": 0.88, + "content": "1 7 . 4 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 134, + 477, + 303, + 491 + ], + "score": 1.0, + "content": "than TOPICPREFIX alone, while reducing", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 304, + 479, + 328, + 490 + ], + "score": 0.89, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 329, + 477, + 351, + 491 + ], + "score": 1.0, + "content": "from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 352, + 478, + 379, + 489 + ], + "score": 0.88, + "content": "2 7 . 6 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 379, + 477, + 390, + 491 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 390, + 478, + 417, + 489 + ], + "score": 0.87, + "content": "2 3 . 6 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 418, + 477, + 505, + 491 + ], + "score": 1.0, + "content": ". Note that the results", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "show consistent improvement across different pivot selection strategies, suggesting that the sentence", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 499, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 323, + 514 + ], + "score": 1.0, + "content": "completion loss is robust. In particular, the simplest", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 323, + 500, + 355, + 511 + ], + "score": 0.91, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 356, + 499, + 505, + 514 + ], + "score": 1.0, + "content": "performs as good as others or even", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 511, + 422, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 422, + 524 + ], + "score": 1.0, + "content": "better in terms of several metrics. Thus we recommend it as the default option.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 680, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 133, + 100, + 475, + 448 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 78, + 504, + 98 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 75, + 502, + 91 + ], + "spans": [ + { + "bbox": [ + 105, + 75, + 496, + 91 + ], + "score": 1.0, + "content": "Table 5: Results for factuality enhanced training. The decoding settings are formatted as: nucleus probability", + "type": "text" + }, + { + "bbox": [ + 497, + 81, + 502, + 89 + ], + "score": 0.62, + "content": "p", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 215, + 100 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 145, + 100 + ], + "score": 1.0, + "content": "decay rate", + "type": "text" + }, + { + "bbox": [ + 145, + 88, + 152, + 97 + ], + "score": 0.6, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 87, + 203, + 100 + ], + "score": 1.0, + "content": ", lower-bound", + "type": "text" + }, + { + "bbox": [ + 203, + 90, + 210, + 97 + ], + "score": 0.64, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 87, + 215, + 100 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 133, + 100, + 475, + 448 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 100, + 475, + 448 + ], + "spans": [ + { + "bbox": [ + 133, + 100, + 475, + 448 + ], + "score": 0.986, + "html": "
Decoding (plλlw)Factual PromptNonfactual Prompt
NEER↓EntailR↑Div.Rep.NEEREntailRDiv.Rep.
Vanilla Pretrained LM (1.3B)
0.952.4%2.9%0.880.2%56.8%2.0%0.890.3%
0.9 10.910.342.1%10.1%0.557.1%46.5%5.6%0.596.4%
Factual Domain-Adaptive Training with Wikipedia (1.3B)
0.952.5%2.8%0.850.2%55.8%2.2%0.860.1%
0.910.9 10.342.7%7.1%0.517.2%48.2%4.9%0.566.0%
TOPICPREFIX (1.3B)
0.9 0.910.910.334.4%4.2%0.840.3%36.2%2.7%0.850.2%
27.6%8.7%0.438.0%30.5%6.1%0.476.9%
TOPICPREFIX + SCROOT (1.3B)
0.9 0.9 10.9 10.332.5% 24.7%6.7% 15.8%0.83 0.401.2% 13.6%34.3% 27.6%4.6% 9.1%0.84 0.441.1% 13.7%
TOPICPREFIX+ SCRANDOM (1.3B)
0.9 0.9 10.9 10.332.0% 23.6%7.9% 17.6%0.81 0.391.2% 14.2%34.2% 26.9%5.5% 9.3%0.83 0.421.1% 13.2%
TOPICPREFIX + SCHALF
0.931.6%7.6% 17.4%0.81 0.381.4% 14.4%33.5% 27.2%5.1% 10.2%0.83 0.421.5% 13.1%
0.9 10.9 10.3 23.6% Vanilla Pretrained LM (530B)
0.933.3%11.8%0.900.1%40.5%7.25%0.920.1%
TOPICPREFIX + SCHALF (530B)
0.9 0.9 10.9 10.318.3% 14.5%19.3% 25.5%0.68 0.330.1% 0.2%21.7% 17.7%13.7% 20.0%0.680.1%
", + "type": "table", + "image_path": "3b71d0d72f989503a027c0504b62706eb90fa0164aed8add6a781931bf6b323e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 133, + 100, + 475, + 216.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 133, + 216.0, + 475, + 332.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 133, + 332.0, + 475, + 448.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 295, + 480 + ], + "score": 1.0, + "content": "decoding on trained 1.3B model, TOPICPREFIX", + "type": "text" + }, + { + "bbox": [ + 296, + 468, + 337, + 478 + ], + "score": 0.55, + "content": "+ \\ S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 467, + 471, + 480 + ], + "score": 1.0, + "content": "can further improve EntailR from", + "type": "text" + }, + { + "bbox": [ + 471, + 467, + 494, + 478 + ], + "score": 0.86, + "content": "8 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 133, + 489 + ], + "score": 0.88, + "content": "1 7 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 477, + 303, + 491 + ], + "score": 1.0, + "content": "than TOPICPREFIX alone, while reducing", + "type": "text" + }, + { + "bbox": [ + 304, + 479, + 328, + 490 + ], + "score": 0.89, + "content": "\\Nu \\mathrm { E } _ { \\mathrm { E R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 477, + 351, + 491 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 352, + 478, + 379, + 489 + ], + "score": 0.88, + "content": "2 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 477, + 390, + 491 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 390, + 478, + 417, + 489 + ], + "score": 0.87, + "content": "2 3 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 477, + 505, + 491 + ], + "score": 1.0, + "content": ". Note that the results", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "show consistent improvement across different pivot selection strategies, suggesting that the sentence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 499, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 323, + 514 + ], + "score": 1.0, + "content": "completion loss is robust. In particular, the simplest", + "type": "text" + }, + { + "bbox": [ + 323, + 500, + 355, + 511 + ], + "score": 0.91, + "content": "S C _ { \\mathrm { H A L F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 499, + 505, + 514 + ], + "score": 1.0, + "content": "performs as good as others or even", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 511, + 422, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 422, + 524 + ], + "score": 1.0, + "content": "better in terms of several metrics. 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NEER↓EntailR↑Div.Rep.NEEREntailRDiv.Rep.
Vanilla Pretrained LM (1.3B)
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TOPICPREFIX + SCROOT (1.3B)
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TOPICPREFIX + SCHALF
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TOPICPREFIX + SCHALF (530B)
0.9 0.9 10.9 10.318.3% 14.5%19.3% 25.5%0.68 0.330.1% 0.2%21.7% 17.7%13.7% 20.0%0.680.1%
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Potential future work would be to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 711, + 285, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 285, + 723 + ], + "score": 1.0, + "content": "reduce the degree of the observed trade-offs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 622, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 70, + 507, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 108, + 89, + 506, + 103 + ], + "spans": [ + { + "bbox": [ + 108, + 89, + 506, + 103 + ], + "score": 1.0, + "content": "[1] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. 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0000000000000000000000000000000000000000..a1f0e909ac297dc22bfa7b785554efaea0f31ac3 --- /dev/null +++ b/parse/dev/T__V3uLix7V/T__V3uLix7V.md @@ -0,0 +1,418 @@ +# REGIONVIT: REGIONAL-TO-LOCAL ATTENTION FOR VISION TRANSFORMERS + +Chun-Fu (Richard) Chen, Rameswar Panda, Quanfu Fan MIT-IBM Watson AI Lab chenrich@us.ibm.com, rpanda@ibm.com, qfan@us.ibm.com + +# ABSTRACT + +Vision transformer (ViT) has recently shown its strong capability in achieving comparable results to convolutional neural networks (CNNs) on image classification. However, vanilla ViT simply inherits the same architecture from the natural language processing directly, which is often not optimized for vision applications. Motivated by this, in this paper, we propose a new architecture that adopts the pyramid structure and employ novel regional-to-local attention rather than global self-attention in vision transformers. More specifically, our model first generates regional tokens and local tokens from an image with different patch sizes, where each regional token is associated with a set of local tokens based on the spatial location. The regional-to-local attention includes two steps: first, the regional self-attention extracts global information among all regional tokens and then the local self-attention exchanges the information among one regional token and the associated local tokens via self-attention. Therefore, even though local self-attention confines the scope in a local region but it can still receive global information. Extensive experiments on four vision tasks, including image classification, object and keypoint detection, semantics segmentation and action recognition, show that our approach outperforms or is on par with state-of-the-art ViT variants including many concurrent works. Our source codes and models are available at https://github.com/IBM/RegionViT. + +# 1 INTRODUCTION + +Transformers (Vaswani et al., 2017) based on self-attention come naturally with the ability to learn long-range dependencies in sequential data. As important as it is to language modeling (Devlin et al., 2019), such ability is also highly desired for many vision tasks where contextual modeling plays a significant role. For this reason, self-attention and transformers have been receiving an increasing attention in the vision community (Bello, 2021; Srinivas et al., 2021; Zhao et al., 2020; Ramachandran et al., 2019a; Bello et al., 2019; Hu et al., 2019; Ramachandran et al., 2019b; Wang et al., 2018). Especially, the recent Vision Transformers (ViT) (Dosovitskiy et al., 2021) demonstrates comparable image classification results against the firmly established and prevalent CNNs in computer vision (He et al., 2016; Tan & Le, 2019; Brock et al., 2021), albeit relying on a huge amount of training data. It has since then led to an explosion of interest in further investigating its potential for a wide variety of vision applications (Wu et al., 2021; Wang et al., 2021; Heo et al., 2021; Zhang et al., 2021a; Li et al., 2021; Graham et al., 2021; Liu et al., 2021; Chu et al., 2021a; Yan et al., 2021; Chen et al., 2021b). + +The ViT inherits the entire architecture from the vanilla transformer (Vaswani et al., 2017), which is designed for natural language processing tasks, and some of those designs thus may not meet the needs of vision tasks. For example, the transformer has an isotropic network structure with a fixed number of tokens and unchanged embedding size, which loses the capability to model the context with different scales and allocates computations at different scales. As opposed to this, a majority of CNNs adopt a popular pyramid architecture to compute multi-scale features efficiently. Recent vision transformers such as PVT (Wang et al., 2021) and PiT (Heo et al., 2021) also follow a similar pyramid structure as CNNs, showing improvement on both computation and memory efficiency as well as on model accuracy. Another critical bottleneck of the transformer is that the self-attention module has a quadratic cost in memory and computation with regard to the sequence length (i.e., the number of tokens). This issue is even worse in ViT as images are 2-D, suggesting a quadratic relationship between the number of tokens and image resolution. As a result, ViT indicates a quadruple complexity w.r.t image resolution. The highly compute- and memory-intensive self-attention makes it challenging to train vision transformer models at fine-grained patch sizes. It also significantly undermines the applications of these models to tasks such as object detection and semantic segmentation, which benefit from or require fine feature information computed from high-resolution images. + +To address the aforementioned computational limitations of vision transformers, in this work, we develop a memoryfriendly and efficient self-attention method for transformer models to reach their promising potential for vision applications. We propose a novel coarse-to-fine mechanism to compute self-attention in a hierarchical way. Specifically, our approach first divides the input image into a group of non-overlapping patches of large size (e.g., $2 8 \times 2 8 )$ ), on which regional tokens are computed via linear projection. Similarly, local tokens are created for each region using a smaller patch size (e.g., $4 \times 4 )$ . We then use a standard transformer to process regional and local tokens separately. To enable communication between the two types of tokens, we first perform self-attention on regional tokens (regional attention) and then jointly attend to the local tokens of each region including their associated regional token (local attention). By doing so, regional tokens pass global contextual information to local tokens efficiently while being able to effectively learn from local tokens themselves. For clarity, we represent this two-stage attention mechanism as Regional-to-Local Attention, or $R 2 L$ attention for short (see Figure 1 for an illustration). Since both regional and local attention involve much fewer tokens, our R2L attention requires substantially less memory than regular global self-attention used in vision transformers. For example, in our default setting, the memory saving using + +![](images/be438c75eaab31cb96439f283128309a198fe8950d097322f0c0e7274132054b.jpg) +Figure 1: Regional-to-Local Attention for Vision Transformers. (a) ViT uses a fixed patch size through the whole network, (b) PVT adopts a pyramid structure to gradually enlarge the patch size in the network. Both ViT and PVT uses all tokens in self-attention, which are computationaland memory-intensive. (c) Our proposed approach combines a pyramid structure with an efficient regional-to-local (R2L) attention mechanism to reduce computation and memory usage. Our approach divides the input image into two groups of tokens, regional tokens of large patch size (red) and local ones of small patch size (black). The two types of tokens communicate efficiently through R2L attention, which jointly attends to the local tokens in the same region and the associated regional token. Note that the numbers denote the patch sizes at each stage of a model. + +R2L attention can be up to as much as $73 \%$ . We demonstrate the effectiveness of our approach on image classification and several downstream vision tasks including object detection and action recognition. + +To summarize, our key contributions in this work are as follows: + +1. We propose a new vision transformer (RegionViT) based on regional-to-local attention to learn both local and global features. Our proposed regional-to-local attention alleviates the overhead of standard global attention (too many tokens) and the weakness of pure local attention (no interaction between regions) used in existing vision transformers. +2. Our regional-to-local attention reduces the memory complexity significantly as compared to standard self-attention, leading to a savings in memory complexity by about $\dot { \mathcal { O } } ( N / M ^ { 2 } )$ , where $M$ is the window size of a region and $N$ is the total number of tokens. This effectively allows us to train a more deep network for better performance while with comparable complexity. +3. Our models outperform or on par with several concurrent works on vision transformer that exploit pyramid structure for image classification. Experiments also demonstrate that our models work well on several downstream classification tasks, including object detection and action recognition. + +# 2 RELATED WORK + +CNNs with Attention. Attention has been widely used for enhancing the features in CNNs, e.g., SENet (Hu et al., 2018) and CBAM (Woo et al., 2018). Various methods have been proposed that combine self-attention with CNNs (Bello, 2021; Srinivas et al., 2021; Zhao et al., 2020; Ramachandran et al., 2019a; Bello et al., 2019; Hu et al., 2019; Ramachandran et al., 2019b; Wang et al., 2018). E.g., SAN (Zhao et al., 2020) and SASA (Ramachandran et al., 2019a) attempts to replace all convolutional layers with local self-attention network. While these works show promising results, their complexity is relatively high due to the self-attention. On the other hand, BoTNet (Srinivas et al., 2021) replaces few convolutional layers with a slightly-modified self-attention to balance computation and accuracy. LambdaNet work (Bello, 2021) uses the approximated self-attention to reduce the overhead of selfattention and make the network efficient. Few works use attention approach to define the regional of interest for the fine-grained visual recognition (Zheng et al., 2020; Ding et al., 2021; Wharton et al., 2021). In contrast, our model utilizes regional-to-local attention to alleviate the workload of global self-attention by local attention while still keeping the global information via regional attention. + +Table 1: Comparison to related works. Most works use non-overlapped windows to group tokens, and then propose the corresponding methods to assure the information exchange among regions. + +
MethodsStructureAttention TypesWindows of Local AttentionHandling Non-overlapped RegionsGlobal Tokens
ViT (Dosovitskiy et al.,2021)IsotropicGlobalN/AN/AN/A
PVT (Wang et al., 2021)PyramidGlobalN/AN/AN/A
Swin (Liu et al.,2021)PyramidLocalStrictly Non-overlappedShifting the windowNone
ViL (Zhang et al., 2021a)PyramidLocal + GlobalOverlappedN/A1One learnable token
Twins (Chu et al., 2021a)PyramidLocal + GlobalNon-overlappedGASubsampled from local tokens
RegionViT (Ours)PyramidLocal + RegionalNon-overlappedRegional-to-local attentionRegional tokens
+ +GA: global attention. 1 : not applicable as ViL used overlapped windows. + +Vision Transformer. ViT (Dosovitskiy et al., 2021) has recently achieved comparable results to CNNs when using a huge amount of data, e.g., JFT300M (Sun et al., 2017). DeiT (Touvron et al., 2020; Wightman, 2019) subsequently proposes an efficient training scheme that allows vision transformer to work on par with CNNs while training only on ImageNet1K (Deng et al., 2009). Despite promising performance of ViT, the architecture is directly borrowed from natural language processing, which might not be suitable for vision applications. Motivated by this, two main line of research works have been developed to improve ViT. One is to enhance different components of the vision transformer (Yuan et al., 2021; Han et al., 2021; Touvron et al., 2021; Jiang et al., 2021) while still using isotropic structure (i.e., fixed token numbers and channel dimension) like ViT, e.g., while T2T-ViT (Yuan et al., 2021) introduces a Tokens-to-Token (T2T) transformation to encode local structure for each token, CrossViT propose a dual-path architecture, each with different scales, to learn multi-scale features (Chen et al., 2021a). CaiT (Touvron et al., 2021) proposes a layer-scalar to train a deeper network for better performance and LV-ViT (Jiang et al., 2021) modified how the model is trained when CutMix (Yun et al., 2019) augmentation is applied on ViT. Another parallel thread for improving vision transformer is in incorporating CNN-like pyramid structure into ViT (Wu et al., 2021; Wang et al., 2021; Heo et al., 2021; Zhang et al., 2021a; Li et al., 2021; Graham et al., 2021; Liu et al., 2021; Chu et al., 2021a; Yan et al., 2021; Chen et al., 2021b). PVT (Wang et al., 2021) and PiT (Heo et al., 2021) introduce the pyramid structure into ViT, which makes them more suitable for objection detection as it can provide multi-scale features. LocalViT (Li et al., 2021) and ConT (Yan et al., 2021) mix convolutions with self-attention to encode locality information. Swin (Liu et al., 2021), ViL (Zhang et al., 2021a) and Twins (Chu et al., 2021a) limited the self-attention into a local region and then propose different methods to allow the interaction among each local region. Our model also utilizes pyramid structure and limits the self-attention in a local region; while we propose the regional-to-local attention to exchange the information among each region efficiently. + +Difference from PVT, Swin, ViL and Twins. Table 1 summarizes the difference between our proposed approach, RegionViT with the closely related works. PVT (Wang et al., 2021) uses pyramid structure but the scope of self-attention is still global. While ViL (Zhang et al., 2021a) limits self-attention into a local region, they adopt overlapping windows as convolutions to process all the tokens. On the other hand, Swin (Liu et al., 2021), Twins (Chu et al., 2021a) and ours adopt non-overlapped windows. While Swin shifts the position of windows alternatively in the consecutive transformer encoder to allow the interaction between regions, Twins subsamples local tokens as the global tokens, and then use the global tokens as the keys for self-attention to achieve the interaction between regions. By contrast, our proposed method utilizes a extra set of regional tokens with regional-to-local attention to perform self-attention on regional tokens only for the global information and then each regional token is sent to the associated local tokens to pass the global information. One major difference from others is that the local self-attention in our approach includes one extra regional token to exchange the global information with local tokens. + +# 3 METHOD + +Our method is built on top of a vision transformer, so we first present a brief background of ViT and then describe our method (RegionViT) on regional-to-local attention for vision transformers. + +![](images/4f51e129978441e6204aba5ef5a836f32d26b9a369736a313e286198ad637a3c.jpg) +Figure 2: Architecture of the proposed RegionViT. Two paths are in our proposed network, including regional tokens and local tokens, and their information is exchanged in the regional-to-local (R2L) transformer encoders. In the end, we average all regional tokens and use it for the classification. The tensor shape here is computed based on that regional tokens take a patch of $2 8 ^ { 2 }$ and local tokens take a patch of $4 ^ { 2 }$ . + +Vision Transformer. As opposed to CNN-based approaches for image classification, ViT is a purely attention-based counterpart, borrowed from NLP. It consists of stacked transformer encoders, each of which contains a multihead self-attention (MSA) and a feed-forward network (FFN) with layer normalization and residual shortcut. To classify an image, ViT first splits it into patches of fixed size, e.g. $1 6 \times 1 6$ , and then transforms them into tokens by linear projection. A class token is additionally prepended to the patch tokens to form the input sequence. Before the token are fed into transformer encoders, a learnable absolute positional embedding is added to each token to learn the position information. At the end of the network, the class token is used as the final feature representation for classification. Mathematically, ViT can be expressed as + +$$ +\begin{array} { r l } & { \mathbf { x } _ { 0 } = [ \mathbf { x } _ { c l s } | | \mathbf { x } _ { p a t c h } ] + \mathbf { x } _ { p o s } } \\ & { \mathbf { y } _ { k } = \mathbf { x } _ { k - 1 } + \mathtt { M S A } \big ( \mathtt { L N } ( \mathbf { x } _ { k - 1 } ) \big ) , \mathbf { x } _ { k } = \mathbf { y } _ { k } + \mathtt { F F N } \big ( \mathtt { L N } \big ( \mathbf { y } _ { k } \big ) \big ) , } \end{array} +$$ + +where $\mathbf { x } _ { c l s } ~ \in ~ \mathbb { R } ^ { 1 \times C }$ and $\mathbf { x } _ { p a t c h } \in \mathbb { R } ^ { N \times C }$ are the class token and patch tokens respectively and $\mathbf { x } _ { p o s } \in \mathbb { R } ^ { ( 1 + N ) \times C }$ is the position embedding, and $\left[ \left| \left| \right. \right] \right.$ denotes the tensor concatenation. $k , N$ and $C$ are the layer index, the number of patch tokens and dimension of the embedding, respectively. + +While the vanilla ViT is ideally capable of learning global interaction among all the patch tokens, the memory complexity of self-attention becomes high when there are many tokens as the complexity is quadratically linear to the length of the input sequence. Moreover, the use of isotropic structure limits the capability of extending the vanilla ViT model to many vision applications that require high-resolution details, e.g., object detection. To address these issues, we propose regional-to-local attention for vision transformer, RegionViT for short. + +An Overview of Our Proposed Approach (RegionViT). Figure 2 illustrates the architecture of the proposed vision transformer, which consists of two tokenization processes that convert an image into regional (upper path) and local tokens (lower path). Each tokenization is a convolution with different patch sizes, e.g., in Figure 2, the patch size of regional tokens is $2 8 ^ { 2 }$ while $4 ^ { 2 }$ is used for local tokens with dimensions projected to $C$ , which means that one regional token covers $7 ^ { 2 }$ local tokens based on the spatial locality, leading to the window size of a local region to $7 ^ { 2 }$ . At stage 1, two sets of tokens are passed through the proposed regional-to-local transformer encoders. However, for the later stages, to balance the computational load and to have feature maps at different resolutions, we deploy a downsampling process to halve the spatial resolution while doubling the channel dimension like CNN on both regional and local tokens before going to the next stage. Finally, at the end of the network, we average the remaining regional tokens as the final embedding for the classification while the detection uses all local tokens at each stage since it provides more fine-grained location information. By having the pyramid structure, the ViT can generate multi-scale features and hence it could be easily extended to more vision applications, e.g., object detection, rather than image classification only. We explain the main components of the regional-to-local transformer encoder in the next section. + +Regional-to-Local Transformer Encoder. Figure 3 shows the proposed regional-to-local (R2L) transformer encoder which includes R2L attention and feed-forward network (FFN). Specifically, within R2L attention, the regional self-attention (RSA) first involves all regional tokens to learn the global information efficiently as the number of regional tokens is few and then local self-attention (LSA) takes the local tokens with the associated regional token to learn local feature while the involved regional token could provide global information at the same time. Both RSA and LSA are multihead self-attention (MSA) but with different input tokens. Finally, the FFN is applied to enhance the features. We also add layer normalization (LN) and residual shortcuts as in standard transformer encoders. Mathematically, given the regional and local tokens, $\mathbf { x } _ { r } ^ { d - 1 }$ and $\mathbf { x } _ { l } ^ { d - 1 }$ as the inputs at layer $d$ , the R2L transformer encoder can be expressed as: + +$$ +\begin{array} { r l } & { \mathbf { y } _ { r } ^ { d } = \mathbf { x } _ { r } ^ { d - 1 } + \mathtt { R S A } \big ( \mathrm { L N } ( \mathbf { x } _ { r } ^ { d - 1 } ) \big ) , \mathbf { y } _ { i , j } ^ { d } = [ \mathbf { y } _ { r _ { i , j } } ^ { d } | | \{ \mathbf { x } _ { l _ { i , j , m , n } } ^ { d - 1 } \} _ { m , n \in M } ] } \\ & { \mathbf { z } _ { i , j } ^ { d } = \mathbf { y } _ { i , j } ^ { d } + \mathtt { L S A } \big ( \mathrm { L N } ( \mathbf { y } _ { i , j } ^ { d } ) \big ) , \mathbf { x } _ { i , j } ^ { d } = \mathbf { z } _ { i , j } ^ { d } + \mathtt { F F N } \big ( \mathrm { L N } ( \mathbf { z } _ { i , j } ^ { d } ) \big ) } \end{array} +$$ + +where $i , j$ are the spatial index with respect to regional tokens while $m , n$ are the index of local token in the window size $\bar { M } ^ { 2 }$ . Note that the input of LSA, $\mathbf { y } _ { i , j } ^ { d }$ , includes one regional token and corresponding local tokens, and thus, the information between local and regional tokens are exchanged; on the other hand, the outputs, $\mathbf { x } _ { r } ^ { d }$ and $\mathbf { x } _ { l } ^ { d }$ , can be extracted from $\mathbf { x } _ { i , j } ^ { d }$ like the top-right equation. + +The RSA exchanges information among all tokens, which covers the context of the whole image; while the LSA combines the features among the tokens belonging to the spatial region, including both regional and local tokens. Since the regions are divided by non-overlapped windows, the RSA is also designed to exchange the information among regions where the LSA takes one regional token and then combines with it the local tokens in the same region. In such a case, all local tokens are still capable of getting global information while being more focused on local neighbors. It is worth noting that the weights are shared be + +![](images/b24588ed075ddb0fab57df5144e35582830e656d60d057cf0a2c111269b41fbc.jpg) +Figure 3: Illustration of Regional-to-Local (R2L) Transformer Encoder. All regional tokens are first passed through regional self-attention (RSA) to exchange the information among regions and then local self-attention (LSA) performs parallel self-attention where each takes one regional token and corresponding local tokens. After that, all the tokens are passed through the feed-forward network and split back to the regional and local tokens. RSA and LSA share the same weights. + +tween RSA and LSA except for the layer normalization; therefore, the number of parameters won’t increase significantly when compared to the standard transformer encoder. With these two attentions, the R2L can effectively and efficiently exchange information among all regional and local tokens. In particular, the self-attention on regional tokens aims to extract high-level information and act as a bridge to pass information of local tokens from one region to other regions. On the other hand, the R2L attention focus on local contextual information with one regional token. + +Moreover, the R2L transformer encoder can further reduce the memory complexity significantly. The memory complexity of a self-attention is $\mathcal { O } ( N ^ { 2 } )$ , where $N$ is the number of local tokens. When a region contains $M$ local tokens, and there are $N / M$ regions, the memory complexity is $\mathcal { O } ( ( M + 1 \bar { ) } ^ { 2 } \times ( N / M ) )$ ; while the complexity from the attention on regional tokens is $\mathcal { O } ( ( N / M ) ^ { 2 } )$ . Hence, the overall complexity becomes $\mathcal { \dot { O } } ( ( M { \dot { } } 1 ) ^ { 2 } \times ( N / M ) + ( N / M ) ^ { 2 } )$ . E.g., the main component of RegionViT is stage 3, where $M$ is 49 and $N$ is 196; therefore, the memory complexity saving is $\sim 7 3 \%$ . The saving of each model and each stage is varied based on the model configuration. + +Relative Position Bias. The locality is an important clue for understanding visual content; therefore, instead of adding absolute position embedding used by the vanilla ViT, we introduce the relative position bias into the attention map of R2L attention since relative position between patches (or pixels) are more important than the absolution position as objects in the images can be placed in an arbitrary way (Ramachandran et al., 2019a; Liu et al., 2021). We only add this bias to the attention between local tokens and not the attention between regional tokens and local tokens. Specifically, for a given pair of local tokens at location, $( x _ { m } , y _ { m } ) , ( x _ { n } , y _ { n } )$ , the attention value $a$ can be expressed as + +$$ +a _ { ( x _ { m } , y _ { m } ) , ( x _ { n } , y _ { n } ) } = \mathtt { s o f t m a x } \left( q _ { ( x _ { m } , y _ { m } ) } k _ { ( x _ { n } , y _ { n } ) } ^ { T } + b _ { ( x _ { m } - x _ { n } , y _ { m } - y _ { n } ) } \right) , +$$ + +where is taken from a learnable parameter $B \in \mathbb { R } ^ { 2 M - 1 \times 2 M - 1 }$ , where $M$ is the m n m n window size of a region. The first term $q _ { ( x _ { m } , y _ { m } ) } k _ { ( x _ { n } , y _ { n } ) } ^ { T }$ is the attention value based on their content. + +With relative position bias, our model does not require the absolute positional embedding as vanilla ViT since this relative position bias helps to encode the position information. + +Downsampling. The spatial and channel dimension are changed along different stages of our network by simply applying $3 \times 3$ depth-wise convolution with stride 2 for halving the resolution and doubling the channel dimension for both regional and local tokens (weights are shared) (Heo et al., 2021). We also test with regular convolutions but it does not improve the accuracy with increased complexity. + +Table 2: Model architectures of RegionViT. For all networks, the dimension per head is 32 and the expanding ratio $r$ in FFN is 4. The patch size of local tokens is always 4 while the patch size of regional tokens is $4 \times M$ . + +
ModelTokenization Local RegionalWindow size (M)Dimension (C)# of Encoders at each stage
RegionViT-Ti3-convlinear7{64,128,256,512}{2,2,8,2}
RegionViT-S3-convlinear7{96,192,384,768}{2,2,8,2}
RegionViT-M1-convlinear7{96,192,384,768}{2,2,14,2}
RegionViT-B1-convlinear7{128,256,512,1024}{2,2,14,2}
+ +1-conv: Conv(out=C, $k { = } 8$ , $s { = } 4$ , $\overline { { { \bf p } { = } 3 } } )$ . 3-conv: Conv(out=C/4, $\overline { { \mathbf { k } = 3 } }$ , $\overline { { \mathsf { s } { = } 2 } }$ , $\mathrm { p } { = } 1$ ) Conv(out=C/2, $\overline { { \mathbf { k } = 3 } }$ , s=2, p=1) Conv(out=C, $\mathrm { k } = 3$ , $\mathrm { s } { = } 1$ , $\mathbf { p } { = } 1 ,$ ), where C is the channel dimension of the first block. LayerNorm and GeLU are added between Conv. + +Input Tokenization. The tokenization for local tokens can be implemented by using a $4 \times 4$ convolution with channel size $C$ and stride 4 (i.e. non-overlapped). However, as pointed out in T2T (Yuan et al., 2021), CrossViT (Chen et al., 2021a), and PiT (Heo et al., 2021), using a stronger but still simple subnetwork for the tokenization could further improve the performance, especially for the smaller models. Thus, we adopt two input tokenizations in our model, one still contains only one convolutional layer and another one contains three convolutional layers (see Table 2 for details). + +Regional Tokens. We adopt a simple approach to generate regional tokens, that is, using linear projection with larger patch sizes in comparison to the patch size for the local tokens. E.g., as shown in Figure 2, the patch size for local tokens is $4 \times 4$ , we simply use patch size $2 8 \times 2 8$ to split the image and then project each patch linearly to generate regional tokens. + +Table 2 shows our RegionViT models with different configurations. By default, the window size is 7 while we also experiment with a larger window size (14), those models are annotated with $^ +$ . + +# 4 EXPERIMENTS + +# 4.1 IMAGE CLASSIFICATION + +Datasets. We use ImageNet1K (Deng et al., 2009) (IN1K) and ImageNet21K (Deng et al., 2009) (IN21K) to validate our method. ImageNet1K contains 1.28 million training images and 50k validation images over 1k classes, and ImageNet21K is a large-scale dataset that consists of around 14 million images over 21,841 classes. We use all images for training and then finetune the model on ImageNet1K. Moreover, we also perform the transfer learning from ImageNet1K to five downstream datasets, including CIFAR10 (Krizhevsky et al., 2009), CIFAR100 (Krizhevsky et al., 2009), IIIPets (Parkhi et al., 2012), StandfordCars (Krause et al., 2013) and ChestXRay8 (Wang et al., 2017). + +Training and Evaluation. We follow DeiT (Touvron et al., 2020) to train our models on IN1K except that we use batch size 4,096 with a base learning rate 0.004 and the warm-up epochs is 50. We adopt the AdamW (Loshchilov & Hutter, 2019) optimizer with cosine learning rate scheduler (Loshchilov & Hutter, 2017). We apply Mixup (Zhang et al., 2018), CutMix (Yun et al., 2019), RandomErasing (Zhong et al., 2020), label smoothing (Szegedy et al., 2016), RandAugment (Cubuk et al., 2020) and instance repetition (Hoffer et al., 2020). During training, we random cropped a $2 2 4 \times 2 2 4$ region and take a $2 2 4 \times 2 2 4$ center crop after resizing the shorter side to 256 for evaluation. We used a similar setting for IN21K and transfer learning, and more details can be found in Section A.1. + +Table 3: Comparisons with recent pyramid-like structure-based ViT models on ImageNet1K. The bold numbers indicate the best number within each section. + +
ModelParams (M)FLOPs (G)Acc. (%)
PiT-XS (Heo et al., 2021)10.61.478.1
ConT-S(Yan et al.,2021)10.11.576.5
PVT-T(Wang et al.,2021)13.21.975.1
ConViT-Ti+ (d'Ascoli et al., 2021)10.02.076.7
Twins-SVT-S (Chu et al.,2021a)24.02.881.7
PiT-S (Heo et al., 2021)23.52.980.9
ConT-M(Yan et al., 2021)19.23.180.2
Twins-PCPVT-S (Chu et al., 2021a)24.13.781.2
PVT-S (Wang et al., 2021)24.53.879.8
RegionViT-Ti13.82.480.4
RegionViT-Ti+14.32.781.5
DeiT-S (Touvron et al., 2020)22.14.679.9
CvT-13 (Wu et al.,2021)20.04.581.6
Swin-T (Liu et al., 2021)29.04.581.3
LocalViT-S (Li et al.,2021)22.44.680.8
ViL-S (Zhang et al., 2021a)24.64.982.0
Visformer-S (Chen et al.,2021b)40.24.982.3
ConViT-S (d'Ascoli et al.,2021)27.05.481.3
NesT-S (Zhang et al., 2021b)17.05.881.5
RegionViT-S30.65.382.6
RegionViT-S+31.35.783.3
+ +
ModelParams (M)FLOPs (G)Acc. (%)
ConT-M(Yan et al., 2021)39.66.481.8
Twins-PCPVT-B (Chu et al.,2021a)43.86.482.7
PVT-M(Wang et al., 2021)44.26.781.2
CvT-21(Wu et al.,2021)32.07.182.5
Twins-SVT-B (Chu et al., 2021a)568.383.2
ViL-M (Zhang et al.,2021a)39.78.783.3
Swin-S (Liu et al., 2021)50.08.783.0
PVT-L (Wang et al.,2021)61.49.881.7
ConViT-S+ (d'Ascoli et al., 2021)48.010.082.2
NesT-S (Zhang et al., 2021b)38.010.483.3
RegionViT-M41.27.483.1
RegionViT-M+42.07.983.4
DeiT-B (Touvron et al.,2020)86.617.681.8
PiT-B (Heo et al., 2021)73.812.582.0
ViL-B (Zhang et al., 2021a)55.713.483.2
Twins-SVT-L(Chu et al.,2021a)99.214.883.7
Swin-B (Liu et al.,2021)88.015.483.5
ConViT-B (d'Ascoli et al.,2021)86.017.082.4
NesT-B (Zhang et al., 2021b)68.017.983.8
RegionViT-B72.713.083.2
RegionViT-B+73.813.683.8
+ +Table 4: Results on IN1K with IN21K and transfer learning. + +
ModelParams (M)FLOPs (G)Acc. (%)
ViL-B55.743.786.0
Swin-L197.0103.987.3
CvT-W24277193.287.7
RegionViT-B+76.542.686.5
+ +(b) Transfer learning on downstream tasks. + +
CIFAR10CIFAR100PetStanfordCarsChestXRay8
DeiT-S99.190.994.991.555.4
DeiT-B99.190.894.491.755.8
RegionViT-S98.990.095.392.857.8
RegionViT-M99.090.895.591.958.3
+ +Table 5: Object detection performance on MS COCO val2017 with $1 \times$ and $3 \times$ schedule. The bold number indicates the best number within the section, and for MaskRCNN, both $\mathsf { A P } ^ { b }$ and $\mathsf { A P } ^ { m }$ are annotated. + +
BackboneParamsFLOPsRetineNet3× APParams (M)FLOPsMaskRCNN
(M)1× AP(G)1× APb |APmAPb3× APm
ResNet5037.723436.339.044.2260.038.034.441.0
ConT-M (Yan et al.,2021)27.0217.239.311140.538.1137.1 1
PVT-S (Wang et al., 2021)34.240.442.244.140.437.843.039.6
ViL-S (Zhang et al., 2021a)35.7252.241.642.945.0174.341.838.543.439.6
Swin-T (Liu et al., 2021)38.524541.543.947.826442.239.146.041.6
Twins-SVT-S (w/PEG) (Chu et al.,2021a)34.320943.045.644.022843.540.346.842.6
RegionViT-S40.8192.642.245.850.1171.342.539.546.342.3
RegionViT-S+41.5204.243.146.950.9182.943.540.447.343.4
RegionViT-S+ w/PEG41.6204.343.946.750.9183.044.240.847.643.4
ResNet10156.731538.540.963.233640.436.442.838.5
ResNeXt101-32x4d56.4319.139.941.462.834041.937.544.039.2
PVT-M(Wang et al., 2021)53.941.943.263.942.039.044.240.5
ViL-M (Zhang et al.,2021a)50.8338.942.943.760.1261.143.439.744.640.7
Swin-S (Liu et al., 2021)59.833544.546.369.135444.840.947.642.8
Twins-SVT-B (w/PEG) (Chu et al., 2021a)67.032245.346.976.334045.241.548.043.0
RegionViT-B83.4308.943.346.192.2287.943.540.147.243.0
RegionViT-B+84.4328.144.246.993.2307.144.541.048.143.5
RegionViT-B+ w/PEG84.5328.244.646.993.2307.245.441.648.343.5
ResNeXt101-64x4d95.547341.01101.949342.838.4
PVT-L (Wang et al., 2021)71.134542.681.036442.939.511
ViL-B (Zhang et al., 2021a)66.7443.044.344.776.1365.145.141.045.741.3
Swin-B (Liu et al.,2021)98.447744.71107.249645.541.3二
Twins-SVT-L (w/ PEG) (Chu et al.,2021a)110.945545.7119.747445.941.61
RegionViT-B+ w/PEGt84.5506.446.148.293.2464.446.342.449.244.5
+ +The reported results of Swin are from Twins as the original paper does not include resutls with ImageNet1K weights. †: input resolution is $\overline { { 8 9 6 \times 1 3 4 4 } }$ + +Results on ImageNet1K. Table 3 shows the results on ImageNet1K where the methods listed all adopt CNN-like pyramid structure into the ViT and are all very recent and concurrent works. For the smaller models (Ti, S), RegionViT achieves a better trade-off between accuracy and complexity (parameters or FLOPs); on the other hand, for the larger models (M and B), it obtains better accuracy while having fewer FLOPs and parameters. The efficiency of RegionViT comes from the proposed R2L transformer encoder, and hence with such efficiency, it enables the network to be wide and deep for better accuracy with comparable complexity. + +Results on ImageNet21K. Table 4a shows that our model achieves a good accuracy-efficiency tradeoff with ImageNet21K for pretraining. RegionViT- $\mathbf { \cdot B + }$ only needs $2 5 \%$ parameters and FLOPs compared to CvT-W24 (Wu et al., 2021) and half parameters and FLOPs to Swin-L (Liu et al., 2021). + +Results on Transfer Learning. Table 4b shows the transfer learning results with ImageNet1K pretrained weights. Our model outperforms DeiT (Touvron et al., 2020) by $2 \sim 3 \%$ on ChestXRay8, which has a larger domain gap from ImageNet1K than other four datasets. We think this is because the hierarchical feature models could provide better generalization for the domain gap compared to DeiT which uses isotropic spatial resolution throughout the whole network. + +# 4.2 OBJECT DETECTION AND SEMANTIC SEGMENTATION + +Dataset. We use MS COCO 2017 to validate our models (Lin et al., 2014) on object detection. COCO 2017 dataset contains 118K images for training and 5K images for validation across 80 categories. We also test our model on keypoint detection and results are shown in Sec. B. On the other hand, we validate our model with semantic segmentation on ADE20K (Zhou et al., 2017). The ADE20K dataset contains $2 0 \mathrm { k }$ images in the training set, 2k images for validation. + +Training and Evaluation. For object detection, we adopt RetinaNet (Lin et al., 2017) and MaskRCNN (He et al., 2017) as our detection framework. We simply replace the backbone with RegionViT, and then output the local tokens at each stages as multi-scale features for the detector. The regional tokens are not used in the detection. Before sending the features to the detector framework, we add new layer normalization layers to normalize the features. We use RegionViT-S and RegionViT-B as the backbone and the weights are initialized from ImageNet1K pretraining. The shorter side and longer side of the input image is resized to 672 and 1,120, respectively. We train our models based on the settings in $1 \times$ and $3 \times$ schedule in Detectron2 (Wu et al., 2019) for object detection. More training details could be found in A.2. For semantic segmentation, we adopt Semantic FPN as the framework (Kirillov et al., 2019) and use RegionViT as the backbone. We initialize models with ImageNet1K weights, and mostly follow the training receipt in Twins’ paper (Chu et al., 2021a). More training details can be found in Sec. A.3. + +Results on Object Detection. Table 5 compares RegionViT with other methods. For the smaller models, RegionViT-S and RegionViT- $^ { S + }$ achieve similar accuracy while being moderately efficient in terms of FLOPs. We find that the position encoding generator (PEG) proposed by (Chu et al., 2021b) could help the detection accuracy substantially, so we also provide the results with PEG, which are also used in Twins (Chu et al., 2021a). For the middle size models with RetinaNet, RegionViT-B are slightly worse than Twins with $1 \times$ schedule; however, our model catches up the performance with $3 \times$ schedule. When comparing the large models under similar FLOPs (RegionViT- $\mathbf { \cdot B + }$ w/ $\mathrm { P E G \dag }$ ), we further train our model with the similar resolution used by other works, and observe that our models outperform all others while being more parameter efficient (models marked with $\dagger$ in Table 5). + +Results on Semantic Segmentation. Table 6 compares ours to Swin (Liu et al., 2021) and Twins. As PEG is useful for object detection, we show the results with PEG. Our models achieved significant improvement with similar FLOPs for both models. This suggests that the proposed R2L attention can effectively model global context. + +Table 6: Performance on semantic segmentation. + +
ModelFLOPs (G)Params (M)mIoU (%)
Swin-T*4631.941.5
Twins-SVT-S3728.343.2
RegionViT-S+ (w/PEG)3735.745.3
Swin-S*7053.245.2
Twins-SVT-B6760.445.3
RegionViT-B+ (w/PEG)6778.347.5
+ +∗ : Numbers are cited from the reproduced results of Twins’ paper. + +# 4.3 ACTION RECOGNITION + +Datasets and Setup. We validate our approach on Kinetics400 (K400) (Kay et al., 2017) and Something-Something V2 (SSV2) (Goyal et al., 2017). We adopt the divided-space-time attention in TimeSformer (Bertasius et al., 2021) as the temporal modeling to perform action recognition experiments. More details could be found in Sec. A.4. + +Results. Table 7 shows that with RegionViT as the backbone, the model can be much more efficient with competitive accuracy to TimeSformer, which uses vanilla ViT as the backbone. RegionViT-M could reduce more than $50 \%$ FLOPs and parameters than the TimeS + +Table 7: Performance on action recognition. + +
ModelFLOPs* (G)Params (M)K400 Acc. (%)SSV2 Acc.(%)
TimeSformer197121.475.859.5
x TimeSformer†197121.477.159.2
RegionViT-S59.442.976.659.7
RegionViT-M83.157.577.659.8
+ +∗ : FLOPs of single crop, †: retrained with the same setting. + +former. This not only shows the importance of spatial modeling in action recognition but also validates that our proposed model can also be extended for efficient action recognition. + +# 4.4 ABLATION STUDY + +We perform the following experiments to verify the effectiveness of different components in RegionViT. All experiments are conducted based on RegionViT-S. + +Regional Tokens. Table 8 shows the results with and without regional tokens on three tasks, and Table A5 includes FLOPs and parameters comparison. For image classification, the regional tokens provide around $0 . 4 \%$ improvement with negligible overhead in both compu + +Table 8: Ablation on regional tokens. + +
Dataset Regional TokensIN1K Acc.MS COCO MaskRCNN RetinaNetADE20K SemanticFPN
(APb/APm) 40.5/37.8(APb) 40.7(mIoU) 42.3
N Y82.2 82.642.5/39.542.243.7
+ +tations $( 6 \% )$ and parameters $( 0 . 7 \% )$ . On the other hand, the regional tokens clearly improve the performance of object detection and semantic segmentation with negligible overhead ( $< 2 \%$ on both FLOPs and parameters). This is because dense prediction tasks require more multi-scale features with global contextual information (provided by the regional tokens) than image classification. + +Downsampling. Table 9 shows different approaches for downsampling the patches. $3 \times 3$ kernel size achieves better results than $2 \times 2$ convolution, because the kernel is non-overlapped with $2 \times 2$ convolution, which limits the interac + +Table 9: Different downsampling approaches. + +
DownsamplingParamsFLOPs (G)IN1K Acc. (%)
2×2 convolution32.15.582.3
3×3 convolution34.15.782.5
3×3 depth-wise conv.30.65.382.6
+ +tion among local tokens. Moreover, using regular convolution does not improve the performance but increases computations and parameters. Thus, we use $3 \times 3$ depthwise convolution for downsampling. + +Weight sharing between RSA and LSA as well as downsampling. Table 10 shows the results with and without weight sharing. As seen from the table, using separated weights for RSA and LSA only slightly improves the accuracy + +but significantly increases the model size. This suggests sharing parameters between RSA and LSA suffices for learning local and global information in our approach. + +Table 10: Weight sharing. + +
Weight sharingParamsFLOPs (G)IN1K Acc. (%)
N40.45.382.7
Y30.65.382.6
+ +Keys in LSA. Table 11 studies the number of regional tokens in LSA. When using all regional tokens in the LSA, it provides the local tokens the possibility to explore all global information. Nonetheless, this model only improves $0 . 1 \%$ but + +Table 11: Keys in LSA. + +
Regional tokensParamsFLOPs (G)IN1K Acc. (%)
All30.66.082.7
Corresponding30.65.382.6
+ +with $13 \%$ more computations, which suggests that when considering the regional tokens, the one associated with the current region is sufficient to achieve good performance. + +Position information. Table 12 compares performance of different combinations of absolute position embedding (APE) and relative position bias (RPB). The models with RPB achieve better accuracy with similar FLOPs and parameters. Although the model with APE could slightly im + +Table 12: Different position information. + +
Abs. pos.Rel. pos.Params (M)FLOPs (G)IN1K Acc. (%)
NN30.65.382.4
YN30.95.382.2
YY30.95.382.7
NY30.65.382.6
+ +prove the performance, it limits the model to run at a fixed resolution, which is not suitable for vision tasks where image size could be varied. Thus, we only adopt the relative position bias in our models. + +Overlapped windows. Table 13 compares the results with and without overlapped windows. The overlapping ratio of neighboring windows is $50 \%$ and hence this model allows more interactions among local tokens. It improves + +Table 13: Overlapped windows. + +
ModelParamsFLOPs (G)IN1K Acc. (%)
overlapped30.618.882.8
non-overlapped30.65.382.6
+ +RegionViT-S by $0 . 2 \%$ but increases the FLOPs by $3 . 5 \times$ . This suggests that the information exchange between the border of windows is sufficiently covered by the proposed R2L attention. + +Comparison between R2L attention and shifted-window attention. Table 14 shows the comparison between shifted-window attention (Liu et al., 2021) and proposed R2L attention. R2L attention outperforms shifted-window + +attention by $0 . 7 \%$ , which shows that our proposed R2L attention can effectively model global information to achieve good performance. + +Table 14: Comparison of different attentions. + +
ModelParamsFLOPs (G)IN1K Acc. (%)
Shifted-window29.04.581.3
R2L32.15.282.0
+ +# 5 CONCLUSION + +In this paper, we propose a new ViT architecture that exploits the pyramid structure used in most of CNNs to provide multi-scale features and hence, the models can be more easily extended to different vision applications, like object detection. Moreover, the proposed regional-to-local attention relaxes the memory overhead of performing self-attention on the fine-grained image tokens by limiting the scope of attention but still keeping the capability to explore global information. Extensive experiments on several standard benchmark datasets well demonstrate that our proposed models outperform or are on par with many concurrent ViT variants on four vision applications, including image classification, object and keypoint detection, semantic segmentation and action recognition. + +# ETHICS STATEMENT + +Our work introduces a memory-friendly and efficient self-attention method for transformer models, which have wide-ranging applications to image classification, object detection, human activity recognition, etc. By improving efficiency, our work can have a positive effect on these applications in society, allowing faster processing. E.g., this could enable faster responses to detected events such as medical emergencies or detecting defects in manufacturing parts. Lower computation costs could also save energy and therefore have a positive impact on the environment. Negative impacts of our research are difficult to predict, however, it shares many of the pitfalls associated with deep classification models. E.g., Image and video classification systems have negative implications on privacy and could be used by malicious actors or governments to infringe on the privacy of citizens. Future research into private and ethical aspects of visual recognition is an important direction. + +# REPRODUCIBILITY STATEMENT + +In order to reproduce our results, we describe the details of the hyperparameters used in our training in Appendix A for all the tasks and we also attach our codes for image classification as part of the supplemental material. We will publicly release all the codes and models after acceptance. In the attached codes, we set all the hyperparameters as their default value, so that users can re-run our experiments with exactly the same hyperparameters. 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Exploring self-attention for image recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. + +Heliang Zheng, Jianlong Fu, Zheng-Jun Zha, Jiebo Luo, and Tao Mei. Learning Rich Part Hierarchies With Progressive Attention Networks for Fine-Grained Image Recognition. IEEE Transactions on Image Processing, 29:476–488, 2020. ISSN 1057-7149. doi: 10.1109/tip.2019.2921876. + +Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang. Random Erasing Data Augmentation. Proceedings of the AAAI Conference on Artificial Intelligence, 34(07):13001–13008, April 2020. + +Bolei Zhou, Hang Zhao, Xavier Puig, Sanja Fidler, Adela Barriuso, and Antonio Torralba. Scene parsing through ade20k dataset. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. + +Table A1: Details of training settings for image classification on ImageNet1K and ImageNet21K. + +
IN1KIN21KFinetune to IN1K@384²Transfer
Batch size4,0964,0961,024768
Epochs300120301,000
OptimizerAdamWAdamWAdamWSGD
Weight Decay0.050.011e-81e-4
Linear-rate Scheduler (Initial LR)Cosine (0.004)Cosine (0.001)Cosine (0.0002)Cosine (0.01)
Warmup Epochs50505
Warmup linear-rateLinear (1e-6)
Scheduler (Initial LR)
Data Aug.RandAugment (m=9,n=2)
Mixup (α)0.8
CutMix (α) Random Erasing1.0 0.25
Instance
Repetition*3
Drop-path0.1
Label Smoothing0.0 0.1
+ +∗: disabled for RegionViT-Ti. + +# APPENDIX + +Summary This appendix contains the following additional details. First, in Sec. A, we describe the training details on image classification, object and keypoint detection, semantic segmentation and action recognition separately. Then, we provide detailed results on object detection and ablation studies. The supplementary materials include our codes to reproduce our results on image classification and the README file in the attached codes (RegionViT Code.zip) also provides the detailed instructions to train and evaluate the networks. + +# A TRAINING DETAILS + +# A.1 IMAGE CLASSIFICATION + +We follow DeiT (Touvron et al., 2020) to train our models on ImageNet1K (Deng et al., 2009) except that we use batch size 4,096 with base learning rate 0.004 and the warm-up epochs is 50. We adopt the AdamW (Loshchilov & Hutter, 2019) optimizer with cosine learning rate scheduler (Loshchilov & Hutter, 2017). We apply Mixup (Zhang et al., 2018), CutMix (Yun et al., 2019), RandomErasing (Zhong et al., 2020), label smoothing (Szegedy et al., 2016), RandAugment (Cubuk et al., 2020) and instance repetition (Hoffer et al., 2020). During training, we randomly crop a $2 2 4 \times 2 2 4$ region and take a $2 2 4 \times 2 2 4$ center crop after resizing the shorter side to 256 for evaluation. While pretraining on ImageNet21K (Deng et al., 2009), we use similar settings with slight modifications. We train the model using 120 epochs with 5 epochs for warmup, weight-decay of 0.01 and base learning rate of 0.001. For transfer learning experiments, we finetune the ImageNet1K pretrained models with 1,000 epochs, batch size of 768, learning rate of 0.01, SGD optimizer, weight decay of 0.0001, and using the same data augmentation in training on ImageNet1K. During evaluation, we resize the shorter side of an image to 256 and then take a $2 2 4 \times 2 2 4$ region at the center and report top-1 accuracy. For finetuning experiments from ImageNet21K, we finetune the models with higher resolution of $3 8 4 \times 3 8 4$ , for 30 epochs with cosine learning rate scheduler, base linear rate of 0.002, weight-decay of 1e-8. Moreover, as it is trained on larger resolution, we adjust the window size $M$ to have the same number of regional tokens at the first stage, e.g., we change the window size to 12 for the the models trained at $2 2 4 \times 2 2 4$ with window size 7. We adopt the bicubic interpolation to upsample the weights of tokenizations and relative position bias. During evaluation, we directly resize the shorter side to 384 and then take center $3 8 4 \times 3 8 4$ crop. We trained the models with 32 GPUs. + +Table A2: Object detection performance on the COCO val2017 with $1 \times$ schedule. The bold number indicates the best number within the section, and for MaskRCNN, both $\mathsf { A P } ^ { b }$ and $\mathbf { A P } ^ { m }$ are annotated. + +
BackboneParams (M)FLOPs (G)AP AP50RetineNet AP75 APsParamsFLOPsMaskRCNN AP5
1APMAPL(M)(G)APbAPAPmAP0APm5
ResNet5037.723436.355.338.619.340.048.844.2260.038.058.641.434.455.136.7
ConT-M(Yan et al.,2021)27.0217.239.359.341.823.143.151.940.5一38.1
PVT-S (Wang et al.,2021)34.240.461.343.025.042.955.744.140.462.943.837.860.140.3
ViL-S (Zhang et al.,2021a)35.7252.241.662.544.124.944.656.245.0174.341.864.145.138.561.141.4
Swin-T(Liu et al.,2021)38.524541.562.144.225.144.955.547.826442.264.646.239.161.642.0
Twins-SVT-S(w/PEG)(Chu etal.,21a)34.320943.064.246.328.046.457.544.022843.566.047.340.363.243.4
RegionViT-S40.8192.642.264.145.127.545.455.350.1171.342.565.846.139.562.842.2
RegionViT-S+41.5204.243.164.846.229.646.656.150.9182.943.566.947.540.463.743.4
RegionViT-S+W/PEG41.6204.343.965.547.328.547.357.950.9183.044.267.348.240.864.144.0
ResNet10131557.738.8
ResNeXt101-32x4d56.7 56.4319.138.5 39.957.8 59.641.2 42.721.4 22.342.6 44.251.1 52.563.233640.4 41.961.1 62.544.236.4 37.559.440.2
PVT-M(Wang et al.,2021)53.941.963.144.325.044.957.662.8 63.934042.064.445.9 45.639.061.642.1
ViL-M(Zhang et al.,021a)50.8338.942.964.045.427.046.157.260.1261.143.465.947.039.762.842.1
Swin-S (Liu et al.,2021)59.833544.565.747.527.448.059.969.135444.866.648.940.963.444.2
Twins-SVT-B(w/PEG)(Chu etal.,221a)67.032245.366.748.128.548.960.676.334045.267.649.341.564.544.8
RegionViT-B83.4308.943.365.246.429.246.457.092.2287.943.566.747.440.163.443.0
RegionViT-B+84.4328.144.266.247.129.247.558.693.2307.144.567.648.741.064.443.9
RegionViT-B+ W/PEG84.5328.244.666.447.629.647.659.093.2307.245.468.449.641.665.244.8
ResNeXt101-64x4d95.547341.060.944.023.945.254.0101.949342.863.847.338.460.641.3
PVT-L (Wang et al., 2021)71.134542.663.745.425.846.058.481.036442.965.046.639.561.942.5
ViL-B (Zhang et al., 2021a)66.7443.044.365.547.128.947.958.376.1365.145.167.249.341.064.344.2
Swin-B (Liu etal.,2021)98.447744.765.949.2111107.249645.51141.311
Twins-SVT-L (w/PEG) (Chu et al.,2021a)110.945545.767.149.2一一一119.747445.9一一41.6 42.466.245.6
RegionViT-B+W/PEGt84.5506.446.168.049.530.549.960.193.2464.446.369.151.2
+ +The reported results of Swin (Liu et al., 2021) are from Twins (Chu et al., 2021a) as the original paper does not include resutls with ImageNet1K weights. †: input resolution is 896×1344. + +# A.2 OBJECT DETECTION AND KEYPOINT DETECTION + +We adopt RetinaNet (Lin et al., 2017) and MaskRCNN (He et al., 2017) as our detection framework and use KeypoinyRCNN for keypoint detection, and we simply replace the backbone with RegionViT, and then output the local tokens at each stages as multi-scale features for the detector. The regional tokens are not used in the detection. Before sending the features to the detector framework, we add new layer normalization layers to normalize the features. We use RegionViT-S and RegionViT-B as the backbone and the weights are initialized from ImageNet1K pretraining. We train our models based on the settings in $1 \times$ schedule in Detectron2 (Wu et al., 2019), i.e., the batch size is 16 and the learning rate is set to 0.0001 which drops $1 0 \times$ at the 60,000-th and 80,000-th iteration. We use AdamW optimizer with weight-decay of 0.05, and resize the shorter and longer side of an image to 672 and 1,120, respectively. We also adopt drop-path when finetuning the detector, with a rate set to 0.2 for both RetinaNet and MaskRCNN. We trained the models with 8 GPUs. When training with longer schedule $( 3 \times )$ , we adopt stronger data augmentation (multi-scale training) used in Swin (Liu et al., 2021) or Twins (Chu et al., 2021a) rather than fixed-size training used in $1 \times$ schedule. + +# A.3 SEMANTIC SEGMENTATION + +We adopt Semantic FPN as our semantic segmentation framework (Kirillov et al., 2019) by replacing the backbone network with RegionViT like the object detection task. We train the models with ImageNet1K pretrained weights. We mostly follow the training receipt in Twins (Chu et al., 2021a) The shorter side of the image is resized to 448 and the longer side won’t exceed 1792, and then we crop a $4 4 8 \times 4 4 8$ region for training. During the evaluation, we take the whole image after resizing as illustrated above. We train the model with AdamW optimizer with learning rate 1e-4 for 80k iterations, and the learning rate is decayed with poly schedule (power is 0.9). The weight decay and drop-path rate is 0.05 and 0.2, respectively. We trained all models with 8 GPUs. + +# A.4 ACTION RECOGNITION + +We adopt the divided-space-time attention in TimeSformer (Bertasius et al., 2021) as the temporal modeling to perform action recognition experiments. More specifically, we plugin the temporal attention before every R2L transformer encoder and finetune the model pretrained with ImageNet1K (instead of ImageNet21K). We use the AdamW optimizer (Loshchilov & Hutter, 2019) with base linear rate of 2e-4 and weight-decay of 0.0001. We apply the same Mixup (Zhang et al., 2018), CutMix (Yun et al., 2019) and drop path (Tan & Le, 2019) in the image classification. For the input, we uniformly sample 8 frames from the whole video and the $2 2 4 \times 2 2 4$ region is cropped after shorter side of images is resized to the range of 256 to 320, resulting the size of the input video as $8 \times 2 2 4 \times 2 2 4$ . During evaluation, we use the same way to sample frames but take $3 2 2 4 \times 2 2 4$ spatial crops (top-left, center and bottom-right) after resizing shorter side of image to 256, and then ensemble the predictions as final prediction for the input video. We trained the models with 16 GPUs. + +Table A3: Object detection performance on the COCO val2017 with $3 \times$ schedule. The bold number indicates the best number within the section, and for MaskRCNN, both $\mathsf { A P } ^ { b }$ and $\mathbf { A P } ^ { m }$ are annotated. + +
BackboneParams (M)FLOPs (G)AP AP50RetineNet AP75|APsParams (M)FLOPs GAPbMaskRCNN
APM|APLAPAP5|APmAPAPm5
ResNet5037.723439.058.441.822.442.851.644.2260.041.061.744.937.158.440.1
PVT-S (Wang etal.,2021)34.2142.262.745.026.245.257.244.124543.065.346.939.962.542.8
ViL-S (Zhang etal.,2021a)35.7252.242.963.845.627.846.456.345.0174.343.464.947.039.662.142.4
Swin-T(Liu et al., 2021)38.524543.964.847.128.447.257.847.826446.068.250.241.665.144.8
Twins-SVT-S (w/PEG)(Chu et al., 2021a)34.320945.667.148.629.849.360.044.022846.869.251.242.666.345.8
RegionViT-S40.8192.645.867.249.230.050.060.550.1171.346.368.850.642.365.545.7
RegionViT-S+41.5204.246.968.350.731.151.062.050.9182.947.369.552.043.166.446.7
RegionViT-S+ w/PEG41.6204.346.768.250.230.750.862.450.9183.047.670.052.043.467.147.0
ResNet10156.731540.944.063.233642.847.138.560.141.3
ResNeXt101-32x4d56.4319.141.460.1 61.044.323.7 23.945.0 45.553.8 53.762.834044.063.2 64.448.039.261.441.9
PVT-M(Wang etal.,021)53.9143.263.846.127.346.358.963.930244.266.048.240.563.143.5
ViL-M(Zhang etal.,2021a)50.8338.943.764.646.427.947.156.960.1261.144.666.348.540.763.843.7
Swin-S (Liuetal.,221)59.833546.367.449.831.150.360.969.135447.669.452.542.866.546.4
Twins-SVT-B (w/ PEG)(Chu et al., 2021a)67.032246.968.050.231.750.361.876.334048.069.552.743.066.846.6
RegionViT-B83.4308.946.167.849.131.550.261.292.2287.947.269.151.743.066.446.5
RegionViT-B+84.4328.146.968.650.130.850.762.693.2307.148.170.252.543.567.147.1
RegionViT-B+ W/PEG84.5328.246.968.350.331.150.562.493.2307.248.370.152.843.567.147.0
ViL-B(Zhang etal.,2021a)66.7443.044.765.547.629.948.058.176.1365.145.767.249.941.364.444.5
RegionViT-B+w/PEGt84.5506.448.269.951.534.351.561.793.2464.449.271.053.744.568.448.3
+ +he reported results of Swin (Liu et al., 2021) are from Twins (Chu et al., 2021a) as the original paper does not include resutls with ImageNet1K weights. †: input resolution is 896×1344. + +Table A4: Performance on person keypoint detection. + +
ModelFLOPs (G)Params (M)bbox AP (%)keypoint AP (%)
ResNet-50137.759.153.664.0
RegionViT-S+ (w/ PEG)172.265.756.066.1
+ +# B MORE DETAILED RESULTS + +We provided more details of some Tables shown in the main paper here. Table A2 and Table A3 include more metrics as compared to Table 5. Table A4 shows the results on keypoint detection. RegionViT outperforms ResNet-50 with moderate increases in FLOPs and parameters. This suggests that RegionViT can model the global context in keypoint detection as well. Table A5 includes complexity comparison for the ablation study on regional tokens as shown in Table 8. + +Table A5: Performance w/ and w/o regional tokens on RegionViT-S. + +
Regional TokensFLOPs (G) Params (M)ImageNet1K Acc.(%)
N5.030.482.2
Y5.330.682.6
MaskRCNNMS COCO (APb/APm)
N168.549.940.5/37.8
Y171.350.142.2/39.0
RetinaNetMS COCO (APb)
N189.840.540.7
Y192.640.841.7
SemanticFPNADE20K (mIoU)
N36.834.742.3
Y37.335.043.7
+ +Table A6: Throughput comparison. + +
ModelAcc. (%)Params (M)FLOPs (G)Throughput (images/sec)
Swin-T81.329.04.51129
Swin-S83.050.08.7710
RegionViT-S82.630.65.3823
RegionViT-M83.141.27.4706
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"text", + "text": "Are we biased on bias? Characterizing social bias research in the ACL community ", + "text_level": 1, + "bbox": [ + 196, + 90, + 801, + 129 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous EMNLP submission ", + "text_level": 1, + "bbox": [ + 361, + 166, + 642, + 183 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 263, + 253, + 339, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent events in business, politics and society have shed light on the importance and potential dangers of Natural Language processing (NLP) in the real world. NLP applications have gained unprecedented popularity not just among scientist and practitioners, but also the general public. As we develop new methodologies and curate new benchmarks and datasets it is more important than ever to consider the implications and societal impact of our work. In this paper, we characterize the landscape of societal bias research within the ACL community and provide a quantitative and qualitative survey by analyzing an categorized corpus of 348 papers. More specifically, we present a definition of social bias based on ethical principals and investigate (i) types of bias, (ii) languages, and (iii) type of paper. We find that there is significantly more work on gender biases and English than other languages. Finally, we discuss the possible causes behind our findings and provide pointers to future opportunities. ", + "bbox": [ + 144, + 279, + 458, + 592 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 117, + 605, + 257, + 620 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Traditionally, the NLP community has focused on ethical debates around privacy (Hovy and Spruit, 2016) ensuring that data is anonymized appropriately. More recently, there has been increased awareness that NLP research has a direct impact on peoples’ lives (Mayfield et al., 2019; Bender and Friedman, 2018). For example, summarization systems can amplify misinformation (Smiley et al., 2017), and sentiment analysis (SA) systems can assign more negative sentiments/emotions based on race and/or gender (Kiritchenko and Mohammad, 2018). While such research used to be more academic (Leidner and Plachouras, 2017), these concerns are having an increasing impact in industry (Schnoebelen, 2017; Jin et al., 2021b) with consequences for users (Prabhumoye et al., 2021). It is well known that language data encodes demographics and biases (Bender and Friedman, 2018). There is a risk that using such data can disclose inappropriate information about particular individuals, as well as undesirable attitudes towards individuals and groups (Hovy and Spruit, 2016; Eckert and Rickford, 2001) and social hierarchies (Blodgett et al., 2020). There are also concerns that systems based on inappropriate data are likely to repeat such biases, and may even amplify them (Bender and Friedman, 2018). In this paper, we survey 348 papers collected from the ACL anthology that focus on social bias and ethics in NLP research. We make three kinds of contributions, where (i) we present a working definition of social bias from a philosophy perspective, (ii) quantify our findings by annotating our corpus of papers and (iii) provide a discussion and pointers for possible future research directions. Through a quantitative analysis of current trends we attempt to answer the following questions: ", + "bbox": [ + 115, + 630, + 487, + 919 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 512, + 253, + 882, + 543 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• What kind of social biases is the ACL community concerned with? • How many languages is bias studied in? • What types of papers are present? ", + "bbox": [ + 532, + 557, + 882, + 621 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2 Related previous surveys ", + "text_level": 1, + "bbox": [ + 512, + 636, + 757, + 652 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There is a considerable literature on social biases in NLP. Much of this work provides guidelines and/or recommendations. One of the first position papers on the topic outlined the need for ethical considerations that go beyond privacy concerns for users, and focus on the social impact of experiments and applications on individuals and (minority) groups Hovy and Spruit (2016). ", + "bbox": [ + 512, + 662, + 884, + 790 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Surveys on social bias emphasize a variety of different aspects, such as embedding representations, data collection and annotation, downstream task performance, metrics and limiting negative impact (Garrido-Muñoz et al., 2021; Mohammad, 2022b; Schnoebelen, 2017). Work by Bender and Friedman (2018); Hovy and Prabhumoye (2021) outline the concept of data statements and sources of bias respectively to aid the research design process. Other research has reviewed how to mitigate bias (Chandrabose et al., 2021; Meade et al., 2022; Balkir et al., 2022), how to teach bias, ethics and privacy to students (Bender et al., 2020; Friedrich and Zesch, 2021), evaluate existing metrics (Czarnowska et al., 2021; Delobelle et al., 2022), handle challenges presented by new laws (e.g., GDPR) (Lewis et al., 2017) and apply existing principles from ethics and privacy to NLP (De Jong et al., 2018; Leidner and Plachouras, 2017; Prabhumoye et al., 2021). Our work follows the precedent established by Blodgett et al. (2020), who used keywords to select papers from the ACL anthology, and then enlarged the sample by following citations to other popular venues (eg AAAI, ICML etc.). Similarly, we also align with Field et al. (2021) who solely focus on papers published at ACL venues but draw on conclusions from NLP papers published elsewhere. Relying on keywords, of course, introduces possibilities for false positives and false negatives. ", + "bbox": [ + 512, + 793, + 882, + 919 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 115, + 85, + 487, + 438 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Scope of this survey In this work, we solely focus on papers published in the ACL anthology to gain a better insight into current trends and popular research approaches for social bias. One limitation of such an approach is that seminal work published at other venues is not reviewed here. In Appendix 7, we provide a table that shows each paper reviewed in this survey. ", + "bbox": [ + 115, + 448, + 487, + 575 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 Defining Bias ", + "text_level": 1, + "bbox": [ + 117, + 588, + 263, + 605 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The dictionary definition of bias is “an inclination or prejudice for or against one person or group, especially in a way considered to be unfair” (Stevenson, 2010). Based on this, defining bias involves an interaction among different components: (i) individuals or categories that determine a group, (ii) attitudes towards this group, and (iii) assessment of this attitude in relation to fairness. Bias comes into existence when a specific attitude is formed, which may or may not be fair. For example, in investigating a disease that is more prevalent among women, use of gender is a relevant variable and, in itself, does not entail a differential attitude. Once a differential attitude is formed however, bias comes into existence and the attitude may be negative or positive. By definition, if a distinctly positive attitude is formed about one variable (e.g., gender), it entails a less favorable attitude about the other gender categories. This does not mean that all bias is necessarily unfair, there are multiple theories and definitions of fairness that are formulated and analyzed in-depth in political philosophy (Lamont and Favor, 2004). The formal principle of equality formulated by Aristotle (Ameriks and Clarke, 2000) states that equals must be treated equally, which is often referred to as ‘the fairness ideal’, but it is neither a prevailing definition nor a useful one in practice. Without identifying relevant features, such a definition would not prevent categories such as race or wealth to be used as variables for differential treatment. However, a fairness approach (Lamont and Favor, 2004) based on equal opportunity might require a differential attitude (i.e., bias) towards a certain category in order to ‘level the playing field’. For example, if women are routinely given worse performance reviews and lower pay for successfully completing the same tasks as men, then there is differential treatment. Meaning women who are as successful as men cannot have the same opportunities. According to this understanding of fairness, a bias towards women would be fair. It is also worth noting that positive or negative discrimination is distinct from bias. While discrimination is about treatment, bias is about attitude. In other words, bias may lead to discrimination (Mateo and Williams, 2020). In this context, the ethical issue about bias is tied to differential treatment of descriptive categories resulting in unfair outcomes. Identifying and dealing with ethical concerns related to bias, must necessarily involve identifying the descriptive categories and the biases against those categories as well as examining whether the said bias is unfair, according to the relevant definition of fairness. ", + "bbox": [ + 115, + 614, + 487, + 920 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 512, + 76, + 882, + 649 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "4 Methodology ", + "text_level": 1, + "bbox": [ + 512, + 669, + 655, + 684 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Following the standard practice mentioned above, we searched the ACL anthology1 in September 2022 for relevant titles and abstracts using the keywords: ethical, ethics, fairness, fair, bias, social, society, societal, social good. For papers, published after September 2022 we manually screened all conference proceedings for the same keywords. One limitation of relying on a keyword search alone is that we might miss any work that refers to a bias directly in the title, for example ‘fatphobia detection in online forums’. ", + "bbox": [ + 512, + 701, + 884, + 876 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Filtering Strategy A total of 1,437 papers were returned by the search; 523 papers were retained after a manual screening of titles and abstracts. We removed duplicates, as well as work not related to bias and/or ethics in NLP. Then we downloaded full papers, and filtered out papers if: (i) the contribution was a talk, demonstration, abstract or keynote, (ii) “bias” was used in the machine learning sense, or (iii) the paper did not focus on social bias. This process produced a collection of 348 papers. ", + "bbox": [ + 115, + 84, + 487, + 244 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Categorization process We identify trends in our corpus by empirically determining a set of five categories, where we fully review each paper manually. We make our corpus alongside with its corresponding categories and labels publicly available2. We focus on four elements for each paper to identify trends, where we identify language investigated, type of bias analyzed and what kind of paper is introduced and which NLP area it belong to. For the type of paper analysis we utilize the authors description of contributions to split the papers in the following categories: ", + "bbox": [ + 115, + 256, + 487, + 449 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Method: In this category of papers, the main focus of the work is to contribute a new method, which includes but is not limited to novel ways to measure or mitigate social bias. ", + "bbox": [ + 137, + 463, + 487, + 526 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Analysis Papers in this category, examine existing datasets, benchmarks, language models, NLP systems or embeddings for bias using social science methods, statistics and mixed methods approaches. For example, authors who have conducted research in this category have explicitly stated that they conduct an analysis or outline a mixed method approach. ", + "bbox": [ + 137, + 539, + 487, + 668 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Surveys and Position papers: This category of papers includes surveys, guides, tutorials, reviews and position papers. ", + "bbox": [ + 137, + 681, + 487, + 728 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Dataset, Benchmarks or Resources: Paper in this category propose new datasets, benchmarks, lexicons, challenge sets and often include some preliminary analysis of the new data either collected through crowd-sourcing. ", + "bbox": [ + 137, + 741, + 487, + 821 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Datasets, Benchmarks and Methods: This is a combination of papers that focus on both introducing a new resource (e.g.: dataset or benchmark) in addition to a new methodology. ", + "bbox": [ + 137, + 834, + 487, + 898 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/396c87474e44adce2b9147b366a4360a106ec0132272dc17032e3a2650b3688c.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: $7 6 , 1 5 \\%$ of the 348 papers are from 2020-2023. " + ], + "table_body": "
YearPapersYearPapers
2010-201611202068
201716202196
201810202279
201946202322
Totals83Totals265
", + "bbox": [ + 557, + 80, + 836, + 156 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "5 Empirical Findings ", + "text_level": 1, + "bbox": [ + 512, + 200, + 710, + 217 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The 348 papers were published between 2010 and 2023. Table 1 shows that there has been considerable growth in interest in bias research. ", + "bbox": [ + 510, + 227, + 882, + 273 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Type of Paper Based on the criteria outlined above, we have found that the majority of papers introduced are Method papers $( 5 7 . 7 5 \\% )$ , followed by Datasets and Benchmarks $( 2 0 . 4 0 \\% )$ , Surveys $( 1 5 . 2 2 \\% )$ , Analysis $( 4 . 3 1 \\% )$ and combined Datasets, Benchmarks and Methods $( 2 . 2 9 \\% )$ . We take a closer look the majority category Methods, where we assigned Method papers into different categories based on the contribution of the paper (see Table 2). ", + "bbox": [ + 512, + 284, + 884, + 443 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Methods We distinguish between bias detection and measurement, as detection does not necessarily measure or remove any kind of bias. We found similarly to (Blodgett et al., 2020) that many papers propose a combination of techniques, hence why we have decided to merge such approaches into the category ‘Measurement and Mitigation’. Many works in Debiasing and Mitigation apply or extend methods such as WEAT (Caliskan et al., 2017) or HARD DB (Bolukbasi et al., 2016) to a specific benchmark or dataset. One negative side effect of this could be that for example in gender bias Measurement/Mitigation, there is no evidence that HardDB can or should be applied to languages with grammatical gender (Sun et al., 2019). There has also been criticism of removing bias (Caliskan et al., 2017), where removing bias (i) only changes how an application or algorithm understands the world but not how it applies the knowledge gained from understanding (‘fairness through blindness’), (ii) could harm meaning and accuracy and (iii) bias can (unintentionally) be introduced through other avenues during the design process. Therefore, simply removing bias is not enough (Chandrabose et al., 2021) and developing new methods requires consistent reflection as bias in NLP systems is never fully inescapable (Waseem et al., 2021). Work by (Hovy, 2015; Kiritchenko and Mohammad, 2018) has found that the some information bias mitigation techniques can be beneficial in improving performance in downstream tasks. The methods in the Miscellaneous category take a different approach to working on bias. For example (Fisher et al., 2020) in including bias sensitive attributes by defining a whitelist of triples for uncontroversial cases. Arguably, one drawback here is that it is up to the person whitelisting to decide what is not controversial and what is. Similarly, (Touileb et al., 2021) and (Wang et al., 2017) use gender and linguistic bias respectively to improve classification results. Touileb et al. (2021) first show how female critics disproportionately give lower ratings to female authors, where removing metadata may have the opposite effect in that it does not help traditionally underrepresented groups in a specific domain. At the same time, many of the methods looking at measuring or removing bias complicate the data and tasks at hand and can lead to the development of systems that are not reliable when used in a more complex context (Talat et al., 2022b). This also applies to the predominant use of intrinsic metrics in bias measurement. These metrics may shed more light on how much bias exists in a dataset/LM, but does not necessarily correlate with performance on downstream tasks and therefore does not show the true harm of bias (Orgad and Belinkov, 2022). Thus, we may run the risk of developing methods for each new dataset or benchmark and missing out on crucial information that shows how bias affects different downstream tasks in different ways. However, documenting and measuring bias in a systematic way is crucial to understanding what harms can be caused in real life situations, so that preventive methods can be developed (Dev et al., 2021b). Current approaches in mitigation and/or measurement methods are evaluated on a variety of NLP areas, including Language Models $( 3 5 . 7 4 \\% )$ , Classification $( 2 2 . 8 5 \\% )$ , NLG $( 1 4 . 7 6 \\% )$ and NLI $( 3 . 3 3 \\% )$ . It is unsurprising that much attention has been paid to embedding representations that are trained on large amounts of text (Kiritchenko and Mohammad, 2018; Mayfield et al., 2019; Talat et al., 2022b). This has the benefit of bias methods being more widely applicable, but it also means that there are distinct limitations when a method is tied to a specific architecture rather than the task/benchmark itself. It means bias measures are no longer comparable in relation to other benchmarks and bias can be introduced at any stage of an NLP system design as it de", + "bbox": [ + 510, + 451, + 882, + 917 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 115, + 85, + 487, + 907 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/d6baa467994d5c3389ca0f0023454c93c80590fa1b1b7e98f38c41251a8b987a.jpg", + "table_caption": [ + "" + ], + "table_footnote": [], + "table_body": "
TypePapers%
Measurement Debiasing /Mitigation98 5246.66 24.76
Combinations of above3215.23
Detection199.04
Generation52.38
Miscellaneous41.9
Total210100.0
", + "bbox": [ + 554, + 80, + 840, + 180 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Table 2: Empirical taxonomy of methods. pends on where and how the final LLM is applied and to which community (Talat et al., 2022b). An important trade-off to consider is the balance between generalizable and context-sensitive methods to measure bias in downstream tasks. There are also other areas that have done work on bias but are not represented as well in this survey, which include but are not limited to Speech Recognition (Kwako et al., 2022; Savoldi et al., 2022), MultiModal NLP (Chen et al., 2020a; Srinivasan and Bisk, 2022a) and Information Extraction (Li et al., 2022b; Sun and Peng, 2021). ", + "bbox": [ + 512, + 195, + 882, + 394 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Social biases In Figure 1 we show the types of biases investigated, where we include all social biases that occur more than once (see diagonal of the matri). Furthermore there are a small number of biases that only occur once and are not shown in the table, such as bias transfer hypothesis (Steed et al., 2022) or dialect bias (Tatman, 2017b). Furthermore, we included the category multiple social biases, where the paper does not explicitly list or describe the specific type of bias investigated (Ghosh et al., 2021; Ramponi and Tonelli, 2022; Mireshghallah and Berg-Kirkpatrick, 2021; Loukina et al., 2019). There also is the intersectional bias category, which shows how different elements of a person’s identity (e.g., gender, race and age) can either be a benefit or disadvantage and lead to compounded discrimination (Crenshaw, 2013). For example, recent work by Lalor et al. (2022) benchmarks a variety of NLP models on different downstream tasks for its performance on intersectional biases and (Câmara et al., 2022) introduce a framework for unisectional and intersectional analysis of sentiment analysis in a multilingual setting. Few works focus on biases other than gender, where (Davidson et al., 2019; Sap et al., 2019; Manzini et al., 2019) look at racial bias and (Hutchinson and Mitchell, 2019; Herold et al., 2022) investigate disability bias. Noticeable is that some biases are not investigated on their own, such as age, religion, sexuality or profession. We included political/media bias in this analysis, if the paper also looks at social bias. For example, debiasing claims that include attitudes towards a group (e.g., sexuality). However, this type of work does not explicitly mention social biases when attitudes or characteristics of the targeted group are only implied. Dayanık and Padó (2020) looks media bias on a immigration dataset (MARDY) but does not mention implied social biases (e.g., nationality or ethnicity). The most frequently combined social biases are gender and race.The most frequently combined social biases are gender and race. In Table 3 we compute residuals between observed joints and predictions on margin, where highlighted in green are highly saturated areas and shades of red show less saturated areas. ", + "bbox": [ + 510, + 404, + 882, + 919 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 115, + 85, + 485, + 309 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Languages We show the languages used in each type of paper, excluding Surveys and Position papers in Figure 2. There are a total of 34 languages, however we leave out any languages that only occur once in the visualization. There are 11 languages not visualized, including Farsi, Urdu, Wolof, Bengali, Armenian, Bengali, Inuktitut, Ukrainian, Hungarian, Indonesian and Lithuanian. English, German, Spanish and Chinese are most commonly used either on their own or in combination with each other. The majority of all papers focus on a single language at a time. Furthermore, the vast majority of LLMs are monolingual and do not encode the cultural variety that naturally occurs within one language, for example non-standard English varieties (Talat et al., 2022b). Therefore it is important to not only document the type of bias investigated, but also contextualize bias within a language’s cultural context, understanding of said bias and document the language itself (Bender Rule (Bender, 2011)). Based on this collection, bias research is heavily biased towards western and Anglo-centric notions of bias and very few works focus on nonEnglish benchmarks (Talat et al., 2022b; Hovy and Spruit, 2016). This proves extremely problematic when English benchmarks are automatically translated, but many of the biases do not hold true in nonWestern cultural contexts. For example, gendered professions do not necessarily translate across every language or culture (Talat et al., 2022b) and many NLP systems trained on written English (e.g., Penn Tree Bank) do not perform well on nonstandard English (Mayfield et al., 2019). From Table 4 we can see the residuals between observed joints showing a clear over-saturation (green) for specific combinations of languages (e.g: English and German or English and Spanish). ", + "bbox": [ + 115, + 321, + 487, + 920 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "6 Discussion ", + "text_level": 1, + "bbox": [ + 512, + 84, + 635, + 99 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Datasets and Benchmarks Previous work (Hovy and Spruit, 2016; Hovy, 2018; Talat et al., 2022b) outlined a number of reasons that may explain why there are so many papers focusing on the same datasets, benchmarks, biases, and languages. In the following section, we highlight some of the elements that may explain why there is an uneven distribution of work and resources. ", + "bbox": [ + 512, + 109, + 884, + 237 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Experimental setup The majority of of work in bias research has focused on using intrinsic bias measurements (bias in internal model representations) and little attention has been paid to extrinsic metrics (Orgad and Belinkov, 2022; Delobelle et al., 2022). A very real consequence of this is that much work does not appropriately describe, contextualize and identify the potential harms that bias has in real-world scenarios (Blodgett et al., 2020). Also bias is inadvertently introduced in intrinsic metrics, where lexicons used to measure bias in one dataset produces very different results in another (Antoniak and Mimno, 2021). Similarly, (Goldfarb-Tarrant et al., 2021) have found that there is no correlation between intrinsic and extrinsic metrics. Another key problem is that often newly proposed datasets are linked to specific metrics, which makes it hard to draw conclusions from individual case studies as many results are not generalizable (Orgad and Belinkov, 2022). Another element impacting metrics is the composition of test data, where (Orgad and Belinkov, 2022) found that test sets often don’t contain balanced examples. However, most metrics are defined over a whole dataset and are therefore sensitive to its composition, which may lead to variability in results. Both metric choice and dataset composition can significantly change the results and conclusions drawn from a downstream task or dataset (Akyürek et al., 2022a). Therefore, it is important to (i) provide motivation for including / excluding a particular metric and describe how it impacts downstream performance, (ii) compare a metric across a variety of datasets and (iii) compare many metrics across individual datasets. ", + "bbox": [ + 547, + 242, + 882, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Funding There are unintended consequences of research that can be traced to how research projects are funded (e.g., governments or military interests), where researchers should be aware that their work has broader impact and can be abused (Hovy and Spruit, 2016). ", + "bbox": [ + 534, + 872, + 882, + 920 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/39ca6d5c5159ffa9ac3916f710618c0913d0652965d2d794b75fa81a432ad4dd.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
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", + "bbox": [ + 115, + 79, + 875, + 388 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/64372056be13c8c957ec2140ff3d8d3e2493a704db3fe2278224e14b15d46ac4.jpg", + "image_caption": [ + "Table 3: Observed joints: number of papers with combinations of languages (ISO 639). biases ", + "Figure 1: A log-scaled heatmap showing the type and frequency of social biases. " + ], + "image_footnote": [], + "bbox": [ + 121, + 428, + 882, + 752 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 152, + 808, + 487, + 854 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• Availability and Overexposure We have found that a small number of papers introduce new datasets or benchmarks (see 5). Creating and curating new datasets as well as benchmarks are often a time-consuming, expensive and long process, where it is oftentimes easier to utilize existing resources to try out new methods (Hovy, 2018). Similarly there is the phenomenon of topic overexposure, where there are waves of seemingly ’popular’ research topics that eventually go out of fashion. This is based on availability heuristic, if people recall a certain event or have knowledge about certain things then it must be important (Hovy and Spruit, 2016). ", + "bbox": [ + 137, + 872, + 487, + 920 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 549, + 807, + 884, + 920 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/3906adbdb3b5a358a8d9c95272e9856fc8e0f511e7afc62b0ed1121ade239ca1.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 942, + 808, + 978, + 919 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/fe7a493744c546c2110411feef807ba5516c435e039e9658b6907aafa2799376.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
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", + "bbox": [ + 115, + 80, + 810, + 356 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1efea812531c21fb81d76842d582ec3a4e7f20e054c6d813c0e065b5df8b6148.jpg", + "image_caption": [ + "Table 4: Residuals between observed joints and predictions based on margins. ", + "Figure 2: A log-scaled heatmap showing the different languages (ISO-639-1) and their frequency of occurrence. " + ], + "image_footnote": [], + "bbox": [ + 122, + 397, + 880, + 703 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e3f85ee35e1f7d2867ff554f66f56e6b5549b65d1fecf5a80be7a0badcf5ae73.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 10, + 753, + 48, + 921 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 152, + 755, + 487, + 834 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Bias We have found a limited focus on specific social biases, where possible causes are rooted in (i) the data that encodes bias by default (Chandrabose et al., 2021), where already available data determines what kind of bias we focus on, (ii) machine learning breakthroughs in NLP has enabled ‘streetlamp science’ and we focus on tasks that can be solved (Hovy, 2018) and (iii) lack of awareness. This has the consequence that difficult tasks are not being tackled and bias remains present in NLP tools. Therefore it is key to raise awareness (BaezaYates, 2018), understand and measure what kind of bias has influence on NLP models and work towards developing solutions that are equitable. Here, we (i) showcase challenges in three frequently researched social biases that have been identified through this survey and (ii) point out opportunities for the future with the aim to raise awareness. ", + "bbox": [ + 115, + 856, + 487, + 919 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 510, + 755, + 885, + 915 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/5cf221f024cf842ddbea096c8791ed9b3fa0654027f7459eb5ef68a5ae68e47a.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 944, + 753, + 983, + 917 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 115, + 85, + 487, + 147 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Gender bias There is a strong emphasis on a binary understanding of gender (Schnoebelen, 2017; Orgad and Belinkov, 2022) and most task have been reduced to a masculine/feminine dichotomy (Savoldi et al., 2021). Initially, this may be perceived as useful to enable research, however it does not capture the reality of the society and world we live in today. For example, in the USA alone over 1.4 million people identify as transgender (Larson, 2017) and 1.2 million identify as non-binary (Williams Institute, 2021). At the same time, there is not only a common misconception that gender and sex are the same (Larson, 2017), but also that sexuality is somehow indicative of either gender or sex. Sexuality refers to a person’s attraction to a sex or gender, but is not a marker of gender/sex itself (Baum and Westheimer, 2015). Therefore, one can not use sex or gender as a predictor or precursor to assuming a person’s sexuality. Talat et al. (2022b) argue that characteristics like sexuality are usually not observable, which can lead to a reliance on hegemonic stereotypes and unnatural language in bias evaluation benchmarks. This leaves plenty of opportunity to start conversations around developing new datasets, benchmarks and methods that are more inclusive and reflect the world we live in (Savoldi et al., 2021; Orgad and Belinkov, 2022). ", + "bbox": [ + 151, + 155, + 487, + 642 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "• Race bias Related work by (Field et al., 2021) provides an excellent overview of the state of the art of race bias research in NLP. In their survey they identify that there are very few datasets and benchmarks and that oftentimes a narrow view of race and racial identity are perpetuated. Additionally, researchers often doesn’t explicitly state if they are focusing on racial bias through downstream tasks such as abusive language detection. Subsequently, currently deployed hate speech or toxicity classifiers mislabel language predominantly used in the African American community as toxic or hate speech when it is not (Dixon et al., 2018; Xia et al., 2020). ", + "bbox": [ + 137, + 653, + 485, + 894 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, this survey mentions a number of social biases that have been mentioned such as religious, age and disability with few papers in Figure 1. It is outside of the scope of this work to address each social bias individually, but we would like to point out that there is a lack of relevant benchmarks, datasets and surveys to make substantial progress in these areas and understand the unique challenges each community faces (individually and at an intersectional-level). Most importantly, we would like to emphasize that this type of future work needs to be deeply grounded in interdisciplinary research and led by diverse teams that connect and engage with relevant communities. ", + "bbox": [ + 136, + 904, + 484, + 919 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 512, + 85, + 882, + 293 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Interdisciplinary research The relationship between language and social hierarchies is far more complex than what current techniques can capture. Therefore new methods need to be grounded in relevant literature outside of NLP (Blodgett et al., 2020), because social bias is a complex issue (Sun et al., 2019). Whilst NLP researchers may be committed to using ethical approaches, they may not necessarily have the required ethical and legal knowledge to do so (Santy et al., 2021). This makes it incredibly important to foster collaboration between disciplines to ensure that historical inequalities and biases are taken into consideration when building new algorithms or systems (Caliskan et al., 2017). ", + "bbox": [ + 512, + 325, + 884, + 564 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Diversity Given the real-life impact of NLP systems and research on people, there is not just a need for diversity in experts working on such systems (Caliskan et al., 2017), but also a need for practitioners and researchers to engage with the affected communities and stakeholders (Blodgett et al., 2020; Fortuna et al., 2021). Therefore, we need to recognize the implicit bias of the people working on different NLP systems and sense-check at different stages how this bias may be reflected in collected data, new benchmarks or models (Savoldi et al., 2021; Hovy and Spruit, 2016). We also need to acknowledge the lack of diversity in teams working on NLP (Schluter, 2018; Savoldi et al., 2021) and work towards more inclusive teams that represent a wide variety of backgrounds and lived experiences (Field et al., 2021). Otherwise, NLP systems continue to represent majorities and we risk the further oppression of already disadvantaged communities (Talat et al., 2022b; Schnoebelen, 2017). ", + "bbox": [ + 512, + 598, + 882, + 919 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Limitations and Ethics Statement ", + "text_level": 1, + "bbox": [ + 117, + 84, + 406, + 99 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we surveyed a collection of papers and identified continued challenges in social bias research. We have created this collection based on a keyword search and outlined how this may not fully reflect all literature on social bias existing in the ACL anthology or other venues. We only used open-access papers in this collection and no human participants were involved in this work. Traditionally, social biases have been investigated in fields such as social sciences, law, or psychology which we have not discussed here. Furthermore, we do not give an analysis of algorithmic or dataset biases (e.g., machine learning, data mining or otherwise) or provided an in-depth review of technical contributions in computational social biases. We are also limited by the resource of the reviewed papers, where substantial contributions to the field have been made outside of ACL venues. Finally, we would like to point out that opportunities and recommendations for future bias research as proposed in section 6 should be considered from a euro- and/ or anglo-centric perspective. There may be a variation depending on the social context, country or culture that works on a specific bias problem. 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In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. ", + "bbox": [ + 510, + 444, + 882, + 510 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang. 2017. Men also like shopping: Reducing gender bias amplification using corpus-level constraints. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 2979–2989. ", + "bbox": [ + 510, + 520, + 882, + 600 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang. 2018. Gender bias in coreference resolution: Evaluation and debiasing methods. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), pages 15–20. ", + "bbox": [ + 510, + 611, + 882, + 702 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Ruiqi Zhong, Yanda Chen, Desmond Patton, Charlotte Selous, and Kathleen Mckeown. 2019. Detecting and reducing bias in a high stakes domain. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 4765–4775. ", + "bbox": [ + 510, + 713, + 882, + 804 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Yang Zhong, Jingfeng Yang, Wei Xu, and Diyi Yang. 2021. Wikibias: Detecting multi-span subjective biases in language. In Findings of the Association for Computational Linguistics: EMNLP 2021, pages 1799–1814. ", + "bbox": [ + 512, + 816, + 882, + 881 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Pei Zhou, Weijia Shi, Jieyu Zhao, Kuan-Hao Huang, Muhao Chen, Ryan Cotterell, and Kai-Wei Chang. ", + "bbox": [ + 512, + 892, + 884, + 919 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "2019. Examining gender bias in languages with grammatical gender. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLPIJCNLP), pages 5276–5284. ", + "bbox": [ + 132, + 85, + 487, + 164 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Xuhui Zhou, Maarten Sap, Swabha Swayamdipta, Yejin Choi, and Noah A Smith. 2021. Challenges in automated debiasing for toxic language detection. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pages 3143–3155. ", + "bbox": [ + 117, + 174, + 485, + 253 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Yi Zhou, Masahiro Kaneko, and Danushka Bollegala. 2022. Sense embeddings are also biased–evaluating social biases in static and contextualised sense embeddings. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1924–1935. ", + "bbox": [ + 117, + 263, + 487, + 342 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Caleb Ziems, Jane Yu, Yi-Chia Wang, Alon Halevy, and Diyi Yang. 2022. The moral integrity corpus: A benchmark for ethical dialogue systems. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 3755–3773. ", + "bbox": [ + 117, + 353, + 487, + 431 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Nadezhda Zueva, Madina Kabirova, and Pavel Kalaidin. 2020. Reducing unintended identity bias in russian hate speech detection. In Proceedings of the Fourth Workshop on Online Abuse and Harms, pages 65–69. ", + "bbox": [ + 115, + 442, + 487, + 494 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "7 Appendix ", + "text_level": 1, + "bbox": [ + 117, + 506, + 231, + 523 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/a382be79d9753e784fe68bdfd7767983a6fa244d8bc1525c59e240348da279ad.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
TypePapers
PaperSurvey/PositionDrugan and Babych (2010), Masthoff(2011), Escartin et al. (2(2016), Hovy and Spruit (2O16), Leidner and Plachouras (2017),et al.(2017), Schnoebelen (2017),Larson (2017),De Jong et a(2018),Mayfield et al. (2019),Sun et al. (2019), Blodgett et al.(2021), Bender et al. (2020), Schoch et al. (2020), Savoldi et (2021), Mishra et al. (2021), Czarnowska et al. (2021), DoughSheng et al. (2021), Hovy and Yang (2021), Santy et al. (2021)(2022b), Chandrabose et al. (2021), Vecchi et al. (2021),Talat et :Balkir et al. (2022), Orgad and Belinkov (2022), Akyurek et al(2022), Mohammad (2022a), Orr et al. (2022), Benotti and Blaet al.(2023),Benotti et al. (2023), Ramesh et al. (2023), Rios el
Dataset,BenchmarkYano et al. (2010), Flekova et al. (2016), Braun et al. (2016) KRudinger et al. (2018), Zhao et al. (2018), Kiritchenko and Mo(2019), Gaut et al. (202O),Friedman et al. (2019), Hitti et al. ((2019), Fan et al. (2019),Liu et al. (2020a), Sap et al. (2020),Louet al. (2021), Lim et al. (2020), Shahid et al. (2020), LazaridoUthus (2020), Gala et al. (2020),Dinan et al. (2020b), Meyer e(2020b), Gangula et al. (2019), Zueva et al. (2020), Nangia et al.Blodgett et al. (2021), Spinde et al. (2021), Zhong et al. (2021),Spgaard (2021), Zhao et al. (2021), Aksenov et al. (2021), KamRenduchintala et al. (2021), Barikeri et al. (2021),Milkowski et :et al. (2022), Ziems et al. (2022), Zhou et al. (2022),Akyirek(2022), Bansal et al. (2022),Smith et al. (2022), Qian et al. (2022et al. (2023), Madnani et al. (2017), Kumar and Bhotia (2020)(2021), Zad et al. (2021), Nozza et al. (2022),Mechura (2022)
MethodsGonen and Goldberg (2019), Wevers (2019), Hovy et al. (2020Nissim et al. (2020),Wang etal. (2021d),Wolfe and Caliskan (2et al. (2021), Herold et al. (2022),Bertsch et al. (2022), RampCao et al. (2022a),Alshahrani et al. (2022) Daems and Hackenbuet al. (2015), Terkik et al. (2016), Tatman (2017b),Wang et alTatman (2017a), Zhao et al. (2017), Long et al. (2018), BalyKloppenburg (2019), Farber et al. (2019), Zhong et al. (2019),and Ungar (2019), Gangula et al. (2019), Karve et al. (2019),and Najafian (20i9), Chaloner and Maldonado (2019), Qian et(2019),Kaneko and Bollegala (2019), Huang et al. (2020a),BDinan et al. (2020a), Bordia and Bowman (2O19),Manzini et al, Schwertmann et al. (2023), Pedersen et al. (2023), Bauer et al. (Kumar et al. (2023),Touileb et al. (2023), Parmar et al. (2022),et al. (2023),Alemany et al. (2023), Martinkovä et al. (2023),(2022), Cao et al. (2022b), Sap et al. (2020), Xia et al. (2020(2022), Orgad et al. (2022), Sen et al. (2022), Gupta et al. (202Gira et al. (2022), Guo et al. (2022), Wang et al. (2022), An et a(2022), Borchers et al. (2022),Touileb et al. (2022), Jentzsch andet al. (2022), Tal et al. (2022),Li et al. (2022b),Limisiewicz aret al. (2022),Ahn et al. (2022), Joniak and Aizawa (2022), Chen(2022), Du et al. (2022),Agrawal et al. (2022), Sesari et al. (0Steed et al. (2022), Schick et al. (2021), Zhou et al. (2021), Kaneetal.(2021),Lucy and Bamman (2021), Jin et al.(2021a),)(2021b), Silva et al. (2021), Sun and Peng (2021), Wang et al. (2,Han et al. (2021), Sen et al. (2021),Ramesh et al. (2021), Suband Oh (2021),Davidson et al. (2019),May et al. (2019), SapPiper (2020), Guo et al. (2020), Vu et al. (2020), Spliethöver aMunro and Morrison (2020), Zhao and Chang (2020), Schmahl(2020), Chen et al. (2020b),Mulsa and Spanakis (2020), ShiCotterell(2020),Liang et al. (2020), Wich et al. (2020a), Jia et a(2020),Dev et al. (2021a), Kumar et al. (2020),Field and Tsvetk(2020a), Stafanovics et al. (2020),Li et al. (2020),Ma et al. (AlKuwatly et al. (2020), Bartl et al. (202O),Liu et al. (2020b), I,Sheng et al. (2020), Gaci et al. (2022), Hutchinson et al. (2020)et al. (2021e),Van Der Wal et al. (2022), Hansen and Spgaard (2Azarpanah and Farhadloo (2021), Jiao and Luo (2021), Touileb,Ciora et al. (2021), Murayama et al. (2021), Gaido et al. (202(2022), Dawkins (2021a) , Gillis (2021),Mireshghallah and BWang et al. (2021a),Malik et al. (2022), Subramanian et al. (202Garimella et al. (2021),Ghosh et al. (2021),Mehrabi et al. (20)
Both aboveGoldfarb-Tarrant et al. (2021), Janghorbani and De Melo (202Troles and Schmid (2021),Liu etal. (2022a),Névéol etal. (202
AnalysisHerzig et al. (2011), Rudinger et al. (2017), Lauscher and GlavaVanderlyn et al. (2021),Falenska and Cetinoglu (2021), Sap et(2021),Bhat et al. (2021), Haroutunian (2022), Qian et al. (201
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It is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 69, + 748, + 291, + 760 + ], + "spans": [ + { + "bbox": [ + 69, + 748, + 291, + 760 + ], + "score": 1.0, + "content": "well known that language data encodes demograph-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 69, + 761, + 290, + 774 + ], + "spans": [ + { + "bbox": [ + 69, + 761, + 290, + 774 + ], + "score": 1.0, + "content": "ics and biases (Bender and Friedman, 2018). 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There are also concerns that systems", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 304, + 295, + 525, + 307 + ], + "spans": [ + { + "bbox": [ + 304, + 295, + 525, + 307 + ], + "score": 1.0, + "content": "based on inappropriate data are likely to repeat such", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 304, + 308, + 525, + 321 + ], + "spans": [ + { + "bbox": [ + 304, + 308, + 525, + 321 + ], + "score": 1.0, + "content": "biases, and may even amplify them (Bender and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 304, + 322, + 527, + 335 + ], + "spans": [ + { + "bbox": [ + 304, + 322, + 527, + 335 + ], + "score": 1.0, + "content": "Friedman, 2018). In this paper, we survey 348 pa-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 336, + 526, + 348 + ], + "spans": [ + { + "bbox": [ + 304, + 336, + 526, + 348 + ], + "score": 1.0, + "content": "pers collected from the ACL anthology that focus", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 349, + 525, + 362 + ], + "spans": [ + { + "bbox": [ + 304, + 349, + 525, + 362 + ], + "score": 1.0, + "content": "on social bias and ethics in NLP research. We make", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 362, + 525, + 376 + ], + "spans": [ + { + "bbox": [ + 304, + 362, + 525, + 376 + ], + "score": 1.0, + "content": "three kinds of contributions, where (i) we present a", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 304, + 375, + 525, + 390 + ], + "spans": [ + { + "bbox": [ + 304, + 375, + 525, + 390 + ], + "score": 1.0, + "content": "working definition of social bias from a philosophy", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 303, + 389, + 526, + 404 + ], + "spans": [ + { + "bbox": [ + 303, + 389, + 526, + 404 + ], + "score": 1.0, + "content": "perspective, (ii) quantify our findings by annotating", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 404, + 525, + 416 + ], + "spans": [ + { + "bbox": [ + 304, + 404, + 525, + 416 + ], + "score": 1.0, + "content": "our corpus of papers and (iii) provide a discussion", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 417, + 527, + 429 + ], + "spans": [ + { + "bbox": [ + 304, + 417, + 527, + 429 + ], + "score": 1.0, + "content": "and pointers for possible future research directions.", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 304, + 430, + 526, + 444 + ], + "spans": [ + { + "bbox": [ + 304, + 430, + 526, + 444 + ], + "score": 1.0, + "content": "Through a quantitative analysis of current trends", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 304, + 444, + 507, + 457 + ], + "spans": [ + { + "bbox": [ + 304, + 444, + 507, + 457 + ], + "score": 1.0, + "content": "we attempt to answer the following questions:", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 53.5 + }, + { + "type": "text", + "bbox": [ + 317, + 469, + 525, + 523 + ], + "lines": [ + { + "bbox": [ + 317, + 468, + 527, + 482 + ], + "spans": [ + { + "bbox": [ + 317, + 468, + 527, + 482 + ], + "score": 1.0, + "content": "• What kind of social biases is the ACL commu-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 325, + 483, + 421, + 496 + ], + "spans": [ + { + "bbox": [ + 325, + 483, + 421, + 496 + ], + "score": 1.0, + "content": "nity concerned with?", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 317, + 495, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 317, + 495, + 505, + 510 + ], + "score": 1.0, + "content": "• How many languages is bias studied in?", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 317, + 509, + 477, + 524 + ], + "spans": [ + { + "bbox": [ + 317, + 509, + 477, + 524 + ], + "score": 1.0, + "content": "• What types of papers are present?", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 64.5 + }, + { + "type": "title", + "bbox": [ + 305, + 535, + 451, + 549 + ], + "lines": [ + { + "bbox": [ + 302, + 533, + 453, + 552 + ], + "spans": [ + { + "bbox": [ + 302, + 533, + 453, + 552 + ], + "score": 1.0, + "content": "2 Related previous surveys", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 67 + }, + { + "type": "text", + "bbox": [ + 305, + 557, + 526, + 665 + ], + "lines": [ + { + "bbox": [ + 304, + 557, + 525, + 569 + ], + "spans": [ + { + "bbox": [ + 304, + 557, + 525, + 569 + ], + "score": 1.0, + "content": "There is a considerable literature on social biases in", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "spans": [ + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "score": 1.0, + "content": "NLP. 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It is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 69, + 748, + 291, + 760 + ], + "spans": [ + { + "bbox": [ + 69, + 748, + 291, + 760 + ], + "score": 1.0, + "content": "well known that language data encodes demograph-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 69, + 761, + 290, + 774 + ], + "spans": [ + { + "bbox": [ + 69, + 761, + 290, + 774 + ], + "score": 1.0, + "content": "ics and biases (Bender and Friedman, 2018). There", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 304, + 212, + 527, + 227 + ], + "spans": [ + { + "bbox": [ + 304, + 212, + 527, + 227 + ], + "score": 1.0, + "content": "is a risk that using such data can disclose inappro-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 227, + 525, + 240 + ], + "spans": [ + { + "bbox": [ + 304, + 227, + 525, + 240 + ], + "score": 1.0, + "content": "priate information about particular individuals, as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 240, + 525, + 253 + ], + "spans": [ + { + "bbox": [ + 304, + 240, + 525, + 253 + ], + "score": 1.0, + "content": "well as undesirable attitudes towards individuals", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 254, + 525, + 267 + ], + "spans": [ + { + "bbox": [ + 304, + 254, + 525, + 267 + ], + "score": 1.0, + "content": "and groups (Hovy and Spruit, 2016; Eckert and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 304, + 267, + 525, + 280 + ], + "spans": [ + { + "bbox": [ + 304, + 267, + 525, + 280 + ], + "score": 1.0, + "content": "Rickford, 2001) and social hierarchies (Blodgett", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 304, + 281, + 525, + 294 + ], + "spans": [ + { + "bbox": [ + 304, + 281, + 525, + 294 + ], + "score": 1.0, + "content": "et al., 2020). There are also concerns that systems", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 304, + 295, + 525, + 307 + ], + "spans": [ + { + "bbox": [ + 304, + 295, + 525, + 307 + ], + "score": 1.0, + "content": "based on inappropriate data are likely to repeat such", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 304, + 308, + 525, + 321 + ], + "spans": [ + { + "bbox": [ + 304, + 308, + 525, + 321 + ], + "score": 1.0, + "content": "biases, and may even amplify them (Bender and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 304, + 322, + 527, + 335 + ], + "spans": [ + { + "bbox": [ + 304, + 322, + 527, + 335 + ], + "score": 1.0, + "content": "Friedman, 2018). In this paper, we survey 348 pa-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 336, + 526, + 348 + ], + "spans": [ + { + "bbox": [ + 304, + 336, + 526, + 348 + ], + "score": 1.0, + "content": "pers collected from the ACL anthology that focus", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 349, + 525, + 362 + ], + "spans": [ + { + "bbox": [ + 304, + 349, + 525, + 362 + ], + "score": 1.0, + "content": "on social bias and ethics in NLP research. We make", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 362, + 525, + 376 + ], + "spans": [ + { + "bbox": [ + 304, + 362, + 525, + 376 + ], + "score": 1.0, + "content": "three kinds of contributions, where (i) we present a", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 304, + 375, + 525, + 390 + ], + "spans": [ + { + "bbox": [ + 304, + 375, + 525, + 390 + ], + "score": 1.0, + "content": "working definition of social bias from a philosophy", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 303, + 389, + 526, + 404 + ], + "spans": [ + { + "bbox": [ + 303, + 389, + 526, + 404 + ], + "score": 1.0, + "content": "perspective, (ii) quantify our findings by annotating", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 404, + 525, + 416 + ], + "spans": [ + { + "bbox": [ + 304, + 404, + 525, + 416 + ], + "score": 1.0, + "content": "our corpus of papers and (iii) provide a discussion", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 417, + 527, + 429 + ], + "spans": [ + { + "bbox": [ + 304, + 417, + 527, + 429 + ], + "score": 1.0, + "content": "and pointers for possible future research directions.", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 304, + 430, + 526, + 444 + ], + "spans": [ + { + "bbox": [ + 304, + 430, + 526, + 444 + ], + "score": 1.0, + "content": "Through a quantitative analysis of current trends", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 304, + 444, + 507, + 457 + ], + "spans": [ + { + "bbox": [ + 304, + 444, + 507, + 457 + ], + "score": 1.0, + "content": "we attempt to answer the following questions:", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 35.5, + "bbox_fs": [ + 68, + 529, + 292, + 774 + ] + }, + { + "type": "text", + "bbox": [ + 305, + 213, + 525, + 457 + ], + "lines": [], + "index": 53.5, + "bbox_fs": [ + 303, + 212, + 527, + 457 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 317, + 469, + 525, + 523 + ], + "lines": [ + { + "bbox": [ + 317, + 468, + 527, + 482 + ], + "spans": [ + { + "bbox": [ + 317, + 468, + 527, + 482 + ], + "score": 1.0, + "content": "• What kind of social biases is the ACL commu-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 325, + 483, + 421, + 496 + ], + "spans": [ + { + "bbox": [ + 325, + 483, + 421, + 496 + ], + "score": 1.0, + "content": "nity concerned with?", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 317, + 495, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 317, + 495, + 505, + 510 + ], + "score": 1.0, + "content": "• How many languages is bias studied in?", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 317, + 509, + 477, + 524 + ], + "spans": [ + { + "bbox": [ + 317, + 509, + 477, + 524 + ], + "score": 1.0, + "content": "• What types of papers are present?", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 64.5, + "bbox_fs": [ + 317, + 468, + 527, + 524 + ] + }, + { + "type": "title", + "bbox": [ + 305, + 535, + 451, + 549 + ], + "lines": [ + { + "bbox": [ + 302, + 533, + 453, + 552 + ], + "spans": [ + { + "bbox": [ + 302, + 533, + 453, + 552 + ], + "score": 1.0, + "content": "2 Related previous surveys", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 67 + }, + { + "type": "text", + "bbox": [ + 305, + 557, + 526, + 665 + ], + "lines": [ + { + "bbox": [ + 304, + 557, + 525, + 569 + ], + "spans": [ + { + "bbox": [ + 304, + 557, + 525, + 569 + ], + "score": 1.0, + "content": "There is a considerable literature on social biases in", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "spans": [ + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "score": 1.0, + "content": "NLP. 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Our work follows the precedent", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 68, + 232, + 291, + 248 + ], + "spans": [ + { + "bbox": [ + 68, + 232, + 291, + 248 + ], + "score": 1.0, + "content": "established by Blodgett et al. 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Sim-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 69, + 288, + 290, + 301 + ], + "spans": [ + { + "bbox": [ + 69, + 288, + 290, + 301 + ], + "score": 1.0, + "content": "ilarly, we also align with Field et al. (2021) who", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 69, + 302, + 290, + 315 + ], + "spans": [ + { + "bbox": [ + 69, + 302, + 290, + 315 + ], + "score": 1.0, + "content": "solely focus on papers published at ACL venues", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 68, + 316, + 291, + 329 + ], + "spans": [ + { + "bbox": [ + 68, + 316, + 291, + 329 + ], + "score": 1.0, + "content": "but draw on conclusions from NLP papers pub-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 68, + 329, + 291, + 343 + ], + "spans": [ + { + "bbox": [ + 68, + 329, + 291, + 343 + ], + "score": 1.0, + "content": "lished elsewhere. 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One limitation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 68, + 431, + 291, + 445 + ], + "spans": [ + { + "bbox": [ + 68, + 431, + 291, + 445 + ], + "score": 1.0, + "content": "of such an approach is that seminal work published", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 69, + 446, + 291, + 459 + ], + "spans": [ + { + "bbox": [ + 69, + 446, + 291, + 459 + ], + "score": 1.0, + "content": "at other venues is not reviewed here. 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Based on this, defining bias involves", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 69, + 572, + 291, + 585 + ], + "spans": [ + { + "bbox": [ + 69, + 572, + 291, + 585 + ], + "score": 1.0, + "content": "an interaction among different components: (i) in-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 69, + 584, + 291, + 599 + ], + "spans": [ + { + "bbox": [ + 69, + 584, + 291, + 599 + ], + "score": 1.0, + "content": "dividuals or categories that determine a group, (ii)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 598, + 291, + 613 + ], + "spans": [ + { + "bbox": [ + 68, + 598, + 291, + 613 + ], + "score": 1.0, + "content": "attitudes towards this group, and (iii) assessment", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 69, + 612, + 290, + 624 + ], + "spans": [ + { + "bbox": [ + 69, + 612, + 290, + 624 + ], + "score": 1.0, + "content": "of this attitude in relation to fairness. 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For example, in in-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 69, + 653, + 290, + 667 + ], + "spans": [ + { + "bbox": [ + 69, + 653, + 290, + 667 + ], + "score": 1.0, + "content": "vestigating a disease that is more prevalent among", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 69, + 667, + 290, + 679 + ], + "spans": [ + { + "bbox": [ + 69, + 667, + 290, + 679 + ], + "score": 1.0, + "content": "women, use of gender is a relevant variable and, in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 69, + 680, + 290, + 692 + ], + "spans": [ + { + "bbox": [ + 69, + 680, + 290, + 692 + ], + "score": 1.0, + "content": "itself, does not entail a differential attitude. Once a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 69, + 693, + 290, + 707 + ], + "spans": [ + { + "bbox": [ + 69, + 693, + 290, + 707 + ], + "score": 1.0, + "content": "differential attitude is formed however, bias comes", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 69, + 707, + 291, + 720 + ], + "spans": [ + { + "bbox": [ + 69, + 707, + 291, + 720 + ], + "score": 1.0, + "content": "into existence and the attitude may be negative or", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 69, + 721, + 292, + 734 + ], + "spans": [ + { + "bbox": [ + 69, + 721, + 292, + 734 + ], + "score": 1.0, + "content": "positive. 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At the same time, many of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 68, + 302, + 291, + 316 + ], + "spans": [ + { + "bbox": [ + 68, + 302, + 291, + 316 + ], + "score": 1.0, + "content": "the methods looking at measuring or removing bias", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 67, + 315, + 291, + 329 + ], + "spans": [ + { + "bbox": [ + 67, + 315, + 291, + 329 + ], + "score": 1.0, + "content": "complicate the data and tasks at hand and can lead", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 67, + 329, + 291, + 343 + ], + "spans": [ + { + "bbox": [ + 67, + 329, + 291, + 343 + ], + "score": 1.0, + "content": "to the development of systems that are not reliable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 69, + 343, + 292, + 357 + ], + "spans": [ + { + "bbox": [ + 69, + 343, + 292, + 357 + ], + "score": 1.0, + "content": "when used in a more complex context (Talat et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 68, + 356, + 291, + 371 + ], + "spans": [ + { + "bbox": [ + 68, + 356, + 291, + 371 + ], + "score": 1.0, + "content": "2022b). 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There are a total of 34 languages,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 68, + 314, + 291, + 328 + ], + "spans": [ + { + "bbox": [ + 68, + 314, + 291, + 328 + ], + "score": 1.0, + "content": "however we leave out any languages that only oc-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 67, + 327, + 292, + 341 + ], + "spans": [ + { + "bbox": [ + 67, + 327, + 292, + 341 + ], + "score": 1.0, + "content": "cur once in the visualization. There are 11 lan-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 67, + 340, + 292, + 356 + ], + "spans": [ + { + "bbox": [ + 67, + 340, + 292, + 356 + ], + "score": 1.0, + "content": "guages not visualized, including Farsi, Urdu, Wolof,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 67, + 354, + 292, + 369 + ], + "spans": [ + { + "bbox": [ + 67, + 354, + 292, + 369 + ], + "score": 1.0, + "content": "Bengali, Armenian, Bengali, Inuktitut, Ukrainian,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 68, + 367, + 292, + 381 + ], + "spans": [ + { + "bbox": [ + 68, + 367, + 292, + 381 + ], + "score": 1.0, + "content": "Hungarian, Indonesian and Lithuanian. English,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 68, + 380, + 291, + 396 + ], + "spans": [ + { + "bbox": [ + 68, + 380, + 291, + 396 + ], + "score": 1.0, + "content": "German, Spanish and Chinese are most commonly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 68, + 394, + 291, + 408 + ], + "spans": [ + { + "bbox": [ + 68, + 394, + 291, + 408 + ], + "score": 1.0, + "content": "used either on their own or in combination with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 68, + 408, + 292, + 423 + ], + "spans": [ + { + "bbox": [ + 68, + 408, + 292, + 423 + ], + "score": 1.0, + "content": "each other. The majority of all papers focus on a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 68, + 421, + 292, + 437 + ], + "spans": [ + { + "bbox": [ + 68, + 421, + 292, + 437 + ], + "score": 1.0, + "content": "single language at a time. Furthermore, the vast ma-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 68, + 435, + 291, + 450 + ], + "spans": [ + { + "bbox": [ + 68, + 435, + 291, + 450 + ], + "score": 1.0, + "content": "jority of LLMs are monolingual and do not encode", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 68, + 448, + 291, + 463 + ], + "spans": [ + { + "bbox": [ + 68, + 448, + 291, + 463 + ], + "score": 1.0, + "content": "the cultural variety that naturally occurs within one", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 68, + 462, + 292, + 477 + ], + "spans": [ + { + "bbox": [ + 68, + 462, + 292, + 477 + ], + "score": 1.0, + "content": "language, for example non-standard English vari-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 68, + 475, + 292, + 491 + ], + "spans": [ + { + "bbox": [ + 68, + 475, + 292, + 491 + ], + "score": 1.0, + "content": "eties (Talat et al., 2022b). Therefore it is important", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 68, + 489, + 292, + 504 + ], + "spans": [ + { + "bbox": [ + 68, + 489, + 292, + 504 + ], + "score": 1.0, + "content": "to not only document the type of bias investigated,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 68, + 503, + 292, + 517 + ], + "spans": [ + { + "bbox": [ + 68, + 503, + 292, + 517 + ], + "score": 1.0, + "content": "but also contextualize bias within a language’s cul-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 68, + 516, + 292, + 531 + ], + "spans": [ + { + "bbox": [ + 68, + 516, + 292, + 531 + ], + "score": 1.0, + "content": "tural context, understanding of said bias and doc-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 68, + 530, + 292, + 544 + ], + "spans": [ + { + "bbox": [ + 68, + 530, + 292, + 544 + ], + "score": 1.0, + "content": "ument the language itself (Bender Rule (Bender,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 68, + 543, + 292, + 558 + ], + "spans": [ + { + "bbox": [ + 68, + 543, + 292, + 558 + ], + "score": 1.0, + "content": "2011)). Based on this collection, bias research is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 68, + 556, + 291, + 571 + ], + "spans": [ + { + "bbox": [ + 68, + 556, + 291, + 571 + ], + "score": 1.0, + "content": "heavily biased towards western and Anglo-centric", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 68, + 571, + 292, + 585 + ], + "spans": [ + { + "bbox": [ + 68, + 571, + 292, + 585 + ], + "score": 1.0, + "content": "notions of bias and very few works focus on non-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 584, + 291, + 598 + ], + "spans": [ + { + "bbox": [ + 68, + 584, + 291, + 598 + ], + "score": 1.0, + "content": "English benchmarks (Talat et al., 2022b; Hovy and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 68, + 598, + 291, + 612 + ], + "spans": [ + { + "bbox": [ + 68, + 598, + 291, + 612 + ], + "score": 1.0, + "content": "Spruit, 2016). This proves extremely problematic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 68, + 610, + 292, + 626 + ], + "spans": [ + { + "bbox": [ + 68, + 610, + 292, + 626 + ], + "score": 1.0, + "content": "when English benchmarks are automatically trans-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 68, + 625, + 292, + 639 + ], + "spans": [ + { + "bbox": [ + 68, + 625, + 292, + 639 + ], + "score": 1.0, + "content": "lated, but many of the biases do not hold true in non-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 68, + 638, + 292, + 653 + ], + "spans": [ + { + "bbox": [ + 68, + 638, + 292, + 653 + ], + "score": 1.0, + "content": "Western cultural contexts. For example, gendered", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 69, + 653, + 292, + 666 + ], + "spans": [ + { + "bbox": [ + 69, + 653, + 292, + 666 + ], + "score": 1.0, + "content": "professions do not necessarily translate across ev-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 67, + 666, + 292, + 680 + ], + "spans": [ + { + "bbox": [ + 67, + 666, + 292, + 680 + ], + "score": 1.0, + "content": "ery language or culture (Talat et al., 2022b) and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 66, + 677, + 293, + 696 + ], + "spans": [ + { + "bbox": [ + 66, + 677, + 293, + 696 + ], + "score": 1.0, + "content": "many NLP systems trained on written English (e.g.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 67, + 692, + 291, + 708 + ], + "spans": [ + { + "bbox": [ + 67, + 692, + 291, + 708 + ], + "score": 1.0, + "content": "Penn Tree Bank) do not perform well on non-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 68, + 706, + 291, + 720 + ], + "spans": [ + { + "bbox": [ + 68, + 706, + 291, + 720 + ], + "score": 1.0, + "content": "standard English (Mayfield et al., 2019). From", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 69, + 720, + 291, + 733 + ], + "spans": [ + { + "bbox": [ + 69, + 720, + 291, + 733 + ], + "score": 1.0, + "content": "Table 4 we can see the residuals between observed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 68, + 734, + 291, + 748 + ], + "spans": [ + { + "bbox": [ + 68, + 734, + 291, + 748 + ], + "score": 1.0, + "content": "joints showing a clear over-saturation (green) for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 68, + 746, + 292, + 762 + ], + "spans": [ + { + "bbox": [ + 68, + 746, + 292, + 762 + ], + "score": 1.0, + "content": "specific combinations of languages (e.g: English", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 68, + 761, + 236, + 775 + ], + "spans": [ + { + "bbox": [ + 68, + 761, + 236, + 775 + ], + "score": 1.0, + "content": "and German or English and Spanish).", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 305, + 71, + 378, + 84 + ], + "lines": [ + { + "bbox": [ + 302, + 69, + 379, + 87 + ], + "spans": [ + { + "bbox": [ + 302, + 69, + 379, + 87 + ], + "score": 1.0, + "content": "6 Discussion", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 305, + 92, + 526, + 200 + ], + "lines": [ + { + "bbox": [ + 304, + 92, + 526, + 105 + ], + "spans": [ + { + "bbox": [ + 304, + 92, + 526, + 105 + ], + "score": 1.0, + "content": "Datasets and Benchmarks Previous work", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 305, + 106, + 527, + 119 + ], + "spans": [ + { + "bbox": [ + 305, + 106, + 527, + 119 + ], + "score": 1.0, + "content": "(Hovy and Spruit, 2016; Hovy, 2018; Talat et al.,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 118, + 527, + 133 + ], + "spans": [ + { + "bbox": [ + 304, + 118, + 527, + 133 + ], + "score": 1.0, + "content": "2022b) outlined a number of reasons that may ex-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 134, + 525, + 146 + ], + "spans": [ + { + "bbox": [ + 304, + 134, + 525, + 146 + ], + "score": 1.0, + "content": "plain why there are so many papers focusing on the", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 146, + 527, + 160 + ], + "spans": [ + { + "bbox": [ + 304, + 146, + 527, + 160 + ], + "score": 1.0, + "content": "same datasets, benchmarks, biases, and languages.", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 303, + 159, + 526, + 174 + ], + "spans": [ + { + "bbox": [ + 303, + 159, + 526, + 174 + ], + "score": 1.0, + "content": "In the following section, we highlight some of the", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 304, + 174, + 525, + 187 + ], + "spans": [ + { + "bbox": [ + 304, + 174, + 525, + 187 + ], + "score": 1.0, + "content": "elements that may explain why there is an uneven", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 187, + 459, + 200 + ], + "spans": [ + { + "bbox": [ + 304, + 187, + 459, + 200 + ], + "score": 1.0, + "content": "distribution of work and resources.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 55.5 + }, + { + "type": "text", + "bbox": [ + 326, + 204, + 525, + 736 + ], + "lines": [ + { + "bbox": [ + 325, + 209, + 527, + 223 + ], + "spans": [ + { + "bbox": [ + 325, + 209, + 527, + 223 + ], + "score": 1.0, + "content": "Experimental setup The majority of of work", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 324, + 223, + 527, + 238 + ], + "spans": [ + { + "bbox": [ + 324, + 223, + 527, + 238 + ], + "score": 1.0, + "content": "in bias research has focused on using intrin-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 325, + 237, + 526, + 250 + ], + "spans": [ + { + "bbox": [ + 325, + 237, + 526, + 250 + ], + "score": 1.0, + "content": "sic bias measurements (bias in internal model", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 325, + 250, + 526, + 265 + ], + "spans": [ + { + "bbox": [ + 325, + 250, + 526, + 265 + ], + "score": 1.0, + "content": "representations) and little attention has been", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 325, + 264, + 527, + 278 + ], + "spans": [ + { + "bbox": [ + 325, + 264, + 527, + 278 + ], + "score": 1.0, + "content": "paid to extrinsic metrics (Orgad and Belinkov,", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 325, + 276, + 527, + 292 + ], + "spans": [ + { + "bbox": [ + 325, + 276, + 527, + 292 + ], + "score": 1.0, + "content": "2022; Delobelle et al., 2022). A very real", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 325, + 291, + 525, + 304 + ], + "spans": [ + { + "bbox": [ + 325, + 291, + 525, + 304 + ], + "score": 1.0, + "content": "consequence of this is that much work does", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 324, + 305, + 527, + 318 + ], + "spans": [ + { + "bbox": [ + 324, + 305, + 527, + 318 + ], + "score": 1.0, + "content": "not appropriately describe, contextualize and", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 325, + 317, + 527, + 333 + ], + "spans": [ + { + "bbox": [ + 325, + 317, + 527, + 333 + ], + "score": 1.0, + "content": "identify the potential harms that bias has in", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 325, + 331, + 527, + 346 + ], + "spans": [ + { + "bbox": [ + 325, + 331, + 527, + 346 + ], + "score": 1.0, + "content": "real-world scenarios (Blodgett et al., 2020).", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 325, + 345, + 527, + 360 + ], + "spans": [ + { + "bbox": [ + 325, + 345, + 527, + 360 + ], + "score": 1.0, + "content": "Also bias is inadvertently introduced in intrin-", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 325, + 360, + 527, + 372 + ], + "spans": [ + { + "bbox": [ + 325, + 360, + 527, + 372 + ], + "score": 1.0, + "content": "sic metrics, where lexicons used to measure", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 324, + 371, + 528, + 387 + ], + "spans": [ + { + "bbox": [ + 324, + 371, + 528, + 387 + ], + "score": 1.0, + "content": "bias in one dataset produces very different re-", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 325, + 385, + 527, + 400 + ], + "spans": [ + { + "bbox": [ + 325, + 385, + 527, + 400 + ], + "score": 1.0, + "content": "sults in another (Antoniak and Mimno, 2021).", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 326, + 400, + 526, + 413 + ], + "spans": [ + { + "bbox": [ + 326, + 400, + 526, + 413 + ], + "score": 1.0, + "content": "Similarly, (Goldfarb-Tarrant et al., 2021) have", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 326, + 414, + 526, + 427 + ], + "spans": [ + { + "bbox": [ + 326, + 414, + 526, + 427 + ], + "score": 1.0, + "content": "found that there is no correlation between in-", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 325, + 425, + 526, + 442 + ], + "spans": [ + { + "bbox": [ + 325, + 425, + 526, + 442 + ], + "score": 1.0, + "content": "trinsic and extrinsic metrics. Another key", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 325, + 440, + 527, + 455 + ], + "spans": [ + { + "bbox": [ + 325, + 440, + 527, + 455 + ], + "score": 1.0, + "content": "problem is that often newly proposed datasets", + "type": "text" + } + ], + "index": 77 + }, + { + "bbox": [ + 325, + 455, + 527, + 467 + ], + "spans": [ + { + "bbox": [ + 325, + 455, + 527, + 467 + ], + "score": 1.0, + "content": "are linked to specific metrics, which makes it", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 324, + 466, + 527, + 482 + ], + "spans": [ + { + "bbox": [ + 324, + 466, + 527, + 482 + ], + "score": 1.0, + "content": "hard to draw conclusions from individual case", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 326, + 481, + 526, + 494 + ], + "spans": [ + { + "bbox": [ + 326, + 481, + 526, + 494 + ], + "score": 1.0, + "content": "studies as many results are not generalizable", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 326, + 495, + 526, + 508 + ], + "spans": [ + { + "bbox": [ + 326, + 495, + 526, + 508 + ], + "score": 1.0, + "content": "(Orgad and Belinkov, 2022). Another element", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 326, + 509, + 526, + 522 + ], + "spans": [ + { + "bbox": [ + 326, + 509, + 526, + 522 + ], + "score": 1.0, + "content": "impacting metrics is the composition of test", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 326, + 522, + 526, + 535 + ], + "spans": [ + { + "bbox": [ + 326, + 522, + 526, + 535 + ], + "score": 1.0, + "content": "data, where (Orgad and Belinkov, 2022) found", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 325, + 534, + 528, + 550 + ], + "spans": [ + { + "bbox": [ + 325, + 534, + 528, + 550 + ], + "score": 1.0, + "content": "that test sets often don’t contain balanced ex-", + "type": "text" + } + ], + "index": 84 + }, + { + "bbox": [ + 326, + 549, + 526, + 562 + ], + "spans": [ + { + "bbox": [ + 326, + 549, + 526, + 562 + ], + "score": 1.0, + "content": "amples. However, most metrics are defined", + "type": "text" + } + ], + "index": 85 + }, + { + "bbox": [ + 325, + 562, + 528, + 577 + ], + "spans": [ + { + "bbox": [ + 325, + 562, + 528, + 577 + ], + "score": 1.0, + "content": "over a whole dataset and are therefore sen-", + "type": "text" + } + ], + "index": 86 + }, + { + "bbox": [ + 326, + 576, + 526, + 589 + ], + "spans": [ + { + "bbox": [ + 326, + 576, + 526, + 589 + ], + "score": 1.0, + "content": "sitive to its composition, which may lead to", + "type": "text" + } + ], + "index": 87 + }, + { + "bbox": [ + 326, + 590, + 526, + 603 + ], + "spans": [ + { + "bbox": [ + 326, + 590, + 526, + 603 + ], + "score": 1.0, + "content": "variability in results. Both metric choice and", + "type": "text" + } + ], + "index": 88 + }, + { + "bbox": [ + 325, + 602, + 526, + 618 + ], + "spans": [ + { + "bbox": [ + 325, + 602, + 526, + 618 + ], + "score": 1.0, + "content": "dataset composition can significantly change", + "type": "text" + } + ], + "index": 89 + }, + { + "bbox": [ + 326, + 617, + 527, + 630 + ], + "spans": [ + { + "bbox": [ + 326, + 617, + 527, + 630 + ], + "score": 1.0, + "content": "the results and conclusions drawn from a", + "type": "text" + } + ], + "index": 90 + }, + { + "bbox": [ + 325, + 628, + 527, + 645 + ], + "spans": [ + { + "bbox": [ + 325, + 628, + 527, + 645 + ], + "score": 1.0, + "content": "downstream task or dataset (Akyürek et al.,", + "type": "text" + } + ], + "index": 91 + }, + { + "bbox": [ + 325, + 644, + 526, + 658 + ], + "spans": [ + { + "bbox": [ + 325, + 644, + 526, + 658 + ], + "score": 1.0, + "content": "2022a). 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Based on this collection, bias research is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 68, + 556, + 291, + 571 + ], + "spans": [ + { + "bbox": [ + 68, + 556, + 291, + 571 + ], + "score": 1.0, + "content": "heavily biased towards western and Anglo-centric", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 68, + 571, + 292, + 585 + ], + "spans": [ + { + "bbox": [ + 68, + 571, + 292, + 585 + ], + "score": 1.0, + "content": "notions of bias and very few works focus on non-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 584, + 291, + 598 + ], + "spans": [ + { + "bbox": [ + 68, + 584, + 291, + 598 + ], + "score": 1.0, + "content": "English benchmarks (Talat et al., 2022b; Hovy and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 68, + 598, + 291, + 612 + ], + "spans": [ + { + "bbox": [ + 68, + 598, + 291, + 612 + ], + "score": 1.0, + "content": "Spruit, 2016). 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For example, gendered", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 69, + 653, + 292, + 666 + ], + "spans": [ + { + "bbox": [ + 69, + 653, + 292, + 666 + ], + "score": 1.0, + "content": "professions do not necessarily translate across ev-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 67, + 666, + 292, + 680 + ], + "spans": [ + { + "bbox": [ + 67, + 666, + 292, + 680 + ], + "score": 1.0, + "content": "ery language or culture (Talat et al., 2022b) and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 66, + 677, + 293, + 696 + ], + "spans": [ + { + "bbox": [ + 66, + 677, + 293, + 696 + ], + "score": 1.0, + "content": "many NLP systems trained on written English (e.g.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 67, + 692, + 291, + 708 + ], + "spans": [ + { + "bbox": [ + 67, + 692, + 291, + 708 + ], + "score": 1.0, + "content": "Penn Tree Bank) do not perform well on non-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 68, + 706, + 291, + 720 + ], + "spans": [ + { + "bbox": [ + 68, + 706, + 291, + 720 + ], + "score": 1.0, + "content": "standard English (Mayfield et al., 2019). From", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 69, + 720, + 291, + 733 + ], + "spans": [ + { + "bbox": [ + 69, + 720, + 291, + 733 + ], + "score": 1.0, + "content": "Table 4 we can see the residuals between observed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 68, + 734, + 291, + 748 + ], + "spans": [ + { + "bbox": [ + 68, + 734, + 291, + 748 + ], + "score": 1.0, + "content": "joints showing a clear over-saturation (green) for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 68, + 746, + 292, + 762 + ], + "spans": [ + { + "bbox": [ + 68, + 746, + 292, + 762 + ], + "score": 1.0, + "content": "specific combinations of languages (e.g: English", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 68, + 761, + 236, + 775 + ], + "spans": [ + { + "bbox": [ + 68, + 761, + 236, + 775 + ], + "score": 1.0, + "content": "and German or English and Spanish).", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 32, + "bbox_fs": [ + 66, + 273, + 293, + 775 + ] + }, + { + "type": "title", + "bbox": [ + 305, + 71, + 378, + 84 + ], + "lines": [ + { + "bbox": [ + 302, + 69, + 379, + 87 + ], + "spans": [ + { + "bbox": [ + 302, + 69, + 379, + 87 + ], + "score": 1.0, + "content": "6 Discussion", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 305, + 92, + 526, + 200 + ], + "lines": [ + { + "bbox": [ + 304, + 92, + 526, + 105 + ], + "spans": [ + { + "bbox": [ + 304, + 92, + 526, + 105 + ], + "score": 1.0, + "content": "Datasets and Benchmarks Previous work", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 305, + 106, + 527, + 119 + ], + "spans": [ + { + "bbox": [ + 305, + 106, + 527, + 119 + ], + "score": 1.0, + "content": "(Hovy and Spruit, 2016; Hovy, 2018; Talat et al.,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 118, + 527, + 133 + ], + "spans": [ + { + "bbox": [ + 304, + 118, + 527, + 133 + ], + "score": 1.0, + "content": "2022b) outlined a number of reasons that may ex-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 134, + 525, + 146 + ], + "spans": [ + { + "bbox": [ + 304, + 134, + 525, + 146 + ], + "score": 1.0, + "content": "plain why there are so many papers focusing on the", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 146, + 527, + 160 + ], + "spans": [ + { + "bbox": [ + 304, + 146, + 527, + 160 + ], + "score": 1.0, + "content": "same datasets, benchmarks, biases, and languages.", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 303, + 159, + 526, + 174 + ], + "spans": [ + { + "bbox": [ + 303, + 159, + 526, + 174 + ], + "score": 1.0, + "content": "In the following section, we highlight some of the", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 304, + 174, + 525, + 187 + ], + "spans": [ + { + "bbox": [ + 304, + 174, + 525, + 187 + ], + "score": 1.0, + "content": "elements that may explain why there is an uneven", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 187, + 459, + 200 + ], + "spans": [ + { + "bbox": [ + 304, + 187, + 459, + 200 + ], + "score": 1.0, + "content": "distribution of work and resources.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 55.5, + "bbox_fs": [ + 303, + 92, + 527, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 326, + 204, + 525, + 736 + ], + "lines": [ + { + "bbox": [ + 325, + 209, + 527, + 223 + ], + "spans": [ + { + "bbox": [ + 325, + 209, + 527, + 223 + ], + "score": 1.0, + "content": "Experimental setup The majority of of work", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 324, + 223, + 527, + 238 + ], + "spans": [ + { + "bbox": [ + 324, + 223, + 527, + 238 + ], + "score": 1.0, + "content": "in bias research has focused on using intrin-", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 325, + 237, + 526, + 250 + ], + "spans": [ + { + "bbox": [ + 325, + 237, + 526, + 250 + ], + "score": 1.0, + "content": "sic bias measurements (bias in internal model", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 325, + 250, + 526, + 265 + ], + "spans": [ + { + "bbox": [ + 325, + 250, + 526, + 265 + ], + "score": 1.0, + "content": "representations) and little attention has been", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 325, + 264, + 527, + 278 + ], + "spans": [ + { + "bbox": [ + 325, + 264, + 527, + 278 + ], + "score": 1.0, + "content": "paid to extrinsic metrics (Orgad and Belinkov,", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 325, + 276, + 527, + 292 + ], + "spans": [ + { + "bbox": [ + 325, + 276, + 527, + 292 + ], + "score": 1.0, + "content": "2022; Delobelle et al., 2022). A very real", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 325, + 291, + 525, + 304 + ], + "spans": [ + { + "bbox": [ + 325, + 291, + 525, + 304 + ], + "score": 1.0, + "content": "consequence of this is that much work does", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 324, + 305, + 527, + 318 + ], + "spans": [ + { + "bbox": [ + 324, + 305, + 527, + 318 + ], + "score": 1.0, + "content": "not appropriately describe, contextualize and", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 325, + 317, + 527, + 333 + ], + "spans": [ + { + "bbox": [ + 325, + 317, + 527, + 333 + ], + "score": 1.0, + "content": "identify the potential harms that bias has in", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 325, + 331, + 527, + 346 + ], + "spans": [ + { + "bbox": [ + 325, + 331, + 527, + 346 + ], + "score": 1.0, + "content": "real-world scenarios (Blodgett et al., 2020).", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 325, + 345, + 527, + 360 + ], + "spans": [ + { + "bbox": [ + 325, + 345, + 527, + 360 + ], + "score": 1.0, + "content": "Also bias is inadvertently introduced in intrin-", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 325, + 360, + 527, + 372 + ], + "spans": [ + { + "bbox": [ + 325, + 360, + 527, + 372 + ], + "score": 1.0, + "content": "sic metrics, where lexicons used to measure", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 324, + 371, + 528, + 387 + ], + "spans": [ + { + "bbox": [ + 324, + 371, + 528, + 387 + ], + "score": 1.0, + "content": "bias in one dataset produces very different re-", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 325, + 385, + 527, + 400 + ], + "spans": [ + { + "bbox": [ + 325, + 385, + 527, + 400 + ], + "score": 1.0, + "content": "sults in another (Antoniak and Mimno, 2021).", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 326, + 400, + 526, + 413 + ], + "spans": [ + { + "bbox": [ + 326, + 400, + 526, + 413 + ], + "score": 1.0, + "content": "Similarly, (Goldfarb-Tarrant et al., 2021) have", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 326, + 414, + 526, + 427 + ], + "spans": [ + { + "bbox": [ + 326, + 414, + 526, + 427 + ], + "score": 1.0, + "content": "found that there is no correlation between in-", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 325, + 425, + 526, + 442 + ], + "spans": [ + { + "bbox": [ + 325, + 425, + 526, + 442 + ], + "score": 1.0, + "content": "trinsic and extrinsic metrics. Another key", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 325, + 440, + 527, + 455 + ], + "spans": [ + { + "bbox": [ + 325, + 440, + 527, + 455 + ], + "score": 1.0, + "content": "problem is that often newly proposed datasets", + "type": "text" + } + ], + "index": 77 + }, + { + "bbox": [ + 325, + 455, + 527, + 467 + ], + "spans": [ + { + "bbox": [ + 325, + 455, + 527, + 467 + ], + "score": 1.0, + "content": "are linked to specific metrics, which makes it", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 324, + 466, + 527, + 482 + ], + "spans": [ + { + "bbox": [ + 324, + 466, + 527, + 482 + ], + "score": 1.0, + "content": "hard to draw conclusions from individual case", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 326, + 481, + 526, + 494 + ], + "spans": [ + { + "bbox": [ + 326, + 481, + 526, + 494 + ], + "score": 1.0, + "content": "studies as many results are not generalizable", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 326, + 495, + 526, + 508 + ], + "spans": [ + { + "bbox": [ + 326, + 495, + 526, + 508 + ], + "score": 1.0, + "content": "(Orgad and Belinkov, 2022). Another element", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 326, + 509, + 526, + 522 + ], + "spans": [ + { + "bbox": [ + 326, + 509, + 526, + 522 + ], + "score": 1.0, + "content": "impacting metrics is the composition of test", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 326, + 522, + 526, + 535 + ], + "spans": [ + { + "bbox": [ + 326, + 522, + 526, + 535 + ], + "score": 1.0, + "content": "data, where (Orgad and Belinkov, 2022) found", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 325, + 534, + 528, + 550 + ], + "spans": [ + { + "bbox": [ + 325, + 534, + 528, + 550 + ], + "score": 1.0, + "content": "that test sets often don’t contain balanced ex-", + "type": "text" + } + ], + "index": 84 + }, + { + "bbox": [ + 326, + 549, + 526, + 562 + ], + "spans": [ + { + "bbox": [ + 326, + 549, + 526, + 562 + ], + "score": 1.0, + "content": "amples. However, most metrics are defined", + "type": "text" + } + ], + "index": 85 + }, + { + "bbox": [ + 325, + 562, + 528, + 577 + ], + "spans": [ + { + "bbox": [ + 325, + 562, + 528, + 577 + ], + "score": 1.0, + "content": "over a whole dataset and are therefore sen-", + "type": "text" + } + ], + "index": 86 + }, + { + "bbox": [ + 326, + 576, + 526, + 589 + ], + "spans": [ + { + "bbox": [ + 326, + 576, + 526, + 589 + ], + "score": 1.0, + "content": "sitive to its composition, which may lead to", + "type": "text" + } + ], + "index": 87 + }, + { + "bbox": [ + 326, + 590, + 526, + 603 + ], + "spans": [ + { + "bbox": [ + 326, + 590, + 526, + 603 + ], + "score": 1.0, + "content": "variability in results. Both metric choice and", + "type": "text" + } + ], + "index": 88 + }, + { + "bbox": [ + 325, + 602, + 526, + 618 + ], + "spans": [ + { + "bbox": [ + 325, + 602, + 526, + 618 + ], + "score": 1.0, + "content": "dataset composition can significantly change", + "type": "text" + } + ], + "index": 89 + }, + { + "bbox": [ + 326, + 617, + 527, + 630 + ], + "spans": [ + { + "bbox": [ + 326, + 617, + 527, + 630 + ], + "score": 1.0, + "content": "the results and conclusions drawn from a", + "type": "text" + } + ], + "index": 90 + }, + { + "bbox": [ + 325, + 628, + 527, + 645 + ], + "spans": [ + { + "bbox": [ + 325, + 628, + 527, + 645 + ], + "score": 1.0, + "content": "downstream task or dataset (Akyürek et al.,", + "type": "text" + } + ], + "index": 91 + }, + { + "bbox": [ + 325, + 644, + 526, + 658 + ], + "spans": [ + { + "bbox": [ + 325, + 644, + 526, + 658 + ], + "score": 1.0, + "content": "2022a). 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This", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 304, + 410, + 527, + 422 + ], + "spans": [ + { + "bbox": [ + 304, + 410, + 527, + 422 + ], + "score": 1.0, + "content": "makes it incredibly important to foster collabora-", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 304, + 424, + 525, + 436 + ], + "spans": [ + { + "bbox": [ + 304, + 424, + 525, + 436 + ], + "score": 1.0, + "content": "tion between disciplines to ensure that historical", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 304, + 437, + 525, + 449 + ], + "spans": [ + { + "bbox": [ + 304, + 437, + 525, + 449 + ], + "score": 1.0, + "content": "inequalities and biases are taken into consideration", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 303, + 449, + 526, + 464 + ], + "spans": [ + { + "bbox": [ + 303, + 449, + 526, + 464 + ], + "score": 1.0, + "content": "when building new algorithms or systems (Caliskan", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 304, + 464, + 362, + 476 + ], + "spans": [ + { + "bbox": [ + 304, + 464, + 362, + 476 + ], + "score": 1.0, + "content": "et al., 2017).", + "type": "text" + } + ], + "index": 77 + } + ], + "index": 70 + }, + { + "type": "text", + "bbox": [ + 305, + 503, + 525, + 773 + ], + "lines": [ + { + "bbox": [ + 304, + 502, + 527, + 518 + ], + "spans": [ + { + "bbox": [ + 304, + 502, + 527, + 518 + ], + "score": 1.0, + "content": "Diversity Given the real-life impact of NLP sys-", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 304, + 518, + 526, + 530 + ], + "spans": [ + { + "bbox": [ + 304, + 518, + 526, + 530 + ], + "score": 1.0, + "content": "tems and research on people, there is not just a", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 304, + 531, + 527, + 543 + ], + "spans": [ + { + "bbox": [ + 304, + 531, + 527, + 543 + ], + "score": 1.0, + "content": "need for diversity in experts working on such sys-", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 304, + 545, + 525, + 557 + ], + "spans": [ + { + "bbox": [ + 304, + 545, + 525, + 557 + ], + "score": 1.0, + "content": "tems (Caliskan et al., 2017), but also a need for", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 303, + 558, + 525, + 570 + ], + "spans": [ + { + "bbox": [ + 303, + 558, + 525, + 570 + ], + "score": 1.0, + "content": "practitioners and researchers to engage with the", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "spans": [ + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "score": 1.0, + "content": "affected communities and stakeholders (Blodgett", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 304, + 586, + 525, + 597 + ], + "spans": [ + { + "bbox": [ + 304, + 586, + 525, + 597 + ], + "score": 1.0, + "content": "et al., 2020; Fortuna et al., 2021). 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This", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 304, + 410, + 527, + 422 + ], + "spans": [ + { + "bbox": [ + 304, + 410, + 527, + 422 + ], + "score": 1.0, + "content": "makes it incredibly important to foster collabora-", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 304, + 424, + 525, + 436 + ], + "spans": [ + { + "bbox": [ + 304, + 424, + 525, + 436 + ], + "score": 1.0, + "content": "tion between disciplines to ensure that historical", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 304, + 437, + 525, + 449 + ], + "spans": [ + { + "bbox": [ + 304, + 437, + 525, + 449 + ], + "score": 1.0, + "content": "inequalities and biases are taken into consideration", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 303, + 449, + 526, + 464 + ], + "spans": [ + { + "bbox": [ + 303, + 449, + 526, + 464 + ], + "score": 1.0, + "content": "when building new algorithms or systems (Caliskan", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 304, + 464, + 362, + 476 + ], + "spans": [ + { + "bbox": [ + 304, + 464, + 362, + 476 + ], + "score": 1.0, + "content": "et al., 2017).", + "type": "text" + } + ], + "index": 77 + } + ], + "index": 70, + "bbox_fs": [ + 303, + 273, + 527, + 476 + ] + }, + { + "type": "text", + "bbox": [ + 305, + 503, + 525, + 773 + ], + "lines": [ + { + "bbox": [ + 304, + 502, + 527, + 518 + ], + "spans": [ + { + "bbox": [ + 304, + 502, + 527, + 518 + ], + "score": 1.0, + "content": "Diversity Given the real-life impact of NLP sys-", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 304, + 518, + 526, + 530 + ], + "spans": [ + { + "bbox": [ + 304, + 518, + 526, + 530 + ], + "score": 1.0, + "content": "tems and research on people, there is not just a", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 304, + 531, + 527, + 543 + ], + "spans": [ + { + "bbox": [ + 304, + 531, + 527, + 543 + ], + "score": 1.0, + "content": "need for diversity in experts working on such sys-", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 304, + 545, + 525, + 557 + ], + "spans": [ + { + "bbox": [ + 304, + 545, + 525, + 557 + ], + "score": 1.0, + "content": "tems (Caliskan et al., 2017), but also a need for", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 303, + 558, + 525, + 570 + ], + "spans": [ + { + "bbox": [ + 303, + 558, + 525, + 570 + ], + "score": 1.0, + "content": "practitioners and researchers to engage with the", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "spans": [ + { + "bbox": [ + 304, + 571, + 525, + 584 + ], + "score": 1.0, + "content": "affected communities and stakeholders (Blodgett", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 304, + 586, + 525, + 597 + ], + "spans": [ + { + "bbox": [ + 304, + 586, + 525, + 597 + ], + "score": 1.0, + "content": "et al., 2020; Fortuna et al., 2021). 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We only used", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 69, + 175, + 290, + 187 + ], + "spans": [ + { + "bbox": [ + 69, + 175, + 290, + 187 + ], + "score": 1.0, + "content": "open-access papers in this collection and no human", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 68, + 188, + 291, + 201 + ], + "spans": [ + { + "bbox": [ + 68, + 188, + 291, + 201 + ], + "score": 1.0, + "content": "participants were involved in this work. Tradition-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 69, + 201, + 290, + 214 + ], + "spans": [ + { + "bbox": [ + 69, + 201, + 290, + 214 + ], + "score": 1.0, + "content": "ally, social biases have been investigated in fields", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 69, + 215, + 290, + 228 + ], + "spans": [ + { + "bbox": [ + 69, + 215, + 290, + 228 + ], + "score": 1.0, + "content": "such as social sciences, law, or psychology which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 68, + 228, + 290, + 241 + ], + "spans": [ + { + "bbox": [ + 68, + 228, + 290, + 241 + ], + "score": 1.0, + "content": "we have not discussed here. 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We are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 68, + 295, + 292, + 311 + ], + "spans": [ + { + "bbox": [ + 68, + 295, + 292, + 311 + ], + "score": 1.0, + "content": "also limited by the resource of the reviewed papers,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 69, + 309, + 290, + 323 + ], + "spans": [ + { + "bbox": [ + 69, + 309, + 290, + 323 + ], + "score": 1.0, + "content": "where substantial contributions to the field have", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 68, + 322, + 290, + 337 + ], + "spans": [ + { + "bbox": [ + 68, + 322, + 290, + 337 + ], + "score": 1.0, + "content": "been made outside of ACL venues. 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PaperSurvey/PositionDrugan and Babych (2010), Masthoff(2011), Escartin et al. (2(2016), Hovy and Spruit (2O16), Leidner and Plachouras (2017),et al.(2017), Schnoebelen (2017),Larson (2017),De Jong et a(2018),Mayfield et al. (2019),Sun et al. (2019), Blodgett et al.(2021), Bender et al. (2020), Schoch et al. (2020), Savoldi et (2021), Mishra et al. (2021), Czarnowska et al. (2021), DoughSheng et al. (2021), Hovy and Yang (2021), Santy et al. (2021)(2022b), Chandrabose et al. (2021), Vecchi et al. (2021),Talat et :Balkir et al. (2022), Orgad and Belinkov (2022), Akyurek et al(2022), Mohammad (2022a), Orr et al. (2022), Benotti and Blaet al.(2023),Benotti et al. (2023), Ramesh et al. (2023), Rios el
Dataset,BenchmarkYano et al. (2010), Flekova et al. (2016), Braun et al. (2016) KRudinger et al. (2018), Zhao et al. (2018), Kiritchenko and Mo(2019), Gaut et al. (202O),Friedman et al. (2019), Hitti et al. ((2019), Fan et al. (2019),Liu et al. (2020a), Sap et al. (2020),Louet al. (2021), Lim et al. (2020), Shahid et al. (2020), LazaridoUthus (2020), Gala et al. (2020),Dinan et al. (2020b), Meyer e(2020b), Gangula et al. (2019), Zueva et al. (2020), Nangia et al.Blodgett et al. (2021), Spinde et al. (2021), Zhong et al. (2021),Spgaard (2021), Zhao et al. (2021), Aksenov et al. (2021), KamRenduchintala et al. (2021), Barikeri et al. (2021),Milkowski et :et al. (2022), Ziems et al. (2022), Zhou et al. (2022),Akyirek(2022), Bansal et al. (2022),Smith et al. (2022), Qian et al. (2022et al. (2023), Madnani et al. (2017), Kumar and Bhotia (2020)(2021), Zad et al. (2021), Nozza et al. (2022),Mechura (2022)
MethodsGonen and Goldberg (2019), Wevers (2019), Hovy et al. (2020Nissim et al. (2020),Wang etal. (2021d),Wolfe and Caliskan (2et al. (2021), Herold et al. (2022),Bertsch et al. (2022), RampCao et al. (2022a),Alshahrani et al. (2022) Daems and Hackenbuet al. (2015), Terkik et al. (2016), Tatman (2017b),Wang et alTatman (2017a), Zhao et al. (2017), Long et al. (2018), BalyKloppenburg (2019), Farber et al. (2019), Zhong et al. (2019),and Ungar (2019), Gangula et al. (2019), Karve et al. (2019),and Najafian (20i9), Chaloner and Maldonado (2019), Qian et(2019),Kaneko and Bollegala (2019), Huang et al. (2020a),BDinan et al. (2020a), Bordia and Bowman (2O19),Manzini et al, Schwertmann et al. (2023), Pedersen et al. (2023), Bauer et al. (Kumar et al. (2023),Touileb et al. (2023), Parmar et al. (2022),et al. (2023),Alemany et al. (2023), Martinkovä et al. (2023),(2022), Cao et al. (2022b), Sap et al. (2020), Xia et al. (2020(2022), Orgad et al. (2022), Sen et al. (2022), Gupta et al. (202Gira et al. (2022), Guo et al. (2022), Wang et al. (2022), An et a(2022), Borchers et al. (2022),Touileb et al. (2022), Jentzsch andet al. (2022), Tal et al. (2022),Li et al. (2022b),Limisiewicz aret al. (2022),Ahn et al. (2022), Joniak and Aizawa (2022), Chen(2022), Du et al. (2022),Agrawal et al. (2022), Sesari et al. (0Steed et al. (2022), Schick et al. (2021), Zhou et al. (2021), Kaneetal.(2021),Lucy and Bamman (2021), Jin et al.(2021a),)(2021b), Silva et al. (2021), Sun and Peng (2021), Wang et al. (2,Han et al. (2021), Sen et al. (2021),Ramesh et al. (2021), Suband Oh (2021),Davidson et al. (2019),May et al. (2019), SapPiper (2020), Guo et al. (2020), Vu et al. (2020), Spliethöver aMunro and Morrison (2020), Zhao and Chang (2020), Schmahl(2020), Chen et al. (2020b),Mulsa and Spanakis (2020), ShiCotterell(2020),Liang et al. (2020), Wich et al. (2020a), Jia et a(2020),Dev et al. (2021a), Kumar et al. (2020),Field and Tsvetk(2020a), Stafanovics et al. (2020),Li et al. (2020),Ma et al. (AlKuwatly et al. (2020), Bartl et al. (202O),Liu et al. (2020b), I,Sheng et al. (2020), Gaci et al. (2022), Hutchinson et al. (2020)et al. (2021e),Van Der Wal et al. (2022), Hansen and Spgaard (2Azarpanah and Farhadloo (2021), Jiao and Luo (2021), Touileb,Ciora et al. (2021), Murayama et al. (2021), Gaido et al. (202(2022), Dawkins (2021a) , Gillis (2021),Mireshghallah and BWang et al. (2021a),Malik et al. (2022), Subramanian et al. (202Garimella et al. (2021),Ghosh et al. (2021),Mehrabi et al. (20)
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TypePapers
PaperSurvey/PositionDrugan and Babych (2010), Masthoff(2011), Escartin et al. (2(2016), Hovy and Spruit (2O16), Leidner and Plachouras (2017),et al.(2017), Schnoebelen (2017),Larson (2017),De Jong et a(2018),Mayfield et al. (2019),Sun et al. (2019), Blodgett et al.(2021), Bender et al. (2020), Schoch et al. (2020), Savoldi et (2021), Mishra et al. (2021), Czarnowska et al. (2021), DoughSheng et al. (2021), Hovy and Yang (2021), Santy et al. (2021)(2022b), Chandrabose et al. (2021), Vecchi et al. (2021),Talat et :Balkir et al. (2022), Orgad and Belinkov (2022), Akyurek et al(2022), Mohammad (2022a), Orr et al. (2022), Benotti and Blaet al.(2023),Benotti et al. (2023), Ramesh et al. (2023), Rios el
Dataset,BenchmarkYano et al. (2010), Flekova et al. (2016), Braun et al. (2016) KRudinger et al. (2018), Zhao et al. (2018), Kiritchenko and Mo(2019), Gaut et al. (202O),Friedman et al. (2019), Hitti et al. ((2019), Fan et al. (2019),Liu et al. (2020a), Sap et al. (2020),Louet al. (2021), Lim et al. (2020), Shahid et al. (2020), LazaridoUthus (2020), Gala et al. (2020),Dinan et al. (2020b), Meyer e(2020b), Gangula et al. (2019), Zueva et al. (2020), Nangia et al.Blodgett et al. (2021), Spinde et al. (2021), Zhong et al. (2021),Spgaard (2021), Zhao et al. (2021), Aksenov et al. (2021), KamRenduchintala et al. (2021), Barikeri et al. (2021),Milkowski et :et al. (2022), Ziems et al. (2022), Zhou et al. (2022),Akyirek(2022), Bansal et al. (2022),Smith et al. (2022), Qian et al. (2022et al. (2023), Madnani et al. (2017), Kumar and Bhotia (2020)(2021), Zad et al. (2021), Nozza et al. (2022),Mechura (2022)
MethodsGonen and Goldberg (2019), Wevers (2019), Hovy et al. (2020Nissim et al. (2020),Wang etal. (2021d),Wolfe and Caliskan (2et al. (2021), Herold et al. (2022),Bertsch et al. (2022), RampCao et al. (2022a),Alshahrani et al. (2022) Daems and Hackenbuet al. (2015), Terkik et al. (2016), Tatman (2017b),Wang et alTatman (2017a), Zhao et al. (2017), Long et al. (2018), BalyKloppenburg (2019), Farber et al. (2019), Zhong et al. (2019),and Ungar (2019), Gangula et al. (2019), Karve et al. (2019),and Najafian (20i9), Chaloner and Maldonado (2019), Qian et(2019),Kaneko and Bollegala (2019), Huang et al. (2020a),BDinan et al. (2020a), Bordia and Bowman (2O19),Manzini et al, Schwertmann et al. (2023), Pedersen et al. (2023), Bauer et al. (Kumar et al. (2023),Touileb et al. (2023), Parmar et al. (2022),et al. (2023),Alemany et al. (2023), Martinkovä et al. (2023),(2022), Cao et al. (2022b), Sap et al. (2020), Xia et al. (2020(2022), Orgad et al. (2022), Sen et al. (2022), Gupta et al. (202Gira et al. (2022), Guo et al. (2022), Wang et al. (2022), An et a(2022), Borchers et al. (2022),Touileb et al. (2022), Jentzsch andet al. (2022), Tal et al. (2022),Li et al. (2022b),Limisiewicz aret al. (2022),Ahn et al. (2022), Joniak and Aizawa (2022), Chen(2022), Du et al. (2022),Agrawal et al. (2022), Sesari et al. (0Steed et al. (2022), Schick et al. (2021), Zhou et al. (2021), Kaneetal.(2021),Lucy and Bamman (2021), Jin et al.(2021a),)(2021b), Silva et al. (2021), Sun and Peng (2021), Wang et al. (2,Han et al. (2021), Sen et al. (2021),Ramesh et al. (2021), Suband Oh (2021),Davidson et al. (2019),May et al. (2019), SapPiper (2020), Guo et al. (2020), Vu et al. (2020), Spliethöver aMunro and Morrison (2020), Zhao and Chang (2020), Schmahl(2020), Chen et al. (2020b),Mulsa and Spanakis (2020), ShiCotterell(2020),Liang et al. (2020), Wich et al. (2020a), Jia et a(2020),Dev et al. (2021a), Kumar et al. (2020),Field and Tsvetk(2020a), Stafanovics et al. (2020),Li et al. (2020),Ma et al. (AlKuwatly et al. (2020), Bartl et al. (202O),Liu et al. (2020b), I,Sheng et al. (2020), Gaci et al. (2022), Hutchinson et al. (2020)et al. (2021e),Van Der Wal et al. (2022), Hansen and Spgaard (2Azarpanah and Farhadloo (2021), Jiao and Luo (2021), Touileb,Ciora et al. (2021), Murayama et al. (2021), Gaido et al. (202(2022), Dawkins (2021a) , Gillis (2021),Mireshghallah and BWang et al. (2021a),Malik et al. (2022), Subramanian et al. (202Garimella et al. (2021),Ghosh et al. (2021),Mehrabi et al. (20)
Both aboveGoldfarb-Tarrant et al. (2021), Janghorbani and De Melo (202Troles and Schmid (2021),Liu etal. (2022a),Névéol etal. (202
AnalysisHerzig et al. (2011), Rudinger et al. (2017), Lauscher and GlavaVanderlyn et al. (2021),Falenska and Cetinoglu (2021), Sap et(2021),Bhat et al. (2021), Haroutunian (2022), Qian et al. (201
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TypePapers
PaperSurvey/PositionDrugan and Babych (2010), Masthoff(2011), Escartin et al. (2(2016), Hovy and Spruit (2O16), Leidner and Plachouras (2017),et al.(2017), Schnoebelen (2017),Larson (2017),De Jong et a(2018),Mayfield et al. (2019),Sun et al. (2019), Blodgett et al.(2021), Bender et al. (2020), Schoch et al. (2020), Savoldi et (2021), Mishra et al. (2021), Czarnowska et al. (2021), DoughSheng et al. (2021), Hovy and Yang (2021), Santy et al. (2021)(2022b), Chandrabose et al. (2021), Vecchi et al. (2021),Talat et :Balkir et al. (2022), Orgad and Belinkov (2022), Akyurek et al(2022), Mohammad (2022a), Orr et al. (2022), Benotti and Blaet al.(2023),Benotti et al. (2023), Ramesh et al. (2023), Rios el
Dataset,BenchmarkYano et al. (2010), Flekova et al. (2016), Braun et al. (2016) KRudinger et al. (2018), Zhao et al. (2018), Kiritchenko and Mo(2019), Gaut et al. (202O),Friedman et al. (2019), Hitti et al. ((2019), Fan et al. (2019),Liu et al. (2020a), Sap et al. (2020),Louet al. (2021), Lim et al. (2020), Shahid et al. (2020), LazaridoUthus (2020), Gala et al. (2020),Dinan et al. (2020b), Meyer e(2020b), Gangula et al. (2019), Zueva et al. (2020), Nangia et al.Blodgett et al. (2021), Spinde et al. (2021), Zhong et al. (2021),Spgaard (2021), Zhao et al. (2021), Aksenov et al. (2021), KamRenduchintala et al. (2021), Barikeri et al. (2021),Milkowski et :et al. (2022), Ziems et al. (2022), Zhou et al. (2022),Akyirek(2022), Bansal et al. (2022),Smith et al. (2022), Qian et al. (2022et al. (2023), Madnani et al. (2017), Kumar and Bhotia (2020)(2021), Zad et al. (2021), Nozza et al. (2022),Mechura (2022)
MethodsGonen and Goldberg (2019), Wevers (2019), Hovy et al. (2020Nissim et al. (2020),Wang etal. (2021d),Wolfe and Caliskan (2et al. (2021), Herold et al. (2022),Bertsch et al. (2022), RampCao et al. (2022a),Alshahrani et al. (2022) Daems and Hackenbuet al. (2015), Terkik et al. (2016), Tatman (2017b),Wang et alTatman (2017a), Zhao et al. (2017), Long et al. (2018), BalyKloppenburg (2019), Farber et al. (2019), Zhong et al. (2019),and Ungar (2019), Gangula et al. (2019), Karve et al. (2019),and Najafian (20i9), Chaloner and Maldonado (2019), Qian et(2019),Kaneko and Bollegala (2019), Huang et al. (2020a),BDinan et al. (2020a), Bordia and Bowman (2O19),Manzini et al, Schwertmann et al. (2023), Pedersen et al. (2023), Bauer et al. (Kumar et al. (2023),Touileb et al. (2023), Parmar et al. (2022),et al. (2023),Alemany et al. (2023), Martinkovä et al. (2023),(2022), Cao et al. (2022b), Sap et al. (2020), Xia et al. (2020(2022), Orgad et al. (2022), Sen et al. (2022), Gupta et al. (202Gira et al. (2022), Guo et al. (2022), Wang et al. (2022), An et a(2022), Borchers et al. (2022),Touileb et al. (2022), Jentzsch andet al. (2022), Tal et al. (2022),Li et al. (2022b),Limisiewicz aret al. (2022),Ahn et al. (2022), Joniak and Aizawa (2022), Chen(2022), Du et al. (2022),Agrawal et al. (2022), Sesari et al. (0Steed et al. (2022), Schick et al. (2021), Zhou et al. (2021), Kaneetal.(2021),Lucy and Bamman (2021), Jin et al.(2021a),)(2021b), Silva et al. (2021), Sun and Peng (2021), Wang et al. (2,Han et al. (2021), Sen et al. (2021),Ramesh et al. (2021), Suband Oh (2021),Davidson et al. (2019),May et al. (2019), SapPiper (2020), Guo et al. (2020), Vu et al. (2020), Spliethöver aMunro and Morrison (2020), Zhao and Chang (2020), Schmahl(2020), Chen et al. (2020b),Mulsa and Spanakis (2020), ShiCotterell(2020),Liang et al. (2020), Wich et al. (2020a), Jia et a(2020),Dev et al. (2021a), Kumar et al. (2020),Field and Tsvetk(2020a), Stafanovics et al. (2020),Li et al. (2020),Ma et al. (AlKuwatly et al. (2020), Bartl et al. (202O),Liu et al. (2020b), I,Sheng et al. (2020), Gaci et al. (2022), Hutchinson et al. (2020)et al. (2021e),Van Der Wal et al. (2022), Hansen and Spgaard (2Azarpanah and Farhadloo (2021), Jiao and Luo (2021), Touileb,Ciora et al. (2021), Murayama et al. (2021), Gaido et al. (202(2022), Dawkins (2021a) , Gillis (2021),Mireshghallah and BWang et al. (2021a),Malik et al. (2022), Subramanian et al. (202Garimella et al. (2021),Ghosh et al. (2021),Mehrabi et al. (20)
Both aboveGoldfarb-Tarrant et al. (2021), Janghorbani and De Melo (202Troles and Schmid (2021),Liu etal. (2022a),Névéol etal. (202
AnalysisHerzig et al. (2011), Rudinger et al. (2017), Lauscher and GlavaVanderlyn et al. (2021),Falenska and Cetinoglu (2021), Sap et(2021),Bhat et al. (2021), Haroutunian (2022), Qian et al. (201
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/dev/null +++ b/parse/dev/n0OeTdNRG0Q/images/fb8c1fd4f288f36e27a0dac2d6d3336fa6a97fd8f903929d9b68210c1531c376.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3d14ae86b712bbaef8861c683affbc5d931d381920ca16df35618d21ce50104d +size 5547 diff --git a/parse/dev/pd1P2eUBVfq/pd1P2eUBVfq.md b/parse/dev/pd1P2eUBVfq/pd1P2eUBVfq.md new file mode 100644 index 0000000000000000000000000000000000000000..3935b5b69c7795afb3f386f88d553cb928d1228c --- /dev/null +++ b/parse/dev/pd1P2eUBVfq/pd1P2eUBVfq.md @@ -0,0 +1,711 @@ +# DIFFUSION MODELS ALREADY HAVE A SEMANTIC LATENT SPACE + +Mingi Kwon, Jaeseok Jeong, Youngjung Uh∗ + +Department of Artificial Intelligence +Yonsei University +Seoul, Republic of Korea +{kwonmingi,jete jeong,yj.uh}@yonsei.ac.kr + +# ABSTRACT + +Diffusion models achieve outstanding generative performance in various domains. Despite their great success, they lack semantic latent space which is essential for controlling the generative process. To address the problem, we propose asymmetric reverse process (Asyrp) which discovers the semantic latent space in frozen pretrained diffusion models. Our semantic latent space, named $h$ -space, has nice properties to accommodate semantic image manipulation: homogeneity, linearity, robustness, and consistency across timesteps. In addition, we introduce a principled design of the generative process for versatile editing and quality boosting by quantifiable measures: editing strength of an interval and quality deficiency at a timestep. Our method is applicable to various architectures $\mathrm { ( D D P M + + }$ , iDDPM, and ADM) and datasets (CelebA-HQ, AFHQ-dog, LSUN-church, LSUNbedroom, and METFACES). Project page: https://kwonminki.github.io/Asyrp/ + +# 1 INTRODUCTION + +In image synthesis, diffusion models have advanced to achieve state-of-the-art performance regarding quality and mode coverage since the introduction of denoising diffusion probabilistic models (Ho et al., 2020). They disrupt images by adding noise through multiple steps of forward process and generate samples by progressive denoising through multiple steps of reverse (i.e., generative) process. Since their deterministic version provides nearly perfect reconstruction of original images (Song et al., 2020a), they are suitable for image editing, which renders target attributes on the real images. However, simply editing the latent variables (i.e., intermediate noisy images) causes degraded results (Kim & Ye, 2021). Instead, they require complicated procedures: providing guidance in the reverse process or finetuning models for an attribute. + +Figure 1(a-c) briefly illustrates the existing approaches. Image guidance mixes the latent variables of the guiding image with unconditional latent variables (Choi et al., 2021; Lugmayr et al., 2022; Meng et al., 2021). Though it provides some control, it is ambiguous to specify which attribute to reflect among the ones in the guide and the unconditional result, and it lacks intuitive control for the magnitude of change. Classifier guidance manipulates images by imposing gradients of a classifier on the latent variables in the reverse process to match the target class (Dhariwal & Nichol, 2021; Avrahami et al., 2022; Liu et al., 2021). It requires training an extra classifier for the latent variables, i.e., noisy images. Furthermore, computing gradients through the classifier during sampling is costly. Finetuning the whole model can steer the resulting images to the target attribute without the above problems (Kim & Ye, 2021). Still, it requires multiple models to reflect multiple descriptions. + +On the other hand, generative adversarial networks (Goodfellow et al., 2020) inherently provide straightforward image editing in their latent space. Given a latent vector for an original image, we can find the direction in the latent space that maximizes the similarity of the resulting image with a target description in CLIP embedding (Patashnik et al., 2021). The latent direction found on one image leads to the same manipulation of other images. However, given a real image, finding its exact latent vector is often challenging and produces unexpected appearance changes. + +![](images/80648f78ecfc7ad5b9897cecadcaf4711f94bdafc585f1c639190f153bb4ad3d.jpg) +Figure 1: Manipulation approaches for diffusion models. (a) Image guidance suffers ambiguity while controlling the generative process. (b) Classifier guidance requires an extra classifier, is hardly editable, degrades quality, or alters the content. (c) DiffusionCLIP requires fine-tuning the whole model. (d) Our method discovers a semantic latent space of a frozen diffusion model. + +It would allow admirable image editing if the diffusion models with the nearly perfect inversion property have such a semantic latent space. Preechakul et al. (2022) introduces an additional input to the reverse diffusion process: a latent vector from an original image embedded by an extra encoder. This latent vector contains the semantics to condition the process. However, it requires training from scratch and does not match with pretrained diffusion models. + +In this paper, we propose an asymmetric reverse process (Asyrp) which discovers the semantic latent space of a frozen diffusion model such that modifications in the space edits attributes of the original images. Our semantic latent space, named $h$ -space, has the properties necessary for editing applications as follows. The same shift in this space results in the same attribute change in all images. Linear changes in this space lead to linear changes in attributes. The changes do not degrade the quality of the resulting images. The changes throughout the timesteps are almost identical to each other for a desired attribute change. Figure 1(d) illustrates some of these properties and $\ S 5 . 3$ provides detailed analyses. To the best of our knowledge, it is the first attempt to discover the semantic latent space in the frozen pretrained diffusion models. Spoiler alert: our semantic latent space is different from the intermediate latent variables in the diffusion process. Moreover, we introduce a principled design of the generative process for versatile editing and quality boosting by quantifiable measures: editing strength of an interval and quality deficiency at a timestep. Extensive experiments demonstrate that our method is generally applicable to various architectures $\mathrm { ( D D P M + + }$ , iDDPM, and ADM) and datasets (CelebA-HQ, AFHQ-dog, LSUN-church, LSUN-bedroom, and METFACES). + +# 2 BACKGROUND + +We briefly describe essential backgrounds. The rest of the related work is deferred to Appendix A. + +# 2.1 DENOISING DIFFUSION PROBABILITY MODEL (DDPM) + +DDPM is a latent variable model that learns a data distribution by denoising noisy images (Ho et al., 2020). The forward process diffuses the data samples through Gaussian transitions parameterized with a Markov process: + +$$ +q \left( \pmb { x } _ { t } \mid \pmb { \dot { x } } _ { t - 1 } \right) = \mathcal { N } \left( \pmb { x } _ { t } ; \sqrt { 1 - \beta _ { t } } \pmb { x } _ { t - 1 } , \beta _ { t } \mathbf { I } \right) = \mathcal { N } \left( \sqrt { \frac { \alpha _ { t } } { \alpha _ { t - 1 } } } \pmb { x } _ { t - 1 } , \left( 1 - \frac { \alpha _ { t } } { \alpha _ { t - 1 } } \right) \pmb { I } \right) , +$$ + +where $\{ \beta _ { t } \} _ { t = 1 } ^ { T }$ is the variance schedule and $\begin{array} { r } { \alpha _ { t } = \prod _ { s = 1 } ^ { t } ( 1 - \beta _ { s } ) } \end{array}$ . Then the reverse process becomes $\begin{array} { r } { p _ { \theta } \left( \pmb { x } _ { 0 : T } \right) : = p \left( \pmb { x } _ { T } \right) \prod _ { t = 1 } ^ { T } p _ { \theta } \left( \pmb { x } _ { t - 1 } \mid \pmb { x } _ { t } \right) } \end{array}$ , starting from $\mathbf { \boldsymbol { x } } _ { T } \sim \mathcal { N } ( 0 , \mathbf { I } )$ with noise predictor $\epsilon _ { t } ^ { \theta }$ : + +$$ +{ \pmb x } _ { t - 1 } = \frac { 1 } { \sqrt { 1 - \beta _ { t } } } \left( { \pmb x } _ { t } - \frac { \beta _ { t } } { \sqrt { 1 - \alpha _ { t } } } { \pmb \epsilon } _ { t } ^ { \theta } \left( { \pmb x } _ { t } \right) \right) + \sigma _ { t } { \pmb z } _ { t } , +$$ + +where $\boldsymbol { z } _ { t } \sim \mathcal { N } ( 0 , \mathbf { I } )$ and $\sigma _ { t } ^ { 2 }$ is a variance of the reverse process which is set to $\sigma _ { t } ^ { 2 } = \beta _ { t }$ by DDPM. + +# 2.2 DENOISING DIFFUSION IMPLICIT MODEL (DDIM) + +DDIM redefines Eq. (1) as $\begin{array} { r } { q _ { \sigma } ( \pmb { x } _ { t - 1 } | \pmb { x } _ { t } , \pmb { x } _ { 0 } ) = \mathcal { N } ( \sqrt { \alpha _ { t - 1 } } \pmb { x } _ { 0 } + \sqrt { 1 - \alpha _ { t - 1 } - \sigma _ { t } ^ { 2 } } \cdot \frac { \pmb { x } _ { t } - \sqrt { \alpha _ { t } } \pmb { x } _ { 0 } } { \sqrt { 1 - \alpha _ { t } } } , \sigma _ { t } ^ { 2 } I ) } \end{array}$ which is a non-Markovian process (Song et al., 2020a). Accordingly, the reverse process becomes + +$$ +\begin{array} { r } { x _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } \underbrace { \left( \frac { x _ { t } - \sqrt { 1 - \alpha _ { t } } \epsilon _ { t } ^ { \theta } \left( x _ { t } \right) } { \sqrt { \alpha _ { t } } } \right) } _ { \mathrm { \normalfont ~ \ " p r e d i c t e d ~ \ b { x } _ 0 ~ \ " ~ } } + \underbrace { \sqrt { 1 - \alpha _ { t - 1 } - \sigma _ { t } ^ { 2 } } \cdot \epsilon _ { t } ^ { \theta } \left( x _ { t } \right) } _ { \mathrm { \normalfont ~ \ " d i r e c t i o n ~ p o i n t i n g ~ t o ~ \ b { x } _ t \cdot \ } } + \underbrace { \sigma _ { t } z _ { t } } _ { \mathrm { \normalfont ~ \mathrm { r a n d o m ~ n o i s e } } } \ , } \end{array} +$$ + +where $\sigma _ { t } = \eta \sqrt { \left( 1 - \alpha _ { t - 1 } \right) / \left( 1 - \alpha _ { t } \right) } \sqrt { 1 - \alpha _ { t } / \alpha _ { t - 1 } }$ . When $\eta = 1$ for all $t$ , it becomes DDPM. As $\eta = 0$ , the process becomes deterministic and guarantees nearly perfect inversion. + +# 2.3 IMAGE MANIPULATION WITH CLIP + +CLIP learns multimodal embeddings with an image encoder $E _ { I }$ and a text encoder $E _ { T }$ whose similarity indicates semantic similarity between images and texts (Radford et al., 2021). Compared to directly minimizing the cosine distance between the edited image and the target description (Patashnik et al., 2021), directional loss with cosine distance achieves homogeneous editing without mode collapse (Gal et al., 2021): + +$$ +\mathcal { L } _ { \mathrm { d i r e c t i o n } } \left( { \boldsymbol { x } } ^ { \mathrm { e d i t } } , { \boldsymbol { y } } ^ { \mathrm { t a r g e t } } ; { \boldsymbol { x } } ^ { \mathrm { s o u r c e } } , { \boldsymbol { y } } ^ { \mathrm { s o u r c e } } \right) : = 1 - \frac { \Delta I \cdot \Delta T } { \| \Delta I \| \| \Delta T \| } , +$$ + +where $\Delta T = E _ { T } \left( y ^ { \mathrm { t a r g e t } } \right) - E _ { T } \left( y ^ { \mathrm { s o u r c e } } \right)$ and $\Delta I = E _ { I } \left( { \pmb x } ^ { \mathrm { e d i t } } \right) - E _ { I } \left( { \pmb x } ^ { \mathrm { s o u r c e } } \right)$ for edited image $\pmb { x } ^ { \mathrm { e d i t } }$ , target description $y ^ { \mathrm { t a r g e t } }$ , original image $\pmb { x } ^ { \mathrm { s o u r c e } }$ , and source description $y ^ { \mathrm { s o u r c e } }$ . We use the prompts ‘smiling face’ and ‘face’ as the target and source descriptions for facial attribute smiling. + +# 3 DISCOVERING SEMANTIC LATENT SPACE IN DIFFUSION MODELS + +This section explains why naive approaches do not work and proposes a new controllable reverse process. Then we describe the techniques for controlling the generative process. Throughout this paper, we use an abbreviated version of Eq. (3): + +$$ +\pmb { x } _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( \pmb { x } _ { t } ) ) + \mathbf { D } _ { t } ( \epsilon _ { t } ^ { \theta } ( \pmb { x } _ { t } ) ) + \sigma _ { t } \boldsymbol { z } _ { t } , +$$ + +where $\mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( { \pmb x } _ { t } ) )$ denotes the predicted $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ and $\mathbf { D } _ { t } ( \epsilon _ { t } ^ { \theta } ( { \pmb x } _ { t } ) )$ denotes the direction pointing to $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ . We omit $\sigma _ { t } z _ { t }$ for brevity, except when $\eta \neq 0$ . We further abbreviate $\mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( { \pmb x } _ { t } ) )$ as $\mathbf { P } _ { t }$ and $\mathbf { D } _ { t } ( \epsilon _ { t } ^ { \theta } ( { \pmb x } _ { t } ) )$ as $\mathbf { D } _ { t }$ when the context clearly specifies the arguments. + +# 3.1 PROBLEM + +We aim to allow semantic latent manipulation of images $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ generated from $\mathbf { \nabla } _ { \mathbf { x } _ { T } }$ given a pretrained and frozen diffusion model. The easiest idea to manipulate $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ is simply updating $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } \mathcal { T } }$ to optimize the directional CLIP loss given text prompts with Eq. (4). However, it leads to distorted images or incorrect manipulation (Kim & Ye, 2021). + +An alternative approach is to shift the noise $\epsilon _ { t } ^ { \theta }$ predicted by the network at each sampling step. However, it does not achieve manipulating $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ because the intermediate changes in $\mathbf { P } _ { t }$ and $\mathbf { D } _ { t }$ cancel out each other resulting in the same $p _ { \theta } ( { \pmb x } _ { 0 : T } )$ , similarly to destructive interference. + +Theorem 1. Let $\epsilon _ { t } ^ { \theta }$ be a predicted noise during the original reverse process at $t$ and $\tilde { \epsilon } _ { t } ^ { \theta }$ be its shifted counterpart. Then, $\Delta \pmb { x } _ { t } = \tilde { \pmb { x } } _ { t - 1 } - \pmb { x } _ { t - 1 }$ is negligible where $\begin{array} { r } { \tilde { { \pmb { x } } } _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } { \bf P } _ { t } ( \tilde { \epsilon } _ { t } ^ { \theta } ( { \pmb { x } } _ { t } ) ) + } \end{array}$ $\mathbf { D } _ { t } ( \tilde { \epsilon } _ { t } ^ { \theta } ( { \pmb x } _ { t } ) )$ . I.e., the shifted terms of $\tilde { \epsilon } _ { t } ^ { \theta }$ in $\mathbf { P } _ { t }$ and $\mathbf { D } _ { t }$ destruct each other in the reverse process. + +Appendix C proves above theorem. Figure 13(a-b) shows that $\tilde { \mathbf { x } } _ { 0 }$ is almost identical to $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ + +# 3.2 ASYMMETRIC REVERSE PROCESS + +In order to break the interference, we propose a new controllable reverse process with asymmetry: + +$$ +\pmb { x } _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } \big ( \tilde { \epsilon } _ { t } ^ { \theta } ( \pmb { x } _ { t } ) \big ) + \mathbf { D } _ { t } \big ( \epsilon _ { t } ^ { \theta } ( \pmb { x } _ { t } ) \big ) , +$$ + +![](images/06ae64f291010710a1b60bf1d063bfc7c5ce461052c3aeb0dadeab995109361f.jpg) +Figure 2: Generative process of Asyrp. The green box on the left illustrates Asyrp which only alters $\mathbf { P } _ { t }$ while preserving $\mathbf { D } _ { t }$ shared by DDIM. The right describes that Asyrp modifies the original reverse process toward the target attribute reflecting the change in $h$ -space. + +i.e., we modify only $\mathbf { P } _ { t }$ by shifting $\epsilon _ { t } ^ { \theta }$ to $\tilde { \epsilon } _ { t } ^ { \theta }$ while preserving $\mathbf { D } _ { t }$ . Intuitively, it modifies the original reverse process according to $\Delta \epsilon _ { t } = \widetilde { \epsilon } _ { t } ^ { \theta } - \widetilde { \epsilon } _ { t } ^ { \theta }$ while it does not alter the direction toward $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ so that ${ \mathbf { \mathcal { x } } } _ { t - 1 }$ follows the original flow $\mathbf { D } _ { t }$ at each sampling step. Figure 2 illustrates the above intuition. + +As in Avrahami et al. (2022), we use the modified $\mathbf { P } _ { t } ^ { \mathrm { e d i t } }$ and the original $\mathbf { P } _ { t } ^ { \mathrm { s o u r c e } }$ as visual inputs for the directional CLIP loss in Eq. (4), and regularize the difference between the modified $\mathbf { P } _ { t } ^ { \mathrm { { e d i t } } }$ and the original $\mathbf { P } _ { t } ^ { \mathrm { s o u r c e } }$ . We find $\begin{array} { r } { \Delta \epsilon = \arg \operatorname* { m i n } _ { \Delta \epsilon } \mathbb { E } _ { t } \mathcal { L } ^ { ( t ) } } \end{array}$ where + +$$ +\begin{array} { r } { \mathcal { L } ^ { ( t ) } = \lambda _ { \mathrm { { C L I P } } } \mathcal { L } _ { \mathrm { d i r e c t i o n } } \left( \mathbf { P } _ { t } ^ { \mathrm { { e d i t } } } , y ^ { \mathrm { { r e f } } } ; \mathbf { P } _ { t } ^ { \mathrm { { s o u r c e } } } , y ^ { \mathrm { { s o u r c e } } } \right) + \lambda _ { \mathrm { { r e c o n } } } \left| \mathbf { P } _ { t } ^ { \mathrm { { e d i t } } } - \mathbf { P } _ { t } ^ { \mathrm { { s o u r c e } } } \right| } \end{array} +$$ + +Although $\Delta \epsilon$ indeed renders the attribute in the $\pmb { x } _ { 0 } ^ { \mathrm { e d i t } }$ , $\epsilon$ -space lacks the necessary properties of the semantic latent space in diffusion models that will be described in the following. + +# 3.3 $h$ -space + +Note that $\epsilon _ { t } ^ { \theta }$ is implemented as U-Net in all state-of-the-art diffusion models. We choose its bottleneck, the deepest feature maps $\boldsymbol { h } _ { t }$ , to control $\epsilon _ { t } ^ { \theta }$ . By design, $h _ { t }$ has smaller spatial resolutions and high-level semantics than $\epsilon _ { t } ^ { \theta }$ . Accordingly, the sampling equation becomes + +$$ +\pmb { x } _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( \pmb { x } _ { t } | \Delta \pmb { h } _ { t } ) ) + \mathbf { D } _ { t } ( \epsilon _ { t } ^ { \theta } ( \pmb { x } _ { t } ) ) + \sigma _ { t } \boldsymbol { z } _ { t } , +$$ + +where $\epsilon _ { t } ^ { \theta } ( \boldsymbol { x } _ { t } | \Delta h _ { t } )$ adds $\Delta { h _ { t } }$ to the original feature maps $h _ { t }$ . The $\Delta \boldsymbol { h } _ { t }$ minimizing the same loss in Eq. (7) with $\dot { \mathbf { P } } _ { t } ( \epsilon _ { t } ^ { \theta } ( \pmb { x } _ { t } | \Delta \pmb { h } _ { t } ) )$ instead of $\mathbf { P } _ { t } ( \tilde { \epsilon } _ { t } ^ { \theta } ( x _ { t } ) )$ successfully manipulates the attributes. + +We observe that $h$ -space in Asyrp has the following properties that others do not have. + +• The same $\Delta h$ leads to the same effect on different samples. +• Linearly scaling $\Delta h$ controls the magnitude of attribute change, even with negative scales. +• Adding multiple $\Delta h$ manipulates the corresponding multiple attributes simultaneously. +• $\Delta h$ preserves the quality of the resulting images without degradation. +• $\Delta { h _ { t } }$ is roughly consistent across different timesteps $t$ . + +The above properties are demonstrated thoroughly in $\ S 5 . 3$ . Appendix D.3 provides details of $h$ -space and suboptimal results from alternative choices. + +# 3.4 IMPLICIT NEURAL DIRECTIONS + +Although $\Delta h$ succeeds in manipulating images, directly optimizing $\Delta \boldsymbol { h } _ { t }$ on multiple timesteps requires many iterations of training with a carefully chosen learning rate and its scheduling. Instead, we define an implicit function $f _ { t } ( h _ { t } )$ which produces $\Delta { h _ { t } }$ for given $h _ { t }$ and $t$ . $\pmb { f } _ { t }$ is implemented as a small neural network with two $1 \times 1$ convolutions concatenating timestep $t$ . See Appendix $\mathrm { E }$ for the details. Accordingly, we optimize the same loss in Eq. (7) with $\mathbf { P } _ { t } ^ { \mathrm { e d i t } } = \mathbf { \bar { P } } _ { t } \bigl ( \epsilon _ { t } ^ { \theta } ( { \pmb x } _ { t } \mathbf { \bar { | } } \mathbf { \bar { \mathbf { f } } } _ { t } ) \bigr )$ . + +Learning $f _ { t }$ is more robust to learning rate settings and converges faster than learning every $\Delta \boldsymbol { h } _ { t }$ . In addition, as $\pmb { f } _ { t }$ learns an implicit function for given timesteps and bottleneck features, it generalizes to unseen timesteps and bottleneck features. The generalization allows us to borrow the accelerated training scheme of DDIM defined on a subsequence $\{ \pmb { x } _ { \tau _ { i } } \} _ { \forall i \in [ 1 , S ] }$ where $\{ \tau _ { i } \}$ is a subsequence of $[ 1 , . . . , T ]$ and $S \textless T$ . Then, we can use the generative process with a custom subsequence $\{ \tilde { \tau } _ { i } \}$ with length $\tilde { S } < T$ through normalization: $\Delta \tilde { h } _ { \tilde { \tau } } = f _ { \tilde { \tau } } ( h _ { \tilde { \tau } } ) { S } / { \tilde { S } }$ . It preserves the amount of $\sum \Delta { { h } _ { t } }$ , $\Delta \tilde { h } _ { \tilde { \tau } } \tilde { S } = \Delta h _ { t } S$ . Therefore, we can use $f _ { t }$ trained on any subsequence for any length of the generative process. See Appendix F for details. We use $\pmb { f } _ { t }$ to get $\Delta \boldsymbol { h } _ { t }$ for all experiments except Figure 6. + +![](images/47dbd6b534787f72c5d6825a64bc954aec392ef479c4aed91b47b808b2fa38d3.jpg) +Figure 3: Intuition for choosing the intervals for editing and quality boosting. We choose the intervals by quantifying two measures (top left inset). The editing strength of an interval $[ T , t ]$ measures its perceptual difference from $T$ until $t$ . We set $[ T , t ]$ to the interval with the smallest editing strength that synthesizes $\mathbf { P } _ { t }$ close to $_ { \textbf { \em x } }$ , i.e., $\mathrm { L P I P S } ( { \pmb x } , { \bf P } _ { t } ) = 0 . 3 3$ . Editing flexibility of an interval $[ t , 0 ]$ measures the potential amount of changes after $t$ . Quality deficiency at $t$ measures the amount of noise in $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ . We set $[ t , 0 ]$ to handle large quality deficiency (i.e., $\mathrm { L P I P S } ( \pmb { x } , \pmb { x } _ { t } ) = 1 . 2 )$ with small editing flexibility. + +# 4 GENERATIVE PROCESS DESIGN + +This section describes the entire editing process, which consists of three phases: editing with Asyrp, traditional denoising, and quality boosting. We design formulas to determine the length of each phase with quantifiable measures. + +# 4.1 EDITING PROCESS WITH ASYRP + +Diffusion models generate the high-level context in the early stage and imperceptible fine details in the later stage (Choi et al., 2022). Likewise, we modify the generative process in the early stage to achieve semantic changes. We refer to the early stage as the editing interval $[ T , t _ { \mathrm { e d i t } } ]$ . + +$\mathrm { L P I P S } ( { \pmb x } , { \pmb P } _ { T } )$ and $\mathrm { L P I P S } ( \pmb { x } , \mathbf { P } _ { t } )$ calculate the perceptual distance between the original image and the predicted image at time steps $T$ and $t$ , respectively. Intuitively, the high-level content is already determined by the predicted terms at the respective timesteps and LPIPS measures the remaining component to be edited through the remaining reverse process. Consequently, we define editing strength of an interval $[ T , t ]$ : + +$$ +\xi _ { t } = \mathrm { L P I P S } ( x , \mathbf { P } _ { T } ) - \mathrm { L P I P S } ( x , \mathbf { P } _ { t } ) +$$ + +indicating the perceptual change from timestep $T$ to $t$ in the original generative process. Figure 3 illustrates $\mathrm { L P I P S } ( { \pmb x } , \cdot )$ for $\mathbf { P } _ { t }$ and $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ with examples and the inset depicts editing strength. The shorter editing interval has the lower $\xi _ { t }$ , and the longer editing interval brings more changes to the resulting images. We seek the shortest editing interval which will bring enough distinguishable changes in the images in general. We empirically find that $t _ { \mathrm { e d i t } }$ with $\mathrm { L P I P S } ( \mathbf { x } , \mathbf { P } _ { t _ { \mathrm { e d i t } } } ) = 0 . \dot { 3 } 3$ builds the shortest editing interval with enough editing strength as $\mathbf { P } _ { t _ { \mathrm { e d i t } } }$ has nearly all visual attributes in $_ { \textbf { \em x } }$ . + +However, some attributes require more visual changes than others, e.g., pixar $>$ smile. For such attributes, we increase the editing strength $\xi _ { t }$ by $\delta = 0 . 3 3 d ( E _ { T } ( y _ { \mathrm { s o u r c e } } ) , E _ { T } ( y _ { \mathrm { t a r g e t } } ) )$ where + +$E _ { T } ( \cdot )$ produces CLIP text embedding, $y _ { ( \cdot ) }$ denotes the descriptions, and $d ( \cdot , \cdot )$ computes the cosine distance between the arguments. Choosing $t _ { \mathrm { e d i t } }$ with $\mathrm { L P I P S } ( \pmb { x } , \mathbf { P } _ { t _ { \mathrm { e d i t } } } ) = 0 . 3 3 - \delta$ expands the editing interval to a suitable length. It consistently produces good results in various settings. The supporting experiments are shown in Appendix G. + +# 4.2 QUALITY BOOSTING WITH STOCHASTIC NOISE INJECTION + +Although DDIM achieves nearly perfect inversion by removing stochasticity $( \eta = 0$ ), Karras et al. (2022) demonstrate that stochasticity improves image quality. Likewise, we inject stochastic noise in the boosting interval $[ t _ { \mathrm { b o o s t } } , 0 ]$ . + +Though the longer boosting interval would achieve higher quality, boosting over excessively long intervals would modify the content. Hence, we want to determine the shortest interval that shows enough quality boosting to guarantee minimal change in the content. We consider the noise in the image as the capacity for the quality boosting and define quality deficiency at $t$ $: \gamma _ { t } = \mathrm { L P I P S } ( \pmb { x } , \pmb { x } _ { t } )$ indicating the amount of noise in $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ compared to the original image. We use $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ instead of $\mathbf { P } _ { t }$ because we consider the actual image rather than the semantics. Figure 3 inset depicts editing flexibility and quality deficiency. We empirically find that $t _ { \mathrm { b o o s t } }$ with $\gamma _ { t _ { \mathrm { b o o s t } } } ~ = ~ 1 . 2$ achieves quality boosting with minimal content change. We confirmed that the editing strength of the intervals $[ t _ { \mathrm { b o o s t } } , 0 ]$ is guaranteed to be less than 0.25. In Figure 3, after $t _ { \mathrm { b o o s t } }$ , $\mathrm { L P I P S } ( { \pmb x } , { \pmb x } _ { t } )$ sharply drops in the original generative process while $\mathrm { L P I P S } ( \pmb { x } , \mathbf { P } _ { t } )$ changes little. Note that most of the quality degradation of the resulting images is caused by DDIM reverse process, not by Asyrp. We use this quality boosting for all experiments except ablation in Appendix $_ \mathrm { H }$ . + +# 4.3 OVERALL PROCESS OF IMAGE EDITING + +Using $t _ { \mathrm { e d i t } }$ and $t _ { \mathrm { b o o s t } }$ determined by the above formulas, we modify the generative process of DDIM with + +$$ +p _ { \theta } ^ { ( t ) } \left( x _ { t - 1 } \mid x _ { t } \right) = \left\{ \begin{array} { l l } { \mathcal { N } \left( \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( x _ { t } | f _ { t } ) ) + \mathbf { D } _ { t } , \sigma _ { t } ^ { 2 } I \right) , \eta = 0 } & { \mathrm { ~ i f ~ } T \geq t \geq t _ { \mathrm { e d i t } } } \\ { \mathcal { N } \left( \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( x _ { t } ) ) + \mathbf { D } _ { t } , \sigma _ { t } ^ { 2 } I \right) , \eta = 0 } & { \mathrm { ~ i f ~ } t _ { \mathrm { e d i t } } > t \geq t _ { \mathrm { b o o s t } } } \\ { \mathcal { N } \left( \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( x _ { t } ) ) + \mathbf { D } _ { t } , \sigma _ { t } ^ { 2 } I \right) , \eta = 1 } & { \mathrm { ~ i f ~ } t _ { \mathrm { b o o s t } } > t } \end{array} \right. +$$ + +The visual overview and comprehensive algorithms of the entire process are in Appendix I. + +# 5 EXPERIMENTS + +In this section, we show the effectiveness of semantic latent editing in $h$ -space with Asyrp on various attributes, datasets and architectures in $\ S \ S . 1$ . Moreover, we provide quantitative results including user study in $\mathrm { ~ \normalfont ~ \ S ~ } 5 . 2$ . Lastly, we provide detailed analyses for the properties of the semantic latent space on $h$ -space and alternatives in $\ S 5 . 3$ . + +Implementation details. We implement our method on various settings: CelebA-HQ (Karras et al., 2018) and LSUN-bedroom/-church (Yu et al., 2015) on $\mathrm { D D P M + + }$ (Song et al., 2020b) (Meng et al., 2021); AFHQ-dog (Choi et al., 2020) on iDDPM (Nichol & Dhariwal, 2021); and METFACES (Karras et al., 2020) on ADM with P2-weighting (Dhariwal & Nichol, 2021) (Choi et al., 2022). Please note that all models are official pretrained checkpoints and are kept frozen. Detailed settings including the coefficients for $\lambda _ { \mathrm { C L I P } }$ and $\lambda _ { \mathrm { { r e c o n } } }$ , and source/target descriptions can be found in Appendix J.1. We train $\pmb { f } _ { t }$ with $S ~ = ~ 4 0$ for 1 epoch using 1000 samples. The real samples are randomly chosen from each dataset for in-domain-like attributes. For out-of-domainlike attributes, we randomly draw 1,000 latent variables $\mathbf { \boldsymbol { x } } _ { T } \sim \mathcal { N } ( \mathbf { \boldsymbol { 0 } } , I )$ . Details are described in Appendix J.2. Training takes about 20 minutes with three RTX 3090 GPUs. All the images in the figures are not used for training. We set $\tilde { S } = 1 , 0 0 0$ for inference. The code is available at https://github.com/kwonminki/Asyrp official + +# 5.1 VERSATILITY OF $h$ -space WITH ASYRP + +Figure 4 shows the effectiveness of our method on various datasets and any existing U-Net based architectures. Our method can synthesize the attributes that are not even included in the training dataset, such as church $ \{ \begin{array} { r l } \end{array} $ department, factory, and temple}. Even for dogs, our method synthesizes smiling Poodle and Yorkshire, the species that barely smile in the dataset. Figure 5 provides results for changing human faces to different identities, painting styles, and ancient primates. More result can be found in Appendix N. Versatility of our method is surprising because we do not alter the models but only shift the bottleneck feature maps in $h$ -space with Asyrp during inference. + +![](images/306d377572a44f6224fe73778efba99d8a0dfab3966d278640038c75118326f5.jpg) +Figure 4: Editing results of Asyrp on various datasets. We conduct experiments on CelebA-HQ, LSUN-church, METFACES, AFHQ-dog, and LSUN-bedroom. + +![](images/a2ba33a9e5710b55fca14f10f93b7fa652ad4c1a82d2bbb67e10456214757d19.jpg) +Figure 5: Editing results of Asyrp for unseen domains in CelebA-HQ dataset. + +Table 1: User study with 80 participants. The details are described in $\ S \ K . 1$ + +
CelebA-HQ in-domainCelebA-HQ unseen-domainLSUN-church
qualityattributeoverallqualityattributeoverallqualityattributediversityoverall
Asyrp (ours)98.36%88.13%94.92%71.56%59.84%63.13%73.19%71.81%87.50%76.81%
DiffusionCLIP1.64%11.88%5.08%28.44%40.16%36.88%26.81%28.19%12.50%23.19%
+ +![](images/66fd5aab2ef8fa4bdecea189af2d5eb233396464cc901dad958ae0cc0424c117.jpg) +Figure 6: Optimization for smiling on $\pmb { h }$ -space and ϵ-space. (a) Optimizing $\Delta { h _ { t } }$ for a sample with smiling results in natural editing while the change due to optimizing $\Delta \epsilon _ { t }$ is relatively small. (b) Applying $\Delta { h _ { t } }$ from (a) to other samples yields the same attribute change while $\Delta \epsilon _ { t }$ distorts the images. The result of $\Delta \epsilon _ { t }$ is the best sample we could find. + +![](images/7bac52c03fecc44b4f929ac853df2555bd33134ce5593830fb81ee5e8c225924.jpg) +Figure 7: Linearity of $\pmb { h }$ -space. Linear interpolation and extrapolation on $h$ -space lead to gradual changes even to the unseen intensity and directions. Right side shows the interpolation results by positive scaling of $\Delta h _ { t } ^ { \mathrm { s m i l i n g } }$ . Left side shows the extrapolation results by negative scaling of $\Delta h _ { t } ^ { s m i l i n g }$ . Note that we did not train $\pmb { f } _ { t }$ for the negative direction. + +# 5.2 QUANTITATIVE COMPARISON + +Considering that our method can be combined with various diffusion models without finetuning, we do not find such a versatile competitor. Nonetheless, we compare Asyrp against DiffusionCLIP using the official code that edits the real images by finetuning the whole model. We asked 80 participants to choose the images with better quality, natural attribute change, and overall preference for given total of 40 sets of original images, ours, and DiffusionCLIP. Table 1 shows that Asyrp outperforms DiffusionCLIP in the all perspectives including the attributes unseen in the training dataset. We list the settings for fair comparison including the questions and example images in Appendix K.1. See $\textrm { \AA K } . 2$ for more evaluation metrics: segmentation consistency (SC) and directional CLIP similarity( $S _ { \mathrm { d i r } } )$ . + +# 5.3 ANALYSIS ON $h$ -space + +We provide detailed analyses to validate the properties of semantic latent space for diffusion models: homogeneity, linearity, robustness, and consistency across timesteps. + +Homogeneity. Figure 6 illustrates homogeneity of $h$ -space compared to $\epsilon$ -space. One $\Delta { h _ { t } }$ optimized for an image results in the same attribute change to other input images. On the other hand, one $\Delta \epsilon _ { t }$ optimized for an image distorts other input images. In Figure 10, applying $\Delta h _ { t } ^ { \mathrm { m e a n } } =$ $\begin{array} { r } { \frac { 1 } { N } \sum \Delta h _ { t } ^ { i } } \end{array}$ produces almost identical results where $i$ indicates indices of $N = 2 0$ random samples. + +Linearity. In Figure 7, we observe that linearly scaling a $\Delta h$ reflects the amount of change in the visual attributes. Surprisingly, it generalizes to negative scales that are not seen during training. Moreover, Figure 8 shows that combinations of different $\Delta h$ ’s yield their combined semantic changes in the resulting images. Appendix N.2 provides mixed interpolation between multiple attributes. + +![](images/ec4e0c1c15035d0be274617d3f54f6a61905cce32ac51db848c37eaa6738a5c4.jpg) +Figure 8: Linear combination. Combining multiple ∆hs leads to combined attribute changes. + +![](images/f16b3b6406f8475789e2643d74a9891b20a10d09c81ef00ec5fdfc72bd1f3e33.jpg) +Figure 9: Ramdom manipulation on $\pmb { h }$ -space and $\epsilon$ -space. (a) Adding random noise with magnitude of ∆hsmilingt and random direction in $h$ -space leads to realistic images with small changes. (b) Adding random noise with magnitude of $\Delta \epsilon _ { t } ^ { \mathrm { s m i l i n g } }$ and random direction in $\epsilon$ -space leads to severely distorted images. (c) Adding random noises with tripled magnitude produce diverse visual changes in realistic images. + +![](images/516dc2690d986f75d0a76feebed8568de3bb5e07882b27922278b8419f1f3e97.jpg) +Figure 10: Consistency on $\pmb { h }$ -space. Results of $\Delta { h _ { t } }$ , $\Delta h _ { t } ^ { \mathrm { m e a n } }$ and $\Delta h ^ { \mathrm { g l o b a l } }$ are almost identical for in-domain samples. However, we choose $\Delta { h _ { t } }$ over others to prevent unexpected small difference in the unseen domain. We provide more results in $\ S \mathrm { { L . 1 } }$ . + +Robustness. Figure 9 compares the effect of adding random noise in $h$ -space and $\epsilon$ -space. The random noises are chosen to be the vectors with random directions and magnitude of the example $\Delta { h _ { t } }$ and $\Delta \epsilon _ { t }$ in Figure 6 on each space. Perturbation in $h$ -space leads to realistic images with a minimal difference or some semantic changes. On the contrary, perturbation in $\epsilon$ -space severely distorts the resulting images. See Appendix D.2 for more analyses. + +Consistency across timesteps. Recall that $\Delta \boldsymbol { h } _ { t }$ for all samples are homogeneous and replacing them by their mea time-invariant $\Delta h _ { t } ^ { \mathrm { m e a n } }$ ults. Interesinstead of , in Figure 10, we observe that addalso yields similar results where $\Delta { h ^ { \mathrm { g l o b a l } } } ~ \dot { = } ~ \frac { 1 } { T _ { e } } \sum _ { t } \Delta { h _ { t } ^ { \mathrm { m e a n } } }$ $\Delta \boldsymbol { h } _ { t }$ $T _ { e }$ denotes the length of the editing interval $[ T , t _ { \mathrm { e d i t } } ]$ . Though we use $\Delta \boldsymbol { h } _ { t }$ to deliver the best quality and manipulation, using $\Delta h _ { t } ^ { \mathrm { m e a n } }$ or even $\Delta h ^ { \mathrm { g l o b a l } }$ with some compromise would be worth trying for simplicity. We report more detail about mean and global direction in Appendix L.1. + +# 6 CONCLUSION + +We proposed a new generative process, Asyrp, which facilitates image editing in a semantic latent space $h$ -space for pretrained diffusion models. $h$ -space has nice properties as in the latent space of GANs: homogeneity, linearity, robustness, and consistency across timesteps. The full editing process is designed to achieve versatile editing and high quality by measuring editing strength and quality deficiency at timesteps. We hope that our approach and detailed analyses help cultivate a new paradigm of image editing in the semantic latent space of diffusion models. 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International Journal of Computer Vision, 127(3):302–321, 2019. + +# Appendix + +# Table of Contents + +A Related work 14 +B More discussion 1 5 +C Proof of Theorem 1 1 5 +D Additional supports for $\pmb { h }$ -space with Asyrp 17 +D.1 Random perturbation on $\epsilon$ -space without Asyrp . 17 +D.2 Robustness and semantics in $h$ -space and $\epsilon$ -space with Asyrp 17 +D.3 Choice of $h$ -space in U-Net 17 +E Implicit neural directions 17 +$\mathbf { F }$ Quality improvements by non-accelerated sampling with scaled $\Delta h _ { t }$ 18 +G Editing strength and editing flexibility 18 +G.1 Editing strength and $t _ { e d i t }$ 18 +G.2 Editing flexibility and $t _ { \mathrm { b o o s t } }$ . 18 + +H Quality boosting 1 9 + +# I Algorithm 2 3 + +# J Training details 2 3 + +J.1 Loss coefficients . 23 +J.2 Training with random sampling instead of the training datasets . 24 + +# K Evaluation + +# 2 4 + +K.1 User study . 24 +K.2 Segmentation consistency and directional CLIP similarity 25 + +# L Directions + +# 26 + +L.1 Global direction 26 +L.2 Compare three methods 29 + +# M Random sampling + +3 0 + +# N More samples + +# + +N.1 ImageNet 32 +N.2 Multi-interpolation 32 +N.3 More results on all datasets 32 + +![](images/ccb938a111ed5b23226f87f2c9601955a9f9a495d880770b80c3a6c0651db3e8.jpg) +Figure 11: Editing dog to smile. We provide high-resolution results of editing real images in the teaser and more. + +# A RELATED WORK + +After Sohl-Dickstein et al. (2015), denoising diffusion probabilistic models (DDPMs) provide a universal approach for generative modeling (Ho et al., 2020). On the other hand, Song et al. (2020b) suggests score-based model and unifies SDEs incorporating diffusion models with score-based models. Subsequent works renovate diffusion models by focusing on architectures, scheduling, weighting, and fast sampling (Nichol & Dhariwal (2021), Karras et al. (2022), Choi et al. (2022), Song et al. (2020a), Watson et al. (2022)). They mainly consider random generation rather than controlled generation. + +In the meantime, Dhariwal & Nichol (2021) introduces classifier guidance not only improving the quality of images but also retrieving specific class of images. Since it can apply any guidance, its variants have emerged (Sehwag et al. (2022), Avrahami et al. (2022), Liu et al. (2021), Nichol et al. (2021)). However it requires a noise-dependent classifier (or any off-the-shelf models) and additional cost to compute gradients for the guidance during its sampling process. The other works try to control the generative process using image-space guidance (Choi et al. (2021), Meng et al. (2021), Lugmayr et al. (2022), Avrahami et al. (2022)). They manipulate resulting images by matching noisy images with target images during the reverse process. Still, it is hard to expect delicate control of the reverse process from the image guidance. Furthermore, Preechakul et al. (2022) introduces an extra encoder which encodes the semantic features of a real image in order to condition the generative process. Although the semantics allow one to control diffusion models, it requires additional training from scratch with the encoder and inherently can not use the other pretrained diffusion models. + +For controllability, Rombach et al. (2022) and Vahdat et al. (2021) apply another approach which adapts VAE (Kingma & Welling, 2013) and autoencoder (Rumelhart et al., 1985) to diffusion models. In spite of their great success in editing, their diffusion models learn the distribution of the learned embeddings in VAE or autoencoder, not the images. Kim & Ye (2021) proposes another strategy: fine-tuning a whole diffusion model for image editing. It shows valid performance but it requires each fine-tuned model corresponding each attribute. + +In comparison, Asyrp enables outstanding manipulation without high computation, specifically designed architectures, or fine-tuning whole models. + +Meanwhile, generative adversarial network Goodfellow et al. (2020) address their latent space for image editing (Ling et al. (2021), Hark ¨ onen et al. (2020), Chefer et al. (2021), Shen et al. (2020), ¨ + +Yuksel et al. (2021), Patashnik et al. (2021), Gal et al. (2021), Dai et al. (2019), Xu et al. (2022)). ¨ However they have to conduct ‘inversion’ to their latent space for real image editing and ‘GAN inversion’ is often challenging and produces unexpected appearance changes. + +On the contrary, Asyrp enables to use latent space of real images by nearly perfect easy inversion of DDIM. + +# B MORE DISCUSSION + +In this section, we discuss the pros and cons of diffusion-model-based and GAN-based methods. +And we provide guidelines for further improvements. + +GAN-based latent manipulation methods Patashnik et al. (2021); Gal et al. (2021)require careful inversion from real images to latent codes for real image editing. On the contrary, our proposed method based on diffusion models has a powerful advantage; the sophisticated inversion method is not necessary. This means that we can obtain the latent code of an arbitrary real image even if the image is not in the trained domain. On the other hand, several inversion methods have been proposed for GANs to obtain the latent of the real image, and the corresponding latent manipulation method should be considered for each inversion method. For example, it is difficult to apply the method of editing in $w$ space to the method of inversion using $w ^ { + }$ space. + +However, GANs have the advantage of fast sampling. In addition, diffusion models have a relatively slow sampling time. Additionally, we have to be aware of the time steps of diffusion models, which is still less well known. + +The advantage of being free from Inversion provides the following milestones. The manipulation in the latent of the diffusion models is the same as the editing in real images. It can be expanded to segmentation, clustering, classification, etc. in $h$ -space for real-world images. + +It would be an interesting research direction to employ previous techniques. Our method can be used in conjunction with gradient guidance methods. Although we do not focus on random sampling, ours works effectively for sampling with stochastic. (See $\ S \mathbf { M } .$ ) It may bring more diverse methods to steer diffusion models. + +$h$ -space in the latent diffusion models such as stable diffusion, is another interesting research direction. The main contribution of our paper is only modifying $\mathbf { P } _ { t }$ while preserving $\mathbf { D } _ { t }$ , and can be adapted with latent diffusion models. However, since the latent meaning may be different due to structural differences, research on this is needed. + +Furthermore, all of the properties of $h$ -space according to the time step has not been fully discussed so far. Research on them can be expected to expand further. + +Limitations. Editing with Asyrp seldom yields changes in overall style or peripheral objects but edits attributes of the main object. Style transfer using frozen diffusion models is our future work. + +Societal impact / Ethics statement. Techniques for high-quality image manipulation such as Asyrp should be accompanied by social and/or technical solutions to prevent abuse. We acknowledge the potential ethical implications that may arise from the use of our image manipulation technique, Asyrp. We advocate for the development and implementation of social and technical solutions to prevent potential abuses such as spreading disinformation or propaganda. We are committed to ensuring fairness and non-discrimination, legal compliance, and research integrity in our work. + +# C PROOF OF THEOREM 1 + +Proof of Theorem 1. Let $\epsilon _ { t } ^ { \theta }$ be a predicted noise during the original reverse process at $t$ and $\tilde { \epsilon } _ { t } ^ { \theta }$ be its shifted counterpart. Then, $\Delta \pmb { x } _ { t } = \tilde { \pmb { x } } _ { t - 1 } - \pmb { x } _ { t - 1 }$ is negligible where $\tilde { \pmb { x } } _ { t - 1 } = \bar { \sqrt { \alpha _ { t - 1 } } } \mathbf { P } _ { t } ( \tilde { \epsilon } _ { t } ^ { \theta } ( \pmb { x } _ { t } \bar { \Big ) } ) +$ $\mathbf { D } _ { t } ( \tilde { \epsilon } _ { t } ^ { \theta } ( { \pmb x } _ { t } ) )$ . + +![](images/278cf3f351f5df6dbbb8f4ac91469563ba5dc86a95e75f069873149e24faa7f4.jpg) +Figure 12: Illustration of Theorem 1. Upper blue line describes applying noise $\tilde { \epsilon } _ { t } = \epsilon _ { t } + \Delta \epsilon _ { t }$ to produce $\mathbf { P } _ { t } ( \tilde { \epsilon } _ { t } ) =$ the shifted predicted $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ . However, the shift due to $\Delta \epsilon _ { t }$ is canceled out by the shift in $\mathbf { D } _ { t } ( \tilde { \epsilon } _ { t } )$ due to $\Delta \epsilon _ { t }$ . As a results, applying $\Delta \epsilon _ { t }$ both on $\mathbf { P } _ { t }$ and $\mathbf { D } _ { t }$ brings identical outputs to the original. + +Define $\tilde { \epsilon } _ { t } ^ { \theta } ( { \bf { x } } _ { t } ) = \epsilon _ { t } ^ { \theta } ( { \bf { x } } _ { t } ) + \Delta \epsilon _ { t }$ , $\{ \beta _ { t } \} _ { t = 1 } ^ { T } = \{ \beta _ { 1 } = \beta _ { \operatorname* { m i n } } , . . . , \beta _ { T } = \beta _ { \operatorname* { m a x } } \}$ , and $\begin{array} { r } { \alpha _ { t } = \prod _ { s = 1 } ^ { t } ( 1 - } \end{array}$ $\beta _ { s , \ }$ ). Note that $\beta _ { \mathrm { m a x } }$ is defined as a small value (e.g., $\beta _ { \mathrm { m a x } } = 0 . 0 0 1 )$ and $\{ \beta _ { t } \} _ { t = 1 } ^ { T }$ are defined by a decreasing schedule from $\beta _ { T } = \beta _ { \mathrm { m a x } }$ to $\beta _ { 1 } = \beta _ { \mathrm { m i n } } \approx 0$ (e.g., $\beta _ { \mathrm { m i n } } = 0 . 0 0 0 0 1 \rangle$ . Then, + +$$ +\tilde { \pmb { x } } _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } \big ( \tilde { \epsilon } _ { t } ^ { \theta } ( \pmb { x } _ { t } ) \big ) + \mathbf { D } _ { t } \big ( \tilde { \epsilon } _ { t } ^ { \theta } ( \pmb { x } _ { t } ) \big ) +$$ + +$$ += \sqrt { \alpha _ { t - 1 } } \left( \frac { x _ { t } - \sqrt { 1 - \alpha _ { t } } \left( \epsilon _ { t } ^ { \theta } ( x _ { t } ) + \Delta \epsilon _ { t } \right) } { \sqrt { \alpha _ { t } } } \right) + \sqrt { 1 - \alpha _ { t - 1 } } \cdot \left( \epsilon _ { t } ^ { ( \theta ) } \left( x _ { t } \right) + \Delta \epsilon _ { t } \right) +$$ + +$$ += \sqrt { \alpha _ { t - 1 } } \mathbf { P } _ { t } ( \epsilon _ { t } ^ { \theta } ( x _ { t } ) ) + \mathbf { D } _ { t } ( \epsilon _ { t } ^ { \theta } ( x _ { t } ) ) - \frac { \sqrt { \alpha _ { t - 1 } } \sqrt { 1 - \alpha _ { t } } } { \sqrt { \alpha _ { t } } } \cdot \Delta \epsilon _ { t } + \sqrt { 1 - \alpha _ { t - 1 } } \cdot \Delta \epsilon _ { t } +$$ + +$$ += x _ { t - 1 } + \left( - \frac { \sqrt { 1 - \alpha _ { t } } } { \sqrt { 1 - \beta _ { t } } } + \sqrt { 1 - \alpha _ { t - 1 } } \right) \cdot \Delta \epsilon _ { t } +$$ + +$$ += x _ { t - 1 } + \left( - \frac { \sqrt { 1 - \alpha _ { t } } } { \sqrt { 1 - \beta _ { t } } } + \frac { \sqrt { 1 - \prod _ { s = 1 } ^ { t - 1 } \left( 1 - \beta _ { s } \right) } \sqrt { 1 - \beta _ { t } } } { \sqrt { 1 - \beta _ { t } } } \right) \cdot \Delta \epsilon _ { t } +$$ + +$$ += x _ { t - 1 } + \left( \frac { \sqrt { 1 - \alpha _ { t } - \beta _ { t } } - \sqrt { 1 - \alpha _ { t } } } { \sqrt { 1 - \beta _ { t } } } \right) \cdot \Delta \epsilon _ { t } \quad \cdot : \alpha _ { t } = \prod _ { s = 1 } ^ { t } \left( 1 - \beta _ { s } \right) +$$ + +$$ +\therefore \Delta x _ { t } = { \tilde { x } } _ { t - 1 } - x _ { t - 1 } = \left( { \frac { { \sqrt { 1 - \alpha _ { t } - \beta _ { t } } } - { \sqrt { 1 - \alpha _ { t } } } } { { \sqrt { 1 - \beta _ { t } } } } } \right) \cdot \Delta \epsilon _ { t } { \mathrm { ~ i s ~ n e g l i g i b l e ~ } } \quad \cdot \cdot \beta _ { t } < \beta _ { m a x } +$$ + +# D ADDITIONAL SUPPORTS FOR $h$ -space WITH ASYRP + +D.1 RANDOM PERTURBATION ON $\epsilon$ -space WITHOUT ASYRP + +In $\ S 3 . 1$ , we argue that if both $\mathbf { P } _ { t }$ and $\mathbf { D } _ { t }$ are shifted, we can not manipulate $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ . In Figure 13, we do not observe the noticeable difference between (a) and (b) which are the result of the original reverse process of DDIM and the one with shifting both terms, respectively. + +# D.2 ROBUSTNESS AND SEMANTICS IN $h$ -space AND $\epsilon$ -space WITH ASYRP + +In $\mathrm { ~ \ S ~ } 3 . 2$ , we also argue that $h$ -space is more robust than $\epsilon$ -space with Asyrp. In Figure 13, we observe that small random noise $z \sim \mathcal { N } ( 0 , \mathbf { I } )$ in $\epsilon$ -space degrades the resulting image without semantic changes (c) and much larger random noise in $h$ -space yields random semantic changes without severe artifacts (d). + +Note that diffusion models are designed as latent variable models with learned Gaussian transitions and the reverse process should also be close to Gaussian. Based on the assumption, we manage $\epsilon _ { t } ^ { \theta }$ to follow the original Gaussian distribution as follows. Adding $z \sim \mathcal { N } ( 0 , \sigma \bar { \bf { I } } )$ to $\epsilon _ { t } ^ { \theta }$ expands the distribution of the predicted noise and may produce distorted images. To preserve the distribution,√ we scale $\tilde { \epsilon } _ { t } ^ { \theta } = ( \epsilon _ { t } ^ { \theta } + z ) / \sqrt { 1 ^ { 2 } + { \sigma } ^ { 2 } }$ . Still, the resulting images are almost identical compared to the original images where $\tilde { \epsilon } _ { t } ^ { \theta } = \epsilon _ { t } ^ { \theta } + z$ as shown in Figure 13. It is no wonder that the scaling does not improve the distorted results since the additive random noise disturbs the denoising operation of the predicted random noise. + +![](images/bb209ea4fa736576560335d535c5528804419b23be9234941f9e22111f467649.jpg) +Figure 13: (a) The reconstructed image by the original DDIM inversion process which is almost indistinguishable to the real image. (b) The result from adding random noise $z$ both on $\mathbf { P } _ { t }$ and $\mathbf { D } _ { t }$ . (a) looks identical to (b). The insets of (b, c) depict the full SSIM image from (a). It shows that simply shifting $\epsilon _ { t } ^ { \theta }$ without Asyrp does not affect the result. (c) Adding $z \sim \mathcal { N }$ to $\epsilon$ -space with Asyrp easily degrades image with little semantic change. (d) Adding $z \sim \mathcal { N }$ to $h$ -space with Asyrp yields random semantic change without image degradation. (e) Correlation between image degradation and noise strength in the two spaces. + +# D.3 CHOICE OF $h$ -space IN U-NET + +As shown in Figure 14, there are many other candidates for $h$ -space in the architecture. Among the layers, we choose the 8th layer, the bridge of the U-Net based architecture. The layer is not influenced by any skip connection, has the smallest spatial dimension with compressed information, and is located just before the upsampling blocks. Thus, we assume that it could possibly be considered as the most suitable latent embedding. To confirm the assumption, we train $\pmb { f } _ { t }$ on the other layers. The results are shown in Figure 15. We carefully tuned the training hyperparameters $\lambda _ { C L I P }$ and $\lambda _ { r e c o n } )$ ) for fair comparison. The 1st to the 6th layers hardly bring visible changes. The 7th and 9th layers bring not only the desired changes but also difficulty in finding optimal hyperparameters. After the 9th layer, the results bear severe artifacts. + +# E IMPLICIT NEURAL DIRECTIONS + +Figure 16 illustrates the neural implicit function $f _ { t }$ . It has only two 1x1 convolution layers with 512 channels. Note that we haven’t explored the network architecture much. + +![](images/3e8c45121a1bfb589a589081725c2d792d7bced359e7ac19f594c2ade31cda78.jpg) +Figure 14: Location of $\pmb { h }$ -space. The U-Net architecture of diffusion models outputs $2 5 6 \times 2 5 6$ images. Each layer is indexed with a number along the operating sequence of the model. The 8th layer is our $h$ -space which is not directly influenced by a skip connection. + +# F QUALITY IMPROVEMENTS BY NON-ACCELERATED SAMPLING WITH SCALED $\Delta h _ { t }$ + +Figure 17 shows the quality improvements by non-accelerated sampling with scaled $\Delta { h _ { t } }$ described in $\ S \ S . 4$ . Even with different number of inference steps, we observe similar changes of an attribute if we preserve the sum of $\Delta \boldsymbol { h } _ { t }$ . This scaling technique allows non-accelerated sampling with 1000 steps for the models trained by accelerated training with 40 steps. Using non-accelerated sampling with scaled $\Delta { h _ { t } }$ leads to the same magnitude of manipulation and higher-quality images. In our experiments, it takes about 1.5 seconds to sample for 40 steps and 40 seconds for 1000 steps. + +# G EDITING STRENGTH AND EDITING FLEXIBILITY + +# G.1 EDITING STRENGTH AND $t _ { e d i t }$ + +Figure 18 shows the results according to $t _ { \mathrm { e d i t } }$ . If $t _ { \mathrm { e d i t } }$ is too high, the length of the editing process becomes too short resulting in insufficient changes. On the contrary, too low $t _ { \mathrm { e d i t } }$ causes excessively unnecessary manipulation from the long editing process. + +We observe that $t _ { \mathrm { e d i t } }$ is one of the important hyperparameters. We argue that the formula for choosing $t _ { \mathrm { e d i t } }$ using editing strength is reasonable because it applies to all five different datasets despite its sensitivity, even though the choice of $\mathrm { L P I P S = 0 . 3 3 }$ is empirical. We provide ablation of thresholds in Figure 22. + +Additionally, these results imply why we need to use sufficiently low $t _ { \mathrm { e d i t } }$ in the unseen domains. +Editing interval with insufficient editing strength struggles to escape from the training domain. + +# G.2 EDITING FLEXIBILITY AND tboost + +Table 2 shows the prompts, $t _ { \mathrm { e d i t } }$ , and $t _ { \mathrm { b o o s t } }$ . Intuitively, the attributes with larger visual changes have smaller cosine similarity and require longer editing interval. Figure 19 shows the average $\mathrm { L P I P S } ( \pmb { x } , \mathbf { P } _ { t } )$ and $\mathrm { L P I P S } ( { \pmb x } , { \pmb x } _ { t } )$ of 100 samples on all datasets. Note that $t _ { \mathrm { e d i t } }$ and $t _ { \mathrm { b o o s t } }$ dif + +![](images/20e18f4b6da57f683105204199ad21e00b13391adc9cd4bf0a95392c4e8d14c8.jpg) +Figure 15: Exhaustive enumeration over the choices for semantic latent space. We show the result of the training data. We observe that the eighth layer ( $h$ -space) of the U-net suits the best for the semantic latent space. + +![](images/174bbae683903d32c4469c85c1da04852c054dd195ebaf00a66e2a411d7df08d.jpg) +Figure 16: Illustration of $\pmb { f } _ { t }$ . We use group norm and swish following DDPM. + +fer across datasets and attributes, and they are chosen by the formulas in $\ S 4 .$ . We provide ablation of $t _ { \mathrm { b o o s t } }$ in Figure 23. + +# H QUALITY BOOSTING + +We validate the effectiveness of our quality boosting (§ 4.2) in the original DDIM process and in Asyrp. + +Smiling with + +![](images/e119805b4e47eb46571e4d9c938e16083540325efbc089de26542ba2e25cbc3d.jpg) +Figure 17: Quality improvements by non-accelerated sampling with scaled $\Delta \boldsymbol { h } _ { t }$ . We observe quality improvements by non-accelerated (1000-step) sampling with scaled $\Delta \boldsymbol { h } _ { t }$ from accelerated (40-step) training described in $\ S 3 . 4$ . Please zoom in for detailed comparison. + +$$ +t _ { e d i t } = 9 0 0 ~ t _ { e d i t } = 8 0 0 ~ t _ { e d i t } = 7 0 0 ~ t _ { e d i t } = 6 0 0 ~ t _ { e d i t } = 5 0 0 ~ t _ { e d i t } = 4 0 0 ~ t _ { e d i t } = 3 0 0 ~ t _ { e d i t } = 2 0 0 ~ t _ { e d i t } = 1 0 0 +$$ + +![](images/5e9f94c3b8ff254f0b1c246dba27d57fe4749c6b53d4da39b435229817caa682.jpg) +Figure 18: Importance of choosing proper $t _ { \mathrm { e d i t } }$ . We explore various $t _ { \mathrm { e d i t } }$ with smiling. Too short editing interval struggles to manipulate attributes. Excessive editing strength results in degraded images. + +
Datasetsrc_txttrg_txtCLIP similaritytedittboost
CelebA-HQ LSUN-churchPerson FaceYoung person0.905515167
Smiling face0.899513167
FaceSad face0.894513167
FaceAngry face0.892512167
FaceTanned face0.886512167
FaceDisgusted face0.880511167
Person HumanPerson with makeup0.875509167
PersonZombie0.868506167
PersonPerson with bald head0.861505167
Person with curly hair0.835499167
Human PersonNeanderthal0.802490167
PersonMark Zuckerberg0.797489167
HumanNicolas Cage0.710461167
PhotoPainting in the style of Pixar0.667446167
PhotoPainting in Modigliani style Self-portrait by Frida Kahlo0.565403167
ChurchGothic Church0.443321167
ChurchTemple0.912 0.898371293
ChurchDepartment store0.841367293
ChurchWooden House349293
Church0.793333293
ChurchAncient traditional Asian tower0.784330293
ChurchRed brick wall Church0.774326293
BedroomFactory Princess Bedroom0.702301293
LSUN-bedroomBedroom Hotel Bedroom0.912370221
AFHQDog0.917371221
DogHappy Dog0.883430167
Sleepy Dog0.866422167
DogAngry Dog0.860419167
Dog DogWolf0.850412167
METFACESPainting of a personYorkshire Terrier0.690302167
Painting of a personPainting of a Sad person Painting of a Smiling person0.921 0.908330 320180 180
Painting of a personPainting of a Disgusted person0.879291180
+ +![](images/d400055c06dce9745d0694a750ea708aaf62c69eec8d8f94c39559d4b4c1588e.jpg) +Table 2: Prompts, CLIP similarity, $t _ { \mathrm { e d i t } }$ , and $t _ { \mathrm { b o o s t } }$ for all attributes in the experiments. +Figure 19: Average LPIPS $( \boldsymbol { \mathbf { \mathit { x } } } , \boldsymbol { \mathbf { \mathit { P } } } _ { t } )$ and LPIPS $( \pmb { x } , \pmb { x } _ { t } )$ of 100 samples on all datasets. + +Figure 20 shows the effect of quality boosting with the original DDIM reverse process. Although the reverse process of DDIM has a nearly-perfect inversion property, we observe some noise by zooming in. Our quality boosting improves the quality of a sample and concurrently keeps nearly + +![](images/1434d974f30676869508e856aafe42dbc606c49aa0ad3a871abe5831d5d8d9ad.jpg) +Figure 20: Ablation study of quality boosting on the original DDIM process without Asyrp. Our quality boosting enhances fine details and prevents images from being noisy in the original DDIM process. Please zoom in for detailed comparison. + +![](images/c3c2bcb78752a1c696fd47a0788206292074dab31e50f5bcc13a4530d4914cd3.jpg) +Figure 21: Ablation study of quality boosting with Asyrp. Our quality boosting enhances fine details and prevents images from being noisy. Note that the source of degradation is DDIM process, not Asyrp, confirmed in Figure 20. Please zoom in for detailed comparison. + +![](images/9105ae3ff594358c3dd1ad873a025ed2c74b6b492fe058518f574f1ebe0cc84f.jpg) +Figure 22: Analyzing the effect of the hyperparameters The figure shows that the calculated parameter works effectively as the maximum boundary of editing while maintaining the quality of the image. + +![](images/067bbe65a525c781959e6f88a9e6ed2322b6ae14dd2301e761c08b57fdce7806.jpg) +Figure 23: Analyzing the effect of the hyperparameters We observe that the quality of results has robustness to $\gamma _ { t _ { \mathrm { b o o s t } } }$ , except for too large $\gamma _ { t _ { \mathrm { b o o s t } } }$ . + +perfect inversion property. We observe that $t _ { \mathrm { b o o s t } }$ is not sensitive, but the larger interval brings the less preservation. + +Figure 21 shows quality improvements by our quality boosting in Asyrp. + +# I ALGORITHM + +Figure 24 illustrates generative process. Algorithm 1 and 2 describe training algorithm and inference algorithm of Asyrp, respectively. + +# J TRAINING DETAILS + +# J.1 LOSS COEFFICIENTS + +Table 3 reports loss coefficients for each attribute. $\lambda _ { \mathrm { { C L I P } } } \mathrm { { s } }$ near 0.8 are suitable for most in-domain attributes. For unseen domains, higher coefficient leads to more noticeable changes. Note that we use $\lambda _ { \mathrm { r e c o n } } = \mathbf { C } \mathbf { L } \mathbf { I } \mathbf { P }$ similarity $^ \ast 3$ which reduces L1 loss when an attribute needs a lot of change. + +![](images/4a4dfe0ce3ef126a67f1e86d022d8d15a8d17d09cceb878b5663ee16091daf9b.jpg) +Figure 24: Overview of our generative process. + +J.2 TRAINING WITH RANDOM SAMPLING INSTEAD OF THE TRAINING DATASETS + +Apparently, for training, inverting real-images can be replaced by random sampling. It refers to using $\mathbf { \boldsymbol { x } } _ { T } \sim \mathcal { N } ( 0 , \mathbf { I } )$ instead of $\begin{array} { r } { \pmb { x } _ { T } = \bar { q } ( \pmb { x } _ { 0 } ) \cdot \bar { \prod } _ { t = 1 } ^ { T } q \left( \pmb { x } _ { t } \mid \bar { \pmb { x } } _ { t - 1 } \right) } \end{array}$ where $\pmb { x } _ { 0 } \sim p _ { d a t a } ( x )$ . It allows us to train Asyrp only with the pretrained network and without extra dataset. Using random samples has tradeoff between preservation of contents and possible amount in editing. It take advantages when a target attribute requires large amount changes. We assume that the inversion of real-image is in the long tail of a Gaussian distribution because of realistic background or detailed clothes. On the other hand, random noise is considered to be closer to the mean of the normal, so it is easier to find directions. It can easily bring larger changes but also easily alter the contents. On the contrary, training with inversion shows the opposite property. + +We train $\pmb { f } _ { t }$ with random sampling for attributes whose identity preservation is not important to take advantage of these properties. The rightmost column in Table 3 shows the choices. + +# K EVALUATION + +# K.1 USER STUDY + +We conduct user study to compare the performance of Asyrp and DiffusionCLIP (Kim & Ye, 2021) on Celeba-HQ (Liu et al., 2015) and LSUN-church (Yu et al., 2015). We use official checkpoints provided by DiffusionCLIP except for some facial attributes whose checkpoint do not exist. We tried our best to tune their hyperparameters following the manual for fair comparison. Example images are shown in Figure 25-27. + +We use smiling and sad for in-domain CelebA-HQ attributes, Pixar and Neanderthal for unseen-domain CelebA-HQ attributes, and department store, ancient, and wooden for LSUN-church. + +In unseen domain and Lsun-church, we use official checkpoints provided by DiffusionCLIP. We also randomly select 8 images for each CelebA-HQ attribute and 12 images for each LSUN-church attribute. + +We observe that DiffusionCLIP works better in changing the holistic style of images. At the same time, it is short of the ability to bring semantic changes and suffers noisy results and a lack of diversity. The problems would be caused by fine-tuning the whole diffusion model. + +We use the following questions for the survey. 1) Quality: Which image quality do you think is better? (clear and less noisy) 2) Attribute: Which image do you think is “Attribute(e.g., Smiling) naturally”? 3) Overall: Which image do you think is better considering the above evaluation criteria? + +# Algorithm 1: Editing(Inference) + +Input: $x _ { 0 }$ (Input image), $\{ f _ { t } ^ { i } \} _ { i = 1 } ^ { M }$ (M Neural implicit functions for M attributes), $\{ c _ { i } \} _ { i = 1 } ^ { M }$ (user defined M scaling coefficients for M attributes), $\epsilon _ { \theta }$ (frozen pretrained model), $S _ { f o r }$ (# of inversion steps), $S _ { g e n }$ (# of inference steps), $t _ { \mathrm { e d i t } }$ (computed from $\ S 4 . 1$ ), tboost (computed from $\ S 4 . 2$ ) 1 Function Editor $( x _ { 0 } , \epsilon _ { \theta } , \{ c _ { i } \} _ { i = 1 } ^ { M } , S _ { f o r } , S _ { g e n } , * )$ : // step 1: Semantic encoding 2 Define $\{ \tau _ { s } \} _ { s = 1 } ^ { S _ { f o r } }$ s.t $\tau _ { 1 } = 0$ , $\tau _ { S _ { f o r } } = T$ 3 for $s = 1 , 2 , . . . , S _ { f o r } - 1$ do 4 $\epsilon \epsilon _ { \theta } ( x _ { \tau _ { s } } , \tau _ { s } )$ 5 $x _ { \tau _ { s + 1 } } = \sqrt { \alpha _ { \tau _ { s } } } x _ { \tau _ { s } } + \sqrt { 1 - \alpha _ { \tau _ { s } } } \epsilon$ 6 $\{ \tilde { \tau } _ { s } \} _ { s = 1 } ^ { S _ { g e n } }$ Mas.t $\tilde { \tau } _ { 1 } = 0$ a, $\tilde { \tau } _ { S _ { e d i t } } = t _ { e d i t }$ , $\tilde { \tau } _ { S _ { n o i s e } } = t _ { \mathrm { b o o s t } }$ and $\tilde { \tau } _ { S _ { g e n } } = T$ 7 $\tilde { x } _ { \tilde { \tau } _ { S _ { g e n } } } = x _ { \tilde { \tau } _ { S _ { f o r } } }$ 8 for $s = S _ { g e n } , S _ { g e n } - 1 , . . . , 2$ do // phase 1: editing 9 if $s \geq S _ { e d i t }$ then 10 Extract feature map $h _ { \tilde { \tau } _ { s } }$ from $\epsilon _ { \theta } ( \tilde { x } _ { \tilde { \tau } _ { s } } )$ 11 $\begin{array} { r l } & { \Delta h _ { \tilde { \tau } _ { s } } = \frac { S _ { f o r } } { S _ { g e n } } ( \sum _ { i = 1 } ^ { M } c _ { i } f _ { \tilde { \tau } _ { s } } ^ { i } ( h _ { \tilde { \tau } _ { s } } ) ) } \\ & { \tilde { \epsilon } = \epsilon _ { \theta } ( \tilde { x } _ { \tilde { \tau } _ { s } } | \Delta h _ { \tilde { \tau } _ { s } } ) } \\ & { \epsilon = \epsilon _ { \theta } ( \tilde { x } _ { \tilde { \tau } _ { s } } ) } \\ & { \sigma _ { \tilde { \tau } _ { s } } = 0 } \end{array}$ 12 13 14 // phase 2: denoising 15 else if $s \geq S _ { n o i s e }$ then 16 ϵ˜ = ϵ = ϵθ(˜xτ˜s ) 17 στ˜ = 0 // phase 3: quality boosting 18 else 19 $\begin{array} { r l } & { \quad \Big \lfloor \tilde { \epsilon } = \epsilon = \epsilon _ { \theta } ( \tilde { x } _ { \tilde { \tau } _ { s } } ) } \\ & { \quad \Big \lfloor \sigma _ { \tilde { \tau } _ { s } } = \sqrt { \left( 1 - \alpha _ { \tilde { \tau } _ { s } - 1 } \right) / \left( 1 - \alpha _ { \tilde { \tau } _ { s } } \right) } \sqrt { 1 - \alpha _ { \tilde { \tau } _ { s } } / \alpha _ { \tilde { \tau } _ { s } - 1 } } } \\ & { \quad z \sim N ( 0 , 1 ) } \\ & { \quad \tilde { x } _ { \tilde { \tau } _ { s - 1 } } = \sqrt { \alpha _ { \tilde { \tau } _ { s - 1 } } } \big ( \frac { \tilde { x } _ { \tilde { \tau } _ { s } } - \sqrt { 1 - \alpha _ { \tilde { \tau } _ { s } } } \tilde { \epsilon } } { \sqrt { \alpha _ { \tilde { \tau } _ { s } } } } \big ) + \sqrt { 1 - \alpha _ { \tilde { \tau } _ { s - 1 } } - \sigma _ { \tilde { \tau } _ { s } } ^ { 2 } } \epsilon + \sigma _ { \tilde { \tau } _ { s } } z } \end{array}$ 20 21 22 + +return $\tilde { x } _ { 0 }$ (manipulated image) + +As for LSUN-church, we provide a set of four images at once and add a question: 3) Diversity: Which group do you think has a more diverse style? 4) Overall: Which image do you think is better considering the above evaluation criteria? + +# K.2 SEGMENTATION CONSISTENCY AND DIRECTIONAL CLIP SIMILARITY + +We compare Asyrp and DiffusionCLIP using directional CLIP similarity $( S _ { \mathrm { d i r } } )$ and segmentationconsistency (SC) following the protocols in DiffusionCLIP in Table 4 and Table 5. A pretrained CLIP (Radford et al., 2021) and segmentation models (Yu et al. (2018); Zhou et al. (2019; 2017); Lee et al. (2020)) are used to compute $S _ { \mathrm { d i r } }$ and SC, respectively. We choose three attributes (smiling, sad, tanned) for CelebA-HQ-in-domain, two attributes (Pixar,Neaderthal) for CelebA-HQunseen-domain and three attributes (department store, ancient, red brick) for LSUNchurch. + +For a fair comparison, we use official checkpoints of DiffusionCLIP and provide scores of the attributes (tanned, red brick) following Kim & Ye (2021). Regarding attributes without the official checkpoints (smiling,sad), we train DiffusionCLIP by ourselves with the official code. + +# Algorithm 2: Training Neural implicit function $f _ { t }$ + +Input: $\epsilon _ { \theta }$ (pretrained attribute), el), (sou $\{ x _ { 0 } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ ages to preco(target text), te), (# $f _ { t }$ (Neural implicitinversion steps), nction of(# of $y _ { s r c }$ $y _ { t a r }$ $S _ { f o r }$ $K$ training epochs), $t _ { \mathrm { e d i t } }$ (computed from $\ S 4 . 1 \}$ , $t _ { \mathrm { b o o s t } }$ (computed from $\ S 4 . 2 )$ ) + +// step 1: Precompute latents +1 Define $\{ \tau _ { s } \} _ { s = 1 } ^ { S _ { f o r } }$ s.t $\tau _ { 1 } = 0 , \tau _ { S _ { f o r } } = T$ +2 for $i = 1 , 2 , . . . , N$ do +3 for $s = 1 , 2 , . . . , S _ { f o r } - 1$ do +4 $\epsilon \epsilon _ { \theta } ( x _ { \tau _ { s } } ^ { ( i ) } , \bar { \tau _ { s } } )$ +5 $\begin{array} { r } { x _ { \tau _ { s + 1 } } ^ { ( i ) } = \sqrt { \alpha _ { \tau _ { s } } } x _ { \tau _ { s } } ^ { ( i ) } + \sqrt { 1 - \alpha _ { \tau _ { s } } } \epsilon } \end{array}$ 6 Save the latent x(i)τSf or + +7 $\tau _ { S _ { e d i t } } = t _ { e d i t }$ // step 2: Update $\pmb { f } _ { t }$ 8 for ep ${ \varkappa } h { = 1 , 2 , . . . , K }$ do + +$$ +\begin{array} { r } { P = \frac { \tilde { x } _ { \tau _ { s } } ^ { ( i ) } - \sqrt { 1 - \alpha _ { \tau _ { s } } } \epsilon _ { \theta } \left( \tilde { x } _ { \tau _ { s } } ^ { ( i ) } | \Delta h _ { \tau _ { s } } \right) } { \sqrt { \alpha _ { \tau _ { s } } } } ; P _ { s r c } = \frac { x _ { \tau _ { s } } ^ { ( i ) } - \sqrt { 1 - \alpha _ { \tau _ { s } } } \epsilon _ { \theta } \left( x _ { \tau _ { s } } ^ { ( i ) } \right) } { \sqrt { \alpha _ { \tau _ { s } } } } } \end{array} +$$ + +$$ +\begin{array} { r l } & { \tilde { x } _ { \tau _ { s - 1 } } ^ { ( i ) } = \sqrt { \alpha _ { \tau _ { s - 1 } } } P + \sqrt { 1 - \alpha _ { \tau _ { s - 1 } } } \epsilon _ { \theta } ( \tilde { x } _ { \tau _ { s } } ^ { ( i ) } ) } \\ & { x _ { \tau _ { s - 1 } } ^ { ( i ) } = \sqrt { \alpha _ { \tau _ { s - 1 } } } P _ { s r c } + \sqrt { 1 - \alpha _ { \tau _ { s - 1 } } } \epsilon _ { \theta } ( x _ { \tau _ { s } } ^ { ( i ) } ) } \end{array} +$$ + +$$ +E _ { t o t a l } \lambda _ { C L I P } L _ { d i r e c t i o n } ( P , y _ { t a r } , P _ { s r c } , y _ { s r c } ) + \lambda _ { r e c o n } | P - P _ { s r c } | +$$ + +We use 100 samples per attribute. Asyrp outperforms DiffusionCLIP on $S _ { \mathrm { d i r } }$ for all attributes. On SC, DiffusionCLIP achieves better or competitive scores. Because DiffusionCLIP manipulates images mostly by focusing on texture or color while preserving structure and shape, it takes advantage of getting higher SC scores. However, in Figure 28, results of smiling show that it is not proper to edit attributes which require structural manipulation. Note that better SC scores do not guarantee better qualitative performance. Results of Pixar also show the similar tendency of each method. We allow more structural changes than DiffusionCLIP while editing. Lower SC of our method comes from desirable structural changes as shown in Figure 28. + +$$ +S _ { \mathrm { d i r } } \left( x _ { \mathrm { g e n } } , y _ { \mathrm { t a r } } ; x _ { \mathrm { r e f } } , y _ { \mathrm { r e f } } \right) : = \frac { \left. \Delta I , \Delta T \right. } { \left\| \Delta I \right\| \left\| \Delta T \right\| } , +$$ + +# L DIRECTIONS + +# L.1 GLOBAL DIRECTION + +Figure 29 and Figure 30 show that the effects of mean direction and global direction are quite similar with $\Delta h _ { t }$ by $\mathbf { \Delta } f _ { t }$ in various attributes. We compute mean direction and global direction from 20 different images. + +We argue that $h$ -space is roughly homogeneous across samples and timesteps. However, we observe that $h$ -space is not completely independent to the conditions especially on unseen-domain (See Figure 29). Note that unseen-domains require a longer editing interval with small $t _ { \mathrm { e d i t } }$ . Therefore, we conjecture that the consistency of $h$ -space decreases at the end of the generative process. It is supported by additional experiments that L2 distance between $\Delta h _ { t }$ and global direction gradually increases along with timesteps. We leave a more detailed analysis on $h$ -space at different timesteps as a future work. + +Table 3: The coefficients range from 0.5 to 0.8 for in-domain attributes. Unseen domains need slightly stronger $\lambda _ { \mathrm { { C L I P } } } \mathrm { { s } }$ . We also report which attributes we train with random noise sampling. The criterion is which attributes require relatively less maintenance of identity. We use $\lambda _ { \mathrm { { r e c o n } } } =$ CLIP similarity $^ \ast 3$ which reduces L1 loss when an attribute needs a lot of change. + +
Datasetsrc_txttrg_txtXCLIPXreconfrom random noise
CelebA-HQPersonYoung person0.80.905*3X
FaceSmiling face0.80.899*3X
FaceSad face0.80.894*3X
FaceAngry face0.80.892*3X
FaceTanned face0.80.886*3X
FaceDisgusted face0.80.880*3X
PersonPerson with makeup0.80.875*3X
Person PhotoPerson with curly hair0.80.835*3X
Self-portrait by Frida Kahlo0.50.44330
HumanZombie0.60.868*30
PersonNicolas Cage0.80.710*30
Human PhotoPainting in the style of Pixar0.80.667*30
HumanPainting in Modigliani style0.80.565*30
LSUN-churchChurchNeanderthal Gothic church1.20.802*30
ChurchTemple0.80.912*30
ChurchDepartment store0.80.898*30
ChurchWooden house0.80.841*30
ChurchAncient traditional Asian tower0.8 0.80.793*30
ChurchRed brick wall Church0.80.784*3 0.774*30
ChurchFactory0.80.702*30
LSUN-bedroomBedroomPrincess bedroom1.50.912*30 0
BedroomHotel bedroom1.50.917*3
AFHQ-dogDogHappy dog0
DogSleepy dog1.50.883*30
Dog1.50.866*30
DogWolf Yorkshire terrier1.70.850*30
METFACESPainting of a personPainting of a Sad person2 1.50.690*3 0.921*30 X
Painting of a personPainting of a Smiling person1.50.908*3X
Painting of a personPainting of a Disgusted person1.50.879*3X
+ +Table 4: Quantitative evaluation on CelebA-HQ. + +
CelebA-HQ-in-domainCelebA-HQ-unseen-domain
SmilingSadTannedPixarNeanderthal
SdirSCSdirSCSdirSCSdirSCSdirSC
Asyrp (ours)0.92189.02%0.96488.90%0.99185.71%0.95676.51%0.80579.03
DiffusionCLIP0.81391.41%0.76089.93%0.88892.85%0.81189.91%0.66181.23%
+ +![](images/c2cf22d51b44b51faa69e1a51e8e027ba84a40344d50621d1a284856a9eb758c.jpg) +Figure 25: Asyrp vs. DiffusionCLIP on CelebA-HQ in-domain attributes. We observe that DiffusionCLIP struggles to change semantic facial attributes. Their official checkpoints do not exist and we ran careful hyperparameter tuning for training. + +![](images/06f561db0257bb962e8c406afb37da80494f09913966d224a12c97e0991f5899.jpg) +Figure 26: Asyrp vs. DiffusionCLIP on CelebA-HQ unseen-domain attributes. We use the official checkpoint provided by DiffusionCLIP. Asyrp works well even for the unseen-domains. + +Table 5: Quantitative evaluation on LSUN-church. + +
LSUN_church
Department storeancientred brick
SdirSCSdirSCSdirSC
Asyrp (ours)0.77857.62%0.94362.65%0.98965.83%
DIffusionCLIP0.66154.50%0.90764.82%0.96465.02%
+ +![](images/e6959cd9188841bfcc00b46aeed46d7800c43970a4f909834f9efccdda44607a.jpg) +Figure 27: Asyrp vs. DiffusionCLIP on LSUN-church. Note that we use the official checkpoint provided by DiffusionCLIP. We observe DiffusionCLIP produces narrow range of styles while Asyrp produces diverse styles. + +# L.2 COMPARE THREE METHODS + +In this section, we compare three methods: implicit neural direction $f _ { t }$ , optimized $\Delta h _ { t }$ , and optimized ∆hglobal. + +Training time $f _ { t } \approx \Delta h ^ { g l o b a l } < \Delta h _ { t }$ + +We have to optimize each $\Delta h _ { t }$ for each time step $t$ . Additionally, it needs specific hyperparameters for each $\Delta h _ { t }$ , e.g., higher learning rates for larger $t$ . On the contrary, time-consuming for $f _ { t }$ is similar to optimizing ∆hglobal. + +# Quality + +$f _ { t }$ and $\Delta h _ { t }$ , where directions can be obtained for each timestep, have the best quality. As can be shown in Figure 10, $\Delta h ^ { g l o b a l }$ is sometimes accompanied by slight differences in hair, etc. + +Extensibility $f _ { t } > \Delta h _ { t } > \Delta h ^ { g l o b a l }$ + +![](images/4bcaa6683a66fe5dc47e975b0ea521a3cef4e214da2c1f2d293b268e362cc228.jpg) +Figure 28: Example segmentations for computing segmentation-consistency (SC). + +$\Delta h _ { t }$ can be obtained from $f _ { t }$ , and $\Delta h ^ { g l o b a l }$ can be obtained by aggregating $\Delta h _ { t }$ . +We opt to use $f _ { t }$ for above three advantages. + +# M RANDOM SAMPLING + +We conduct extra experiments: generating images with target attributes using Asyrp not from inversion but from random Gaussian noises. As a consequence, the generative process can be used for conditional random sampling. We provide the results in Figure 31. However, it is beyond the scope of this paper. + +![](images/03e1b521e960c51738ecc1ae3c6697bcf45be2371c9a8ab319c4542be8d594bf.jpg) +Figure 29: We compute mean direction and global direction from 20 other images on CelebAHQ. The effect of mean direction and global direction are quite similar with $\Delta h _ { t }$ by $f _ { t }$ at diverse attributes. + +![](images/94795dd3dafb5d37e50068f7124646c932ba91d53b723f68b82fcc14beab1bf8.jpg) +Figure 30: We obtain mean direction and global direction from 20 other images on LSUN-church. The effect of mean direction and global direction are quite similar with $\Delta h _ { t }$ by $f _ { t }$ at diverse attributes. + +![](images/6eb1c92e476116c392feb38047e039fb60ec3d76570c709ea5ec47a32b19d996.jpg) +Figure 31: Uncurated random sampling. We generate images from random noise with Asyrp. Although we do not focus on these results, Asyrp can be used for conditional sampling. + +# N MORE SAMPLES + +# N.1 IMAGENET + +We conduct extra experiments: editing images with target class using Asyrp with ImageNet pretrained model. We verified that models trained on large datasets, such as ImageNet, can be edited using Asyrp. However, we also observed that in this case the latent space is not partitioned by classes. For an orange, we have different latents for a single orange, for many oranges, for a cross-section of cut orange, and for a single piece of orange. Therefore, we learned the implicit function by collecting similar images to find the direction. + +![](images/774272204ff4ad42b215ddad0f822037c3a70d6b9b02276bd89a0e47fea88d21.jpg) +Figure 32: Result of Asyrp in ImageNet. The result shows that Asyrp works even in ImageNet dataset. + +# N.2 MULTI-INTERPOLATION + +Figure 33 provides mixed interpolation between multiple attributes. We observe that any interpolation with any attribute is possible. + +# N.3 MORE RESULTS ON ALL DATASETS + +We provide more results on CelebA-HQ (Figure 34), LSUN-church (Figure 35), AFHQ, LSUNbedroom, METFACES (Figure 36). + +![](images/b9bf01755a1cba0600b8b0a808d391a7b78c8c875350410552a6a08c44d09b8a.jpg) +$\Delta h$ of smiling and young. + +![](images/e936bdacea2d549661e4189967aeb6d92e07c0baab9c7e9b2fce8aea9a2adbf3.jpg) + +![](images/377b9520a7229fe39d63b39397e4df0928ea29df4c7531e407e9ec506cc84359.jpg) +Figure 34: We provide more results on CelebA-HQ dataset. + +![](images/23c8ad00b208a9a83c86351a6572870e2502164c093437e52becf10d8cf7b5ae.jpg) +Figure 35: We provide more results on LSUN-church dataset. + +![](images/01bb6a5ce5257428dc08aae084bd10e960340eda2d2141d0c31b80453e60a4df.jpg) +Figure 36: We provide more results on AFHQ, LSUN-bedroom and METFACES datasets respectively. \ No newline at end of file diff --git a/parse/dev/r9b6T088_75/r9b6T088_75_layout.pdf 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/dev/null +++ b/parse/dev/rB6TpjAuSRy/images/d3641562ebb4ad395663893572fec1b3f07e6fa21ff0b6fd101d549ef74c18b4.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9eb34cedc143be2021f844f45caf9a79eb254aca98500b6efdc905abdaf56111 +size 14301 diff --git a/parse/dev/sP1fo2K9DFG/sP1fo2K9DFG.md b/parse/dev/sP1fo2K9DFG/sP1fo2K9DFG.md new file mode 100644 index 0000000000000000000000000000000000000000..6804fc6ce1695f839005e44c52377756445afb0d --- /dev/null +++ b/parse/dev/sP1fo2K9DFG/sP1fo2K9DFG.md @@ -0,0 +1,544 @@ +# IS CONDITIONAL GENERATIVE MODELING ALL YOU NEED FOR DECISION-MAKING? + +Anurag Ajay∗ †§¶, Yilun Du \*§¶, Abhi Gupta $\ast \ddag \mathfrak { S } \ P$ , Joshua Tenenbaum¶, Tommi Jaakkola‡§¶, Pulkit Agrawal†§¶ + +Improbable AI Lab† +Operations Research Center‡ +Computer Science and Artificial Intelligence Lab§ +Massachusetts Institute of Technology¶ + +# ABSTRACT + +Recent improvements in conditional generative modeling have made it possible to generate high-quality images from language descriptions alone. We investigate whether these methods can directly address the problem of sequential decisionmaking. We view decision-making not through the lens of reinforcement learning (RL), but rather through conditional generative modeling. To our surprise, we find that our formulation leads to policies that can outperform existing offline RL approaches across standard benchmarks. By modeling a policy as a returnconditional diffusion model, we illustrate how we may circumvent the need for dynamic programming and subsequently eliminate many of the complexities that come with traditional offline RL. We further demonstrate the advantages of modeling policies as conditional diffusion models by considering two other conditioning variables: constraints and skills. Conditioning on a single constraint or skill during training leads to behaviors at test-time that can satisfy several constraints together or demonstrate a composition of skills. Our results illustrate that conditional generative modeling is a powerful tool for decision-making. + +# 1 INTRODUCTION + +Over the last few years, conditional generative modeling has yielded impressive results in a range of domains, including high-resolution image generation from text descriptions (DALL-E, ImageGen) (Ramesh et al., 2022; Saharia et al., 2022), language generation (GPT) (Brown et al., 2020), and step-by-step solutions to math problems (Minerva) (Lewkowycz et al., 2022). The success of generative models in countless domains motivates us to apply them to decision-making. Conveniently, there exists a wide body of research on recovering high-performing policies from data logged by already operational systems (Kostrikov et al., 2022; Kumar et al., 2020; Walke et al., 2022). This is particularly useful in real-world settings where interacting with the environment is not always possible, and exploratory decisions can have fatal consequences (Dulac-Arnold et al., 2021). With access to such offline datasets, the problem of decision-making reduces to learning a probabilistic model of trajectories, a setting where generative models have already found success. + +In offline decision-making, we aim to recover optimal reward-maximizing trajectories by stitching together sub-optimal reward-labeled trajectories in the training dataset. Prior works (Kumar et al., 2020; Kostrikov et al., 2022; Wu et al., 2019; Kostrikov et al., 2021; Dadashi et al., 2021; Ajay et al., 2020; Ghosh et al., 2022) have tackled this problem with reinforcement learning (RL) that uses dynamic programming for trajectory stitching. To enable dynamic programming, these works learn a value function that estimates the discounted sum of rewards from a given state. However, value function estimation is prone to instabilities due to function approximation, off-policy learning, and bootstrapping together, together known as the deadly triad (Sutton & Barto, 2018). Furthermore, to stabilize value estimation in offline regime, these works rely on heuristics to keep the policy within the dataset distribution. These challenges make it difficult to scale existing offline RL algorithms. + +![](images/9dac5e5f723a941f045f73b69f622ccae89b7291e5b8f4eed92c65b334c4c613.jpg) +Figure 1: Decision Making using Conditional Generative Modeling. Framing decision making as a conditional generative modeling problem allows us to maximize rewards, satisfy constraints and compose skills. + +In this paper, we ask if we can perform dynamic programming to stitch together sub-optimal trajectories to obtain an optimal trajectory without relying on value estimation. Since conditional diffusion generative models can generate novel data points by composing training data (Saharia et al., 2022), we leverage it for trajectory stitching in offline decision-making. Given a dataset of reward-labeled trajectories, we adapt diffusion models (Sohl-Dickstein et al., 2015) to learn a return-conditional trajectory model. During inference, we use classifier-free guidance with lowtemperature sampling, which we hypothesize to implicitly perform dynamics programming, to capture the best behaviors in the dataset and glean return maximizing trajectories (see Appendix A). Our straightforward conditional generative modeling formulation outperforms existing approaches on standard D4RL tasks (Fu et al., 2020). + +Viewing offline decision-making through the lens of conditional generative modeling allows going beyond conditioning on returns (Figure 1). Consider an example (detailed in Appendix A) where a robot with linear dynamics navigates an environment containing two concentric circles (Figure 2). We are given a dataset of state-action trajectories of the robot, each satisfying one of two constraints: (i) the final position of the robot is within the larger circle, and (ii) the final position of the robot is outside the smaller circle. With conditional diffusion modeling, we can use the datasets to learn a constraintconditioned model that can generate trajectories satisfying any set of constraints. During inference, the learned trajectory model can merge constraints from the dataset and generate trajectories that satisfy the combined constraint. Figure 2 shows that the constraint-conditioned model can generate trajectories such that the final position of the robot lies between the concentric circles. + +![](images/3f638c6b9477f25a868e5f8e34ac18c10ed64b5789def600b6b4750a38198356.jpg) +Figure 2: Illustrative example. We visualize the 2d robot navigation environment and the constraints satisfied by the trajectories in the dataset derived from the environment. We show the ability of the conditional diffusion model to generate trajectories that satisfy the combined constraints. + +Here, we demonstrate the benefits of modeling policies as conditional generative models. First, conditioning on constraints allows policies to not only generate behaviors satisfying individual constraints but also generate novel behaviors by flexibly combining constraints at test time. Further, conditioning on skills allows policies to not only imitate individual skills but also generate novel behaviors by composing those skills. We instantiate this idea with a state-sequence based diffusion probabilistic model (Ho et al., 2020) called Decision Diffuser, visualized in Figure 1. In summary, our contributions include (i) illustrating conditional generative modeling as an effective tool in offline decision making, (ii) using classifier-free guidance with low-temperature sampling, instead of dynamic programming, to get return-maximizing trajectories and, (iii) leveraging the framework of conditional generative modeling to combine constraints and compose skills during inference flexibly. + +# 2 BACKGROUND + +# 2.1 REINFORCEMENT LEARNING + +We formulate the sequential decision-making problem as a discounted Markov Decision Process (MDP) defined by the tuple $\langle \rho _ { 0 } , S , { \mathcal { A } } , { \mathcal { T } } , { \mathcal { R } } , { \gamma } \rangle$ , where $\rho _ { 0 }$ is the initial state distribution, $s$ and $\mathcal { A }$ are state and action spaces, $\mathcal { T } : \mathcal { S } \times \mathcal { A } \mathcal { S }$ is the transition function, $\mathcal { R } : \mathcal { S } \times \mathcal { A } \times \mathcal { S } \mathbb { R }$ gives the reward at any transition and $\gamma \in [ 0 , 1 )$ is a discount factor. The agent acts with a stochastic policy $\pi :$ ${ \cal S } \to \Delta _ { \cal A }$ , generating a sequence of state-action-reward transitions or trajectory $\tau : = ( s _ { k } , a _ { k } , r _ { k } ) _ { k \geq 0 }$ with probability $p _ { \pi } ( \tau )$ and return $\begin{array} { r } { R ( \tau ) : = \sum _ { k \geq 0 } \gamma ^ { k } r _ { k } } \end{array}$ . The standard objective in RL is to find a return-maximizing policy $\begin{array} { r } { \pi ^ { * } = \arg \operatorname* { m a x } _ { \pi } \mathbb { E } _ { \tau \sim p _ { \pi } } \overline { { [ R ( \tau ) ] } } } \end{array}$ . + +Temporal Difference Learning TD methods (Fujimoto et al., 2018; Lillicrap et al., 2015) estimate $Q ^ { * } ( s , a ) : = \mathbb { E } _ { \tau \sim p _ { \pi ^ { * } } } [ R ( \tau ) | s _ { 0 } = s , a _ { 0 } = a ]$ , the return achieved under the optimal policy $\pi ^ { * }$ when starting in state $s$ and taking action $a$ , with a parameterized $Q$ -function. This requires minimizing the following TD loss: + +$$ +\mathcal { L } _ { \mathrm { T D } } ( \theta ) : = \mathbb { E } _ { ( s , a , r , s ^ { \prime } ) \in \mathcal { D } } [ ( r + \gamma \operatorname* { m a x } _ { a ^ { \prime } \in \mathcal { A } } Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) - Q _ { \theta } ( s , a ) ) ^ { 2 } ] +$$ + +Continuous action spaces further require learning a parametric policy $\pi _ { \phi } ( a | s )$ that plays the role of the maximizing action in equation 1. This results in a policy objective that must be maximized: + +$$ +\mathcal { I } ( \phi ) : = \mathbb { E } _ { s \in \mathcal { D } , a \sim \pi _ { \phi } ( . | s ) } [ Q ( s , a ) ] +$$ + +Here, the dataset of transitions $\mathcal { D }$ evolves as the agent interacts with the environment and both $Q _ { \theta }$ and $\pi _ { \phi }$ are trained together. These methods make use of function approximation, off-policy learning, and bootstrapping, leading to several instabilities in practice (Sutton, 1988; Van Hasselt et al., 2018). + +Offline RL requires finding a return-maximizing policy from a fixed dataset of transitions collected by an unknown behavior policy $\mu$ (Levine et al., 2020). Using TD-learning naively causes the state visitation distribution $d ^ { \pi _ { \phi } } ( s )$ to move away from the distribution of the dataset $d ^ { \mu } ( s )$ . In turn, the policy $\pi _ { \phi }$ begins to take actions that are substantially different from those already seen in the data. Offline RL algorithms resolve this distribution-shift by imposing a constraint of the form $D ( d ^ { \pi _ { \phi } } | | d ^ { \mu } )$ , where $D$ is some divergence metric, directly in the TD-learning procedure. The constrained optimization problem now demands additional implementation heuristics to achieve any reasonable performance (Kumar et al., 2021). The Decision Diffuser, in comparison, doesn’t have any of these disadvantages. It does not require estimating any kind of $Q$ -function, thereby sidestepping TD methods altogether. It also does not face the risk of distribution-shift as generative models are trained with maximum-likelihood estimation. + +# 2.2 DIFFUSION PROBABILISTIC MODELS + +Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a specific type of generative model that learn the data distribution $q ( { \pmb x } )$ from a dataset $\mathcal { D } : = \{ \pmb { x } ^ { i } \} _ { 0 \leq i < M }$ . They have been used most notably for synthesizing high-quality images from text descriptions (Saharia et al., 2022; Nichol et al., 2021). Here, the data-generating procedure is modelled with a predefined forward noising process $q ( \pmb { x } _ { k + 1 } | \pmb { x } _ { k } ) : = \mathcal { N } ( \pmb { x } _ { k + 1 } ; \sqrt { \alpha _ { k } } \pmb { x } _ { k } , ( 1 - \alpha _ { k } ) \pmb { I } )$ and a trainable reverse process $p _ { \theta } ( \pmb { x } _ { k - 1 } | \pmb { x } _ { k } ) : =$ $\mathcal { N } ( \pmb { x } _ { k - 1 } | \mu _ { \theta } ( \pmb { x } _ { k } , k ) , \Sigma _ { k } )$ , where $\mathcal { N } ( \boldsymbol { \mu } , \boldsymbol { \Sigma } )$ denotes a Gaussian distribution with mean $\mu$ and variance $\Sigma$ , $\alpha _ { k } \in \mathbb { R }$ determines the variance schedule, $\scriptstyle { \pmb { x } } _ { 0 } : = { \pmb { x } }$ is a sample, ${ \pmb x } _ { 1 } , { \pmb x } _ { 2 } , . . . , { \pmb x } _ { K - 1 }$ are the latents, and $\pmb { x } _ { K } \sim \mathcal { N } ( \mathbf { 0 } , \pmb { I } )$ for carefully chosen $\alpha _ { k }$ and long enough $K$ . Starting with Gaussian noise, samples are then iteratively generated through a series of ”denoising” steps. + +Although a tractable variational lower-bound on $\log p _ { \theta }$ can be optimized to train diffusion models, Ho et al. (2020) propose a simplified surrogate loss: + +$$ +\mathcal { L } _ { \mathrm { d e n o i s e } } ( \theta ) : = \mathbb { E } _ { k \sim [ 1 , K ] , \boldsymbol { x } _ { 0 } \sim q , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) } [ | | \epsilon - \epsilon _ { \theta } ( \boldsymbol { x } _ { k } , k ) | | ^ { 2 } ] +$$ + +The predicted noise $\boldsymbol { \epsilon } _ { \boldsymbol { \theta } } ( \boldsymbol { x } _ { k } , k )$ , parameterized with a deep neural network, estimates the noise $\epsilon \sim$ $\mathcal { N } ( 0 , I )$ added to the dataset sample $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ to produce noisy $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ . This is equivalent to predicting the mean of $p _ { \theta } ( \pmb { x } _ { k - 1 } | \pmb { x } _ { k } )$ since $\mu _ { \boldsymbol { \theta } } ( \mathbf { \boldsymbol { x } } _ { k } , k )$ can be calculated as a function of $\epsilon _ { \theta } ( \pmb { x } _ { k } , k )$ (Ho et al., 2020). + +Guided Diffusion Modelling the conditional data distribution $q ( { \pmb x } | { \pmb y } )$ makes it possible to generate samples with attributes of the label $\textbf { { y } }$ . The equivalence between diffusion models and scorematching (Song et al., 2021), which shows $\epsilon _ { \theta } ( x _ { k } , \bar { k } ) \propto \nabla _ { x _ { k } } \log p ( \pmb { x } _ { k } )$ , leads to two kinds of methods for conditioning: classifier-guided (Nichol & Dhariwal, 2021) and classifier-free (Ho & Salimans, 2022). The former requires training an additional classifier $p _ { \phi } ( \pmb { y } | \pmb { x } _ { k } )$ on noisy data so that samples may be generated at test-time with the perturbed noise $\epsilon _ { \theta } ( \pmb { x } _ { k } , k ) - \omega \sqrt { 1 - \bar { \alpha } _ { k } } \nabla _ { \pmb { x } _ { k } } \log p ( \pmb { y } | \pmb { x } _ { k } )$ , where $\omega$ is referred to as the guidance scale. The latter does not separately train a classifier but modifies the original training setup to learn both a conditional $\epsilon _ { \theta } ( \pmb { x } _ { k } , \pmb { y } , k )$ and an unconditional $\epsilon _ { \theta } ( \pmb { x } _ { k } , k )$ model for the noise. The unconditional noise is represented, in practice, as the conditional noise $\epsilon _ { \theta } ( \pmb { x } _ { k } , 0 , k )$ where a dummy value $\varnothing$ takes the place of $\textbf { { y } }$ . The perturbed noise $\epsilon _ { \theta } ( { \pmb x } _ { k } , k ) + \omega ( \epsilon _ { \theta } ( { \pmb x } _ { k } , { \pmb y } , k ) - \epsilon _ { \theta } ( { \pmb x } _ { k } , k ) )$ is used to later generate samples. + +# 3 GENERATIVE MODELING WITH THE DECISION DIFFUSER + +It is useful to solve RL from offline data, both without relying on TD-learning and without risking distribution-shift. To this end, we formulate sequential decision-making as the standard problem of + +![](images/5193fd80ab094ae9c4154b23239d26a5d2b68886d0126e16c31679f9afe8be80.jpg) +Figure 3: Planning with Decision Diffuser. Given the current state $s _ { t }$ and conditioning, Decision Diffuser uses classifier-free guidance with low-temperature sampling to generate a sequence of future states. It then uses inverse dynamics to extract and execute the action $a _ { t }$ that leads to the immediate future state $s _ { t + 1 }$ . + +conditional generative modeling: + +$$ +\operatorname* { m a x } _ { \theta } \mathbb { E } _ { \tau \sim \mathcal { D } } [ \log p _ { \theta } ( \pmb { x } _ { 0 } ( \tau ) | \pmb { y } ( \tau ) ) ] +$$ + +Our goal is to estimate the conditional data distribution with $p _ { \theta }$ so we can later generate portions of a trajectory $\pmb { x } _ { 0 } ( \tau )$ from information $\pmb { y } ( \tau )$ about it. Examples of $\textbf { { y } }$ could include the return under the trajectory, the constraints satisfied by the trajectory, or the skill demonstrated in the trajectory. We construct our generative model according to the conditional diffusion process: + +$$ +\begin{array} { r l } { q ( \pmb { x } _ { k + 1 } ( \tau ) | \pmb { x } _ { k } ( \tau ) ) , } & { { } p _ { \theta } ( \pmb { x } _ { k - 1 } ( \tau ) | \pmb { x } _ { k } ( \tau ) , \pmb { y } ( \tau ) ) } \end{array} +$$ + +As usual, $q$ represents the forward noising process while $p _ { \theta }$ the reverse denoising process. In the following, we discuss how we may use diffusion for decision making. First, we discuss the modeling choices for diffusion in Section 3.1. Next, we discuss how we may utilize classifier-free guidance to capture the best aspects of trajectories in Section 3.2. We then discuss the different behaviors that may be implemented with conditional diffusion models in Section 3.3. Finally, we discuss practical training details of our approach in Section 3.4. + +# 3.1 DIFFUSING OVER STATES + +In images, the diffusion process is applied across all pixel values in an image. Na¨ıvely, it would therefore be natural to apply a similar process to model the state and actions of a trajectory. However, in the reinforcement learning setting, directly modeling actions using a diffusion process has several practical issues. First, while states are typically continuous in nature in RL, actions are more varied, and are often discrete in nature. Furthermore, sequences over actions, which are often represented as joint torques, tend to be more high-frequency and less smooth, making them much harder to predict and model (Tedrake, 2022). Due to these practical issues, we choose to diffuse only over states, as defined below: + +$$ +\pmb { x } _ { k } ( \tau ) : = ( s _ { t } , s _ { t + 1 } , . . . , s _ { t + H - 1 } ) _ { k } +$$ + +Here, $k$ denotes the timestep in the forward process and $t$ denotes the time at which a state was visited in trajectory $\tau$ . Moving forward, we will view ${ \pmb x } _ { k } ( \tau )$ as a noisy sequence of states from a trajectory of length $H$ . We represent ${ \pmb x } _ { k } ( \tau )$ as a two-dimensional array with one column for each timestep of the sequence. + +Acting with Inverse-Dynamics. Sampling states from a diffusion model is not enough for defining a controller. A policy can, however, be inferred from estimating the action $a _ { t }$ that led the state $s _ { t }$ to $s _ { t + 1 }$ for any timestep $t$ in ${ \pmb x } _ { 0 } ( \tau )$ . Given two consecutive states, we generate an action according to the inverse dynamics model (Agrawal et al., 2016; Pathak et al., 2018): + +$$ +a _ { t } : = f _ { \phi } ( s _ { t } , s _ { t + 1 } ) +$$ + +Note that the same offline data used to train the reverse process $p _ { \theta }$ can also be used to learn $f _ { \phi }$ . We illustrate in Table 2 how the design choice of directly diffusing state distributions, with an inverse dynamics model to predict action, significantly improves performance over diffusing across both states and actions jointly. Furthermore, we empirically compare and analyze when to use inverse dynamics and when to diffuse over actions in Appendix F. + +# 3.2 PLANNING WITH CLASSIFIER-FREE GUIDANCE + +Given a diffusion model representing the different trajectories in a dataset, we next discuss how we may utilize the diffusion model for planning. To use the model for planning, it is necessary to additionally condition the diffusion process on characteristics $\pmb { y } ( \tau )$ . One approach could be to train a classifier $p _ { \phi } ( \pmb { y } ( \tau ) | \pmb { x } _ { k } ( \tau ) )$ to predict $\pmb { y } ( \tau )$ from noisy trajectories ${ \pmb x } _ { k } ( \tau )$ . In the case that $\pmb { y } ( \tau )$ represents the return under a trajectory, this would require estimating a $Q$ -function, which requires a separate, complex dynamic programming procedure. + +One approach to avoid dynamic programming is to directly train a conditional diffusion model conditioned on the returns $\pmb { y } ( \tau )$ in the offline dataset. However, as our dataset consists of a set of sub-optimal trajectories, the conditional diffusion model will be polluted by such sub-optimal behaviors. To circumvent this issue, we utilize classifier-free guidance (Ho & Salimans, 2022) with low-temperature sampling, to extract high-likelihood trajectories in the dataset. We find that such trajectories correspond to the best set of behaviors in the dataset. For a detailed discussion comparing Q-function guidance and classifier-free guidance, please refer to Appendix K. Formally, to implement classifier free guidance, a $\pmb { x } _ { 0 } ( \tau )$ is sampled by starting with Gaussian noise ${ \pmb x } _ { K } ( \tau )$ and refining ${ \pmb x } _ { k } ( \tau )$ into $\pmb { x } _ { k - 1 } ( \tau )$ at each intermediate timestep with the perturbed noise: + +$$ +\hat { \epsilon } : = \epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , \mathcal { O } , k ) + \omega ( \epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , { \pmb y } ( \tau ) , k ) - \epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , \mathcal { O } , k ) ) , +$$ + +where the scalar $\omega$ applied to $( \epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , { \pmb y } ( \tau ) , k ) - \epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , \emptyset , k ) )$ seeks to augment and extract the best portions of trajectories in the dataset that exhibit $\pmb { y } ( \tau )$ . With these ingredients, sampling from the Decision Diffuser becomes similar to planning in RL. First, we observe a state in the environment. Next, we sample states later into the horizon with our diffusion process conditioned on $\textbf { { y } }$ and history of last $C$ states observed. Finally, we identify the action that should be taken to reach the most immediate predicted state with our inverse dynamics model. This procedure repeats in a standard receding-horizon control loop described in Algorithm 1 and visualized in Figure 3. + +# 3.3 CONDITIONING BEYOND RETURNS + +So far we have not explicitly defined the conditioning variable $\pmb { y } ( \tau )$ . Though we have mentioned that it can be the return under a trajectory, we may also consider guiding our diffusion process towards sequences of states that satisfy relevant constraints or demonstrate specific behavior. + +Maximizing Returns To generate trajectories that maximize return, we condition the noise model on the return of a trajectory so $\begin{array} { r } { \epsilon _ { \theta } ( \bar { \mathbf { x } } _ { k } ( \tau ) , \mathbf { \boldsymbol { y } } ( \tau ) , k ) : = \epsilon _ { \theta } ( \mathbf { \Delta x } _ { k } ( \tau ) , R ( \tau ) , k ) } \end{array}$ . These returns are normalized to keep $R ( \tau ) \in [ 0 , 1 ]$ . Sampling a high return trajectory amounts to conditioning on $R ( \tau ) = 1$ . Note that we do not make use of any $Q$ -values, which would then require dynamic programming. + +Satisfying Constraints Trajectories may satisfy a variety of constraints, each represented by the set $\mathcal { C } _ { i }$ , such as reaching a specific goal, visiting states in a particular order, or avoiding parts of the state space. To generate trajectories satisfying a given constraint $\mathcal { C } _ { i }$ , we condition the noise model on a one-hot encoding so that $\begin{array} { r } { \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \pmb { y } ( \tau ) , \bar { k } ) : = \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \mathbb { 1 } ( \tau \in \mathcal { C } _ { i } ) , k ) } \end{array}$ . Although we train with an offline dataset in which trajectories satisfy only one of the available constraints, at inference we can satisfy several constraints together. + +Composing Skills A skill $i$ can be specified from a set of demonstrations $B _ { i }$ . To generate trajectories that demonstrate a given skill, we condition the noise model on a one-hot encoding so that $\epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , { \pmb y } ( \tau ) , k ) : = \epsilon _ { \theta } ( \bar { { \pmb x } } _ { k } ( \tau ) , \mathbb { 1 } ( \tau \in \mathcal { B } _ { i } ) , k )$ . Although we train with individual skills, we may further compose these skills together during inference. + +Assuming we have learned the data distributions $q ( { \pmb x } _ { 0 } ( \tau ) | { \pmb y } ^ { 1 } ( \tau ) ) , \dots , q ( { \pmb x } _ { 0 } ( \tau ) | { \pmb y } ^ { n } ( \tau ) )$ for $n$ different conditioning variables, we can sample from the composed data distribution $q ( \pmb { x } _ { 0 } ( \tau ) | \pmb { y } ^ { 1 } ( \tau ) , \dots , \pmb { y } ^ { n } ( \bar { \tau } ) )$ using the perturbed noise (Liu et al., 2022): + +$$ +\hat { \epsilon } : = \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \emptyset , k ) + \omega \sum _ { i = 1 } ^ { n } ( \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \pmb { y } ^ { i } ( \tau ) , k ) - \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \emptyset , k ) ) +$$ + +This property assumes that $\{ \pmb { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n }$ are conditionally independent given the state trajectory $\pmb { x } _ { 0 } ( \tau )$ . However, we empirically observe that this assumption doesn’t have to be strictly satisfied as long as the composition of conditioning variables is feasible. For more detailed discussion, please refer to Appendix D. We use this property to compose more than one constraint or skill together at test-time. We also show how Decision Diffuser can avoid particular constraint or skill (NOT) in Appendix J. + +# Algorithm 1 Conditional Planning with the Decision Diffuser + +![](images/978ee750ffe38aa619ac11373331980f674e6924fad8991ee08cfb832205fc9a.jpg) +Figure 4: Results Overview. Decision Diffuser performs better than both TD learning (CQL) and Behavorial Cloning (BC) across D4RL locomotion tasks, D4RL Kitchen tasks and Kuka Block Stacking tasks (single constraint) using only a conditional generative modeling objective. For performance metric, we use normalized average returns (Fu et al., 2020) for D4RL tasks (Locomotion and Kitchen) and success rate for Block Stacking. + +# 3.4 TRAINING THE DECISION DIFFUSER + +The Decision Diffuser, our conditional generative model for decision-making, is trained in a supervised manner. Given a dataset $\mathcal { D }$ of trajectories, each labeled with the return it achieves, the constraint that it satisfies, or the skill that it demonstrates, we simultaneously train the reverse diffusion process $p _ { \theta }$ , parameterized through the noise model $\epsilon _ { \theta }$ , and the inverse dynamics model $f _ { \phi }$ with the following loss: + +$\begin{array} { r } { \natural ( \theta , \phi ) : = \mathbb { E } _ { k , \tau \in \mathcal { D } , \beta \sim \mathrm { B e n } ( p ) } [ \| \epsilon - \epsilon _ { \theta } ( { x } _ { k } ( \tau ) , ( 1 - \beta ) y ( \tau ) + \beta \mathcal { O } , k ) \| ^ { 2 } ] + \mathbb { E } _ { ( s , a , s ^ { \prime } ) \in \mathcal { D } } [ \| a - f _ { \phi } ( s , s ^ { \prime } ) \| ^ { 2 } ] } \end{array}$ For each trajectory $\tau$ , we first sample noise $\epsilon \sim \mathcal { N } ( 0 , I )$ and a timestep $k \sim \mathcal { U } \{ 1 , \dots , K \}$ . Then, we construct a noisy array of states ${ \pmb x } _ { k } ( \tau )$ and finally predict the noise as $\hat { \epsilon } _ { \boldsymbol { \theta } } : = \epsilon _ { \boldsymbol { \theta } } ( \pmb { x } _ { k } ( \tau ) , \pmb { y } ( \tau ) , k )$ . Note that with probability $p$ we ignore the conditioning information and the inverse dynamics is trained with individual transitions rather than trajectories. + +Architecture We parameterize $\epsilon _ { \theta }$ with a temporal U-Net architecture, a neural network consisting of repeated convolutional residual blocks (Janner et al., 2022). This effectively treats a sequence of states ${ \pmb x } _ { k } ( \tau )$ as an image where the height represents the dimension of a single state and the width denotes the length of the trajectory. We encode the conditioning information $\pmb { y } ( \tau )$ as either a scalar or a one-hot vector and project it into a latent variable $z \in \mathbb { R } ^ { h }$ with a multi-layer perceptron (MLP). When $\pmb { y } ( \tau ) = 0$ , we zero out the entries of $z$ . We also parameterize the inverse dynamics $f _ { \phi }$ with an MLP. For implementation details, please refer to the Appendix B. + +Low-temperature Sampling In the denoising step of Algorithm 1, we compute $\mu _ { k - 1 }$ and $\Sigma _ { k - 1 }$ from a noisy sequence of states and a predicted noise. We find that sampling $x _ { k - 1 } \sim$ $\mathcal { N } ( \mu _ { k - 1 } , \alpha \Sigma _ { k - 1 } )$ where the variance is scaled by $\alpha \in [ 0 , 1 )$ leads to better quality sequences (corresponding to sampling lower temperature samples). For a proper ablation study, please refer to Appendix C. + +# 4 EXPERIMENTS + +In this section, we explore the efficacy of the Decision Diffuser on a variety of decision-making tasks (performance illustrated in Figure 4). In particular, we evaluate (1) the ability to recover effective RL policies from offline data, (2) the ability to generate behavior that satisfies multiple sets of constraints, (3) the ability compose multiple different skills together. In addition, we empirically justify use of classifier-free guidance, low-temperature sampling (Appendix C), and inverse dynamics (Appendix F) and test the robustness of Decision Diffuser to stochastic dynamics (Appendix G). + +# 4.1 OFFLINE REINFORCEMENT LEARNING + +Setup We first test whether the Decision Diffuser can generate return-maximizing trajectories. To test this, we train a state diffusion process and inverse dynamics model on publicly available + +D4RL datasets (Fu et al., 2020). We compare with existing offline RL methods, including modelfree algorithms like CQL (Kumar et al., 2020) and IQL (Kostrikov et al., 2022), and model-based algorithms such as trajectory transformer (TT, Janner et al. (2021)) and MoReL (Kidambi et al., 2020). We also compare with sequence-models like the Decision Transformer (DT) (Chen et al. (2021) and diffusion models like Diffuser (Janner et al., 2022). + +
DatasetEnvironmentBCCQLIQLDTTTMOReLDiffuserDD
Med-ExpertHalfCheetah55.291.686.786.89553.379.890.6 ±1.3
Med-ExpertHopper52.5105.491.5107.6110.0108.7107.2111.8 ±1.8
Med-ExpertWalker2d107.5108.8109.6108.1101.995.6108.4108.8 ±1.7
MediumHalfCheetah42.644.047.442.646.942.144.249.1 ±1.0
MediumHopper52.958.566.367.661.195.458.579.3 ±3.6
MediumWalker2d75.372.578.374.07977.879.782.5±1.4
Med-ReplayHalfCheetah36.645.544.236.641.940.242.239.3 ±4.1
Med-ReplayHopper18.19594.782.791.593.696.8100±0.7
Med-ReplayWalker2d26.077.273.966.682.649.861.275±4.3
Average51.977.67774.778.972.975.381.8
MixedKitchen51.552.4511、1165 ±2.8
PartialKitchen3850.146.3--==57 ±2.5
Average44.851.248.7-1--61
+ +Results Across different offline RL tasks, we find that the Decision Diffuser is either competitive or outperforms many offline RL baselines (Table 1). It also outperforms Diffuser and sequence modeling approaches, such as Decision Transformer and Trajectory Transformer. The difference between Decision Diffuser and other methods becomes even more significant on harder D4RL Kitchen tasks which require long-term credit assignment. + +To convey the importance of classifier-free guidance, we also compare with the baseline CondDiffuser, which diffuses over both state and action sequences as in Diffuser without classifierguidance. In Table 2, we observe that CondDiffuser improves over Diffuser in 2 out of 3 environments. Decision Diffuser further improves over CondDiffuser, performing better across all 3 environments. We conclude that learning the inverse dynamics is a good alternative to diffusing over actions. We further empirically analyze when to use inverse dynamics and when to diffuse over actions in Appendix F. We also compare against CondMLPDiffuser, a policy where the current action is denoised according to a diffusion process conditioned on both the state and return. We see that CondMLPDiffuser performs the worst amongst diffusion models. Till now, we mainly tested on offline RL tasks that have deterministic (or near deterministic) environment dynamics. Hence, we test the robustness of Decision Diffuser to stochastic dynamics and compare it to Diffuser and CQL as we vary the stochasticity in environment dynamics, in Appendix G. Finally, we analyze the runtime characteristics of Decision Diffuser in Appendix E. + +# 4.2 CONSTRAINT SATISFACTION + +Setup We next evaluate how well we can generate trajectories that satisfy a set of constraints using the Kuka Block Stacking environment (Janner et al., 2022) visualized in Figure 5. In this domain, there are four blocks which can be stacked as a single tower or rearranged into several towers. A constraint like BlockHeigh $\mathsf { \Lambda } = ( i ) > \mathsf { B l o c k H e i g h t } ( j )$ requires that block $i$ be placed above block $j$ . We train the Decision Diffuser from 10, 000 expert demonstrations each satisfying one of these constraints. We randomize the positions of these blocks and consider two tasks at inference: sampling trajectories that satisfy a single constraint seen before in the dataset or satisfy a group of constraints for which demonstrations were never provided. In the latter, we ask the Decision Diffuser to generate trajectories so BlockHeight $( i ) >$ BlockHeight $( j ) >$ BlockHeight $( k )$ for three of the four blocks $i , j , k$ . For more details, please refer to Appendix H. + +Results In both the stacking and rearrangement settings, Decision Diffuser satisfies single constraints with greater success rate than Diffuser (Table 3). We also compare with BCQ (Fujimoto et al., 2019) and CQL (Kumar et al., 2020), but they consistently fail to stack or rearrange the blocks leading to a 0.0 success rate. Unlike these baselines, our method can just as effectively satisfy several constraints together according to Equation 9. For a visualization of these generated trajectories, please see the website https://anuragajay.github.io/decision-diffuser/. + +
Hopper-*Diffuser CondDiffuser CondMLPDiffuser Decision Diffuser
Med-Expert107.6111.3105.6111.8 ±1.6
Medium58.566.354.179.3 ±3.6
Med-Replay96.876.566.5100 ±0.7
+ +Table 2: Ablations. Using classifier-free guidance with Diffuser, resulting in CondDiffuser, improves performance in 2 (out of 3) environments. Additionally, using inverse dynamics for action prediction in Decision Diffuser improves performance in all 3 environments. CondMLPDiffuser, that diffuses over current action given the current state and the target return, doesn’t perform as well. + +
EnvironmentDiffuserDD
Single Constraint - Stacking45.6 ±3.158.0 ±3.1
Single Constraint - Rearrangement58.9 ±3.462.7 ±3.1
Single Constraint Average52.360.4
Multiple Constraints - Stacking160.3 ±3.1
Multiple Constraints - Rearrangement67.2 ±3.1
Multiple Constraints Average163.8
+ +![](images/52d2870f8dae13bbc0f1462a3db00c970f194d62fa9583c2df125b201342dd4a.jpg) +Figure 5: Kuka Block Stacking task. +Table 3: Block Stacking through Constraint Minimization. Decision Diffuser (DD) improves over Diffuser in terms of the success rate of generating trajectories satisfying a set of block-stacking constraints. It can also flexibly combine multiple constraints during test time. We report the mean success rate and the standard error over 5 random seeds. + +# 4.3 SKILL COMPOSITION + +Setup Finally, we look at how to compose different skills together. We consider the Unitree-gorunning environment (Margolis & Agrawal, 2022), where a quadruped robot can be found running with various gaits, like bounding, pacing, and trotting. We explore if it is possible to generate trajectories that transition between these gaits after only training on individual gaits. For each gait, we collect a dataset of 2500 demonstrations on which we train Decision Diffuser. + +Results During testing, we use the noise model of our reverse diffusion process according to equation 9 to sample trajectories of the quadruped robot with entirely new running behavior. Figure 6 shows a trajectory that begins with bounding but ends with pacing. Appendix I provides additional visualizations of running gaits being composed together. Although it visually appears that trajectories generated with the Decision Diffuser contain more than one gait, we would like to quantify exactly how well different gaits can be composed. To this end, we train a classifier to predict at every time-step or frame in a trajectory the running gait of the quadruped (i.e. bound, pace, or trott). We reuse the demonstrations collected for training the Decision Diffuser to also train this classifier, where our inputs are defined as robot joint states over a fixed period of time (i.e. state sub-sequences of length 10) and the label is the gait demonstrated in this sequence. The complete details of our gait classification procedure can be found in Appendix I. + +
ConditionTrottPaceBound
Only Bound0.81.098.2
Only Pace1.497.70.9
Bound + Pace1.438.560.1
+ +![](images/8874639fd20e50d9304875bc9a9d40220fd33875e237e4b977f7e557544d6fc5.jpg) +Figure 7: Classifying Running Gaits. A classifier predicts the running gait of the quadruped at every timestep. On trajectories generated by conditioning on a single skill, like only bounding or pacing, the classifier predicts the respective gait with largest probability. When conditioned on both skills, some timesteps are classified as bounding while others as pacing. + +We use our running gait classifier in two ways: to evaluate how the behavior of the quadruped changes over the course of a single, generated trajectory and to measure how often each gait emerges over several generated trajectories. In the former, we first sample three trajectories from the Decision Diffuser conditioned either on the bounding gait, the pacing gait, or both. For every trajectory, we separately plot the classification probability of each gait over the length of the sequence. As shown in the plots of Figure 7, the classifier predicts bound and pace respectively to be the most likely running gait in trajectories sampled with this condition. When the trajectory is generated by conditioning on both gaits, the classifier transitions between predicting one gait with largest probability to the other. In fact, there are several instances where the behavior of the quadruped switches between bounding and pacing according to the classifier. This is consistent with the visualizations reported in Figure 6. In the table depicted in Figure 7, we consider 1000 trajectories generated with the Decision Diffuser when conditioned on one or both of the gaits as listed. We record the fraction of time that the quadruped’s running gait was classified as either trott, pace, or bound. It turns out that the classifier identifies the behavior as bounding for $3 8 . 5 \%$ of the time and as pacing for the other $6 0 . 1 \%$ when trajectories are sampled by composing both gaits. This corroborates the fact that the Decision Diffuser can indeed compose running behaviors despite only being trained on individual gaits. + +![](images/ae5902e9ec2fa29c05349103c337c534e1a146b5725255f610070875b832cd24.jpg) +Figure 6: Composing Movement Skills. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. + +# 5 RELATED WORK + +Diffusion Models Diffusion Models is proficient in learning generative models of image and text data (Saharia et al., 2022; Nichol et al., 2021; Nichol & Dhariwal, 2021). It formulates the data sampling process as an iterative denoising procedure (Sohl-Dickstein et al., 2015; Ho et al., 2020). The denoising procedure can be alternatively interpreted as parameterizing the gradients of the data distribution (Song et al., 2021) optimizing the score matching objective (Hyvarinen ¨ , 2005) and thus as a Energy-Based Model (Du & Mordatch, 2019; Nijkamp et al., 2019; Grathwohl et al., 2020). To generate data samples (eg: images) conditioned on some additional information (eg:text), prior works (Nichol & Dhariwal, 2021) have learned a classifier to facilitate the conditional sampling. More recent works (Ho & Salimans, 2022) have argued to leverage gradients of an implicit classifier, formed by the difference in score functions of a conditional and an unconditional model, to facilitate conditional sampling. The resulting classifier-free guidance has been shown to generate better conditional samples than classifier-based guidance. Recent works have also used diffusion models to imitate human behavior (Pearce et al., 2023) and to parameterize policy in offline RL (Wang et al., 2022). Janner et al. (2022) generate trajectories consisting of states and actions with an unconditional diffusion model, therefore requiring a trained reward function on noisy state-action pairs. At inference, the estimated reward function guides the reverse diffusion process towards samples of high-return trajectories. In contrast, we do not train reward functions or diffusion processes separately, but rather model the trajectories in our dataset with a single, conditional generative model. This ensures that the sampling procedure of the learned diffusion process is the same at inference as it is during training. + +Reward Conditioned Policies Prior works (Kumar et al., 2019; Schmidhuber, 2019; Emmons et al., 2021; Chen et al., 2021) have studied learning of reward conditioned policies via reward conditioned behavioral cloning. Chen et al. (2021) used a transformer (Vaswani et al., 2017) to model the reward conditioned policies and obtained a performance competitive with offline RL approaches. Emmons et al. (2021) obtained similar performance as Chen et al. (2021) without using a transformer policy but relied on careful capacity tuning of MLP policy. In contrast, Decision Diffuser can also model constraints or skills and their resulting compositions. + +# 6 DISCUSSION + +We propose Decision Diffuser, a conditional generative model for sequential decision making. It frames offline sequential decision making as conditional generative modeling and sidesteps the need of reinforcement learning, thereby making the decision making pipeline simpler. By sampling for high returns, it is able to capture the best behaviors in the dataset and outperforms existing offline RL approaches on standard D4RL benchmarks. In addition to returns, it can also be conditioned on constraints or skills and can generate novel behaviors by flexibly combining constraints or composing skills during test time. In this work, we focused on offline sequential decision making, thus circumventing the need for exploration. Using ideas from Zheng et al. (2022), future works could look into online fine-tuning of Decision Diffuser by leveraging entropy of the state-sequence model for exploration. While our work focused on state based environments, it can be extended to image based environments by performing the diffusion in latent space, rather than observation space, as done in Rombach et al. (2022). For a detailed discussion on limitations of Decision Diffuser, please refer to Appendix L. + +# ACKNOWLEDGEMENTS + +The authors would like to thank Ofir Nachum, Anthony Simeonov and Richard Li for their helpful feedback on an earlier draft of the work; Jay Whang and Ge Yang for discussions on classifierfree guidance; Gabe Margolis for helping with unitree experiments; Micheal Janner for providing visualization code for Kuka block stacking; and the members of Improbable AI Lab for discussions and helpful feedback. We thank MIT Supercloud and the Lincoln Laboratory Supercomputing Center for providing compute resources. This research was supported by an NSF graduate fellowship, a DARPA Machine Common Sense grant, a MURI grant, an MIT-IBM grant, and ARO W911NF-21-1- 0097. + +This research was also partly sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19- 2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes, notwithstanding any copyright notation herein. + +# AUTHOR CONTRIBUTIONS + +Anurag Ajay conceived the framework of viewing decision-making as conditional diffusion generative modeling, implemented the Decision Diffuser algorithm, ran experiments on Offline RL and Skill Composition, and helped in paper writing. + +Yilun Du helped in conceiving the framework of viewing decision-making as conditional diffusion generative modeling, ran experiments on Constraint Satisfaction, helped in paper writing and advised Anurag. + +Abhi Gupta helped in running experiments on Offline RL and Skill Composition, participated in research discussions, and played the leading role in paper writing and making figures. + +Joshua Tenenbaum participated in research discussions. + +Tommi Jaakkola participated in research discussions and suggested the experiment of classifying running gaits. + +Pulkit Agrawal was involved in research discussions, suggested experiments related to dynamic programming, provided feedback on writing, positioning of the work, and overall advising. + +# REFERENCES + +Pulkit Agrawal, Ashvin V Nair, Pieter Abbeel, Jitendra Malik, and Sergey Levine. 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We analyze the importance of low temperature sampling in Section C, further explain composition of conditioning variable in Section D, discuss the run-time characteristics of decision diffuser in Section E, discuss when to use inverse dynamics in Section F and analyze robustness of Decision Diffuser to stochastic dynamics in Section G. Finally, we provide details of the Kuka Block Stacking environment in Section H and the Unitree environment in Section I. + +# A ILLUSTRATIVE EXAMPLES + +# A.1 IMPLICIT DYNAMIC PROGRAMMING + +![](images/06c49faf819f369ba151ad9d3ff002a905c14d4f9d446bec58e52f70a39c1d4b.jpg) +Figure A1: Illustrative example. We demonstrate the ability of Decision Diffuser to stitch together suboptimal trajectories in training dataset to obtain (near) optimal trajectories, thereby implicitly performing dynamic programming in Maze2D-open environment from Fu et al. (2020). + +We empirically demonstrate the ability of Decision Diffuser to perform implicit dynamic programming in Maze2D-open environment from Fu et al. (2020). The task in Maze2D-open environment is to reach point C and the reward is negative distance from point C. The training dataset consists of 500 trajectories from point A to point B and 500 trajectories from point B to point C. The maximum trajectory length is 50. During test time, the agent starts from point A and needs to reach point C as quickly as possible. As shown in Figure A1, Decision Diffuser can stitch trajectories in training dataset to form trajectories that goes from point A to point B in (near) straight lines. + +# A.2 CONSTRAINT COMBINATION + +Setup In linear system robot navigation, Decision Diffuser is trained on 1000 expert trajectories either satisfying the constraint $\lVert s _ { T } \rVert \leq R$ $R = 1$ ) or the constraint $\| s _ { T } \| \geq r$ $\mathit { r } = 0 . 7$ ). Here, $s _ { T } = [ x _ { T } , y _ { T } ]$ represents the final robot state in a trajectory, specifying its final 2d position. The maximum trajectory length is 50. During test time, Decision Diffuser is asked to generate trajectories satisfying $\| s _ { T } \| \le R$ and $\| s _ { T } \| \geq r$ to test its ability to satisfy single constraints. Furthermore, Decision Diffuser is also asked to generate trajectories satisfying $r \leq \left\lceil | s _ { T } | | \leq R \right\rceil$ to test its ability to satisfy combined constraints. + +Results Figure 2 shows that Decision Diffuser learns to generate trajectories perfectly (i.e. with $1 0 0 \%$ success rate) satisfying single constraints in linear system robot navigation. Furthermore, it learns to generate trajectories satisfying the composed constraint in linear system robot navigation with $9 1 . 3 \hat { \% } ( \pm 2 . 6 \% )$ accuracy where the standard error is calculated over 5 random seeds. + +# B HYPERPARAMETER AND ARCHITECTURAL DETAILS + +In this section, we describe various architectural and hyperparameter details: + +• We represent the noise model $\epsilon _ { \theta }$ with a temporal U-Net (Janner et al., 2022), consisting of a U-Net structure with 6 repeated residual blocks. Each block consisted of two temporal convolutions, each followed by group norm (Wu & He, 2018), and a final Mish nonlinearity (Misra, 2019). Timestep and condition embeddings, both 128-dimensional vectors, are produced by separate 2-layered MLP (with 256 hidden units and Mish nonlinearity) and are concatenated together before getting added to the activations of the first temporal convolution within each block. We borrow the code for temporal U-Net from https://github.com/jannerm/diffuser. + +• We represent the inverse dynamics $f _ { \phi }$ with a 2-layered MLP with 512 hidden units and ReLU activations. + +• We represent the gait classifier with a 3-layered MLP with 1024 hidden units and ReLU activations. + +• We train $\epsilon _ { \theta }$ and $f _ { \phi }$ using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of $2 e - 4$ and batch size of 32 for 2e6 train steps. + +• We train the gait classifier using the Adam optimizer with a learning rate of $2 e - 4$ and batch size of 64 for 1e6 train steps. + +• We choose the probability $p$ of removing the conditioning information to be 0.25. + +• We use $K = 1 0 0$ diffusion steps. + +• We use a planning horizon $H$ of 100 in all the D4RL locomotion tasks, 56 in D4RL kitchen tasks, 128 in Kuka block stacking, 56 in unitree-go-running tasks, 50 in the illustrative example and 60 in Block push tasks. + +• We use a guidance scale $s \in \{ 1 . 2 , 1 . 4 , 1 . 6 , 1 . 8 \}$ but the exact choice varies by task. + +• We choose $\alpha = 0 . 5$ for low temperature sampling. + +• We choose context length $C = 2 0$ + +# C IMPORTANCE OF LOW TEMPERATURE SAMPLING + +In Algorithm 1, we compute $\mu _ { k - 1 }$ and $\Sigma _ { k - 1 }$ from a noisy sequence of states and predicted noise. We find that sampling $x _ { k - 1 } \sim { \mathcal { N } } ( \mu _ { k - 1 } , \alpha \Sigma _ { k - 1 } )$ (where $\alpha \in [ 0 , 1 )$ ) with a reduced variance produces high-likelihood state sequences. We refer to this as low-temperature sampling. To empirically show its importance, we compare performances of Decision Diffuser with different values of $\alpha$ (Table A1). We show that low temperature sampling ( $\alpha = 0 . 5$ ) gives the best average returns. However, reducing the $\alpha$ to 0 eliminates the entropy in sampling and leads to lower returns. On the other hand, $\alpha = 1 . 0$ leads to a higher variance in terms of returns of the trajectories. + +
Decision DiffuserHopper-Medium-Expert
α=0104.3 ± 0.7
α= 0.5111.8 ±1.6
α= 1.0107.1 ± 3.5
+ +Table A1: Low-temperature sampling $( \alpha = 0 . 5 )$ ) allows us to get high return trajectories consistently. While $\alpha = 1 . 0$ leads to a higher variance in returns of the trajectories, $\alpha = 0 . 0$ eliminates entropy in the sampling and leads to lower returns. + +# D COMPOSING CONDITIONING VARIABLES + +In this section, we detail how Decision Diffuser trained with different conditioning variables $\{ \pmb { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n }$ composes these conditioning variables together. It learns the denoising model $\epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , { \pmb y } ^ { i } ( \tau ) , k )$ for a given conditioning variable $\mathbf { \dot { \mathbf { y } } } ^ { i } ( \tau )$ . From the derivations outlined in prior works (Luo, 2022; Song et al., 2021), we know that $\nabla _ { \pmb { x } _ { k } ( \tau ) } \log q ( \pmb { x } _ { k } ( \tau ) | \pmb { y } ^ { i } ( \tau ) ) \propto$ $- \epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , { \pmb y } ^ { i } ( \tau ) , k )$ . Therefore, each conditional trajectory distribution $\{ q ( \pmb { x } _ { k } ( \tau ) | \pmb { y } ^ { i } ( \tau ) ) \} _ { i = 1 } ^ { n }$ can be modelled with a single denoising model $\epsilon _ { \theta }$ that conditions on the respective variable $y ^ { i } ( \tau )$ . + +In order to compose $n$ different conditioning variables (i.e. skills or constraints), we would like to model $q ( \pmb { x } _ { k } ( \hat { \tau } ) | \{ \pmb { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } )$ . We assume that $\{ y ^ { i } ( \tau ) \} _ { i = 1 } ^ { n }$ are conditionally independent given + +${ \pmb x } _ { k } ( \tau )$ . Thus, we can factorize as follows: + +$$ +\begin{array} { r l } { \displaystyle q ( \mathbf { x } _ { k } ( \tau ) | \{ \boldsymbol { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } ) \propto q ( \mathbf { x } _ { k } ( \tau ) ) \displaystyle \prod _ { i = 1 } ^ { n } \frac { q ( \alpha _ { k } ( \tau ) | \boldsymbol { y } ^ { i } ( \tau ) ) } { q ( \alpha _ { k } ( \tau ) ) } } & { ( \mathrm { B a y e s ~ R u l e } ) } \\ { \displaystyle \Rightarrow \log q ( \alpha _ { k } ( \tau ) | \{ \boldsymbol { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } ) \propto \log q ( \alpha _ { k } ( \tau ) ) + \displaystyle \sum _ { i = 1 } ^ { n } ( \log q ( \alpha _ { k } ( \tau ) | \boldsymbol { y } ^ { i } ( \tau ) ) - \log q ( \alpha _ { k } ( \tau ) ) ) } \\ { \displaystyle \Rightarrow \nabla _ { \mathbf { x } _ { k } ( \tau ) } \log q ( x _ { k } ( \tau ) | \{ \boldsymbol { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } ) = \nabla _ { \mathbf { x } _ { k } ( \tau ) } \log q ( \alpha _ { k } ( \tau ) ) } \\ & { \displaystyle + \sum _ { i = 1 } ^ { n } ( \nabla _ { \mathbf { x } _ { k } ( \tau ) } \log q ( \alpha _ { k } ( \tau ) ) y ^ { i } ( \tau ) ) - \nabla _ { \mathbf { x } _ { k } ( \tau ) } \log q ( \alpha _ { k } ( \tau ) ) ) } \\ { \displaystyle \Rightarrow \epsilon _ { \theta } ( \alpha _ { k } ( \tau ) , \{ \boldsymbol { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } , k ) = \epsilon _ { \theta } ( \alpha _ { k } ( \tau ) , \boldsymbol { \mathcal { O } } , k ) + \displaystyle \sum _ { i = 1 } ^ { n } ( \epsilon _ { \theta } ( \alpha _ { k } ( \tau ) , \boldsymbol { y } ^ { i } ( \tau ) , k ) - \epsilon _ { \theta } ( x _ { k } ( \tau ) , \boldsymbol { \mathcal { O } } , k ) ) } \end{array} +$$ + +Using the above equations, we can sample from $q ( \pmb { x } _ { 0 } ( \tau ) | \{ \pmb { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } )$ with classifier free guidance using the perturbed noise: + +$$ +\begin{array} { r l } & { \hat { \boldsymbol { \epsilon } } : = \boldsymbol { \epsilon } _ { \boldsymbol { \theta } } ( \mathbf { x } _ { k } ( \tau ) , \mathcal { O } , \boldsymbol { k } ) + \omega ( \epsilon _ { \boldsymbol { \theta } } ( \mathbf { x } _ { k } ( \tau ) , \{ \boldsymbol { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n } , \boldsymbol { k } ) - \epsilon _ { \boldsymbol { \theta } } ( \mathbf { x } _ { k } ( \tau ) , \mathcal { O } , \boldsymbol { k } ) ) } \\ & { \quad = \epsilon _ { \boldsymbol { \theta } } ( \mathbf { x } _ { k } ( \tau ) , \mathcal { O } , \boldsymbol { k } ) + \omega \displaystyle \sum _ { i = 1 } ^ { n } ( \epsilon _ { \boldsymbol { \theta } } ( \mathbf { x } _ { k } ( \tau ) , \boldsymbol { y } ^ { i } ( \tau ) , \boldsymbol { k } ) - \epsilon _ { \boldsymbol { \theta } } ( \mathbf { x } _ { k } ( \tau ) , \mathcal { O } , \boldsymbol { k } ) ) } \end{array} +$$ + +We use the perturbed noise to compose skills or combine constraints at test time. This derivation was borrowed from Liu et al. (2022) and is presented here for completeness. + +While the composition of conditioning variables $\{ \pmb { y } ^ { i } ( \tau ) \} _ { i = 1 } ^ { n }$ requires them to be conditionally independent given the state trajectory $\pmb { x } _ { 0 } ( \tau )$ , we empirically observe that this condition doesn’t have to be strictly satisfied. However, we require composition of conditioning variables to be feasible (i.e. $\exists x _ { 0 } ( \tau )$ that satisfies all the conditioning variables). When the composition is infeasible, Decision Diffuser produces trajectories with incoherent behavior, as expected. This is best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. + +Requirements on the dataset First, the dataset should have a diverse set of demonstrations that shows different ways of satisfying each conditioning variable $y ^ { i } ( \tau )$ . This would allow Decision Diffuser to learn diverse ways of satisfying each conditioning variable $y ^ { i } ( \tau )$ . Since we use inverse dynamics to extract actions from the predicted state trajectory $\pmb { x } _ { 0 } ( \tau )$ , we assume that the state trajectory ${ \pmb x } _ { 0 } ( \tau )$ resulting from the composition of different conditioning variables contains consecutive state pairs $\left( { { s _ { t } } , { s _ { t + 1 } } } \right)$ that come from the same distribution that generated the demonstration dataset. Otherwise, inverse dynamics can give erroneous predictions. + +# E RUNTIME CHARACTERISTIC OF DECISION DIFFUSER + +We analyze the runtime characteristics of Decision Diffuser in this section. After training the Decision Diffuser on trajectories from the D4RL Hopper-Medium-Expert dataset, we plan in the corresponding environment according to Algorithm 1. Every action taken in the environment requires running 100 reverse diffusion steps to generate a state sequence taking on average 1.26s in wall-clock time. We can improve the run-time of planning by warm-starting the state diffusion as suggested in Janner et al. (2022). Here, we start with a generated state sequence (from the previous environment step), run forward diffusion for a fixed number of steps, and finally run the same number of reverse diffusion steps from the partially noised state sequence to generate another state sequence. Warm-starting in this way allows us to decrease the number of denoising steps to 40 (0.48s on average) without any loss in performance, to 20 (0.21s on average) with minimal loss in performance, and to 5 with less than $2 0 \%$ loss in performance (0.06s on average). We demonstrate the trade-off between performance, measured by normalized average return achieved in the environment, and planning time, measured in wall-clock time after warm-starting the reverse diffusion process, in Figure A2. + +![](images/582a316558824d8218903d31c84d2ea265682dcdd6126b4d53b0d942b675b615.jpg) +Figure A2: Performance vs planning time. We visualize the trade-off between performance, measured by normalized average return achieved in the environment, and planning time, measured in wall-clock time after warm-starting the reverse diffusion process. + +
EnvironmentBCCondDiffuserDecision Diffuser
Position Control57.3 ±1.287.3 ±3.187.8 ±2.8
Torque Control55.2 ±1.571.8 ±3.484.7 ±2.2
+ +![](images/b7e5f34bba8b05d1a3659b787a4b3d2616878cdfbc9e2f790df9d0e668b1b9ef.jpg) +Figure A3: Block push environment. + +Table A2: Block pushing with different controls. Decision Diffuser and CondDiffuser perform similarly when the agent uses position control. However, when the agent uses torque control, CondDiffuser performs worse than Decision Diffuser given it’s harder to diffuse over non-smooth action trajectories. We use the success rate of the red cube reaching the green circle as the performance metric. We report the mean success rate and the standard error over 5 random seeds. + +# F WHEN TO USE INVERSE DYNAMICS? + +In this section, we try to analyze further when using inverse dynamics is better than diffusing over actions. Table 2 showed that Decision Diffuser outperformed CondDiffuser on 3 hopper environment, thereby suggesting that inverse dynamics is a better alternative to diffusing over actions. Our intuition was that sequences over actions, represented as joint torques in our environments, tend to be more high-frequency and less smooth, thus making it harder for the diffusion model to predict (Kingma et al., 2021). We now try to verify this intuition empirically. + +Setup We choose Block Push environment adapted from Gupta et al. (2018) where the goal is to push the red cube to the green circle. When the red cube reaches the green circle, the agent gets a reward of $+ 1$ . The state space is 10-dimensional consisting of joint angles (3) and velocities (3) of the gripper, COM of the gripper (2) and position of the red cube (2). The green circle’s position is fixed and at an initial distance of 0.5 from COM of the gripper. The red cube (of size 0.03) is initially at a distance of 0.1 from COM of the gripper and at an angle $\theta$ sampled from $\mathcal { U } ( - \pi / 4 , \pi / 4 )$ at the start of every episode. The task horizon is 60 timesteps. + +There are 2 control types: (i) torque control, where the agent needs to specify joint torques (3 dimensional) and (ii) position control where the agent needs to specify the position change of COM of the gripper and the angular change in gripper’s orientation $( \Delta x , \Delta y , \Delta \phi )$ (3 dimensional). While action trajectories from position control are smooth, the action trajectories from torque control have higher frequency components. + +Offline dataset collection To collect the offline data, we use Soft Actor-Critic (SAC) (Haarnoja et al., 2018) first to train an expert policy for 1 million environment steps. We then use 1 million environment transitions as our offline dataset, which contains expert trajectories collected towards the end of the training and random action trajectories collected at the beginning of the training. We collect 2 datasets, one for each control type. + +Results Table A2 shows that Decision Diffuser and CondDiffuser perform similarly when the agent uses position control. This is because action trajectories resulting from position control are smoother and hence easier to model with diffusion. However, when the agent uses torque control, CondDiffuser performs worse than Decision Diffuser, given the action trajectories have higher frequency components and hence are harder to model with diffusion. + +# G ROBUSTNESS TO STOCHASTIC DYNAMICS + +
pBCDecision DiffuserDiffuserCQL
0.0055.2±1.584.7±2.272.4±1.473.2±2.3
0.0549.3±3.677.3±3.163.2±2.961.8±3.7
0.1025.8±3.853.2±4.152.3±4.651.2±4.3
0.1515.1±4.341.3±4.941.6±5.142.2±5.5
+ +Table A3: Robustness to stochastic dynamics. Decision Diffuser’s performance suffers when stochasticity is introduced in dynamics function. While it still outperforms Diffuser and CQL when $p = 0 . 0 5$ , its performance becomes similar to that of Diffuser and CQL for higher $p$ values. We use the success rate of the red cube reaching the green circle as the performance metric. We report the mean success rate and the standard error over 5 random seeds. + +We empirically analyze robustness of Decision Diffuser to stochasticity in dynamics function. + +Setup We use Block Push environment, described in Appendix F, with torque control. However, we inject stochasticity into the environment dynamics. For every environment step, we either sample a random action from $\mathcal { U } ( [ - 1 , - 1 , - 1 ] , [ 1 , 1 , \dot { 1 } ] )$ with probability $p$ or execute the action given by the policy with probability $( 1 - p )$ . We use $p \in \{ 0 , 0 . 0 5 , 0 . 1 , 0 . 1 5 \}$ in our experiments. + +Offline dataset collection We collect separate offline datasets for different block push environments, each characterized by a different value of $p$ . Each offline dataset consists of 1 million environment transitions collected using the method described in Appendix F. + +Results Table A3 characterizes how the performance of BC, Decision Diffuser, Diffuser, and CQL changes with increasing stochasticity in the environment dynamics. We observe that the Decision Diffuser outperforms Diffuser and CQL for $p = 0 . 0 5$ , however all methods including the Decision Diffuser settle to a similar performance for larger values of $p$ . + +Several works (Paster et al., 2022; Yang et al., 2022) have shown that the performance of returnconditioned policies suffers as the stochasticity in environment dynamics increases. This is because the return-conditioned policies aren’t able to distinguish between high returns from good actions and high returns from environment stochasticity. Hence, these return-conditioned policies can learn sub-optimal actions that got associated with high-return trajectories in the dataset due to environment stochasticity. Given Decision diffuser uses return conditioning to generate actions in offline RL, its performance also suffers when stochasticity in environment dynamics increases. + +Some recent works (Yang et al., 2022; Villaflor et al., 2022) address the above issue by learning a latent model for future states and then conditioning the policy on predicted latent future states rather than returns. Conditioning Decision Diffuser on future state information, rather than returns, would make it more robust to stochastic dynamics and could be an interesting avenue for future works. + +# H KUKA BLOCK STACKING + +In the Kuka blocking stacking environment, the underlying goal is to stack a set of blocks on top of each other. Models have trained on a set of demonstration data, where a set of 4 blocks are sequentially stacked on top of each other to form a block tower. + +We construct state-space plans of length 128. Following (Janner et al., 2022), we utilize a close-loop controller to generate actions for each state in our state-space plan (controlling the 7 degrees of freedom in joints). The total maximum trajectory length plan in Kuka block stacking is 384. We detail differences between the two consider conditional stacking environments below: + +• Stacking In the stacking environment, at test time we wish to again construct a tower of four blocks. + +• Rearrangement In the rearrangement environment, at test time wish to stack blocks in a configuration where a set of blocks are above a second set. This set of stack-place relations may not precisely correspond to a single block tower (can instead construct two block towers), making this environment an out-of-distribution challenge. + +In addition to Diffuser (Janner et al., 2022), we used goal-conditioned variants of CQL (Kumar et al., 2020) and BCQ (Fujimoto et al., 2019) as baselines for the block stacking and rearrangement with single constraint. However, they get a success rate of 0.0. + +# I UNITREE GO RUNNING + +We consider Unitree-go-running environment (Margolis & Agrawal, 2022) where a quadruped robot runs in 3 different gaits: bounding, pacing, and trotting. The state space is 56 dimensional, the action space is 12 dimensional, and the maximum trajectory length is 250. + +As described in Section 4.3, we train Decision Diffuser on expert trajectories demonstrating individual gaits. During testing, we compose the noise model of our reverse diffusion process according to equation 9. This allows us to sample trajectories of the quadruped robot with entirely new running behavior. Figures A4,A5,A6 shows the ability of Decision Diffuser to imitate bounding, trotting and pacing and their combinations. + +# I.1 QUANTITATIVE VERIFICATION OF COMPOSITION + +We now try to quantitatively verify whether the trajectories resulting from composition of 2 gaits does indeed contain only those 2 gaits. + +Setup We learn a gait classifier that takes in a sub-sequence of states (of length 10) and predicts the gait-ID. It is represented by a 3-layered MLP with 1024 hidden units and ReLU activations that concatenates the sub-sequence of states (of length 10) into a single vector of dimension 560 before taking it in as an input. We train the gait classifier on the demonstration dataset. To ensure that the learned classifier can predict gait-ID on trajectories generated by the composition of skills, we use MixUp-style (Zhang et al., 2017) data augmentation during training. We create a synthetic subsequence of length 10 by concatenating two sampled sub-sequence (from the demonstration dataset) of length $l _ { i }$ and $l _ { j }$ (where $l _ { i } + l _ { j } = 1 0$ ) from gaits with $\operatorname { I D } i$ and $j$ and give it a label $\frac { l _ { i } } { l _ { i } + l _ { j } }$ one-ho $( i ) +$ $\frac { l _ { j } } { l _ { i } + l _ { j } }$ one-hot $( j )$ . During training, we sample a sub-sequence from the demonstration dataset with $7 0 \%$ probability and a sythenthic sub-sequence with $3 0 \%$ probability. We train the classifier for $2 e 6$ train steps with a learning rate of $2 e - 4$ and a batch size of 64. + +Results Figures A4,A5,A6 show that the classifier’s prediction is consistent with the visualized composed trajectories. Furthermore, we use Decision diffuser to act in the environment and generate 1000 trott trajectories, 1000 pace trajectories, 1000 bound trajectories, and 1000 composed trajectories for each possible pair of individual gaits. We then evaluate the learned gait classifier on these trajectories and compute the percentage of timesteps a particular gait has the highest probability. From Figures A4,A5,A6, we can see that if trajectories are generated by the composition of two gaits, then those two gaits will have the two highest probabilities across different timesteps in those trajectories. + +# I.2 A SIMPLE BASELINE FOR COMPOSITION + +Let one-hot $( i )$ and one- $\cdot \mathrm { h o t } ( j )$ represent two different gaits that can be generated using noise models $\epsilon _ { \theta } ( { \pmb x } _ { k } ( \tau ) , \mathrm { o n e - h o t } ( i ) , k )$ and $\begin{array} { r } { \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \mathrm { o n e - h o t } ( j ) , k ) } \end{array}$ respectively. To compose these gaits, we compose the above-mentioned noise models using equation 9. As an alternative, we see if the noise model $\epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \mathrm { o n e - h o t } ( i ) + \mathrm { o n e - h o t } ( j ) , k )$ can lead to composed gaits. However, we observe that $\epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau )$ , one- $\mathbf { \cdot h o t } ( i ) + \mathbf { o n e - h o t } ( j ) , k )$ catastrophically fail to generate any gait (see videos at https://anuragajay.github.io/decision-diffuser/). This happens because the condition variable one- $\mathsf { 1 o t } ( i ) + \mathsf { o n e { \mathrm { - } } h o t } ( j )$ was never seen by the noise model $\epsilon _ { \theta }$ during training. + +![](images/c6eca9326f1240e24c5bc046689af995227c5ce856ee5e3438368121d3c1e16c.jpg) +Figure A4: Composing Trott and Pace. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. + +![](images/718ee11d8a554aad4ad0463dc91824091eb322e60b29c2deeafd23463aa4b13e.jpg) +Figure A5: Composing Trott and Bound. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. + +# J NOT COMPOSITIONS WITH DECISION DIFFUSER + +Decision diffuser can also support ”NOT” composition. Suppose we wanted to sample from $q ( \pmb { x } _ { 0 } ( \tau ) | \mathrm { N O T } \ y ^ { j } ( \tau ) )$ . Let $\{ y ^ { i } ( \tau ) \} _ { i = 1 } ^ { n }$ be the set of all conditioning variables. Then, following derivations from Liu et al. (2022) and using $\beta = 1$ , we can sample from $q ( \pmb { x } _ { 0 } ( \tau ) | \mathrm { N O T } \ y ^ { j } ( \tau ) )$ using the perturbed noise: + +$$ +\begin{array} { l } { \hat { \epsilon } : = \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \emptyset , k ) + \omega ( \displaystyle \sum _ { i \neq j } ( \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \pmb { y } ^ { i } ( \tau ) , k ) - \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \emptyset , k ) ) } \\ { - \left( \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \pmb { y } ^ { j } ( \tau ) , k ) - \epsilon _ { \theta } ( \pmb { x } _ { k } ( \tau ) , \emptyset , k ) ) \right) } \end{array} +$$ + +We demonstrate the ability of Decision Diffuser to support ”NOT” composition by using it to satisfy constraint of type BlockHeight $( i ) >$ BlockHeight $( j )$ AND (NOT BlockHeight $. ( j ) ~ >$ BlockHeight $( i )$ ) in Kuka block stacking task, as visualized in videos at https://anuragajay.github.io/decision-diffuser/. As the Decision Diffuser does not provide an explicit density estimate for each skill, it can’t natively support OR composition. + +![](images/79557db21666a3824496a144965288975b56e26dcafa64a8c6e3a89d15f4a7a3.jpg) +Figure A6: Composing Bound and Pace. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. + +# K COMPARING Q-FUNCTION GUIDED DIFFUSION AND CLASSIFIER-FREE GUIDED DIFFUSION + +Classifier-free guided diffusion and Q-value guided diffusion are theoretically equivalent. However, as noted in several works (Nichol et al., 2021; Ho & Salimans, 2022; Saharia et al., 2022), classifier-free guidance performs better than classifier guidance (i.e. Q function guidance in our case) in practice. This is due to following reasons: + +• Classifier-guided diffusion models learns an unconditional diffusion model along with a classifier (Q-function in our case) and uses gradients from the classifier to perform conditional sampling. However, the unconditional diffusion model doesn’t need to focus on conditional modeling during training and only cares about conditional generation during testing after it has been trained. In contrast, classifier-free guidance relies on conditional diffusion model to estimate gradients of the implicit classifier. Since the conditional diffusion model, learned when using classifier-free guidance, focuses on conditional modeling during train time, it performs better in conditional generation during test time. Q function trained on an offline dataset can erroneously predict high Q values for out-ofdistribution actions given any state. This problem has been extensively studied in offline RL literature (Kumar et al., 2020; Fujimoto et al., 2019; Levine et al., 2020). In online RL, this issue is automatically corrected when the policy acts in the environment, thinking an action to be good but then receives a low reward for it. In offline RL, this issue can’t be corrected easily; hence, the learned Q-function can often guide the diffusion model towards out-of-distribution actions that might be sub-optimal. In contrast, classifier-free guidance circumvents the issue of learning a Q-function and directly conditions the diffusion model on returns. Hence, classifier-free guidance doesn’t suffer due to errors in learned Q-functions and hence performs better than Q-function guided diffusion. + +# L LIMITATIONS OF DECISION DIFFUSER + +We summarize the limitations of Decision Diffuser: + +• No partial observability Decision Diffuser works with fully observable MDPs. Naive extensions to partially observed MDPs (POMDPs) may cause self-delusions (Ortega et al., 2021) in Decision Diffuser. Hence, extending Decision Diffuser to POMDPs could be an exciting avenue for future work. + +• Inability to explore the environment and update itself in online setting In this work, we focused on offline sequential decision making, thus circumventing the need for exploration. Using ideas from Zheng et al. (2022), future works could look into online fine-tuning of Decision Diffuser by leveraging entropy of the state-sequence model for exploration. + +• Experiments on only state-based environments While our work focused on state based environments, it can be extended to image based environments by performing the diffusion in latent space, rather than observation space, as done in Rombach et al. (2022). + +• Only AND and NOT compositions are supported Since Decision Diffuser does not provide an explicit density estimate for each condition variable, it can’t natively support OR composition. + +• Performance degradation in environments with stochastic dynamics In environments with highly stochastic dynamics, Decision Diffuser loses its advantage and performs similarly to Diffuser and CQL. To tackle environments with stochastic dynamics, recent works (Yang et al., 2022; Villaflor et al., 2022) propose learning a latent model for future states and then conditioning the policy on predicted latent future states rather than returns. Conditioning Decision Diffuser on future state information, rather than returns, would make it more robust to stochastic dynamics and could be an interesting avenue for future works. + +• Performance in limited data regime Since diffusion models are prone to overfitting in case of limited data, Decision Diffuser is also prone to overfitting in limited data regime. \ No newline at end of file diff --git a/parse/dev/sP1fo2K9DFG/sP1fo2K9DFG_content_list.json b/parse/dev/sP1fo2K9DFG/sP1fo2K9DFG_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..dc0e0953679e49fcb36fa50657077d6b3c75e0b8 --- /dev/null +++ b/parse/dev/sP1fo2K9DFG/sP1fo2K9DFG_content_list.json @@ -0,0 +1,2912 @@ +[ + { + "type": "text", + "text": "IS CONDITIONAL GENERATIVE MODELING ALL YOU NEED FOR DECISION-MAKING? ", + "text_level": 1, + "bbox": [ + 176, + 99, + 825, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anurag Ajay∗ †§¶, Yilun Du \\*§¶, Abhi Gupta $\\ast \\ddag \\mathfrak { S } \\ P$ , Joshua Tenenbaum¶, Tommi Jaakkola‡§¶, Pulkit Agrawal†§¶ ", + "bbox": [ + 187, + 169, + 836, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Improbable AI Lab† \nOperations Research Center‡ \nComputer Science and Artificial Intelligence Lab§ \nMassachusetts Institute of Technology¶ ", + "bbox": [ + 187, + 200, + 516, + 257 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 295, + 544, + 309 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent improvements in conditional generative modeling have made it possible to generate high-quality images from language descriptions alone. We investigate whether these methods can directly address the problem of sequential decisionmaking. We view decision-making not through the lens of reinforcement learning (RL), but rather through conditional generative modeling. To our surprise, we find that our formulation leads to policies that can outperform existing offline RL approaches across standard benchmarks. By modeling a policy as a returnconditional diffusion model, we illustrate how we may circumvent the need for dynamic programming and subsequently eliminate many of the complexities that come with traditional offline RL. We further demonstrate the advantages of modeling policies as conditional diffusion models by considering two other conditioning variables: constraints and skills. Conditioning on a single constraint or skill during training leads to behaviors at test-time that can satisfy several constraints together or demonstrate a composition of skills. Our results illustrate that conditional generative modeling is a powerful tool for decision-making. ", + "bbox": [ + 233, + 329, + 766, + 536 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 571, + 336, + 587 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Over the last few years, conditional generative modeling has yielded impressive results in a range of domains, including high-resolution image generation from text descriptions (DALL-E, ImageGen) (Ramesh et al., 2022; Saharia et al., 2022), language generation (GPT) (Brown et al., 2020), and step-by-step solutions to math problems (Minerva) (Lewkowycz et al., 2022). The success of generative models in countless domains motivates us to apply them to decision-making. Conveniently, there exists a wide body of research on recovering high-performing policies from data logged by already operational systems (Kostrikov et al., 2022; Kumar et al., 2020; Walke et al., 2022). This is particularly useful in real-world settings where interacting with the environment is not always possible, and exploratory decisions can have fatal consequences (Dulac-Arnold et al., 2021). With access to such offline datasets, the problem of decision-making reduces to learning a probabilistic model of trajectories, a setting where generative models have already found success. ", + "bbox": [ + 174, + 594, + 825, + 747 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In offline decision-making, we aim to recover optimal reward-maximizing trajectories by stitching together sub-optimal reward-labeled trajectories in the training dataset. Prior works (Kumar et al., 2020; Kostrikov et al., 2022; Wu et al., 2019; Kostrikov et al., 2021; Dadashi et al., 2021; Ajay et al., 2020; Ghosh et al., 2022) have tackled this problem with reinforcement learning (RL) that uses dynamic programming for trajectory stitching. To enable dynamic programming, these works learn a value function that estimates the discounted sum of rewards from a given state. However, value function estimation is prone to instabilities due to function approximation, off-policy learning, and bootstrapping together, together known as the deadly triad (Sutton & Barto, 2018). Furthermore, to stabilize value estimation in offline regime, these works rely on heuristics to keep the policy within the dataset distribution. These challenges make it difficult to scale existing offline RL algorithms. ", + "bbox": [ + 174, + 753, + 825, + 892 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/9dac5e5f723a941f045f73b69f622ccae89b7291e5b8f4eed92c65b334c4c613.jpg", + "image_caption": [ + "Figure 1: Decision Making using Conditional Generative Modeling. Framing decision making as a conditional generative modeling problem allows us to maximize rewards, satisfy constraints and compose skills. " + ], + "image_footnote": [], + "bbox": [ + 302, + 99, + 692, + 176 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we ask if we can perform dynamic programming to stitch together sub-optimal trajectories to obtain an optimal trajectory without relying on value estimation. Since conditional diffusion generative models can generate novel data points by composing training data (Saharia et al., 2022), we leverage it for trajectory stitching in offline decision-making. Given a dataset of reward-labeled trajectories, we adapt diffusion models (Sohl-Dickstein et al., 2015) to learn a return-conditional trajectory model. During inference, we use classifier-free guidance with lowtemperature sampling, which we hypothesize to implicitly perform dynamics programming, to capture the best behaviors in the dataset and glean return maximizing trajectories (see Appendix A). Our straightforward conditional generative modeling formulation outperforms existing approaches on standard D4RL tasks (Fu et al., 2020). ", + "bbox": [ + 173, + 222, + 825, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Viewing offline decision-making through the lens of conditional generative modeling allows going beyond conditioning on returns (Figure 1). Consider an example (detailed in Appendix A) where a robot with linear dynamics navigates an environment containing two concentric circles (Figure 2). We are given a dataset of state-action trajectories of the robot, each satisfying one of two constraints: (i) the final position of the robot is within the larger circle, and (ii) the final position of the robot is outside the smaller circle. With conditional diffusion modeling, we can use the datasets to learn a constraintconditioned model that can generate trajectories satisfying any set of constraints. During inference, the learned trajectory model can merge constraints from the dataset and generate trajectories that satisfy the combined constraint. Figure 2 shows that the constraint-conditioned model can generate trajectories such that the final position of the robot lies between the concentric circles. ", + "bbox": [ + 173, + 367, + 826, + 506 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/3f638c6b9477f25a868e5f8e34ac18c10ed64b5789def600b6b4750a38198356.jpg", + "image_caption": [ + "Figure 2: Illustrative example. We visualize the 2d robot navigation environment and the constraints satisfied by the trajectories in the dataset derived from the environment. We show the ability of the conditional diffusion model to generate trajectories that satisfy the combined constraints. " + ], + "image_footnote": [], + "bbox": [ + 336, + 507, + 661, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Here, we demonstrate the benefits of modeling policies as conditional generative models. First, conditioning on constraints allows policies to not only generate behaviors satisfying individual constraints but also generate novel behaviors by flexibly combining constraints at test time. Further, conditioning on skills allows policies to not only imitate individual skills but also generate novel behaviors by composing those skills. We instantiate this idea with a state-sequence based diffusion probabilistic model (Ho et al., 2020) called Decision Diffuser, visualized in Figure 1. In summary, our contributions include (i) illustrating conditional generative modeling as an effective tool in offline decision making, (ii) using classifier-free guidance with low-temperature sampling, instead of dynamic programming, to get return-maximizing trajectories and, (iii) leveraging the framework of conditional generative modeling to combine constraints and compose skills during inference flexibly. ", + "bbox": [ + 173, + 633, + 826, + 772 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 174, + 780, + 326, + 795 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 800, + 416, + 814 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We formulate the sequential decision-making problem as a discounted Markov Decision Process (MDP) defined by the tuple $\\langle \\rho _ { 0 } , S , { \\mathcal { A } } , { \\mathcal { T } } , { \\mathcal { R } } , { \\gamma } \\rangle$ , where $\\rho _ { 0 }$ is the initial state distribution, $s$ and $\\mathcal { A }$ are state and action spaces, $\\mathcal { T } : \\mathcal { S } \\times \\mathcal { A } \\mathcal { S }$ is the transition function, $\\mathcal { R } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } \\mathbb { R }$ gives the reward at any transition and $\\gamma \\in [ 0 , 1 )$ is a discount factor. The agent acts with a stochastic policy $\\pi :$ ${ \\cal S } \\to \\Delta _ { \\cal A }$ , generating a sequence of state-action-reward transitions or trajectory $\\tau : = ( s _ { k } , a _ { k } , r _ { k } ) _ { k \\geq 0 }$ with probability $p _ { \\pi } ( \\tau )$ and return $\\begin{array} { r } { R ( \\tau ) : = \\sum _ { k \\geq 0 } \\gamma ^ { k } r _ { k } } \\end{array}$ . The standard objective in RL is to find a return-maximizing policy $\\begin{array} { r } { \\pi ^ { * } = \\arg \\operatorname* { m a x } _ { \\pi } \\mathbb { E } _ { \\tau \\sim p _ { \\pi } } \\overline { { [ R ( \\tau ) ] } } } \\end{array}$ . ", + "bbox": [ + 173, + 821, + 825, + 925 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Temporal Difference Learning TD methods (Fujimoto et al., 2018; Lillicrap et al., 2015) estimate $Q ^ { * } ( s , a ) : = \\mathbb { E } _ { \\tau \\sim p _ { \\pi ^ { * } } } [ R ( \\tau ) | s _ { 0 } = s , a _ { 0 } = a ]$ , the return achieved under the optimal policy $\\pi ^ { * }$ when starting in state $s$ and taking action $a$ , with a parameterized $Q$ -function. This requires minimizing the following TD loss: ", + "bbox": [ + 173, + 103, + 825, + 159 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2bf532cfc6885cbf314f179956efa8d115c87c8434197ca19e0caa295e22639a.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { T D } } ( \\theta ) : = \\mathbb { E } _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } } [ ( r + \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } \\in \\mathcal { A } } Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) - Q _ { \\theta } ( s , a ) ) ^ { 2 } ]\n$$", + "text_format": "latex", + "bbox": [ + 295, + 162, + 700, + 186 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Continuous action spaces further require learning a parametric policy $\\pi _ { \\phi } ( a | s )$ that plays the role of the maximizing action in equation 1. This results in a policy objective that must be maximized: ", + "bbox": [ + 174, + 190, + 823, + 219 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/04b40983b847436144e6f160dc15a472f7e55a555678def1dbc3c0712123ce7d.jpg", + "text": "$$\n\\mathcal { I } ( \\phi ) : = \\mathbb { E } _ { s \\in \\mathcal { D } , a \\sim \\pi _ { \\phi } ( . | s ) } [ Q ( s , a ) ]\n$$", + "text_format": "latex", + "bbox": [ + 388, + 222, + 609, + 241 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, the dataset of transitions $\\mathcal { D }$ evolves as the agent interacts with the environment and both $Q _ { \\theta }$ and $\\pi _ { \\phi }$ are trained together. These methods make use of function approximation, off-policy learning, and bootstrapping, leading to several instabilities in practice (Sutton, 1988; Van Hasselt et al., 2018). ", + "bbox": [ + 174, + 251, + 826, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Offline RL requires finding a return-maximizing policy from a fixed dataset of transitions collected by an unknown behavior policy $\\mu$ (Levine et al., 2020). Using TD-learning naively causes the state visitation distribution $d ^ { \\pi _ { \\phi } } ( s )$ to move away from the distribution of the dataset $d ^ { \\mu } ( s )$ . In turn, the policy $\\pi _ { \\phi }$ begins to take actions that are substantially different from those already seen in the data. Offline RL algorithms resolve this distribution-shift by imposing a constraint of the form $D ( d ^ { \\pi _ { \\phi } } | | d ^ { \\mu } )$ , where $D$ is some divergence metric, directly in the TD-learning procedure. The constrained optimization problem now demands additional implementation heuristics to achieve any reasonable performance (Kumar et al., 2021). The Decision Diffuser, in comparison, doesn’t have any of these disadvantages. It does not require estimating any kind of $Q$ -function, thereby sidestepping TD methods altogether. It also does not face the risk of distribution-shift as generative models are trained with maximum-likelihood estimation. ", + "bbox": [ + 173, + 295, + 825, + 449 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 DIFFUSION PROBABILISTIC MODELS ", + "text_level": 1, + "bbox": [ + 178, + 458, + 468, + 472 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a specific type of generative model that learn the data distribution $q ( { \\pmb x } )$ from a dataset $\\mathcal { D } : = \\{ \\pmb { x } ^ { i } \\} _ { 0 \\leq i < M }$ . They have been used most notably for synthesizing high-quality images from text descriptions (Saharia et al., 2022; Nichol et al., 2021). Here, the data-generating procedure is modelled with a predefined forward noising process $q ( \\pmb { x } _ { k + 1 } | \\pmb { x } _ { k } ) : = \\mathcal { N } ( \\pmb { x } _ { k + 1 } ; \\sqrt { \\alpha _ { k } } \\pmb { x } _ { k } , ( 1 - \\alpha _ { k } ) \\pmb { I } )$ and a trainable reverse process $p _ { \\theta } ( \\pmb { x } _ { k - 1 } | \\pmb { x } _ { k } ) : =$ $\\mathcal { N } ( \\pmb { x } _ { k - 1 } | \\mu _ { \\theta } ( \\pmb { x } _ { k } , k ) , \\Sigma _ { k } )$ , where $\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )$ denotes a Gaussian distribution with mean $\\mu$ and variance $\\Sigma$ , $\\alpha _ { k } \\in \\mathbb { R }$ determines the variance schedule, $\\scriptstyle { \\pmb { x } } _ { 0 } : = { \\pmb { x } }$ is a sample, ${ \\pmb x } _ { 1 } , { \\pmb x } _ { 2 } , . . . , { \\pmb x } _ { K - 1 }$ are the latents, and $\\pmb { x } _ { K } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\pmb { I } )$ for carefully chosen $\\alpha _ { k }$ and long enough $K$ . Starting with Gaussian noise, samples are then iteratively generated through a series of ”denoising” steps. ", + "bbox": [ + 173, + 477, + 825, + 604 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Although a tractable variational lower-bound on $\\log p _ { \\theta }$ can be optimized to train diffusion models, Ho et al. (2020) propose a simplified surrogate loss: ", + "bbox": [ + 173, + 609, + 823, + 637 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/bd78f527a1f1f8d43ca1aecd368174189ff5bd78b1018ae19a032b1c0623c57a.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { d e n o i s e } } ( \\theta ) : = \\mathbb { E } _ { k \\sim [ 1 , K ] , \\boldsymbol { x } _ { 0 } \\sim q , \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , I ) } [ | | \\epsilon - \\epsilon _ { \\theta } ( \\boldsymbol { x } _ { k } , k ) | | ^ { 2 } ]\n$$", + "text_format": "latex", + "bbox": [ + 312, + 640, + 686, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The predicted noise $\\boldsymbol { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } _ { k } , k )$ , parameterized with a deep neural network, estimates the noise $\\epsilon \\sim$ $\\mathcal { N } ( 0 , I )$ added to the dataset sample $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ to produce noisy $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ . This is equivalent to predicting the mean of $p _ { \\theta } ( \\pmb { x } _ { k - 1 } | \\pmb { x } _ { k } )$ since $\\mu _ { \\boldsymbol { \\theta } } ( \\mathbf { \\boldsymbol { x } } _ { k } , k )$ can be calculated as a function of $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } , k )$ (Ho et al., 2020). ", + "bbox": [ + 178, + 664, + 823, + 707 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Guided Diffusion Modelling the conditional data distribution $q ( { \\pmb x } | { \\pmb y } )$ makes it possible to generate samples with attributes of the label $\\textbf { { y } }$ . The equivalence between diffusion models and scorematching (Song et al., 2021), which shows $\\epsilon _ { \\theta } ( x _ { k } , \\bar { k } ) \\propto \\nabla _ { x _ { k } } \\log p ( \\pmb { x } _ { k } )$ , leads to two kinds of methods for conditioning: classifier-guided (Nichol & Dhariwal, 2021) and classifier-free (Ho & Salimans, 2022). The former requires training an additional classifier $p _ { \\phi } ( \\pmb { y } | \\pmb { x } _ { k } )$ on noisy data so that samples may be generated at test-time with the perturbed noise $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } , k ) - \\omega \\sqrt { 1 - \\bar { \\alpha } _ { k } } \\nabla _ { \\pmb { x } _ { k } } \\log p ( \\pmb { y } | \\pmb { x } _ { k } )$ , where $\\omega$ is referred to as the guidance scale. The latter does not separately train a classifier but modifies the original training setup to learn both a conditional $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } , \\pmb { y } , k )$ and an unconditional $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } , k )$ model for the noise. The unconditional noise is represented, in practice, as the conditional noise $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } , 0 , k )$ where a dummy value $\\varnothing$ takes the place of $\\textbf { { y } }$ . The perturbed noise $\\epsilon _ { \\theta } ( { \\pmb x } _ { k } , k ) + \\omega ( \\epsilon _ { \\theta } ( { \\pmb x } _ { k } , { \\pmb y } , k ) - \\epsilon _ { \\theta } ( { \\pmb x } _ { k } , k ) )$ is used to later generate samples. ", + "bbox": [ + 173, + 708, + 826, + 864 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 GENERATIVE MODELING WITH THE DECISION DIFFUSER ", + "text_level": 1, + "bbox": [ + 176, + 869, + 681, + 886 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It is useful to solve RL from offline data, both without relying on TD-learning and without risking distribution-shift. To this end, we formulate sequential decision-making as the standard problem of ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/5193fd80ab094ae9c4154b23239d26a5d2b68886d0126e16c31679f9afe8be80.jpg", + "image_caption": [ + "Figure 3: Planning with Decision Diffuser. Given the current state $s _ { t }$ and conditioning, Decision Diffuser uses classifier-free guidance with low-temperature sampling to generate a sequence of future states. It then uses inverse dynamics to extract and execute the action $a _ { t }$ that leads to the immediate future state $s _ { t + 1 }$ . " + ], + "image_footnote": [], + "bbox": [ + 271, + 99, + 725, + 244 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "conditional generative modeling: ", + "bbox": [ + 174, + 306, + 392, + 320 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5839cb65e52e459da7b8716946f055e40e525790aaf7cc5eb376c9f196100d62.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\mathbb { E } _ { \\tau \\sim \\mathcal { D } } [ \\log p _ { \\theta } ( \\pmb { x } _ { 0 } ( \\tau ) | \\pmb { y } ( \\tau ) ) ]\n$$", + "text_format": "latex", + "bbox": [ + 390, + 330, + 607, + 353 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our goal is to estimate the conditional data distribution with $p _ { \\theta }$ so we can later generate portions of a trajectory $\\pmb { x } _ { 0 } ( \\tau )$ from information $\\pmb { y } ( \\tau )$ about it. Examples of $\\textbf { { y } }$ could include the return under the trajectory, the constraints satisfied by the trajectory, or the skill demonstrated in the trajectory. We construct our generative model according to the conditional diffusion process: ", + "bbox": [ + 173, + 363, + 825, + 419 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4a2aedb6b54591ec48f654546dd7beae0401fc8c1a4821d171893e3898154bc3.jpg", + "text": "$$\n\\begin{array} { r l } { q ( \\pmb { x } _ { k + 1 } ( \\tau ) | \\pmb { x } _ { k } ( \\tau ) ) , } & { { } p _ { \\theta } ( \\pmb { x } _ { k - 1 } ( \\tau ) | \\pmb { x } _ { k } ( \\tau ) , \\pmb { y } ( \\tau ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 429, + 660, + 446 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As usual, $q$ represents the forward noising process while $p _ { \\theta }$ the reverse denoising process. In the following, we discuss how we may use diffusion for decision making. First, we discuss the modeling choices for diffusion in Section 3.1. Next, we discuss how we may utilize classifier-free guidance to capture the best aspects of trajectories in Section 3.2. We then discuss the different behaviors that may be implemented with conditional diffusion models in Section 3.3. Finally, we discuss practical training details of our approach in Section 3.4. ", + "bbox": [ + 174, + 455, + 825, + 540 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 DIFFUSING OVER STATES ", + "text_level": 1, + "bbox": [ + 174, + 547, + 390, + 561 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In images, the diffusion process is applied across all pixel values in an image. Na¨ıvely, it would therefore be natural to apply a similar process to model the state and actions of a trajectory. However, in the reinforcement learning setting, directly modeling actions using a diffusion process has several practical issues. First, while states are typically continuous in nature in RL, actions are more varied, and are often discrete in nature. Furthermore, sequences over actions, which are often represented as joint torques, tend to be more high-frequency and less smooth, making them much harder to predict and model (Tedrake, 2022). Due to these practical issues, we choose to diffuse only over states, as defined below: ", + "bbox": [ + 173, + 564, + 826, + 676 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/efff52266db41ee034265912e4c7f6655709a26573a8d48807a02ed9a034d46d.jpg", + "text": "$$\n\\pmb { x } _ { k } ( \\tau ) : = ( s _ { t } , s _ { t + 1 } , . . . , s _ { t + H - 1 } ) _ { k }\n$$", + "text_format": "latex", + "bbox": [ + 388, + 679, + 609, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, $k$ denotes the timestep in the forward process and $t$ denotes the time at which a state was visited in trajectory $\\tau$ . Moving forward, we will view ${ \\pmb x } _ { k } ( \\tau )$ as a noisy sequence of states from a trajectory of length $H$ . We represent ${ \\pmb x } _ { k } ( \\tau )$ as a two-dimensional array with one column for each timestep of the sequence. ", + "bbox": [ + 173, + 704, + 825, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Acting with Inverse-Dynamics. Sampling states from a diffusion model is not enough for defining a controller. A policy can, however, be inferred from estimating the action $a _ { t }$ that led the state $s _ { t }$ to $s _ { t + 1 }$ for any timestep $t$ in ${ \\pmb x } _ { 0 } ( \\tau )$ . Given two consecutive states, we generate an action according to the inverse dynamics model (Agrawal et al., 2016; Pathak et al., 2018): ", + "bbox": [ + 173, + 767, + 825, + 824 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d6688a1fe6c467d6c5495ff3e7909d6a6aaedc5b107b9bc358307fbdf2656291.jpg", + "text": "$$\na _ { t } : = f _ { \\phi } ( s _ { t } , s _ { t + 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 437, + 833, + 560, + 851 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that the same offline data used to train the reverse process $p _ { \\theta }$ can also be used to learn $f _ { \\phi }$ . We illustrate in Table 2 how the design choice of directly diffusing state distributions, with an inverse dynamics model to predict action, significantly improves performance over diffusing across both states and actions jointly. Furthermore, we empirically compare and analyze when to use inverse dynamics and when to diffuse over actions in Appendix F. ", + "bbox": [ + 174, + 859, + 825, + 931 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 PLANNING WITH CLASSIFIER-FREE GUIDANCE", + "text_level": 1, + "bbox": [ + 173, + 103, + 542, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Given a diffusion model representing the different trajectories in a dataset, we next discuss how we may utilize the diffusion model for planning. To use the model for planning, it is necessary to additionally condition the diffusion process on characteristics $\\pmb { y } ( \\tau )$ . One approach could be to train a classifier $p _ { \\phi } ( \\pmb { y } ( \\tau ) | \\pmb { x } _ { k } ( \\tau ) )$ to predict $\\pmb { y } ( \\tau )$ from noisy trajectories ${ \\pmb x } _ { k } ( \\tau )$ . In the case that $\\pmb { y } ( \\tau )$ represents the return under a trajectory, this would require estimating a $Q$ -function, which requires a separate, complex dynamic programming procedure. ", + "bbox": [ + 173, + 121, + 825, + 205 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "One approach to avoid dynamic programming is to directly train a conditional diffusion model conditioned on the returns $\\pmb { y } ( \\tau )$ in the offline dataset. However, as our dataset consists of a set of sub-optimal trajectories, the conditional diffusion model will be polluted by such sub-optimal behaviors. To circumvent this issue, we utilize classifier-free guidance (Ho & Salimans, 2022) with low-temperature sampling, to extract high-likelihood trajectories in the dataset. We find that such trajectories correspond to the best set of behaviors in the dataset. For a detailed discussion comparing Q-function guidance and classifier-free guidance, please refer to Appendix K. Formally, to implement classifier free guidance, a $\\pmb { x } _ { 0 } ( \\tau )$ is sampled by starting with Gaussian noise ${ \\pmb x } _ { K } ( \\tau )$ and refining ${ \\pmb x } _ { k } ( \\tau )$ into $\\pmb { x } _ { k - 1 } ( \\tau )$ at each intermediate timestep with the perturbed noise: ", + "bbox": [ + 173, + 210, + 825, + 337 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/25d4496eecae08b1222f19972ee9ea51012b605922dabff02464df366d671eed.jpg", + "text": "$$\n\\hat { \\epsilon } : = \\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , \\mathcal { O } , k ) + \\omega ( \\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , { \\pmb y } ( \\tau ) , k ) - \\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , \\mathcal { O } , k ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 343, + 714, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the scalar $\\omega$ applied to $( \\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , { \\pmb y } ( \\tau ) , k ) - \\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , \\emptyset , k ) )$ seeks to augment and extract the best portions of trajectories in the dataset that exhibit $\\pmb { y } ( \\tau )$ . With these ingredients, sampling from the Decision Diffuser becomes similar to planning in RL. First, we observe a state in the environment. Next, we sample states later into the horizon with our diffusion process conditioned on $\\textbf { { y } }$ and history of last $C$ states observed. Finally, we identify the action that should be taken to reach the most immediate predicted state with our inverse dynamics model. This procedure repeats in a standard receding-horizon control loop described in Algorithm 1 and visualized in Figure 3. ", + "bbox": [ + 173, + 367, + 825, + 465 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 CONDITIONING BEYOND RETURNS ", + "text_level": 1, + "bbox": [ + 174, + 473, + 457, + 488 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "So far we have not explicitly defined the conditioning variable $\\pmb { y } ( \\tau )$ . Though we have mentioned that it can be the return under a trajectory, we may also consider guiding our diffusion process towards sequences of states that satisfy relevant constraints or demonstrate specific behavior. ", + "bbox": [ + 174, + 491, + 823, + 532 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Maximizing Returns To generate trajectories that maximize return, we condition the noise model on the return of a trajectory so $\\begin{array} { r } { \\epsilon _ { \\theta } ( \\bar { \\mathbf { x } } _ { k } ( \\tau ) , \\mathbf { \\boldsymbol { y } } ( \\tau ) , k ) : = \\epsilon _ { \\theta } ( \\mathbf { \\Delta x } _ { k } ( \\tau ) , R ( \\tau ) , k ) } \\end{array}$ . These returns are normalized to keep $R ( \\tau ) \\in [ 0 , 1 ]$ . Sampling a high return trajectory amounts to conditioning on $R ( \\tau ) = 1$ . Note that we do not make use of any $Q$ -values, which would then require dynamic programming. ", + "bbox": [ + 173, + 535, + 825, + 606 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Satisfying Constraints Trajectories may satisfy a variety of constraints, each represented by the set $\\mathcal { C } _ { i }$ , such as reaching a specific goal, visiting states in a particular order, or avoiding parts of the state space. To generate trajectories satisfying a given constraint $\\mathcal { C } _ { i }$ , we condition the noise model on a one-hot encoding so that $\\begin{array} { r } { \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\pmb { y } ( \\tau ) , \\bar { k } ) : = \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\mathbb { 1 } ( \\tau \\in \\mathcal { C } _ { i } ) , k ) } \\end{array}$ . Although we train with an offline dataset in which trajectories satisfy only one of the available constraints, at inference we can satisfy several constraints together. ", + "bbox": [ + 173, + 609, + 825, + 693 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Composing Skills A skill $i$ can be specified from a set of demonstrations $B _ { i }$ . To generate trajectories that demonstrate a given skill, we condition the noise model on a one-hot encoding so that $\\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , { \\pmb y } ( \\tau ) , k ) : = \\epsilon _ { \\theta } ( \\bar { { \\pmb x } } _ { k } ( \\tau ) , \\mathbb { 1 } ( \\tau \\in \\mathcal { B } _ { i } ) , k )$ . Although we train with individual skills, we may further compose these skills together during inference. ", + "bbox": [ + 174, + 695, + 825, + 752 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assuming we have learned the data distributions $q ( { \\pmb x } _ { 0 } ( \\tau ) | { \\pmb y } ^ { 1 } ( \\tau ) ) , \\dots , q ( { \\pmb x } _ { 0 } ( \\tau ) | { \\pmb y } ^ { n } ( \\tau ) )$ for $n$ different conditioning variables, we can sample from the composed data distribution $q ( \\pmb { x } _ { 0 } ( \\tau ) | \\pmb { y } ^ { 1 } ( \\tau ) , \\dots , \\pmb { y } ^ { n } ( \\bar { \\tau } ) )$ using the perturbed noise (Liu et al., 2022): ", + "bbox": [ + 174, + 758, + 825, + 801 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/387eca885b4e7a9ca4a4a7e4e308a743d03ad7f9ae1155a2384c8112667bea94.jpg", + "text": "$$\n\\hat { \\epsilon } : = \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\emptyset , k ) + \\omega \\sum _ { i = 1 } ^ { n } ( \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\pmb { y } ^ { i } ( \\tau ) , k ) - \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\emptyset , k ) )\n$$", + "text_format": "latex", + "bbox": [ + 267, + 808, + 727, + 849 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This property assumes that $\\{ \\pmb { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n }$ are conditionally independent given the state trajectory $\\pmb { x } _ { 0 } ( \\tau )$ . However, we empirically observe that this assumption doesn’t have to be strictly satisfied as long as the composition of conditioning variables is feasible. For more detailed discussion, please refer to Appendix D. We use this property to compose more than one constraint or skill together at test-time. We also show how Decision Diffuser can avoid particular constraint or skill (NOT) in Appendix J. ", + "bbox": [ + 174, + 856, + 826, + 928 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 Conditional Planning with the Decision Diffuser ", + "text_level": 1, + "bbox": [ + 236, + 114, + 562, + 126 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/978ee750ffe38aa619ac11373331980f674e6924fad8991ee08cfb832205fc9a.jpg", + "image_caption": [ + "Figure 4: Results Overview. Decision Diffuser performs better than both TD learning (CQL) and Behavorial Cloning (BC) across D4RL locomotion tasks, D4RL Kitchen tasks and Kuka Block Stacking tasks (single constraint) using only a conditional generative modeling objective. For performance metric, we use normalized average returns (Fu et al., 2020) for D4RL tasks (Locomotion and Kitchen) and success rate for Block Stacking. " + ], + "image_footnote": [], + "bbox": [ + 238, + 127, + 758, + 348 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 TRAINING THE DECISION DIFFUSER ", + "text_level": 1, + "bbox": [ + 178, + 405, + 465, + 420 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The Decision Diffuser, our conditional generative model for decision-making, is trained in a supervised manner. Given a dataset $\\mathcal { D }$ of trajectories, each labeled with the return it achieves, the constraint that it satisfies, or the skill that it demonstrates, we simultaneously train the reverse diffusion process $p _ { \\theta }$ , parameterized through the noise model $\\epsilon _ { \\theta }$ , and the inverse dynamics model $f _ { \\phi }$ with the following loss: ", + "bbox": [ + 174, + 426, + 825, + 497 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\natural ( \\theta , \\phi ) : = \\mathbb { E } _ { k , \\tau \\in \\mathcal { D } , \\beta \\sim \\mathrm { B e n } ( p ) } [ \\| \\epsilon - \\epsilon _ { \\theta } ( { x } _ { k } ( \\tau ) , ( 1 - \\beta ) y ( \\tau ) + \\beta \\mathcal { O } , k ) \\| ^ { 2 } ] + \\mathbb { E } _ { ( s , a , s ^ { \\prime } ) \\in \\mathcal { D } } [ \\| a - f _ { \\phi } ( s , s ^ { \\prime } ) \\| ^ { 2 } ] } \\end{array}$ For each trajectory $\\tau$ , we first sample noise $\\epsilon \\sim \\mathcal { N } ( 0 , I )$ and a timestep $k \\sim \\mathcal { U } \\{ 1 , \\dots , K \\}$ . Then, we construct a noisy array of states ${ \\pmb x } _ { k } ( \\tau )$ and finally predict the noise as $\\hat { \\epsilon } _ { \\boldsymbol { \\theta } } : = \\epsilon _ { \\boldsymbol { \\theta } } ( \\pmb { x } _ { k } ( \\tau ) , \\pmb { y } ( \\tau ) , k )$ . Note that with probability $p$ we ignore the conditioning information and the inverse dynamics is trained with individual transitions rather than trajectories. ", + "bbox": [ + 174, + 500, + 826, + 580 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Architecture We parameterize $\\epsilon _ { \\theta }$ with a temporal U-Net architecture, a neural network consisting of repeated convolutional residual blocks (Janner et al., 2022). This effectively treats a sequence of states ${ \\pmb x } _ { k } ( \\tau )$ as an image where the height represents the dimension of a single state and the width denotes the length of the trajectory. We encode the conditioning information $\\pmb { y } ( \\tau )$ as either a scalar or a one-hot vector and project it into a latent variable $z \\in \\mathbb { R } ^ { h }$ with a multi-layer perceptron (MLP). When $\\pmb { y } ( \\tau ) = 0$ , we zero out the entries of $z$ . We also parameterize the inverse dynamics $f _ { \\phi }$ with an MLP. For implementation details, please refer to the Appendix B. ", + "bbox": [ + 173, + 583, + 825, + 681 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Low-temperature Sampling In the denoising step of Algorithm 1, we compute $\\mu _ { k - 1 }$ and $\\Sigma _ { k - 1 }$ from a noisy sequence of states and a predicted noise. We find that sampling $x _ { k - 1 } \\sim$ $\\mathcal { N } ( \\mu _ { k - 1 } , \\alpha \\Sigma _ { k - 1 } )$ where the variance is scaled by $\\alpha \\in [ 0 , 1 )$ leads to better quality sequences (corresponding to sampling lower temperature samples). For a proper ablation study, please refer to Appendix C. ", + "bbox": [ + 173, + 684, + 825, + 753 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 761, + 328, + 776 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we explore the efficacy of the Decision Diffuser on a variety of decision-making tasks (performance illustrated in Figure 4). In particular, we evaluate (1) the ability to recover effective RL policies from offline data, (2) the ability to generate behavior that satisfies multiple sets of constraints, (3) the ability compose multiple different skills together. In addition, we empirically justify use of classifier-free guidance, low-temperature sampling (Appendix C), and inverse dynamics (Appendix F) and test the robustness of Decision Diffuser to stochastic dynamics (Appendix G). ", + "bbox": [ + 174, + 786, + 825, + 869 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 OFFLINE REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 873, + 480, + 887 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Setup We first test whether the Decision Diffuser can generate return-maximizing trajectories. To test this, we train a state diffusion process and inverse dynamics model on publicly available ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "D4RL datasets (Fu et al., 2020). We compare with existing offline RL methods, including modelfree algorithms like CQL (Kumar et al., 2020) and IQL (Kostrikov et al., 2022), and model-based algorithms such as trajectory transformer (TT, Janner et al. (2021)) and MoReL (Kidambi et al., 2020). We also compare with sequence-models like the Decision Transformer (DT) (Chen et al. (2021) and diffusion models like Diffuser (Janner et al., 2022). ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/0a2d0b726961f161d351afaa00b88a1d4c6293180230e7ffd5e79853da686c2d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DatasetEnvironmentBCCQLIQLDTTTMOReLDiffuserDD
Med-ExpertHalfCheetah55.291.686.786.89553.379.890.6 ±1.3
Med-ExpertHopper52.5105.491.5107.6110.0108.7107.2111.8 ±1.8
Med-ExpertWalker2d107.5108.8109.6108.1101.995.6108.4108.8 ±1.7
MediumHalfCheetah42.644.047.442.646.942.144.249.1 ±1.0
MediumHopper52.958.566.367.661.195.458.579.3 ±3.6
MediumWalker2d75.372.578.374.07977.879.782.5±1.4
Med-ReplayHalfCheetah36.645.544.236.641.940.242.239.3 ±4.1
Med-ReplayHopper18.19594.782.791.593.696.8100±0.7
Med-ReplayWalker2d26.077.273.966.682.649.861.275±4.3
Average51.977.67774.778.972.975.381.8
MixedKitchen51.552.4511、1165 ±2.8
PartialKitchen3850.146.3--==57 ±2.5
Average44.851.248.7-1--61
", + "bbox": [ + 236, + 189, + 759, + 368 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results Across different offline RL tasks, we find that the Decision Diffuser is either competitive or outperforms many offline RL baselines (Table 1). It also outperforms Diffuser and sequence modeling approaches, such as Decision Transformer and Trajectory Transformer. The difference between Decision Diffuser and other methods becomes even more significant on harder D4RL Kitchen tasks which require long-term credit assignment. ", + "bbox": [ + 174, + 433, + 825, + 503 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To convey the importance of classifier-free guidance, we also compare with the baseline CondDiffuser, which diffuses over both state and action sequences as in Diffuser without classifierguidance. In Table 2, we observe that CondDiffuser improves over Diffuser in 2 out of 3 environments. Decision Diffuser further improves over CondDiffuser, performing better across all 3 environments. We conclude that learning the inverse dynamics is a good alternative to diffusing over actions. We further empirically analyze when to use inverse dynamics and when to diffuse over actions in Appendix F. We also compare against CondMLPDiffuser, a policy where the current action is denoised according to a diffusion process conditioned on both the state and return. We see that CondMLPDiffuser performs the worst amongst diffusion models. Till now, we mainly tested on offline RL tasks that have deterministic (or near deterministic) environment dynamics. Hence, we test the robustness of Decision Diffuser to stochastic dynamics and compare it to Diffuser and CQL as we vary the stochasticity in environment dynamics, in Appendix G. Finally, we analyze the runtime characteristics of Decision Diffuser in Appendix E. ", + "bbox": [ + 173, + 510, + 825, + 690 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 CONSTRAINT SATISFACTION ", + "text_level": 1, + "bbox": [ + 176, + 691, + 413, + 705 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Setup We next evaluate how well we can generate trajectories that satisfy a set of constraints using the Kuka Block Stacking environment (Janner et al., 2022) visualized in Figure 5. In this domain, there are four blocks which can be stacked as a single tower or rearranged into several towers. A constraint like BlockHeigh $\\mathsf { \\Lambda } = ( i ) > \\mathsf { B l o c k H e i g h t } ( j )$ requires that block $i$ be placed above block $j$ . We train the Decision Diffuser from 10, 000 expert demonstrations each satisfying one of these constraints. We randomize the positions of these blocks and consider two tasks at inference: sampling trajectories that satisfy a single constraint seen before in the dataset or satisfy a group of constraints for which demonstrations were never provided. In the latter, we ask the Decision Diffuser to generate trajectories so BlockHeight $( i ) >$ BlockHeight $( j ) >$ BlockHeight $( k )$ for three of the four blocks $i , j , k$ . For more details, please refer to Appendix H. ", + "bbox": [ + 173, + 712, + 825, + 851 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results In both the stacking and rearrangement settings, Decision Diffuser satisfies single constraints with greater success rate than Diffuser (Table 3). We also compare with BCQ (Fujimoto et al., 2019) and CQL (Kumar et al., 2020), but they consistently fail to stack or rearrange the blocks leading to a 0.0 success rate. Unlike these baselines, our method can just as effectively satisfy several constraints together according to Equation 9. For a visualization of these generated trajectories, please see the website https://anuragajay.github.io/decision-diffuser/. ", + "bbox": [ + 174, + 854, + 825, + 938 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/257506580b52a6d5d4748e37a00af4828bf9dfd022b9e7bd55dc454a5663de22.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Hopper-*Diffuser CondDiffuser CondMLPDiffuser Decision Diffuser
Med-Expert107.6111.3105.6111.8 ±1.6
Medium58.566.354.179.3 ±3.6
Med-Replay96.876.566.5100 ±0.7
", + "bbox": [ + 326, + 102, + 671, + 148 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/fb01f37be7d7902c1eccd3141e1effc0540e8f4dcdc17360b00d08dd994e4fd5.jpg", + "table_caption": [ + "Table 2: Ablations. Using classifier-free guidance with Diffuser, resulting in CondDiffuser, improves performance in 2 (out of 3) environments. Additionally, using inverse dynamics for action prediction in Decision Diffuser improves performance in all 3 environments. CondMLPDiffuser, that diffuses over current action given the current state and the target return, doesn’t perform as well. " + ], + "table_footnote": [], + "table_body": "
EnvironmentDiffuserDD
Single Constraint - Stacking45.6 ±3.158.0 ±3.1
Single Constraint - Rearrangement58.9 ±3.462.7 ±3.1
Single Constraint Average52.360.4
Multiple Constraints - Stacking160.3 ±3.1
Multiple Constraints - Rearrangement67.2 ±3.1
Multiple Constraints Average163.8
", + "bbox": [ + 431, + 219, + 751, + 323 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/52d2870f8dae13bbc0f1462a3db00c970f194d62fa9583c2df125b201342dd4a.jpg", + "image_caption": [ + "Figure 5: Kuka Block Stacking task. ", + "Table 3: Block Stacking through Constraint Minimization. Decision Diffuser (DD) improves over Diffuser in terms of the success rate of generating trajectories satisfying a set of block-stacking constraints. It can also flexibly combine multiple constraints during test time. We report the mean success rate and the standard error over 5 random seeds. " + ], + "image_footnote": [], + "bbox": [ + 246, + 261, + 372, + 353 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 SKILL COMPOSITION ", + "text_level": 1, + "bbox": [ + 174, + 392, + 362, + 406 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Setup Finally, we look at how to compose different skills together. We consider the Unitree-gorunning environment (Margolis & Agrawal, 2022), where a quadruped robot can be found running with various gaits, like bounding, pacing, and trotting. We explore if it is possible to generate trajectories that transition between these gaits after only training on individual gaits. For each gait, we collect a dataset of 2500 demonstrations on which we train Decision Diffuser. ", + "bbox": [ + 173, + 411, + 825, + 481 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results During testing, we use the noise model of our reverse diffusion process according to equation 9 to sample trajectories of the quadruped robot with entirely new running behavior. Figure 6 shows a trajectory that begins with bounding but ends with pacing. Appendix I provides additional visualizations of running gaits being composed together. Although it visually appears that trajectories generated with the Decision Diffuser contain more than one gait, we would like to quantify exactly how well different gaits can be composed. To this end, we train a classifier to predict at every time-step or frame in a trajectory the running gait of the quadruped (i.e. bound, pace, or trott). We reuse the demonstrations collected for training the Decision Diffuser to also train this classifier, where our inputs are defined as robot joint states over a fixed period of time (i.e. state sub-sequences of length 10) and the label is the gait demonstrated in this sequence. The complete details of our gait classification procedure can be found in Appendix I. ", + "bbox": [ + 173, + 484, + 825, + 637 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/32c5686d1598622b99760125a23a8c4046cff1a0b7089a79a049bd8f9292a57e.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ConditionTrottPaceBound
Only Bound0.81.098.2
Only Pace1.497.70.9
Bound + Pace1.438.560.1
", + "bbox": [ + 550, + 664, + 818, + 732 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/8874639fd20e50d9304875bc9a9d40220fd33875e237e4b977f7e557544d6fc5.jpg", + "image_caption": [ + "Figure 7: Classifying Running Gaits. A classifier predicts the running gait of the quadruped at every timestep. On trajectories generated by conditioning on a single skill, like only bounding or pacing, the classifier predicts the respective gait with largest probability. When conditioned on both skills, some timesteps are classified as bounding while others as pacing. " + ], + "image_footnote": [], + "bbox": [ + 173, + 651, + 537, + 747 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use our running gait classifier in two ways: to evaluate how the behavior of the quadruped changes over the course of a single, generated trajectory and to measure how often each gait emerges over several generated trajectories. In the former, we first sample three trajectories from the Decision Diffuser conditioned either on the bounding gait, the pacing gait, or both. For every trajectory, we separately plot the classification probability of each gait over the length of the sequence. As shown in the plots of Figure 7, the classifier predicts bound and pace respectively to be the most likely running gait in trajectories sampled with this condition. When the trajectory is generated by conditioning on both gaits, the classifier transitions between predicting one gait with largest probability to the other. In fact, there are several instances where the behavior of the quadruped switches between bounding and pacing according to the classifier. This is consistent with the visualizations reported in Figure 6. In the table depicted in Figure 7, we consider 1000 trajectories generated with the Decision Diffuser when conditioned on one or both of the gaits as listed. We record the fraction of time that the quadruped’s running gait was classified as either trott, pace, or bound. It turns out that the classifier identifies the behavior as bounding for $3 8 . 5 \\%$ of the time and as pacing for the other $6 0 . 1 \\%$ when trajectories are sampled by composing both gaits. This corroborates the fact that the Decision Diffuser can indeed compose running behaviors despite only being trained on individual gaits. ", + "bbox": [ + 173, + 813, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/ae5902e9ec2fa29c05349103c337c534e1a146b5725255f610070875b832cd24.jpg", + "image_caption": [ + "Figure 6: Composing Movement Skills. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. " + ], + "image_footnote": [], + "bbox": [ + 334, + 101, + 663, + 161 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 219, + 825, + 330 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 333, + 346, + 348 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Diffusion Models Diffusion Models is proficient in learning generative models of image and text data (Saharia et al., 2022; Nichol et al., 2021; Nichol & Dhariwal, 2021). It formulates the data sampling process as an iterative denoising procedure (Sohl-Dickstein et al., 2015; Ho et al., 2020). The denoising procedure can be alternatively interpreted as parameterizing the gradients of the data distribution (Song et al., 2021) optimizing the score matching objective (Hyvarinen ¨ , 2005) and thus as a Energy-Based Model (Du & Mordatch, 2019; Nijkamp et al., 2019; Grathwohl et al., 2020). To generate data samples (eg: images) conditioned on some additional information (eg:text), prior works (Nichol & Dhariwal, 2021) have learned a classifier to facilitate the conditional sampling. More recent works (Ho & Salimans, 2022) have argued to leverage gradients of an implicit classifier, formed by the difference in score functions of a conditional and an unconditional model, to facilitate conditional sampling. The resulting classifier-free guidance has been shown to generate better conditional samples than classifier-based guidance. Recent works have also used diffusion models to imitate human behavior (Pearce et al., 2023) and to parameterize policy in offline RL (Wang et al., 2022). Janner et al. (2022) generate trajectories consisting of states and actions with an unconditional diffusion model, therefore requiring a trained reward function on noisy state-action pairs. At inference, the estimated reward function guides the reverse diffusion process towards samples of high-return trajectories. In contrast, we do not train reward functions or diffusion processes separately, but rather model the trajectories in our dataset with a single, conditional generative model. This ensures that the sampling procedure of the learned diffusion process is the same at inference as it is during training. ", + "bbox": [ + 174, + 353, + 826, + 616 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Reward Conditioned Policies Prior works (Kumar et al., 2019; Schmidhuber, 2019; Emmons et al., 2021; Chen et al., 2021) have studied learning of reward conditioned policies via reward conditioned behavioral cloning. Chen et al. (2021) used a transformer (Vaswani et al., 2017) to model the reward conditioned policies and obtained a performance competitive with offline RL approaches. Emmons et al. (2021) obtained similar performance as Chen et al. (2021) without using a transformer policy but relied on careful capacity tuning of MLP policy. In contrast, Decision Diffuser can also model constraints or skills and their resulting compositions. ", + "bbox": [ + 174, + 619, + 825, + 717 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 724, + 312, + 739 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose Decision Diffuser, a conditional generative model for sequential decision making. It frames offline sequential decision making as conditional generative modeling and sidesteps the need of reinforcement learning, thereby making the decision making pipeline simpler. By sampling for high returns, it is able to capture the best behaviors in the dataset and outperforms existing offline RL approaches on standard D4RL benchmarks. In addition to returns, it can also be conditioned on constraints or skills and can generate novel behaviors by flexibly combining constraints or composing skills during test time. In this work, we focused on offline sequential decision making, thus circumventing the need for exploration. Using ideas from Zheng et al. (2022), future works could look into online fine-tuning of Decision Diffuser by leveraging entropy of the state-sequence model for exploration. While our work focused on state based environments, it can be extended to image based environments by performing the diffusion in latent space, rather than observation space, as done in Rombach et al. (2022). For a detailed discussion on limitations of Decision Diffuser, please refer to Appendix L. ", + "bbox": [ + 173, + 743, + 826, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 367, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The authors would like to thank Ofir Nachum, Anthony Simeonov and Richard Li for their helpful feedback on an earlier draft of the work; Jay Whang and Ge Yang for discussions on classifierfree guidance; Gabe Margolis for helping with unitree experiments; Micheal Janner for providing visualization code for Kuka block stacking; and the members of Improbable AI Lab for discussions and helpful feedback. We thank MIT Supercloud and the Lincoln Laboratory Supercomputing Center for providing compute resources. This research was supported by an NSF graduate fellowship, a DARPA Machine Common Sense grant, a MURI grant, an MIT-IBM grant, and ARO W911NF-21-1- 0097. ", + "bbox": [ + 174, + 135, + 826, + 246 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This research was also partly sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19- 2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes, notwithstanding any copyright notation herein. ", + "bbox": [ + 174, + 253, + 825, + 351 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "AUTHOR CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 372, + 395, + 387 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Anurag Ajay conceived the framework of viewing decision-making as conditional diffusion generative modeling, implemented the Decision Diffuser algorithm, ran experiments on Offline RL and Skill Composition, and helped in paper writing. ", + "bbox": [ + 174, + 405, + 825, + 448 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Yilun Du helped in conceiving the framework of viewing decision-making as conditional diffusion generative modeling, ran experiments on Constraint Satisfaction, helped in paper writing and advised Anurag. ", + "bbox": [ + 176, + 454, + 821, + 496 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Abhi Gupta helped in running experiments on Offline RL and Skill Composition, participated in research discussions, and played the leading role in paper writing and making figures. ", + "bbox": [ + 171, + 503, + 823, + 531 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Joshua Tenenbaum participated in research discussions. ", + "bbox": [ + 173, + 539, + 547, + 553 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Tommi Jaakkola participated in research discussions and suggested the experiment of classifying running gaits. ", + "bbox": [ + 173, + 559, + 821, + 588 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pulkit Agrawal was involved in research discussions, suggested experiments related to dynamic programming, provided feedback on writing, positioning of the work, and overall advising. 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", + "bbox": [ + 173, + 828, + 823, + 857 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Appendix ", + "text_level": 1, + "bbox": [ + 174, + 99, + 272, + 119 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this appendix, we discuss details of the illustrative examples in Section A. Next, we discuss hyperparameters and architectural details in Section B. We analyze the importance of low temperature sampling in Section C, further explain composition of conditioning variable in Section D, discuss the run-time characteristics of decision diffuser in Section E, discuss when to use inverse dynamics in Section F and analyze robustness of Decision Diffuser to stochastic dynamics in Section G. Finally, we provide details of the Kuka Block Stacking environment in Section H and the Unitree environment in Section I. ", + "bbox": [ + 173, + 131, + 825, + 228 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A ILLUSTRATIVE EXAMPLES ", + "text_level": 1, + "bbox": [ + 176, + 247, + 429, + 263 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 IMPLICIT DYNAMIC PROGRAMMING ", + "text_level": 1, + "bbox": [ + 174, + 280, + 468, + 295 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/06c49faf819f369ba151ad9d3ff002a905c14d4f9d446bec58e52f70a39c1d4b.jpg", + "image_caption": [ + "Figure A1: Illustrative example. We demonstrate the ability of Decision Diffuser to stitch together suboptimal trajectories in training dataset to obtain (near) optimal trajectories, thereby implicitly performing dynamic programming in Maze2D-open environment from Fu et al. (2020). " + ], + "image_footnote": [], + "bbox": [ + 338, + 303, + 660, + 395 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We empirically demonstrate the ability of Decision Diffuser to perform implicit dynamic programming in Maze2D-open environment from Fu et al. (2020). The task in Maze2D-open environment is to reach point C and the reward is negative distance from point C. The training dataset consists of 500 trajectories from point A to point B and 500 trajectories from point B to point C. The maximum trajectory length is 50. During test time, the agent starts from point A and needs to reach point C as quickly as possible. As shown in Figure A1, Decision Diffuser can stitch trajectories in training dataset to form trajectories that goes from point A to point B in (near) straight lines. ", + "bbox": [ + 174, + 459, + 825, + 558 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 CONSTRAINT COMBINATION ", + "text_level": 1, + "bbox": [ + 176, + 571, + 416, + 587 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Setup In linear system robot navigation, Decision Diffuser is trained on 1000 expert trajectories either satisfying the constraint $\\lVert s _ { T } \\rVert \\leq R$ $R = 1$ ) or the constraint $\\| s _ { T } \\| \\geq r$ $\\mathit { r } = 0 . 7$ ). Here, $s _ { T } = [ x _ { T } , y _ { T } ]$ represents the final robot state in a trajectory, specifying its final 2d position. The maximum trajectory length is 50. During test time, Decision Diffuser is asked to generate trajectories satisfying $\\| s _ { T } \\| \\le R$ and $\\| s _ { T } \\| \\geq r$ to test its ability to satisfy single constraints. Furthermore, Decision Diffuser is also asked to generate trajectories satisfying $r \\leq \\left\\lceil | s _ { T } | | \\leq R \\right\\rceil$ to test its ability to satisfy combined constraints. ", + "bbox": [ + 173, + 594, + 826, + 691 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Results Figure 2 shows that Decision Diffuser learns to generate trajectories perfectly (i.e. with $1 0 0 \\%$ success rate) satisfying single constraints in linear system robot navigation. Furthermore, it learns to generate trajectories satisfying the composed constraint in linear system robot navigation with $9 1 . 3 \\hat { \\% } ( \\pm 2 . 6 \\% )$ accuracy where the standard error is calculated over 5 random seeds. ", + "bbox": [ + 174, + 695, + 825, + 751 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B HYPERPARAMETER AND ARCHITECTURAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 771, + 637, + 786 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we describe various architectural and hyperparameter details: ", + "bbox": [ + 173, + 801, + 673, + 816 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• We represent the noise model $\\epsilon _ { \\theta }$ with a temporal U-Net (Janner et al., 2022), consisting of a U-Net structure with 6 repeated residual blocks. Each block consisted of two temporal convolutions, each followed by group norm (Wu & He, 2018), and a final Mish nonlinearity (Misra, 2019). Timestep and condition embeddings, both 128-dimensional vectors, are produced by separate 2-layered MLP (with 256 hidden units and Mish nonlinearity) and are concatenated together before getting added to the activations of the first temporal convolution within each block. We borrow the code for temporal U-Net from https://github.com/jannerm/diffuser. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• We represent the inverse dynamics $f _ { \\phi }$ with a 2-layered MLP with 512 hidden units and ReLU activations. ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We represent the gait classifier with a 3-layered MLP with 1024 hidden units and ReLU activations. ", + "bbox": [ + 173, + 143, + 823, + 159 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We train $\\epsilon _ { \\theta }$ and $f _ { \\phi }$ using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of $2 e - 4$ and batch size of 32 for 2e6 train steps. ", + "bbox": [ + 174, + 170, + 823, + 199 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We train the gait classifier using the Adam optimizer with a learning rate of $2 e - 4$ and batch size of 64 for 1e6 train steps. ", + "bbox": [ + 173, + 210, + 825, + 239 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We choose the probability $p$ of removing the conditioning information to be 0.25. ", + "bbox": [ + 176, + 251, + 720, + 266 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We use $K = 1 0 0$ diffusion steps. ", + "bbox": [ + 174, + 277, + 405, + 292 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We use a planning horizon $H$ of 100 in all the D4RL locomotion tasks, 56 in D4RL kitchen tasks, 128 in Kuka block stacking, 56 in unitree-go-running tasks, 50 in the illustrative example and 60 in Block push tasks. ", + "bbox": [ + 176, + 304, + 826, + 345 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We use a guidance scale $s \\in \\{ 1 . 2 , 1 . 4 , 1 . 6 , 1 . 8 \\}$ but the exact choice varies by task. ", + "bbox": [ + 176, + 357, + 730, + 373 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We choose $\\alpha = 0 . 5$ for low temperature sampling. ", + "bbox": [ + 173, + 383, + 517, + 400 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• We choose context length $C = 2 0$ ", + "bbox": [ + 174, + 411, + 415, + 425 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C IMPORTANCE OF LOW TEMPERATURE SAMPLING", + "text_level": 1, + "bbox": [ + 173, + 454, + 611, + 469 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In Algorithm 1, we compute $\\mu _ { k - 1 }$ and $\\Sigma _ { k - 1 }$ from a noisy sequence of states and predicted noise. We find that sampling $x _ { k - 1 } \\sim { \\mathcal { N } } ( \\mu _ { k - 1 } , \\alpha \\Sigma _ { k - 1 } )$ (where $\\alpha \\in [ 0 , 1 )$ ) with a reduced variance produces high-likelihood state sequences. We refer to this as low-temperature sampling. To empirically show its importance, we compare performances of Decision Diffuser with different values of $\\alpha$ (Table A1). We show that low temperature sampling ( $\\alpha = 0 . 5$ ) gives the best average returns. However, reducing the $\\alpha$ to 0 eliminates the entropy in sampling and leads to lower returns. On the other hand, $\\alpha = 1 . 0$ leads to a higher variance in terms of returns of the trajectories. ", + "bbox": [ + 173, + 491, + 826, + 589 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/e59f79eab126e624fbc836f6cebf586440a9d6ef00746eabcad19e1c336e7e3d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Decision DiffuserHopper-Medium-Expert
α=0104.3 ± 0.7
α= 0.5111.8 ±1.6
α= 1.0107.1 ± 3.5
", + "bbox": [ + 346, + 609, + 648, + 678 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table A1: Low-temperature sampling $( \\alpha = 0 . 5 )$ ) allows us to get high return trajectories consistently. While $\\alpha = 1 . 0$ leads to a higher variance in returns of the trajectories, $\\alpha = 0 . 0$ eliminates entropy in the sampling and leads to lower returns. ", + "bbox": [ + 173, + 688, + 826, + 729 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D COMPOSING CONDITIONING VARIABLES ", + "text_level": 1, + "bbox": [ + 174, + 765, + 545, + 781 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we detail how Decision Diffuser trained with different conditioning variables $\\{ \\pmb { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n }$ composes these conditioning variables together. It learns the denoising model $\\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , { \\pmb y } ^ { i } ( \\tau ) , k )$ for a given conditioning variable $\\mathbf { \\dot { \\mathbf { y } } } ^ { i } ( \\tau )$ . From the derivations outlined in prior works (Luo, 2022; Song et al., 2021), we know that $\\nabla _ { \\pmb { x } _ { k } ( \\tau ) } \\log q ( \\pmb { x } _ { k } ( \\tau ) | \\pmb { y } ^ { i } ( \\tau ) ) \\propto$ $- \\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , { \\pmb y } ^ { i } ( \\tau ) , k )$ . Therefore, each conditional trajectory distribution $\\{ q ( \\pmb { x } _ { k } ( \\tau ) | \\pmb { y } ^ { i } ( \\tau ) ) \\} _ { i = 1 } ^ { n }$ can be modelled with a single denoising model $\\epsilon _ { \\theta }$ that conditions on the respective variable $y ^ { i } ( \\tau )$ . ", + "bbox": [ + 173, + 801, + 825, + 888 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In order to compose $n$ different conditioning variables (i.e. skills or constraints), we would like to model $q ( \\pmb { x } _ { k } ( \\hat { \\tau } ) | \\{ \\pmb { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } )$ . We assume that $\\{ y ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n }$ are conditionally independent given ", + "bbox": [ + 171, + 895, + 823, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "${ \\pmb x } _ { k } ( \\tau )$ . Thus, we can factorize as follows: ", + "bbox": [ + 173, + 103, + 447, + 118 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/9538bdc8a2e5be9ba6de812f3c83201e00416af038f79d46a42db9796578e664.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle q ( \\mathbf { x } _ { k } ( \\tau ) | \\{ \\boldsymbol { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } ) \\propto q ( \\mathbf { x } _ { k } ( \\tau ) ) \\displaystyle \\prod _ { i = 1 } ^ { n } \\frac { q ( \\alpha _ { k } ( \\tau ) | \\boldsymbol { y } ^ { i } ( \\tau ) ) } { q ( \\alpha _ { k } ( \\tau ) ) } } & { ( \\mathrm { B a y e s ~ R u l e } ) } \\\\ { \\displaystyle \\Rightarrow \\log q ( \\alpha _ { k } ( \\tau ) | \\{ \\boldsymbol { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } ) \\propto \\log q ( \\alpha _ { k } ( \\tau ) ) + \\displaystyle \\sum _ { i = 1 } ^ { n } ( \\log q ( \\alpha _ { k } ( \\tau ) | \\boldsymbol { y } ^ { i } ( \\tau ) ) - \\log q ( \\alpha _ { k } ( \\tau ) ) ) } \\\\ { \\displaystyle \\Rightarrow \\nabla _ { \\mathbf { x } _ { k } ( \\tau ) } \\log q ( x _ { k } ( \\tau ) | \\{ \\boldsymbol { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } ) = \\nabla _ { \\mathbf { x } _ { k } ( \\tau ) } \\log q ( \\alpha _ { k } ( \\tau ) ) } \\\\ & { \\displaystyle + \\sum _ { i = 1 } ^ { n } ( \\nabla _ { \\mathbf { x } _ { k } ( \\tau ) } \\log q ( \\alpha _ { k } ( \\tau ) ) y ^ { i } ( \\tau ) ) - \\nabla _ { \\mathbf { x } _ { k } ( \\tau ) } \\log q ( \\alpha _ { k } ( \\tau ) ) ) } \\\\ { \\displaystyle \\Rightarrow \\epsilon _ { \\theta } ( \\alpha _ { k } ( \\tau ) , \\{ \\boldsymbol { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } , k ) = \\epsilon _ { \\theta } ( \\alpha _ { k } ( \\tau ) , \\boldsymbol { \\mathcal { O } } , k ) + \\displaystyle \\sum _ { i = 1 } ^ { n } ( \\epsilon _ { \\theta } ( \\alpha _ { k } ( \\tau ) , \\boldsymbol { y } ^ { i } ( \\tau ) , k ) - \\epsilon _ { \\theta } ( x _ { k } ( \\tau ) , \\boldsymbol { \\mathcal { O } } , k ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 133, + 839, + 321 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Using the above equations, we can sample from $q ( \\pmb { x } _ { 0 } ( \\tau ) | \\{ \\pmb { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } )$ with classifier free guidance using the perturbed noise: ", + "bbox": [ + 173, + 334, + 821, + 364 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a809c3861b59cffd688cf17ec377461efa3936753cc310b6ef48ca1eb981f9d5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\hat { \\boldsymbol { \\epsilon } } : = \\boldsymbol { \\epsilon } _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { k } ( \\tau ) , \\mathcal { O } , \\boldsymbol { k } ) + \\omega ( \\epsilon _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { k } ( \\tau ) , \\{ \\boldsymbol { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n } , \\boldsymbol { k } ) - \\epsilon _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { k } ( \\tau ) , \\mathcal { O } , \\boldsymbol { k } ) ) } \\\\ & { \\quad = \\epsilon _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { k } ( \\tau ) , \\mathcal { O } , \\boldsymbol { k } ) + \\omega \\displaystyle \\sum _ { i = 1 } ^ { n } ( \\epsilon _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { k } ( \\tau ) , \\boldsymbol { y } ^ { i } ( \\tau ) , \\boldsymbol { k } ) - \\epsilon _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { k } ( \\tau ) , \\mathcal { O } , \\boldsymbol { k } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 375, + 735, + 439 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use the perturbed noise to compose skills or combine constraints at test time. This derivation was borrowed from Liu et al. (2022) and is presented here for completeness. ", + "bbox": [ + 174, + 458, + 821, + 487 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "While the composition of conditioning variables $\\{ \\pmb { y } ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n }$ requires them to be conditionally independent given the state trajectory $\\pmb { x } _ { 0 } ( \\tau )$ , we empirically observe that this condition doesn’t have to be strictly satisfied. However, we require composition of conditioning variables to be feasible (i.e. $\\exists x _ { 0 } ( \\tau )$ that satisfies all the conditioning variables). When the composition is infeasible, Decision Diffuser produces trajectories with incoherent behavior, as expected. This is best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. ", + "bbox": [ + 173, + 493, + 825, + 578 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Requirements on the dataset First, the dataset should have a diverse set of demonstrations that shows different ways of satisfying each conditioning variable $y ^ { i } ( \\tau )$ . This would allow Decision Diffuser to learn diverse ways of satisfying each conditioning variable $y ^ { i } ( \\tau )$ . Since we use inverse dynamics to extract actions from the predicted state trajectory $\\pmb { x } _ { 0 } ( \\tau )$ , we assume that the state trajectory ${ \\pmb x } _ { 0 } ( \\tau )$ resulting from the composition of different conditioning variables contains consecutive state pairs $\\left( { { s _ { t } } , { s _ { t + 1 } } } \\right)$ that come from the same distribution that generated the demonstration dataset. Otherwise, inverse dynamics can give erroneous predictions. ", + "bbox": [ + 173, + 580, + 825, + 678 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E RUNTIME CHARACTERISTIC OF DECISION DIFFUSER ", + "text_level": 1, + "bbox": [ + 174, + 705, + 642, + 723 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We analyze the runtime characteristics of Decision Diffuser in this section. After training the Decision Diffuser on trajectories from the D4RL Hopper-Medium-Expert dataset, we plan in the corresponding environment according to Algorithm 1. Every action taken in the environment requires running 100 reverse diffusion steps to generate a state sequence taking on average 1.26s in wall-clock time. We can improve the run-time of planning by warm-starting the state diffusion as suggested in Janner et al. (2022). Here, we start with a generated state sequence (from the previous environment step), run forward diffusion for a fixed number of steps, and finally run the same number of reverse diffusion steps from the partially noised state sequence to generate another state sequence. Warm-starting in this way allows us to decrease the number of denoising steps to 40 (0.48s on average) without any loss in performance, to 20 (0.21s on average) with minimal loss in performance, and to 5 with less than $2 0 \\%$ loss in performance (0.06s on average). We demonstrate the trade-off between performance, measured by normalized average return achieved in the environment, and planning time, measured in wall-clock time after warm-starting the reverse diffusion process, in Figure A2. ", + "bbox": [ + 173, + 742, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/582a316558824d8218903d31c84d2ea265682dcdd6126b4d53b0d942b675b615.jpg", + "image_caption": [ + "Figure A2: Performance vs planning time. We visualize the trade-off between performance, measured by normalized average return achieved in the environment, and planning time, measured in wall-clock time after warm-starting the reverse diffusion process. " + ], + "image_footnote": [], + "bbox": [ + 334, + 99, + 663, + 315 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/1299108cf864d28cd5fd5354a22cc2b848692c2a21bde63eba1dbe87607eda12.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
EnvironmentBCCondDiffuserDecision Diffuser
Position Control57.3 ±1.287.3 ±3.187.8 ±2.8
Torque Control55.2 ±1.571.8 ±3.484.7 ±2.2
", + "bbox": [ + 372, + 372, + 800, + 426 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/b7e5f34bba8b05d1a3659b787a4b3d2616878cdfbc9e2f790df9d0e668b1b9ef.jpg", + "image_caption": [ + "Figure A3: Block push environment. " + ], + "image_footnote": [], + "bbox": [ + 174, + 386, + 338, + 489 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table A2: Block pushing with different controls. Decision Diffuser and CondDiffuser perform similarly when the agent uses position control. However, when the agent uses torque control, CondDiffuser performs worse than Decision Diffuser given it’s harder to diffuse over non-smooth action trajectories. We use the success rate of the red cube reaching the green circle as the performance metric. We report the mean success rate and the standard error over 5 random seeds. ", + "bbox": [ + 372, + 438, + 797, + 526 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F WHEN TO USE INVERSE DYNAMICS? ", + "text_level": 1, + "bbox": [ + 174, + 542, + 509, + 559 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section, we try to analyze further when using inverse dynamics is better than diffusing over actions. Table 2 showed that Decision Diffuser outperformed CondDiffuser on 3 hopper environment, thereby suggesting that inverse dynamics is a better alternative to diffusing over actions. Our intuition was that sequences over actions, represented as joint torques in our environments, tend to be more high-frequency and less smooth, thus making it harder for the diffusion model to predict (Kingma et al., 2021). We now try to verify this intuition empirically. ", + "bbox": [ + 174, + 582, + 825, + 666 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Setup We choose Block Push environment adapted from Gupta et al. (2018) where the goal is to push the red cube to the green circle. When the red cube reaches the green circle, the agent gets a reward of $+ 1$ . The state space is 10-dimensional consisting of joint angles (3) and velocities (3) of the gripper, COM of the gripper (2) and position of the red cube (2). The green circle’s position is fixed and at an initial distance of 0.5 from COM of the gripper. The red cube (of size 0.03) is initially at a distance of 0.1 from COM of the gripper and at an angle $\\theta$ sampled from $\\mathcal { U } ( - \\pi / 4 , \\pi / 4 )$ at the start of every episode. The task horizon is 60 timesteps. ", + "bbox": [ + 174, + 672, + 825, + 770 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "There are 2 control types: (i) torque control, where the agent needs to specify joint torques (3 dimensional) and (ii) position control where the agent needs to specify the position change of COM of the gripper and the angular change in gripper’s orientation $( \\Delta x , \\Delta y , \\Delta \\phi )$ (3 dimensional). While action trajectories from position control are smooth, the action trajectories from torque control have higher frequency components. ", + "bbox": [ + 174, + 777, + 825, + 847 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Offline dataset collection To collect the offline data, we use Soft Actor-Critic (SAC) (Haarnoja et al., 2018) first to train an expert policy for 1 million environment steps. We then use 1 million environment transitions as our offline dataset, which contains expert trajectories collected towards the end of the training and random action trajectories collected at the beginning of the training. We collect 2 datasets, one for each control type. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Results Table A2 shows that Decision Diffuser and CondDiffuser perform similarly when the agent uses position control. This is because action trajectories resulting from position control are smoother and hence easier to model with diffusion. However, when the agent uses torque control, CondDiffuser performs worse than Decision Diffuser, given the action trajectories have higher frequency components and hence are harder to model with diffusion. ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "G ROBUSTNESS TO STOCHASTIC DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 199, + 555, + 215 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/9a10b8466b781048f4175dc18a2454a22f02774d489b7c634b57535433222a26.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
pBCDecision DiffuserDiffuserCQL
0.0055.2±1.584.7±2.272.4±1.473.2±2.3
0.0549.3±3.677.3±3.163.2±2.961.8±3.7
0.1025.8±3.853.2±4.152.3±4.651.2±4.3
0.1515.1±4.341.3±4.941.6±5.142.2±5.5
", + "bbox": [ + 299, + 247, + 696, + 327 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Table A3: Robustness to stochastic dynamics. Decision Diffuser’s performance suffers when stochasticity is introduced in dynamics function. While it still outperforms Diffuser and CQL when $p = 0 . 0 5$ , its performance becomes similar to that of Diffuser and CQL for higher $p$ values. We use the success rate of the red cube reaching the green circle as the performance metric. We report the mean success rate and the standard error over 5 random seeds. ", + "bbox": [ + 173, + 338, + 826, + 401 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We empirically analyze robustness of Decision Diffuser to stochasticity in dynamics function. ", + "bbox": [ + 174, + 412, + 785, + 428 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Setup We use Block Push environment, described in Appendix F, with torque control. However, we inject stochasticity into the environment dynamics. For every environment step, we either sample a random action from $\\mathcal { U } ( [ - 1 , - 1 , - 1 ] , [ 1 , 1 , \\dot { 1 } ] )$ with probability $p$ or execute the action given by the policy with probability $( 1 - p )$ . We use $p \\in \\{ 0 , 0 . 0 5 , 0 . 1 , 0 . 1 5 \\}$ in our experiments. ", + "bbox": [ + 173, + 430, + 825, + 487 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Offline dataset collection We collect separate offline datasets for different block push environments, each characterized by a different value of $p$ . Each offline dataset consists of 1 million environment transitions collected using the method described in Appendix F. ", + "bbox": [ + 176, + 489, + 823, + 531 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Results Table A3 characterizes how the performance of BC, Decision Diffuser, Diffuser, and CQL changes with increasing stochasticity in the environment dynamics. We observe that the Decision Diffuser outperforms Diffuser and CQL for $p = 0 . 0 5$ , however all methods including the Decision Diffuser settle to a similar performance for larger values of $p$ . ", + "bbox": [ + 174, + 535, + 825, + 590 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Several works (Paster et al., 2022; Yang et al., 2022) have shown that the performance of returnconditioned policies suffers as the stochasticity in environment dynamics increases. This is because the return-conditioned policies aren’t able to distinguish between high returns from good actions and high returns from environment stochasticity. Hence, these return-conditioned policies can learn sub-optimal actions that got associated with high-return trajectories in the dataset due to environment stochasticity. Given Decision diffuser uses return conditioning to generate actions in offline RL, its performance also suffers when stochasticity in environment dynamics increases. ", + "bbox": [ + 174, + 597, + 825, + 695 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Some recent works (Yang et al., 2022; Villaflor et al., 2022) address the above issue by learning a latent model for future states and then conditioning the policy on predicted latent future states rather than returns. Conditioning Decision Diffuser on future state information, rather than returns, would make it more robust to stochastic dynamics and could be an interesting avenue for future works. ", + "bbox": [ + 176, + 702, + 825, + 758 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "H KUKA BLOCK STACKING ", + "text_level": 1, + "bbox": [ + 176, + 782, + 419, + 799 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In the Kuka blocking stacking environment, the underlying goal is to stack a set of blocks on top of each other. Models have trained on a set of demonstration data, where a set of 4 blocks are sequentially stacked on top of each other to form a block tower. ", + "bbox": [ + 174, + 818, + 825, + 861 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We construct state-space plans of length 128. Following (Janner et al., 2022), we utilize a close-loop controller to generate actions for each state in our state-space plan (controlling the 7 degrees of freedom in joints). The total maximum trajectory length plan in Kuka block stacking is 384. We detail differences between the two consider conditional stacking environments below: ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• Stacking In the stacking environment, at test time we wish to again construct a tower of four blocks. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Rearrangement In the rearrangement environment, at test time wish to stack blocks in a configuration where a set of blocks are above a second set. This set of stack-place relations may not precisely correspond to a single block tower (can instead construct two block towers), making this environment an out-of-distribution challenge. ", + "bbox": [ + 174, + 138, + 825, + 194 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In addition to Diffuser (Janner et al., 2022), we used goal-conditioned variants of CQL (Kumar et al., 2020) and BCQ (Fujimoto et al., 2019) as baselines for the block stacking and rearrangement with single constraint. However, they get a success rate of 0.0. ", + "bbox": [ + 174, + 207, + 825, + 250 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "I UNITREE GO RUNNING ", + "text_level": 1, + "bbox": [ + 176, + 271, + 398, + 287 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We consider Unitree-go-running environment (Margolis & Agrawal, 2022) where a quadruped robot runs in 3 different gaits: bounding, pacing, and trotting. The state space is 56 dimensional, the action space is 12 dimensional, and the maximum trajectory length is 250. ", + "bbox": [ + 174, + 305, + 823, + 348 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "As described in Section 4.3, we train Decision Diffuser on expert trajectories demonstrating individual gaits. During testing, we compose the noise model of our reverse diffusion process according to equation 9. This allows us to sample trajectories of the quadruped robot with entirely new running behavior. Figures A4,A5,A6 shows the ability of Decision Diffuser to imitate bounding, trotting and pacing and their combinations. ", + "bbox": [ + 174, + 354, + 825, + 424 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "I.1 QUANTITATIVE VERIFICATION OF COMPOSITION ", + "text_level": 1, + "bbox": [ + 173, + 444, + 545, + 458 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We now try to quantitatively verify whether the trajectories resulting from composition of 2 gaits does indeed contain only those 2 gaits. ", + "bbox": [ + 174, + 469, + 823, + 498 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Setup We learn a gait classifier that takes in a sub-sequence of states (of length 10) and predicts the gait-ID. It is represented by a 3-layered MLP with 1024 hidden units and ReLU activations that concatenates the sub-sequence of states (of length 10) into a single vector of dimension 560 before taking it in as an input. We train the gait classifier on the demonstration dataset. To ensure that the learned classifier can predict gait-ID on trajectories generated by the composition of skills, we use MixUp-style (Zhang et al., 2017) data augmentation during training. We create a synthetic subsequence of length 10 by concatenating two sampled sub-sequence (from the demonstration dataset) of length $l _ { i }$ and $l _ { j }$ (where $l _ { i } + l _ { j } = 1 0$ ) from gaits with $\\operatorname { I D } i$ and $j$ and give it a label $\\frac { l _ { i } } { l _ { i } + l _ { j } }$ one-ho $( i ) +$ $\\frac { l _ { j } } { l _ { i } + l _ { j } }$ one-hot $( j )$ . During training, we sample a sub-sequence from the demonstration dataset with $7 0 \\%$ probability and a sythenthic sub-sequence with $3 0 \\%$ probability. We train the classifier for $2 e 6$ train steps with a learning rate of $2 e - 4$ and a batch size of 64. ", + "bbox": [ + 173, + 501, + 826, + 666 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Results Figures A4,A5,A6 show that the classifier’s prediction is consistent with the visualized composed trajectories. Furthermore, we use Decision diffuser to act in the environment and generate 1000 trott trajectories, 1000 pace trajectories, 1000 bound trajectories, and 1000 composed trajectories for each possible pair of individual gaits. We then evaluate the learned gait classifier on these trajectories and compute the percentage of timesteps a particular gait has the highest probability. From Figures A4,A5,A6, we can see that if trajectories are generated by the composition of two gaits, then those two gaits will have the two highest probabilities across different timesteps in those trajectories. ", + "bbox": [ + 173, + 670, + 825, + 781 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "I.2 A SIMPLE BASELINE FOR COMPOSITION", + "text_level": 1, + "bbox": [ + 176, + 800, + 486, + 813 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let one-hot $( i )$ and one- $\\cdot \\mathrm { h o t } ( j )$ represent two different gaits that can be generated using noise models $\\epsilon _ { \\theta } ( { \\pmb x } _ { k } ( \\tau ) , \\mathrm { o n e - h o t } ( i ) , k )$ and $\\begin{array} { r } { \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\mathrm { o n e - h o t } ( j ) , k ) } \\end{array}$ respectively. To compose these gaits, we compose the above-mentioned noise models using equation 9. As an alternative, we see if the noise model $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\mathrm { o n e - h o t } ( i ) + \\mathrm { o n e - h o t } ( j ) , k )$ can lead to composed gaits. However, we observe that $\\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau )$ , one- $\\mathbf { \\cdot h o t } ( i ) + \\mathbf { o n e - h o t } ( j ) , k )$ catastrophically fail to generate any gait (see videos at https://anuragajay.github.io/decision-diffuser/). This happens because the condition variable one- $\\mathsf { 1 o t } ( i ) + \\mathsf { o n e { \\mathrm { - } } h o t } ( j )$ was never seen by the noise model $\\epsilon _ { \\theta }$ during training. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/c6eca9326f1240e24c5bc046689af995227c5ce856ee5e3438368121d3c1e16c.jpg", + "image_caption": [ + "Figure A4: Composing Trott and Pace. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. " + ], + "image_footnote": [], + "bbox": [ + 173, + 101, + 826, + 321 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/718ee11d8a554aad4ad0463dc91824091eb322e60b29c2deeafd23463aa4b13e.jpg", + "image_caption": [ + "Figure A5: Composing Trott and Bound. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. " + ], + "image_footnote": [], + "bbox": [ + 173, + 390, + 838, + 611 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "J NOT COMPOSITIONS WITH DECISION DIFFUSER ", + "text_level": 1, + "bbox": [ + 173, + 690, + 604, + 708 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Decision diffuser can also support ”NOT” composition. Suppose we wanted to sample from $q ( \\pmb { x } _ { 0 } ( \\tau ) | \\mathrm { N O T } \\ y ^ { j } ( \\tau ) )$ . Let $\\{ y ^ { i } ( \\tau ) \\} _ { i = 1 } ^ { n }$ be the set of all conditioning variables. Then, following derivations from Liu et al. (2022) and using $\\beta = 1$ , we can sample from $q ( \\pmb { x } _ { 0 } ( \\tau ) | \\mathrm { N O T } \\ y ^ { j } ( \\tau ) )$ using the perturbed noise: ", + "bbox": [ + 174, + 724, + 825, + 781 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f64ef7f63313279d0fe7890652b4ffe3953e1b4e7a222366a273bc1f6af03c7b.jpg", + "text": "$$\n\\begin{array} { l } { \\hat { \\epsilon } : = \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\emptyset , k ) + \\omega ( \\displaystyle \\sum _ { i \\neq j } ( \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\pmb { y } ^ { i } ( \\tau ) , k ) - \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\emptyset , k ) ) } \\\\ { - \\left( \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\pmb { y } ^ { j } ( \\tau ) , k ) - \\epsilon _ { \\theta } ( \\pmb { x } _ { k } ( \\tau ) , \\emptyset , k ) ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 782, + 732, + 842 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We demonstrate the ability of Decision Diffuser to support ”NOT” composition by using it to satisfy constraint of type BlockHeight $( i ) >$ BlockHeight $( j )$ AND (NOT BlockHeight $. ( j ) ~ >$ BlockHeight $( i )$ ) in Kuka block stacking task, as visualized in videos at https://anuragajay.github.io/decision-diffuser/. As the Decision Diffuser does not provide an explicit density estimate for each skill, it can’t natively support OR composition. ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/79557db21666a3824496a144965288975b56e26dcafa64a8c6e3a89d15f4a7a3.jpg", + "image_caption": [ + "Figure A6: Composing Bound and Pace. Decision Diffuser can imitate individual running gaits using expert demonstrations and compose multiple different skills together during test time. The results are best illustrated by videos viewable at https://anuragajay.github.io/decision-diffuser/. " + ], + "image_footnote": [], + "bbox": [ + 173, + 99, + 836, + 321 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "K COMPARING Q-FUNCTION GUIDED DIFFUSION AND CLASSIFIER-FREE GUIDED DIFFUSION ", + "text_level": 1, + "bbox": [ + 176, + 404, + 789, + 436 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Classifier-free guided diffusion and Q-value guided diffusion are theoretically equivalent. However, as noted in several works (Nichol et al., 2021; Ho & Salimans, 2022; Saharia et al., 2022), classifier-free guidance performs better than classifier guidance (i.e. Q function guidance in our case) in practice. This is due to following reasons: ", + "bbox": [ + 176, + 455, + 825, + 511 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Classifier-guided diffusion models learns an unconditional diffusion model along with a classifier (Q-function in our case) and uses gradients from the classifier to perform conditional sampling. However, the unconditional diffusion model doesn’t need to focus on conditional modeling during training and only cares about conditional generation during testing after it has been trained. In contrast, classifier-free guidance relies on conditional diffusion model to estimate gradients of the implicit classifier. Since the conditional diffusion model, learned when using classifier-free guidance, focuses on conditional modeling during train time, it performs better in conditional generation during test time. Q function trained on an offline dataset can erroneously predict high Q values for out-ofdistribution actions given any state. This problem has been extensively studied in offline RL literature (Kumar et al., 2020; Fujimoto et al., 2019; Levine et al., 2020). In online RL, this issue is automatically corrected when the policy acts in the environment, thinking an action to be good but then receives a low reward for it. In offline RL, this issue can’t be corrected easily; hence, the learned Q-function can often guide the diffusion model towards out-of-distribution actions that might be sub-optimal. In contrast, classifier-free guidance circumvents the issue of learning a Q-function and directly conditions the diffusion model on returns. Hence, classifier-free guidance doesn’t suffer due to errors in learned Q-functions and hence performs better than Q-function guided diffusion. ", + "bbox": [ + 217, + 525, + 825, + 781 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "L LIMITATIONS OF DECISION DIFFUSER ", + "text_level": 1, + "bbox": [ + 174, + 805, + 522, + 821 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We summarize the limitations of Decision Diffuser: ", + "bbox": [ + 174, + 839, + 511, + 854 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• No partial observability Decision Diffuser works with fully observable MDPs. Naive extensions to partially observed MDPs (POMDPs) may cause self-delusions (Ortega et al., 2021) in Decision Diffuser. Hence, extending Decision Diffuser to POMDPs could be an exciting avenue for future work. ", + "bbox": [ + 220, + 867, + 825, + 922 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Inability to explore the environment and update itself in online setting In this work, we focused on offline sequential decision making, thus circumventing the need for exploration. Using ideas from Zheng et al. (2022), future works could look into online fine-tuning of Decision Diffuser by leveraging entropy of the state-sequence model for exploration. ", + "bbox": [ + 217, + 103, + 825, + 160 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• Experiments on only state-based environments While our work focused on state based environments, it can be extended to image based environments by performing the diffusion in latent space, rather than observation space, as done in Rombach et al. (2022). ", + "bbox": [ + 218, + 165, + 823, + 207 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• Only AND and NOT compositions are supported Since Decision Diffuser does not provide an explicit density estimate for each condition variable, it can’t natively support OR composition. ", + "bbox": [ + 218, + 210, + 823, + 252 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• Performance degradation in environments with stochastic dynamics In environments with highly stochastic dynamics, Decision Diffuser loses its advantage and performs similarly to Diffuser and CQL. To tackle environments with stochastic dynamics, recent works (Yang et al., 2022; Villaflor et al., 2022) propose learning a latent model for future states and then conditioning the policy on predicted latent future states rather than returns. Conditioning Decision Diffuser on future state information, rather than returns, would make it more robust to stochastic dynamics and could be an interesting avenue for future works. ", + "bbox": [ + 217, + 257, + 825, + 354 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• Performance in limited data regime Since diffusion models are prone to overfitting in case of limited data, Decision Diffuser is also prone to overfitting in limited data regime. 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DatasetEnvironmentBCCQLIQLDTTTMOReLDiffuserDD
Med-ExpertHalfCheetah55.291.686.786.89553.379.890.6 ±1.3
Med-ExpertHopper52.5105.491.5107.6110.0108.7107.2111.8 ±1.8
Med-ExpertWalker2d107.5108.8109.6108.1101.995.6108.4108.8 ±1.7
MediumHalfCheetah42.644.047.442.646.942.144.249.1 ±1.0
MediumHopper52.958.566.367.661.195.458.579.3 ±3.6
MediumWalker2d75.372.578.374.07977.879.782.5±1.4
Med-ReplayHalfCheetah36.645.544.236.641.940.242.239.3 ±4.1
Med-ReplayHopper18.19594.782.791.593.696.8100±0.7
Med-ReplayWalker2d26.077.273.966.682.649.861.275±4.3
Average51.977.67774.778.972.975.381.8
MixedKitchen51.552.4511、1165 ±2.8
PartialKitchen3850.146.3--==57 ±2.5
Average44.851.248.7-1--61
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ConditionTrottPaceBound
Only Bound0.81.098.2
Only Pace1.497.70.9
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EnvironmentDiffuserDD
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Single Constraint - Rearrangement58.9 ±3.462.7 ±3.1
Single Constraint Average52.360.4
Multiple Constraints - Stacking160.3 ±3.1
Multiple Constraints - Rearrangement67.2 ±3.1
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Decision DiffuserHopper-Medium-Expert
α=0104.3 ± 0.7
α= 0.5111.8 ±1.6
α= 1.0107.1 ± 3.5
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EnvironmentBCCondDiffuserDecision Diffuser
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We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 470, + 348 + ], + "score": 1.0, + "content": "used the MNIST dataset to obtain image classification accuracy and to assess the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 346, + 398, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 398, + 359 + ], + "score": 1.0, + "content": "robustness of our rewarding system to varying target responses.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 385, + 205, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 208, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 208, + 401 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "For many years, deep convolutional neural network (DCNN) has dominated the field of computer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 424, + 504, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 504, + 435 + ], + "score": 1.0, + "content": "vision and object recognition Goodfellow et al. (2016); LeCun et al. (2015). Although novel meth-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "ods, such as visual transformers Carion et al. (2020) and very recent MLP-based models Tatsunami", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "& Taki (2021) are threatening its reign, CNN is still the most popular architecture employed for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "solving visual tasks. However, CNNs lack biological plausibility. First of all, neuron activations in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "an artificial neural network (ANN) are static real-numbered values, that are modeled by differen-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "tiable, non-linear activation functions. This is in contrast to biological neurons that use discrete, and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "mostly sparse spike trains to transmit information between each other, and in addition to the rate", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "of spikes (spatial encoding), they also use spike timing to encode information temporally Tavanaei", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "et al. (2019). Therefore, a spiking neural network (SNN) is more akin to the neural networks in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "brain. Spiking neural networks also require fewer labeled data and operations, which makes them", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 339, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 339, + 546 + ], + "score": 1.0, + "content": "compatible with energy-efficient neuromorphic hardware.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 108, + 550, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 107, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Secondly, the brain is incapable of error backpropagation, as done in traditional ANNs. One issue", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "with error backpropagation in ANNs is the weight transport problem, i.e., the fact that weight con-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "nectivity in feedforward and feedback directions is symmetric Liao et al. (2016); Bartunov et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "(2018). Additionally, error feedback propagation that does not affect neural activity is not compliant", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "with the feedback mechanisms that biological neurons use for communication Lillicrap et al. (2020).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "Furthermore, although convolutional neural networks has shown great potential in solving any", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "translation-invariant task, its use of weight sharing is biologically problematic. There is no empirical", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "support for explicit weight sharing in the brain Pogodin et al. (2021). However, local connections be-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "tween neurons is biologically plausible, since neurons in the biological visual system exploit them to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "have local visual receptive fields Gregor & LeCun (2010). To be compatible with this fact, we also", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "used a locally-connected scheme without explicit weight sharing to design our network. Despite", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "the biological nature of local connections, they mostly underperform convolution-based methods", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "with weight sharing in the visual domain, especially on large-scale datasets Bartunov et al. 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Studies are being done to bridge", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "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 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 97 + ], + "score": 1.0, + "content": "BIOLCNET: REWARD-MODULATED LOCALLY CON-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 97, + 393, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 97, + 393, + 118 + ], + "score": 1.0, + "content": "NECTED SPIKING NEURAL NETWORKS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 112, + 136, + 245, + 159 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 214, + 468, + 357 + ], + "lines": [ + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "score": 1.0, + "content": "Recent studies have shown that convolutional neural networks (CNNs) are not", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 224, + 470, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 470, + 239 + ], + "score": 1.0, + "content": "the only feasible solution for image classification. Furthermore, weight sharing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 236, + 470, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 236, + 470, + 249 + ], + "score": 1.0, + "content": "and backpropagation used in CNNs do not correspond to the mechanisms present", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 248, + 470, + 260 + ], + "spans": [ + { + "bbox": [ + 141, + 248, + 470, + 260 + ], + "score": 1.0, + "content": "in the biological visual system. To propose a more biologically plausible so-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 258, + 470, + 271 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 470, + 271 + ], + "score": 1.0, + "content": "lution, we designed a locally connected spiking neural network (SNN) trained", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 269, + 470, + 282 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 470, + 282 + ], + "score": 1.0, + "content": "using spike-timing-dependent plasticity (STDP) and its reward-modulated vari-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 281, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 142, + 281, + 470, + 292 + ], + "score": 1.0, + "content": "ant (R-STDP) learning rules. The use of spiking neurons and local connections", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "score": 1.0, + "content": "along with reinforcement learning (RL) led us to the nomenclature BioLCNet for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 302, + 470, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 470, + 314 + ], + "score": 1.0, + "content": "our proposed architecture. Our network consists of a rate-coded input layer fol-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "score": 1.0, + "content": "lowed by a locally connected hidden layer and a decoding output layer. A spike", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "score": 1.0, + "content": "population-based voting scheme is adopted for decoding in the output layer. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 470, + 348 + ], + "score": 1.0, + "content": "used the MNIST dataset to obtain image classification accuracy and to assess the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 346, + 398, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 398, + 359 + ], + "score": 1.0, + "content": "robustness of our rewarding system to varying target responses.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 213, + 470, + 359 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 385, + 205, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 208, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 208, + 401 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "For many years, deep convolutional neural network (DCNN) has dominated the field of computer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 424, + 504, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 504, + 435 + ], + "score": 1.0, + "content": "vision and object recognition Goodfellow et al. (2016); LeCun et al. (2015). Although novel meth-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "ods, such as visual transformers Carion et al. (2020) and very recent MLP-based models Tatsunami", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "& Taki (2021) are threatening its reign, CNN is still the most popular architecture employed for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "solving visual tasks. However, CNNs lack biological plausibility. First of all, neuron activations in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "an artificial neural network (ANN) are static real-numbered values, that are modeled by differen-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "tiable, non-linear activation functions. This is in contrast to biological neurons that use discrete, and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "mostly sparse spike trains to transmit information between each other, and in addition to the rate", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "of spikes (spatial encoding), they also use spike timing to encode information temporally Tavanaei", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "et al. (2019). Therefore, a spiking neural network (SNN) is more akin to the neural networks in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "brain. Spiking neural networks also require fewer labeled data and operations, which makes them", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 339, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 339, + 546 + ], + "score": 1.0, + "content": "compatible with energy-efficient neuromorphic hardware.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 413, + 505, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 550, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 107, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Secondly, the brain is incapable of error backpropagation, as done in traditional ANNs. One issue", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "with error backpropagation in ANNs is the weight transport problem, i.e., the fact that weight con-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "nectivity in feedforward and feedback directions is symmetric Liao et al. (2016); Bartunov et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "(2018). Additionally, error feedback propagation that does not affect neural activity is not compliant", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "with the feedback mechanisms that biological neurons use for communication Lillicrap et al. (2020).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 550, + 505, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "Furthermore, although convolutional neural networks has shown great potential in solving any", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "translation-invariant task, its use of weight sharing is biologically problematic. There is no empirical", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "support for explicit weight sharing in the brain Pogodin et al. (2021). However, local connections be-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "tween neurons is biologically plausible, since neurons in the biological visual system exploit them to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "have local visual receptive fields Gregor & LeCun (2010). To be compatible with this fact, we also", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "used a locally-connected scheme without explicit weight sharing to design our network. Despite", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "the biological nature of local connections, they mostly underperform convolution-based methods", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "with weight sharing in the visual domain, especially on large-scale datasets Bartunov et al. (2018).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "This weaker performance may be mainly attributed to the smaller number of parameters and better", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "generalization in CNNs. Fewer parameters in CNNs would also require less memory and computa-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "tional cost, and would lead to faster training Poggio et al. (2017). Studies are being done to bridge", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 609, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "the performance gap between convolutional and locally-connected networks Lillicrap et al. (2020);", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 200, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 200, + 106 + ], + "score": 1.0, + "content": "Bartunov et al. (2018).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "Noting the above considerations, in this paper, we are proposing BioLCNet, a reward-modulated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "locally-connected spiking neural network. Our network is trained using the unsupervised spike-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "timing-dependent plasticity and its semi-supervised variant reward-modulated STDP. The input im-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 157 + ], + "score": 1.0, + "content": "ages are encoded proportional to the pixels intensity using Poisson rate-coding that converts intensity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "to average neuron firing rate in Hertz. In the output layer, there are neuronal groups for each class", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "label, and decision making is based on aggregated number of spikes during the decision period.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 280, + 189 + ], + "score": 1.0, + "content": "Our novel dynamic reward prediction error", + "type": "text" + }, + { + "bbox": [ + 281, + 177, + 306, + 187 + ], + "score": 0.53, + "content": "( R P E )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "mechanism exploits strongly supported empirical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "findings to improve classification performance. We test the classification capabilities of our network", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "with different sets of hyperparameters on the MNIST dataset LeCun et al. (1999). We also conduct a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "classical conditioning experiment to prove the effectiveness of our decoding scheme and rewarding", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 160, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 160, + 231 + ], + "score": 1.0, + "content": "mechanisms.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 211, + 265 + ], + "lines": [ + { + "bbox": [ + 104, + 251, + 213, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 251, + 213, + 268 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "Neuroscientists and deep learning researchers have long been searching for more biologically plau-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "sible deep learning approaches in terms of neuronal characteristics, learning rules, and connection", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "types. Regarding neuronal characteristics, researchers have turned to biological neuronal models and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "spiking neural networks. The vanishing performance gap between deep neural netwroks (DNNs)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "and SNNs, and the compatibility of SNNs with neuromorphic hardware and online on-chip training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "Schemmel et al. (2010) has piqued the interest of researchers Mozafari et al. (2019). For compre-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "hensive reviews on deep learning in spiking neural networks, see Tavanaei et al. (2019); Pfeiffer &", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 357, + 159, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 159, + 370 + ], + "score": 1.0, + "content": "Pfeil (2018).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "Spiking neurons are activated by discrete input spike trains. This differs from artificial neurons used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "in an ANN that have differentiable activation functions and can easily employ backpropagation and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "gradient-based optimization. There are works that use gradient-based methods with SNNs Kherad-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "pisheh & Masquelier (2020); Wu et al. (2018); Neftci et al. (2019); Bellec et al. (2020) and some of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "them have achieved great performances. On the other hand, many works in this area use derivations", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "of the Hebbian learning rule where changes in connection weights depend on the activities of the pre", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "and post-synaptic neurons Hebb (1949). Spike-timing-dependent plasticity (STDP) and its variants,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "apply asymmetric weight updates based on the temporal activities of neurons. Normal STDP re-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "quires an external read-out for classification Mozafari et al. (2018), and have been applied to image", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "reconstruction and classification tasks by many researchers. Some have employed fully-connected", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "architectures Beyeler et al. (2013); Tavanaei & Maida (2015); Allred & Roy (2016), while others", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "used convolutional layers for feature extraction Masquelier & Thorpe (2007); Panda & Roy (2016);", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "Kheradpisheh et al. (2016; 2018). Reward-modulated STDP (R-STDP) uses a reward (or punish-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "ment) signal to directly modulate the STDP weight change, and can be used to decode the output", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "without an external cue. Izhikevich (2007) solved the distal reward problem in reinforcement learn-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "score": 1.0, + "content": "ing by using a version of R-STDP with decaying eligibility traces that gives recent spiking activity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "more importance. Around the same time, Florian (2007) showed that R-STDP can be employed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "to solve a simple XOR task with both rate and temporal encoding of the output. Also, Caporale &", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Dan (2008) used R-STDP to generate specific spiking patterns in the output of their spiking network.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Historically, R-STDP was first adopted with temporal (rank-order) encoding for image classification", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Mozafari et al. (2018). They employed a convolutional architecture based on Masquelier & Thorpe", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "(2007) and a time-to-first-spike decoding scheme. An extended architecture was later developed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "which had multiple hidden layers Mozafari et al. (2019). The use of R-STDP with Poisson rate-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "coding has been mostly limited to fully-connected architectures for solving reinforcement learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "robot navigation tasks Shim & Li (2017); Bing et al. (2019). To our knowledge, image recognition", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 457, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 457, + 663 + ], + "score": 1.0, + "content": "problems have not yet been addressed by combining R-STDP and rate-based encoding.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "The most prevalent architectures used for image classification in deep learning with both DNNs", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "and SNNs are based on convolutional layers and weight sharing. However, there are arguments", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "against the biological plausibility of these approaches Bartunov et al. (2018); Pogodin et al. (2021).", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Locally connected (LC) networks are an alternative to the convolutional ones. Illing et al. 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(2020);", + "type": "text" + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 93, + 200, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 200, + 106 + ], + "score": 1.0, + "content": "Bartunov et al. (2018).", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "Noting the above considerations, in this paper, we are proposing BioLCNet, a reward-modulated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "locally-connected spiking neural network. 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We test the classification capabilities of our network", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "with different sets of hyperparameters on the MNIST dataset LeCun et al. (1999). We also conduct a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "classical conditioning experiment to prove the effectiveness of our decoding scheme and rewarding", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 160, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 160, + 231 + ], + "score": 1.0, + "content": "mechanisms.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 110, + 506, + 231 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 252, + 211, + 265 + ], + "lines": [ + { + "bbox": [ + 104, + 251, + 213, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 251, + 213, + 268 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "Neuroscientists and deep learning researchers have long been searching for more biologically plau-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "sible deep learning approaches in terms of neuronal characteristics, learning rules, and connection", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "types. Regarding neuronal characteristics, researchers have turned to biological neuronal models and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "spiking neural networks. The vanishing performance gap between deep neural netwroks (DNNs)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "and SNNs, and the compatibility of SNNs with neuromorphic hardware and online on-chip training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "Schemmel et al. (2010) has piqued the interest of researchers Mozafari et al. (2019). For compre-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "hensive reviews on deep learning in spiking neural networks, see Tavanaei et al. (2019); Pfeiffer &", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 357, + 159, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 159, + 370 + ], + "score": 1.0, + "content": "Pfeil (2018).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 281, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "Spiking neurons are activated by discrete input spike trains. This differs from artificial neurons used", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "in an ANN that have differentiable activation functions and can easily employ backpropagation and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "gradient-based optimization. There are works that use gradient-based methods with SNNs Kherad-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "pisheh & Masquelier (2020); Wu et al. (2018); Neftci et al. (2019); Bellec et al. (2020) and some of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "them have achieved great performances. On the other hand, many works in this area use derivations", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "of the Hebbian learning rule where changes in connection weights depend on the activities of the pre", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "and post-synaptic neurons Hebb (1949). Spike-timing-dependent plasticity (STDP) and its variants,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "apply asymmetric weight updates based on the temporal activities of neurons. Normal STDP re-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "quires an external read-out for classification Mozafari et al. (2018), and have been applied to image", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "reconstruction and classification tasks by many researchers. Some have employed fully-connected", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "architectures Beyeler et al. (2013); Tavanaei & Maida (2015); Allred & Roy (2016), while others", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "used convolutional layers for feature extraction Masquelier & Thorpe (2007); Panda & Roy (2016);", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "Kheradpisheh et al. (2016; 2018). Reward-modulated STDP (R-STDP) uses a reward (or punish-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "ment) signal to directly modulate the STDP weight change, and can be used to decode the output", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "without an external cue. Izhikevich (2007) solved the distal reward problem in reinforcement learn-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "score": 1.0, + "content": "ing by using a version of R-STDP with decaying eligibility traces that gives recent spiking activity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "more importance. Around the same time, Florian (2007) showed that R-STDP can be employed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "to solve a simple XOR task with both rate and temporal encoding of the output. Also, Caporale &", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Dan (2008) used R-STDP to generate specific spiking patterns in the output of their spiking network.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Historically, R-STDP was first adopted with temporal (rank-order) encoding for image classification", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Mozafari et al. (2018). They employed a convolutional architecture based on Masquelier & Thorpe", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "(2007) and a time-to-first-spike decoding scheme. An extended architecture was later developed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "which had multiple hidden layers Mozafari et al. (2019). The use of R-STDP with Poisson rate-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "coding has been mostly limited to fully-connected architectures for solving reinforcement learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "robot navigation tasks Shim & Li (2017); Bing et al. (2019). To our knowledge, image recognition", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 457, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 457, + 663 + ], + "score": 1.0, + "content": "problems have not yet been addressed by combining R-STDP and rate-based encoding.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 374, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "The most prevalent architectures used for image classification in deep learning with both DNNs", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "and SNNs are based on convolutional layers and weight sharing. However, there are arguments", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "against the biological plausibility of these approaches Bartunov et al. (2018); Pogodin et al. (2021).", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Locally connected (LC) networks are an alternative to the convolutional ones. Illing et al. (2019)", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "show that shallow networks with localized connectivity and receptive fields perform much better", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "than fully-connected networks on the MNIST benchmark. However, Bartunov et al. (2018) showed", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "that the lower generalization of LC networks compared to CNNs results in their underperforming", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "CNNs in most image classification tasks, and prevents their scalability to larger datasets such as", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "ImageNet Deng et al. (2009). Very recently, Pogodin et al. (2021) proposed bio-inspired dynamic", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "weight sharing and adding lateral connections to locally-connected layers to achieve the same reg-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "ularization goals of weight sharing and normal convolutional filters. The first work to integrate a", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "locally-connected (LC) layer into an SNN Saunders et al. (2019) used a network with no hidden", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "layers where the rate-coded input is passed to the output layer via local connections. They exploited", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "recurrent inhibitory connections similar to the ones employed by Diehl & Cook (2015) to simu-", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "late a winner-take-all (WTA) inhibition mechanism in their output. Their learning rule is STDP, and", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "score": 1.0, + "content": "therefore an external readout, in this case n-gram voting, is required for classification. Their network", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 388, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 388, + 205 + ], + "score": 1.0, + "content": "scheme was inspiring in designing our locally connected hidden layer.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 665, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "that the lower generalization of LC networks compared to CNNs results in their underperforming", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "CNNs in most image classification tasks, and prevents their scalability to larger datasets such as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "ImageNet Deng et al. (2009). Very recently, Pogodin et al. (2021) proposed bio-inspired dynamic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "weight sharing and adding lateral connections to locally-connected layers to achieve the same reg-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "ularization goals of weight sharing and normal convolutional filters. The first work to integrate a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "locally-connected (LC) layer into an SNN Saunders et al. (2019) used a network with no hidden", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "layers where the rate-coded input is passed to the output layer via local connections. They exploited", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "recurrent inhibitory connections similar to the ones employed by Diehl & Cook (2015) to simu-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "late a winner-take-all (WTA) inhibition mechanism in their output. Their learning rule is STDP, and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 195 + ], + "score": 1.0, + "content": "therefore an external readout, in this case n-gram voting, is required for classification. Their network", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 388, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 388, + 205 + ], + "score": 1.0, + "content": "scheme was inspiring in designing our locally connected hidden layer.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 220, + 169, + 232 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 172, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 172, + 235 + ], + "score": 1.0, + "content": "3 THEORY", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 504, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "In this section, we will outline the theoretical foundations underlying our proposed method. Specif-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "ically, the dynamics of the spiking neuronal model, the learning rules used, and the connection type", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 282, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 282, + 279 + ], + "score": 1.0, + "content": "employed in our network will be described.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 293, + 266, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 268, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 268, + 306 + ], + "score": 1.0, + "content": "3.1 ADAPTIVE LIF NEURON MODEL", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 504, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "The famous leaky and integrate fire neuronal model is governed by the following differential equa-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 325, + 215, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 215, + 337 + ], + "score": 1.0, + "content": "tion Gerstner et al. 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Equation (1)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 404, + 397 + ], + "score": 1.0, + "content": "dictates that the neuron potential exponentially decays to a constant value", + "type": "text" + }, + { + "bbox": [ + 404, + 385, + 426, + 396 + ], + "score": 0.89, + "content": "u _ { r e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "over time. When a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "pre-synaptic neuron fires (spikes), it generates a current that reaches its post-synaptic neurons. 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After firing, the neuron’s potential resets to a constant", + "type": "text" + }, + { + "bbox": [ + 383, + 419, + 409, + 428 + ], + "score": 0.88, + "content": "u _ { r e s e t }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 416, + 506, + 430 + ], + "score": 1.0, + "content": "and will not be affected", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 426, + 430, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 394, + 442 + ], + "score": 1.0, + "content": "by any input current for a period of time known as the refractory period", + "type": "text" + }, + { + "bbox": [ + 395, + 428, + 425, + 440 + ], + "score": 0.89, + "content": "( \\Delta t _ { r e f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 426, + 430, + 442 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 504, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 398, + 458 + ], + "score": 1.0, + "content": "A variant of the LIF model uses adaptive firing thresholds. 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A popular variant of STDP that integrates reinforcement learning into", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 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 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": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 105, + 81, + 507, + 205 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 220, + 169, + 232 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 172, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 172, + 235 + ], + "score": 1.0, + "content": "3 THEORY", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 504, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "In this section, we will outline the theoretical foundations underlying our proposed method. Specif-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "ically, the dynamics of the spiking neuronal model, the learning rules used, and the connection type", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 282, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 282, + 279 + ], + "score": 1.0, + "content": "employed in our network will be described.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 245, + 505, + 279 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 293, + 266, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 268, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 268, + 306 + ], + "score": 1.0, + "content": "3.1 ADAPTIVE LIF NEURON MODEL", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 504, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "The famous leaky and integrate fire neuronal model is governed by the following differential equa-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 325, + 215, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 215, + 337 + ], + "score": 1.0, + "content": "tion Gerstner et al. 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A popular variant of STDP that integrates reinforcement learning into", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "the learning mechanism of spiking neural networks is reward-modulated STDP (also known as R-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "STDP or MSTDP Florian (2007)). In R-STDP, a global reward or punishment signal, which can be", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "a function of time, is generated as the result of the network’s activity or task performance. 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Using", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "a notation similar to Florian (2007), to mathematically formulate both STDP and R-STDP, we can", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "define the spike train of a pre-synaptic neuron as the sum of Dirac functions over the spikes of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 199, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 199, + 151 + ], + "score": 1.0, + "content": "post-synaptic neurons,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 147, + 357, + 169 + ], + "lines": [ + { + "bbox": [ + 254, + 147, + 357, + 169 + ], + "spans": [ + { + "bbox": [ + 254, + 147, + 357, + 169 + ], + "score": 0.94, + "content": "\\Phi ( t ) = \\sum _ { \\mathcal { F } _ { i } } \\delta ( t - t _ { i } ^ { f } ) .", + "type": "interline_equation", + "image_path": "d3ee8d74757c6e6ac5c52ac0e9f43a1dd5a7a59d031868045f28fc39784237b6.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 254, + 147, + 357, + 169 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 133, + 187 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 172, + 143, + 186 + ], + "score": 0.89, + "content": "t _ { i } ^ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 172, + 236, + 187 + ], + "score": 1.0, + "content": "is the firing time of the", + "type": "text" + }, + { + "bbox": [ + 236, + 173, + 249, + 184 + ], + "score": 0.88, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 172, + 471, + 187 + ], + "score": 1.0, + "content": "post-syanptic neuron. 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The update rule for normal STDP can also be written as,", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 420, + 345, + 445 + ], + "lines": [ + { + "bbox": [ + 266, + 420, + 345, + 445 + ], + "spans": [ + { + "bbox": [ + 266, + 420, + 345, + 445 + ], + "score": 0.95, + "content": "\\frac { d w _ { i j } ( t ) } { d t } = \\gamma \\xi _ { i j } ( t ) .", + "type": "interline_equation", + "image_path": "2c2e9dc2608bb21368be24cff57f8e57b3fd2fc89cdb0a9f74e12028475845bb.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 266, + 420, + 345, + 445 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "Based on Equation (9), we note that R-STDP updates only take effect when a non-zero modulation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 461, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 228, + 473 + ], + "score": 1.0, + "content": "signal is received at time step", + "type": "text" + }, + { + "bbox": [ + 228, + 461, + 233, + 470 + ], + "score": 0.55, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 461, + 504, + 473 + ], + "score": 1.0, + "content": ". 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The layer is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 156, + 270 + ], + "score": 1.0, + "content": "divided into", + "type": "text" + }, + { + "bbox": [ + 156, + 258, + 168, + 268 + ], + "score": 0.85, + "content": "n _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 255, + 261, + 270 + ], + "score": 1.0, + "content": "neuronal groups where", + "type": "text" + }, + { + "bbox": [ + 262, + 258, + 273, + 268 + ], + "score": 0.87, + "content": "n _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "is the number of classes related to the task. 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The predicted label for a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "given test sample is the class whose group has the most number of spikes aggregated over the deci-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 288, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 154, + 304 + ], + "score": 1.0, + "content": "sion period", + "type": "text" + }, + { + "bbox": [ + 154, + 290, + 179, + 301 + ], + "score": 0.89, + "content": "( T _ { d e c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 288, + 506, + 304 + ], + "score": 1.0, + "content": ". This decoding layer is trained using reinforcement learning and R-STDP during", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 184, + 314 + ], + "score": 1.0, + "content": "the learning period", + "type": "text" + }, + { + "bbox": [ + 185, + 301, + 217, + 312 + ], + "score": 0.89, + "content": "( T _ { l e a r n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 299, + 505, + 314 + ], + "score": 1.0, + "content": "based on the modulation signal generated by the rewarding mechanism.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "We designed two different rewarding mechanisms, static and dynamic reward prediction error", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "score": 1.0, + "content": "(RPE). In the static mechanism, we use a fixed reward or punishment signal for the whole learning", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 332, + 435, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 135, + 347 + ], + "score": 1.0, + "content": "period", + "type": "text" + }, + { + "bbox": [ + 136, + 334, + 167, + 345 + ], + "score": 0.87, + "content": "( T _ { l e a r n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 332, + 354, + 347 + ], + "score": 1.0, + "content": "based on the prediction of the network for the", + "type": "text" + }, + { + "bbox": [ + 354, + 333, + 367, + 343 + ], + "score": 0.88, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 332, + 435, + 347 + ], + "score": 1.0, + "content": "training sample,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 351, + 399, + 379 + ], + "lines": [ + { + "bbox": [ + 206, + 351, + 399, + 379 + ], + "spans": [ + { + "bbox": [ + 206, + 351, + 399, + 379 + ], + "score": 0.93, + "content": "r _ { i } = \\left\\{ \\begin{array} { c } { { 1 : \\ p r e d i c t e d l a b e l = t a r g e t l a b e l } } \\\\ { { - 1 : \\qquad o t h e r w i s e } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "7e383164a30af869dc72efc9e012acf32e5be7f8450075b6f4dc2dbac4c4e301.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 351, + 399, + 365.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 206, + 365.0, + 399, + 379.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 504, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 243, + 397 + ], + "score": 1.0, + "content": "The second mechanism, dynamic", + "type": "text" + }, + { + "bbox": [ + 243, + 385, + 263, + 395 + ], + "score": 0.47, + "content": "R P E", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "is based on the reward prediction error theory in reinforce-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "ment learning. According to this theory, the dopaminergic neurons in the brain release dopamine", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "proportional to the difference between the actual reward and the expected reward (not solely based", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "on the actual reward) Schultz et al. (1997); Sutton & Barto (2018). 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During", + "type": "text" + }, + { + "bbox": [ + 424, + 594, + 443, + 605 + ], + "score": 0.9, + "content": "T _ { d e c }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "time steps, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "decoding layer accumulates the number of spikes received by each neuronal group to determine the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "predicted label. 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Fig. 3a visualizes the filters learned after", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "2000 iterations for 100 filters of size 15 with a stride of 4 applied to the input images. This fast", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "convergence is an evidence showing the strength of STDP learning. 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This map shows the post-synaptic neurons corresponding to the relevant", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "features activate, and suppress the other neurons in accordance with the WTA inhibition mechanism.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 266, + 135 + ], + "score": 1.0, + "content": "In normal STDP, the LTP learning rate", + "type": "text" + }, + { + "bbox": [ + 266, + 122, + 294, + 133 + ], + "score": 0.89, + "content": "( \\eta _ { p o s t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 120, + 468, + 135 + ], + "score": 1.0, + "content": "is usually chosen larger than the LTD rate", + "type": "text" + }, + { + "bbox": [ + 469, + 122, + 493, + 133 + ], + "score": 0.87, + "content": "( \\eta _ { p r e } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 120, + 506, + 135 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "suppress the random firing of neurons that triggers many LTD updates during the early stages of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "training. However, this may become problematic in the later stages, and the weights may grow too", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "large. Therefore, in practice, different mechanisms, such as weight clipping and normalization are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "used to prevent the weights running amok. In this work, we clipped the weights to stay in the range", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 107, + 176, + 127, + 188 + ], + "score": 0.28, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 176, + 506, + 188 + ], + "score": 1.0, + "content": ". We also employed the normalization technique used by Saunders et al. 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The layer is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 156, + 270 + ], + "score": 1.0, + "content": "divided into", + "type": "text" + }, + { + "bbox": [ + 156, + 258, + 168, + 268 + ], + "score": 0.85, + "content": "n _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 255, + 261, + 270 + ], + "score": 1.0, + "content": "neuronal groups where", + "type": "text" + }, + { + "bbox": [ + 262, + 258, + 273, + 268 + ], + "score": 0.87, + "content": "n _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "is the number of classes related to the task. 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The predicted label for a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "given test sample is the class whose group has the most number of spikes aggregated over the deci-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 288, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 154, + 304 + ], + "score": 1.0, + "content": "sion period", + "type": "text" + }, + { + "bbox": [ + 154, + 290, + 179, + 301 + ], + "score": 0.89, + "content": "( T _ { d e c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 288, + 506, + 304 + ], + "score": 1.0, + "content": ". This decoding layer is trained using reinforcement learning and R-STDP during", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 184, + 314 + ], + "score": 1.0, + "content": "the learning period", + "type": "text" + }, + { + "bbox": [ + 185, + 301, + 217, + 312 + ], + "score": 0.89, + "content": "( T _ { l e a r n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 299, + 505, + 314 + ], + "score": 1.0, + "content": "based on the modulation signal generated by the rewarding mechanism.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "We designed two different rewarding mechanisms, static and dynamic reward prediction error", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "score": 1.0, + "content": "(RPE). In the static mechanism, we use a fixed reward or punishment signal for the whole learning", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 332, + 435, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 135, + 347 + ], + "score": 1.0, + "content": "period", + "type": "text" + }, + { + "bbox": [ + 136, + 334, + 167, + 345 + ], + "score": 0.87, + "content": "( T _ { l e a r n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 332, + 354, + 347 + ], + "score": 1.0, + "content": "based on the prediction of the network for the", + "type": "text" + }, + { + "bbox": [ + 354, + 333, + 367, + 343 + ], + "score": 0.88, + "content": "i ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 332, + 435, + 347 + ], + "score": 1.0, + "content": "training sample,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 245, + 506, + 347 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 351, + 399, + 379 + ], + "lines": [ + { + "bbox": [ + 206, + 351, + 399, + 379 + ], + "spans": [ + { + "bbox": [ + 206, + 351, + 399, + 379 + ], + "score": 0.93, + "content": "r _ { i } = \\left\\{ \\begin{array} { c } { { 1 : \\ p r e d i c t e d l a b e l = t a r g e t l a b e l } } \\\\ { { - 1 : \\qquad o t h e r w i s e } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "7e383164a30af869dc72efc9e012acf32e5be7f8450075b6f4dc2dbac4c4e301.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 351, + 399, + 365.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 206, + 365.0, + 399, + 379.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 504, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 243, + 397 + ], + "score": 1.0, + "content": "The second mechanism, dynamic", + "type": "text" + }, + { + "bbox": [ + 243, + 385, + 263, + 395 + ], + "score": 0.47, + "content": "R P E", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "is based on the reward prediction error theory in reinforce-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "ment learning. According to this theory, the dopaminergic neurons in the brain release dopamine", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "proportional to the difference between the actual reward and the expected reward (not solely based", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "on the actual reward) Schultz et al. (1997); Sutton & Barto (2018). 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During", + "type": "text" + }, + { + "bbox": [ + 424, + 594, + 443, + 605 + ], + "score": 0.9, + "content": "T _ { d e c }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "time steps, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "decoding layer accumulates the number of spikes received by each neuronal group to determine the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "predicted label. Afterwards, the modulation signal is generated and the decoding layer weights are", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 626, + 289, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 215, + 641 + ], + "score": 1.0, + "content": "updated using R-STDP for", + "type": "text" + }, + { + "bbox": [ + 216, + 627, + 242, + 638 + ], + "score": 0.9, + "content": "T _ { l e a r n }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 626, + 289, + 641 + ], + "score": 1.0, + "content": "time steps.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 527, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "When training the LC layer, we observed that after a specific number of iterations (training samples),", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "the weights of this layer converge and remain constant. Fig. 3a visualizes the filters learned after", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "2000 iterations for 100 filters of size 15 with a stride of 4 applied to the input images. This fast", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "convergence is an evidence showing the strength of STDP learning. 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This map shows the post-synaptic neurons corresponding to the relevant", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "features activate, and suppress the other neurons in accordance with the WTA inhibition mechanism.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 644, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "The network is implemented using PyTorch Paszke et al. 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ParameterValue
Uthro-52 (mV)
Urest,Ureset-65 (mV)
go0.05 (mV)
Tg10 (ms)
△tref5(ms)
Tm20(ms)
fmax128(Hz)
hin,Win22
nout[100,500,1000]
chlc[25,50,100,250]
k[11,13, 15,17]
S[2, 3,4]
Tadapt,Tdec,Tlearn256 (ms)
(Npre,Npost)STDP(0.0001,0.01)
(npre,post)R-STDP(0.1,0.1)
21
nrpe[(static),0.075,0.125,0.175,0.25]
a0.9
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Cnorm0.25
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The full list", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 649, + 271, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 271, + 661 + ], + "score": 1.0, + "content": "of hyperparameters are given in Table 1.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Considering the hyperparameters mentioned in Table 1, we report in Table 2, the classification accu-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "racy on the whole MNIST test set (10000 samples) for four hyperparameter configurations chosen", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "based on the highest test accuracy obtained after conducting a grid search. 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ParameterValue
Uthro-52 (mV)
Urest,Ureset-65 (mV)
go0.05 (mV)
Tg10 (ms)
△tref5(ms)
Tm20(ms)
fmax128(Hz)
hin,Win22
nout[100,500,1000]
chlc[25,50,100,250]
k[11,13, 15,17]
S[2, 3,4]
Tadapt,Tdec,Tlearn256 (ms)
(Npre,Npost)STDP(0.0001,0.01)
(npre,post)R-STDP(0.1,0.1)
21
nrpe[(static),0.075,0.125,0.175,0.25]
a0.9
Winh-100
Cnorm0.25
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Some of the hyperparameters were fixed and others were subject to grid search. The full list", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 649, + 271, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 271, + 661 + ], + "score": 1.0, + "content": "of hyperparameters are given in Table 1.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 627, + 505, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Considering the hyperparameters mentioned in Table 1, we report in Table 2, the classification accu-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "racy on the whole MNIST test set (10000 samples) for four hyperparameter configurations chosen", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "based on the highest test accuracy obtained after conducting a grid search. The number of neurons", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "and synapses for each model are also reported in this table. The final models were all trained using", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "10000 training samples from the MNIST training set. Using more training samples did not improve", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "the classification performance as can be observed from Fig. 4. The mean and standard deviations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "reported are estimated from ten independent runs. In addition to the RL-based models, another clas-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "sification approach was employed. In this approach, for each training sample, we create a feature", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 332, + 386 + ], + "score": 1.0, + "content": "vector containing the number of spikes aggregated over", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 332, + 373, + 359, + 384 + ], + "score": 0.91, + "content": "T _ { l e a r n }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 360, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "time steps for every filter in the LC", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "layer. We use these feature vectors to train a support vector machine (SVM) classifier. The SVM", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "results are also obtained by training on 10000 training samples, and testing on the whole MNIST test", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "set. The SVM test results for two different hyperparameter configurations are reported in Table 2", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "and are compared to the RL-based results. The best performance of SVM and RL-based classifica-", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "tion are 87.50, and 76.40 respectively. Table3 compares the MNIST test performance obtained by", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 438, + 453, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 453, + 451 + ], + "score": 1.0, + "content": "different SNN approaches along with the bio-plausibility criteria to which they adhere.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 129, + 112, + 482, + 166 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 2: MNIST test dataset accuracies obtained by four different sets of hyper-parameters; the test", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 204, + 91, + 406, + 104 + ], + "spans": [ + { + "bbox": [ + 204, + 91, + 406, + 104 + ], + "score": 1.0, + "content": "accuracies are averaged over ten independent runs", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 129, + 112, + 482, + 166 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 112, + 482, + 166 + ], + "spans": [ + { + "bbox": [ + 129, + 112, + 482, + 166 + ], + "score": 0.977, + "html": "
Parameters[k,s,nrpe,nout]nneuronsnsynapsesTest accuracySVM test accuracy
[13,3,0.025,100]170043040061.30 ±3.1487.5±1.32
[15,4,0.175,1000]188449000075.00 ±2.6883.3±1.74
[15,4,0.125,1000]188449000076.40 ±2.4383.3±1.74
[15,4,(static),100]98413000068.8±2.8783.3±1.74
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PaperEncodingArchitectureBio-plausibility criteriaAcc.
BioLCNet (proposed, RL)rate-basedLocally connected+DenseSTDP, RL, LC76.40
BioLCNet (proposed,SVM)rate-basedLocally connectedSTDP, LC87.5
Beyeler et al. (2013)rate-basedDenseSTDP91.60
Diehl& Cook (2015)rate-basedDenseSTDP95.00
Tavanaei &Maida (2015)rate-basedDenseSTDP75.93
Allred & Roy (2016)rate-basedDenseSTDP86.59
Kheradpisheh et al. (2018)rank-orderConvolutionalSTDP98.40
Saunders et al. (2018)rate-basedConvolutionalSTDP84.23
Lee et al. (2018)rate-basedConvolutionalSTDP91.1
Mozafari et al. (2019)rank-orderConvolutionalSTDP, RL97.2
Saunders et al. (2019)rate-basedLocally connectedSTDP, LC95.07
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Two", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "important observations can be made from Table 2. First, the classification accuracy has a positive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 642 + ], + "score": 1.0, + "content": "correlation with the filter size, and the number of neurons in the decoding layer. 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Parameters[k,s,nrpe,nout]nneuronsnsynapsesTest accuracySVM test accuracy
[13,3,0.025,100]170043040061.30 ±3.1487.5±1.32
[15,4,0.175,1000]188449000075.00 ±2.6883.3±1.74
[15,4,0.125,1000]188449000076.40 ±2.4383.3±1.74
[15,4,(static),100]98413000068.8±2.8783.3±1.74
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PaperEncodingArchitectureBio-plausibility criteriaAcc.
BioLCNet (proposed, RL)rate-basedLocally connected+DenseSTDP, RL, LC76.40
BioLCNet (proposed,SVM)rate-basedLocally connectedSTDP, LC87.5
Beyeler et al. (2013)rate-basedDenseSTDP91.60
Diehl& Cook (2015)rate-basedDenseSTDP95.00
Tavanaei &Maida (2015)rate-basedDenseSTDP75.93
Allred & Roy (2016)rate-basedDenseSTDP86.59
Kheradpisheh et al. (2018)rank-orderConvolutionalSTDP98.40
Saunders et al. (2018)rate-basedConvolutionalSTDP84.23
Lee et al. (2018)rate-basedConvolutionalSTDP91.1
Mozafari et al. (2019)rank-orderConvolutionalSTDP, RL97.2
Saunders et al. (2019)rate-basedLocally connectedSTDP, LC95.07
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The heat maps in Fig. 5 visualize the weights of the output layer through the training.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "score": 1.0, + "content": "The reward adaptability of an RL agent is critical because in many real-world problems the envi-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 499, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 510 + ], + "score": 1.0, + "content": "ronment is non-stationary. 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ParameterValue
Uthro-52 (mV)
Urest,Ureset-65 (mV)
go0.05 (mV)
Tg10 (ms)
△tref5(ms)
Tm20(ms)
fmax128(Hz)
hin,Win22
nout[100,500,1000]
chlc[25,50,100,250]
k[11,13, 15,17]
S[2, 3,4]
Tadapt,Tdec,Tlearn256 (ms)
(Npre,Npost)STDP(0.0001,0.01)
(npre,post)R-STDP(0.1,0.1)
21
nrpe[(static),0.075,0.125,0.175,0.25]
a0.9
Winh-100
Cnorm0.25
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PaperEncodingArchitectureBio-plausibility criteriaAcc.
BioLCNet (proposed, RL)rate-basedLocally connected+DenseSTDP, RL, LC76.40
BioLCNet (proposed,SVM)rate-basedLocally connectedSTDP, LC87.5
Beyeler et al. (2013)rate-basedDenseSTDP91.60
Diehl& Cook (2015)rate-basedDenseSTDP95.00
Tavanaei &Maida (2015)rate-basedDenseSTDP75.93
Allred & Roy (2016)rate-basedDenseSTDP86.59
Kheradpisheh et al. (2018)rank-orderConvolutionalSTDP98.40
Saunders et al. (2018)rate-basedConvolutionalSTDP84.23
Lee et al. (2018)rate-basedConvolutionalSTDP91.1
Mozafari et al. (2019)rank-orderConvolutionalSTDP, RL97.2
Saunders et al. (2019)rate-basedLocally connectedSTDP, LC95.07
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Parameters[k,s,nrpe,nout]nneuronsnsynapsesTest accuracySVM test accuracy
[13,3,0.025,100]170043040061.30 ±3.1487.5±1.32
[15,4,0.175,1000]188449000075.00 ±2.6883.3±1.74
[15,4,0.125,1000]188449000076.40 ±2.4383.3±1.74
[15,4,(static),100]98413000068.8±2.8783.3±1.74
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