diff --git a/parse/train/3k20LAiHYL2/3k20LAiHYL2.md b/parse/train/3k20LAiHYL2/3k20LAiHYL2.md new file mode 100644 index 0000000000000000000000000000000000000000..63b88409d035479259fc3fc2fff1eeb847ed5ce1 --- /dev/null +++ b/parse/train/3k20LAiHYL2/3k20LAiHYL2.md @@ -0,0 +1,337 @@ +# PRE-TRAINING TEXT-TO-TEXT TRANSFORMERS FOR CONCEPT-CENTRIC COMMON SENSE + +Wangchunshu Zhou1∗, Dong-Ho Lee2∗, Ravi Kiran Selvam2, Seyeon Lee2, Bill Yuchen $\mathbf { L i n ^ { 2 } }$ , Xiang Ren2 +1 Beihang University 2 University of Southern California +zhouwangchunshu@buaa.edu.cn, {dongho.lee, xiangren}@usc.edu + +# ABSTRACT + +Pre-trained language models (PTLM) have achieved impressive results in a range of natural language understanding (NLU) and generation (NLG) tasks. However, current pre-training objectives such as masked token prediction (for BERT-style PTLMs) and masked span infilling (for T5-style PTLMs) do not explicitly model the relational commonsense knowledge about everyday concepts, which is crucial to many downstream tasks that need common sense to understand or generate. To augment PTLMs with concept-centric commonsense knowledge, in this paper, we propose both generative and contrastive objectives for learning common sense from the text, and use them as intermediate self-supervised learning tasks for incrementally pre-training PTLMs (before task-specific fine-tuning on downstream datasets). Furthermore, we develop a joint pre-training framework to unify generative and contrastive objectives so that they can mutually reinforce each other. Extensive experimental results show that our method, concept-aware language model $( { \bf C A L M } ) ^ { 1 }$ , can pack more commonsense knowledge into the parameters of a pre-trained text-to-text transformer without relying on external knowledge graphs, yielding better performance on both NLU and NLG tasks. We show that while only incrementally pre-trained on a relatively small corpus for a few steps, CALM outperforms baseline methods by a consistent margin and even comparable with some larger PTLMs, which suggests that CALM can serve as a general, “plug-and-play” method for improving the commonsense reasoning ability of a PTLM. + +# 1 INTRODUCTION + +Pre-trained language models (PLTMs) such as BERT (Devlin et al., 2018) and T5 (Raffel et al., 2019) have revolutionized the field of NLP, yielding impressive performance on various conventional natural language understanding (NLU) and generation (NLG) tasks. BERT and its novel variants such as RoBERTa (Liu et al., 2019) and ALBERT (Lan et al., 2019) capture syntactical and semantic knowledge mainly from the pre-training task of masked language modeling, while T5-style models such as BART (Lewis et al., 2019) instead focus on masked span infilling tasks. Though yielding better performance on many downstream tasks, these pre-training objectives, however, do not explicitly guide the models to reason with concept-centric commonsense knowledge from language, including the relation and composition of daily concepts in our lives. This leaves room for equipping current PTLMs with richer commonsense reasoning ability. + +For example, consider a multi-choice question “What do you fill with ink to write notes on a piece of copy paper? (A) fountain pen (B) pencil case (C) printer (D) notepad”. The current state-of-the-art question answering model, UnifiedQA (Khashabi et al., 2020), which was fine-tuned on T5-large with multiple datasets, still predicts ‘(C) printer’ as its answer. The model may be overly sensitive to the co-occurrence between phrases in question sentence like ‘ink’ and ‘copy paper’ and the answer choice ‘printer’, but fails to reason with the concept-centric knowledge that ‘fountain pen’ is a writing instrument that needs to be filled with ‘ink’. Such mistake in commonsense reasoning becomes a bottleneck for current PTLMs (Davis & Marcus, 2015). Towards augmenting PTLMs with more knowledge, prior works mainly focus on training larger models (Brown et al., 2020), adding specific architectures to exploit external knowledge (Peters et al., 2019), or incorporating knowledge bases for pre-training (Xiong et al., 2020). In this paper, we instead look to explicitly teach pre-trained models to write and reason with common concepts through novel pre-training strategies. + +We present two kinds of self-supervised pre-training tasks: concept-to-sentence generation (C2S) and concept order recovering (COR). C2S trains the pre-trained model to compose (“write") sentences given a set of concepts, and expects the generated sentences to be fluent and plausible in terms of commonsense. COR aims to teach models to detect and revise a corrupted sentence with incorrect ordering of concepts. As illustrated in Figure 1, both tasks require a pre-trained model to recall relevant commonsense facts about the concepts and to understand the underlying commonsense relations between them. Both of the proposed objectives can explicitly encourage the model to capture the relational concept-centric commonsense knowledge and perform compositional reasoning. + +Specifically, we need a generative pre-training objective to encourage models to capture this generative commonsense reasoning ability, so that models can learn to generate sentences with commonsense knowledge for both C2S and COR. Also, to teach modes to distinguish truth sentences from less plausible ones, we need to teach models with discriminative commonsense through contrastive self-training. To unify both generative and contrastive objectives within a joint learning framework so that the model can learn both generative and discriminative commonsense knowledge at the same time, we propose to use the sentences generated by the model itself as the distractors and train the model to distinguish the generated sentences from real sentences. In this way, the model is forced to acquire new commonsense knowledge in order to distinguish the distractors generated by itself, which probably exploit the knowledge the model already possesses. Therefore, the model is trained to iteratively improve upon itself in a self-play fashion. We share all the parameters between the generator (trained with the generative objective) and the discriminator (trained with the contrastive objective), then train multiple objectives with different prefixes. Compared to previous works (Peters et al., 2019; Li et al., 2019; Xiong et al., 2020) that utilize external knowledge bases like Wikidata or ConceptNet, our approach can directly improve the generative and discriminative commonsense reasoning ability of PTLMs at the same time without relying on external knowledge bases. + +To evaluate the effectiveness of our proposed method, we apply our method in an intermediate-task transfer learning setting (Pruksachatkun et al., 2020) based on the pre-trained T5-base model to train a Concept-Aware Language Model (CALM). While only continually pre-trained on a small dataset for a relatively fewer number of updates (compared to conventional pre-training), CALM consistently outperforms T5-base on four commonsense-related NLU datasets (i.e., COMMONSENSEQA, OPENBOOKQA, PIQA, and ANLI) and COMMONGEN, a commonsense-related NLG dataset. Our results and careful ablation studies demonstrate the potential of our method to serve as a “plug-and-play” method for any pre-trained text-to-text transformer before fine-tuning on commonsense-related tasks. To the best of our knowledge, our work is the first to investigate concept-centric self-supervised objectives that improve both generative and discriminative commonsense reasoning ability of a pre-trained language model. + +# 2 SELF-SUPERVISED OBJECTIVES FOR CONCEPT-CENTRIC LEARNING + +In this section, we first describe the proposed generative and contrastive objectives used for improving the commonsense reasoning ability of pre-trained text-to-text transformers. Then, we introduce the joint learning framework which unifies the proposed self-supervised objectives and learn a unified text-to-text transformer based on pre-trained models such as T5. + +# 2.1 GENERATIVE OBJECTIVES + +Similar to many other pre-training tasks such as masked language modeling, we aim to teach models to recover original sentences from corrupted inputs, which is often regarded as a denoising process. We propose two generative self-supervised pre-training objectives: concept-to-sentence generation (C2S) and concept order recovering (COR). + +Concept Extraction. Given an input $\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \ldots , x _ { n } ]$ , we first conduct part-of-speech tagging with Spacy for the sentence and extract Verb, Noun, and Proper Nouns from the sentence to use as concepts2. Next, we form concept-sets $\mathcal { C } = [ v _ { 1 } , v _ { 2 } , \dotsc , v _ { p } , n _ { 1 } , n _ { 2 } \dotsc , n _ { q } ]$ where $v _ { i }$ and $n _ { i }$ denotes the i-th verb or noun/proper noun concept (token) in $\mathbf { x }$ . We denote $\mathcal { C } _ { v }$ and $\mathcal { C } _ { n }$ as the set of verb and noun/proper noun concepts respectively in $\mathcal { C }$ . (i.e. $\mathcal { C } _ { v } = [ v _ { 1 } , v _ { 2 } , \ldots , v _ { p } ]$ and $\mathcal { C } _ { n } = [ n _ { 1 } , n _ { 2 } , \ldots , n _ { q } ] .$ ) + +![](images/71120918b9f339ca61128329519fd7917e09ca81e0765b3859051f9b3cc76d5b.jpg) +Figure 1: Two self-supervised pre-training objectives that teach text-to-text transformers with generative common sense: (1) Concept-to-Sentence Generation (C2S) pre-trains the model to recover the original sentence with a shuffled concept set, e.g., {forward, Simpson, ignore, information, prosecutor $\} $ “The information was forwarded to Simpson’s prosecutors, but it was ignored.” (2) Concept Order Recovering (COR), similarly, teaches the model to correct the mispositioned concepts in the original sentence. For example, the concepts (stops, fights, bar, drives, performance), are randomly reordered in the input, while the model should recover the original sentence. + +Concept-to-Sentence Generation (C2S). The concept-to-sentence generation (C2S) objective requires the text-to-text transformer to recover the original sentence given only a few unordered keywords of the sentence. Specifically, given a sentence, we shuffle the extracted concept-set $\mathcal { C }$ to create the perturbed source sequence and train the model to generate the original sentence with a prefix (denoted as ${ < } \mathbf { C } { \boldsymbol { 2 } } \mathbf { s } >$ ) as described in Fig. 1. Formally, the C2S objective can be formulated as: + +$$ +L _ { c 2 s } = \mathbb { E } \Big ( \sum _ { i = 1 } ^ { n } - \log p ( x _ { i } | < c 2 \mathsf { s } > ; \mathsf { P E R M U T E } ( \mathcal { C } ) ; x _ { 1 : i - 1 } ) \Big ) +$$ + +where the PERMUTE() function randomly shuffle the concepts in the concept-set. This objective requires the model to construct an acceptable commonsense sentence by adhering to and reasoning over the commonsense relations between the given concepts. Therefore, relational commonsense knowledge is implicitly injected into the parameters of the model. The C2S objective is motivated by the task proposed in Lin et al. (2020). Compared to their work, the concept-set used in C2S covers more concepts such as named entities, while the original task only includes the concepts appearing in ConceptNet. We apply the task in a general domain and as a pre-training objective, instead of merely serving as an evaluation task. + +Concept Order Recovering (COR). As for the concept order recovering (COR) objective, we shuffle the order of concept in a sentence and train the model to recover the original sentence. As illustrated in Figure 1, given an input sentence “tree grows on the apple,”, the models would shuffle the concepts including “tree”, “grow”, and “apple” to recover the original sentence “apple grows on the tree.” The noise introduced by concept shuffling is different from that by traditional self-supervised objectives like mask language modeling and mask span prediction because the corrupted source sentences are in general complete (i.e., no tokens or spans are masked) and grammatically correct, while not acceptable in terms of commonsense because the order and relation between concepts are shuffled. By training the model to detect and correct the disorder of concepts in a sentence, the model is expected to acquire some relational commonsense knowledge like “apple generally grows on a tree” instead of “tree grows on an apple.” + +Formally, the COR objective can be formulated as: + +$$ +L _ { c o r } = \mathbb { E } \Big ( \sum _ { i = 1 } ^ { n } - \log p ( x _ { i } | < \mathrm { c o r } > ; \mathrm { C O N C E P T - P E R M U T E } ( \mathbf { x } , \mathcal { C } ) ; x _ { 1 : i - 1 } ) \Big ) , +$$ + +where ${ < } \mathsf { C O T } >$ is the prefix for the COR objective illustrated in Figure 1. The function CONCEPTPERMUTE() permutes the order between concepts in the same category (i.e. noun or verb) in the sentence, which can be formally defined as: + +$$ +\begin{array} { r } { \mathrm { \scriptscriptstyle { N C E P T - P E R M U T E } } ( \mathbf { x } , \mathcal { C } ) = [ x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } , \ldots , x _ { n } ^ { \prime } ] \mathrm { \ w h e r e \ } x _ { i } ^ { \prime } = \left\{ \begin{array} { l l } { x _ { i } } & { x _ { i } \notin \mathcal C } \\ { \mathrm { P E R M U T E } ( \mathcal { C } _ { v } ) [ j ] } & { x _ { i } = v _ { j } } \\ { \mathrm { P E R M U T E } ( \mathcal { C } _ { n } ) [ j ] } & { x _ { i } = n _ { j } } \end{array} \right. } \end{array} +$$ + +Our proposed objectives require the model to capture the relational commonsense knowledge between concepts and perform relational (COR) and compositional (C2S) commonsense reasoning in order to successfully reconstruct the original sentence. Therefore, the model is encouraged to acquire conceptcentric commonsense knowledge more effectively. In contrast, conventional pre-training objectives like masked language modeling and masked span infilling mainly focus on general token-level co-occurrence patterns and thus are less effective for learning commonsense knowledge. + +# 2.2 CONTRASTIVE OBJECTIVE + +The contrastive objective encourages the pretrained model to distinguish the real sentence from a distractor sentence: a sentence that is similar to the real sentence, generally grammatically correct, but may not follow common sense. We expect it to improve the pre-trained model’s discriminative commonsense reasoning ability so that the model’s performance on commonsensereasoning-discriminative tasks, like CommonsenseQA, can be improved. We formulate the contrastive objective as a Generative QA task: we take the concatenation of a prefix $ (question / context), the real sentence $x$ (answer), and the distractor $x ^ { \prime }$ (distractor) as the input and train the model to output the real sentence $x$ . Formally, we have the loss function of the contrastive objective defined as: + +# Generative QA + +![](images/4bcd87719b77670a28c639b496395502d69cf89639fb487bb622f6fbb338a228.jpg) +Figure 2: Overview of Contrastive self-supervised pre-training objectives. Generative QA style contrastive objective requires the model to distinguish truth sentences from less plausible ones. + +$$ +L _ { c o n t } = \mathbb { E } \big ( - \log p ( x | < \mathrm { c o n t } > ; \mathrm { P E R M U T E } ( x ; x ^ { \prime } ) ) \big ) , +$$ + +where the prefix $ is described in Figure 2. The distractor $x ^ { \prime }$ is either constructed by concept shuffling as described previously (i.e. $x ^ { \prime } =$ CONCEPT-PERMUTE $( \mathbf { x } , { \mathcal { C } } ) ,$ ) when used independently, or generated by a generator trained with the aforementioned generative objectives when used in the joint training framework, which will be described in the next section. + +# 3 JOINT TRAINING WITH GENERATIVE AND CONTRASTIVE OBJECTIVES + +The aforementioned generative and contrastive selfsupervised objectives can be applied independently or simply combined in a multi-task learning fashion. We argue that these two objectives can mutually reinforce each other: the generated sentences from the generative objective can help the contrastive module learn to distinguish commonsense sentences from less plausible ones. + +Therefore, we propose a joint training framework to unify generative objectives and contrastive objectives by using the generator to produce distractors for learning towards contrastive objective. + +Specifically, we have a generator $G _ { \theta }$ (trained with the generative objectives) and a discriminator $D _ { \phi }$ (trained with the contrastive objective). Given an input sentence $\mathbf { x }$ , we first use the method for either C2S or COR to produce the corrupted source sequence $\mathbf { x } ^ { \prime }$ . Then, we use the generator $G _ { \theta }$ trained with the corresponding objective to generate the recovered sentence $\mathbf { x } ^ { \prime \prime } = G _ { \theta } ( \bar { x } ^ { \prime } )$ . We then take $\mathbf { x } ^ { \prime \prime }$ as the distractor to train the discriminator $D _ { \phi }$ with the contrastive objective. The loss function of the proposed joint training framework consists of two parts: the first part is the loss of generative objectives, which is identical to the loss described in Eq.(1) and Eq.(2) and is used to update the generator $G _ { \theta }$ . The second part is the loss of the contrastive objective as described in Eq.(4), which can be formulated as: + +
Algorithm Aware Language Model (CALM).1: Pre-training Concept-
Input:Text-to-Text Transformer Tθ,Text corpus X=[x1,x2,...,xn]. repeat
foreach xi∈Xdo Extract the concept-set Ci; Construct the distractor sentence
x'=CONCEPT-PERMUTE(Xi,Ci); Update Tθ with Eq.(1,2, 4);
until maximum iterations reached; repeat
for each xi ∈Xdo
Update Tθwith Eq.(7)
until maximum iterationsreached;
+ +![](images/f5d2db4a0787731254436813ae341d1d74b47389aca725862fbd62b722ed5e9f.jpg) +Figure 3: Proposed Joint Training Framework. Given an input sentence $x$ (“She was the first woman to hold the position.”), we extract concept-set $\mathcal { C }$ (woman, hold, position). Given $x$ and $\mathcal { C }$ , we produce corrupted source sequence $x ^ { \prime }$ either for C2S and COR. The generator trained with the corresponding objective recovers sentences as distractors $x ^ { \prime \prime }$ to the discriminator. The discriminator is trained to distinguish truth sentences from randomly selected distractor among two objectives. Parameters between the generator and discriminator are shared. + +$$ +\begin{array} { r l } & { L _ { c o n t \_ j o i n t \_ c 2 s } = \mathbb { E } \big ( - \log \mathrm { D } _ { \phi } ( y | < \mathrm { c o n t } > ; x ; \mathbf { G } _ { \theta } ( < \mathrm { c } 2 \mathsf { s } > ; \mathsf { P E R M U T E } ( \mathcal { C } ) ) \big ) } \\ & { L _ { c o n t \_ j o i n t \_ c o r r } = \mathbb { E } \big ( - \log \mathrm { D } _ { \phi } ( y | < \mathrm { c o n t } > ; x ; \mathbf { G } _ { \theta } ( < \mathrm { c o r } > ; \mathrm { C o n c E P T } \cdot \mathrm { P E R M U T E } ( \mathbf { x } , \mathcal { C } ) ) \big ) } \end{array} +$$ + +where $L _ { c o n t \_ j o i n t \_ c 2 s }$ and $L _ { c o n t \_ j o i n t \_ c o r }$ is the contrastive loss with the distractor generated with either the C2S or the COR objective and $y$ is the original sentence. We then have the overall objective for the joint training framework defined as : + +$$ +L _ { j o i n t } = ( L _ { c 2 s } + L _ { c o r } ) + \beta ( L _ { c o n t \_ j o i n t \_ c 2 s } + L _ { c o n t \_ j o i n t \_ c o r } ) . +$$ + +$L _ { c 2 s }$ and $L _ { c o r }$ are defined in Eq.(1) and Eq.(2) respectively and $\beta$ is a hyperparameter controlling the relative weight between the generative and contrastive objectives. Note that since we would like to inject both generative and discriminative commonsense reasoning ability into the parameters of a single text-to-text transformer, we share the parameters between the generator $G _ { \theta }$ and the discriminator $D _ { \phi }$ . + +Finally, we describe the overall procedure to apply the proposed self-supervised objectives and the joint training framework on a pre-trained text-to-text transformer. We apply a two-stage training strategy. During the first stage, we apply our proposed generative and contrastive objectives individually on the model in a multi-task learning fashion with different prefixes. This provides a good starting point for the second stage where the joint training framework is applied. We summarize the workflow of our method in Algorithm 1. + +# 4 EXPERIMENTS + +In this section, motivated by the observation of Pruksachatkun et al. (2020) that tasks requiring commonsense reasoning ability generally serve as good intermediate task, we test our method in the intermediate task transfer setting. Specifically, we initialize our model with T5-base, a pre-trained text-to-text transformer model, and training the model with our proposed method as intermediate task before fine-tuning and target downstream tasks. Another reason for adopting this setting is because we expect our method to serve as a “plug-and-play” method that can be applied to any pre-trained text-to-text transformer by simply continually training for a few steps. + +Details for Pre-training and Fine-tuning CALM is continually pre-trained with our proposed self-supervised objectives as intermediate tasks based on the pre-trained T5-base model following the setting in Pruksachatkun et al. (2020). We randomly sample 500K sentences from the English Wikipedia corpus3, which is used for pre-training BERT and its variants, as the source dataset for our proposed self-supervised objectives which serve as intermediate tasks. We then fine-tune the CALM on each downstream task individually and report the average performance of three runs with different random seeds for fine-tuning on each dataset since the performance is sensitive to different random seeds. Training details and hyperparameter settings are presented in Appendix A.1 and A.2. + +Datasets We consider five commonsense benchmark datasets as target tasks. We categorize these datasets into discriminative and generative tasks. Discriminative tasks are classification tasks while generative tasks are text generation tasks. We consider four datasets for discriminative task: CommonsenseQA (Talmor et al., 2018), OpenbookQA (Mihaylov et al., 2018), PIQA (Bisk et al., 2020), aNLI (Bhagavatula et al., 2019) and one dataset for generative task: CommonGEN (Lin et al., 2020). Details on datasets are discussed in Appendix A.3. + +Compared Methods We compare our model with following models continually trained with different intermediate tasks based on the pre-trained T5-base model: (1) T5-base is the pre-trained T5-base model without continually training on any intermediate task. (2) T5-base w/ additional epochs is continually pre-trained using the original pre-training objective of T5 with additional training steps. The total number of additional training steps is equal to that of our final model. (3) T5-base $\mathbf { \Gamma } + \mathbf { S S M }$ is continual pre-trained with a variant of the salient span masking objective (Guu et al., 2020; Roberts et al., 2020) objective that masks text spans of concepts extracted with POS tagging instead of named entities extracted by a pre-trained NER model, which makes it more focused on concepts. (4) CALM(Generative-Only) is continually pre-trained with the proposed generative objectives including concept-to-sentence generation(C2S) and concept order recovering(COR) as intermediate tasks. (5) CALM(Contrastive-Only) is continually pre-trained with the proposed contrastive objective as described in section 2.2 using the distractor generated by concept shuffling. (6) CALM(Mix-only) is continually pre-trained with both the generative objectives and the contrastive objective, combined with a multi-task learning fashion with identical weights for each objective as the intermediate task. (7) CALM (w/o Mix warmup) is continually pre-trained with the joint training objective described in Eq (7) directly from the pre-trained T5-base model. (8) CALM is our main model trained as described in Algorithm 1. The difference between CALM and CALM (Joint) is that the former is initialized by the CALM(Mix). We also include the performance of the BERT-base model and two knowledge enhanced PTLMs that have similar architecture to BERT-base. + +Evaluation Metrics For discriminative tasks, we choose accuracy as our metric following other conventional question answering tasks. For generative tasks, we report automated metrics including BLEU (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), CIDEr (Vedantam et al., 2015), and SPICE (Anderson et al., 2016) following the leaderboard of COMMONGEN (Lin et al., 2020). Results for COMMONGEN are on the test set and others are on the official development set. We tune the hyperparameters based on the models’ performance on a in-house split dev set. + +# 4.1 EXPERIMENTAL RESULTS + +The result is presented in Table 1. First, we can see that our CALM model consistently and significantly (with p-value $< 0 . 0 1$ ) outperforms the backbone T5-base model on all five datasets by a margin range from 1.5 to 2.9 accuracy on discriminative tasks and $1 . 5 / 0 . 6$ BLEU/SPICE score on CommonGEN. This is an impressive result since we are only performing intermediate training on a relatively small dataset for only around $2 0 \mathrm { k }$ updates. It demonstrates the potential of our method for serving as a “plug-and-play” method for packing more commonsense knowledge into a pre-trained text-to-text transformer. Table 3 also shows that CALM performs comparably with several large-size PTLMs like BART, T5-large, and GPT-2 on the COMMONGEN dataset. The performance is worse than KG-BART (Liu et al., 2020), the current state-of-the-art on COMMONGEN, which is a contemporary work that exploits external knowledge bases as additional information, and is based on a larger backbone(i.e., BART (Lewis et al., 2019)). + +In addition, we can observe that both the proposed generative and contrastive objective outperforms the backbone T5-base model, as well as its variants that continually pre-trained with the original masked span prediction objective and the concept-specific salient span masking scheme, when applied independently. Note that we find the variant of salient span masking that focuses on concept is not + +Table 1: Experimental results on commonsense reasoning datasets. The first group of models are baselines. The models in the middle group and last group except the CALM model are trained with the proposed objectives independently and the final CALM model is trained by joint training. Best models are bold and second best ones are underlined within each metric. + +
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-base53.08(±0.16)57.60(±0.8)64.86(±0.52)61.88(±0.56)11--
ERNIE54.06(±0.12)58.90(±0.9)66.47(±0.58)63.04(±0.46)1
KnowBERT53.88(±0.15)58.50(±0.8)66.61(±0.63)63.18(±0.52)1-1-
T5-base61.88(±0.08)58.20(±1.0)68.14(±0.73)61.10(±0.38)24.9031.2012.9932.40
T5-base + cont. pretraining61.92(±0.45)58.10(±0.9)68.19(±0.77)61.15(±0.52)25.1031.0013.1232.40
T5-base + SSM62.08(±0.41)58.30(±0.8)68.27(±0.71)61.25(±0.51)25.2031.2013.2832.40
CALM (Generative-Only)62.28(±0.36)58.90(±0.4)68.91(±0.88)60.95(±0.46)25.8031.2013.8132.60
CALM (Contrastive-Only)62.73(±0.41)59.30(±0.3)70.67(±0.98)61.35(±0.06)25.5031.2013.5832.60
CALM(w/o Mix warmup)62.18(±0.48)59.00(±0.5)69.21(±0.57)61.25(±0.55)25.8031.2013.7732.60
CALM (Mix-only)63.02(±0.47)60.40(±0.4)70.07(±0.98)62.79(±0.55)26.0031.2013.8232.80
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
+ +
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-large57.06(±0.12)60.40(±0.6)67.08(±0.61)66.75(±0.61)==
T5-large69.81(±1.02)61.40(±1.0)72.19(±1.09)75.54(±1.22)28.6030.1014.9631.60
CALM-large (Mix-only)70.26(±0.23)62.50(±1.0)73.70(±1.09)75.99(±1.26)29.2031.3015.2433.10
CALM-large71.31(±0.04)66.00(±1.0)75.11(±1.65)77.12(±0.34)29.5031.9015.6133.20
RoBERTa-large471.81(±0.25)63.90(±0.8)76.90(±0.62)82.35(±0.54)111-
+ +Table 2: Experimental results on large model. Comparison between large models of other PTLMs and CALM. Best models are bold and second best ones are underlined within each metric. + +very effective. We suspect this is because the resulting training data would be somewhat similar to the original text infilling objective because concepts are very common in the corpus and we only train for a few steps. The combination of the generative and contrastive objectives (i.e., CALM(Mix-only)) yields further improvement upon the model trained independently with either generative or contrastive objectives. Also, we find that the CALM model consistently outperforms CALM(Mix), demonstrating the effectiveness of the proposed joint training framework. Applying joint training directly on top of a pre-trained model (i.e., CALM(w/o Mix warmup)) does not work very well, demonstrating the necessity of applying mixed training to initialize the model before starting joint training. + +To further confirm the effectiveness of our approach, we also apply our method to continually pre-train T5-large with the same data and number of training steps. We then compare the performance of the resulting model with that of the original T5-large model in Table 10. We find that both the proposed training objectives and the joint training framework consistently and significantly (with p-value $< 0 . 0 1$ ) improve upon the original T5-large, showing our approach is effective for models with different sizes. Our model also outperforms BERT-large by a large margin. + +
MethodsParamsCommonGEN
BLEU-4METEORCIDErSPICE
GPT-2 (Radford et al.,2019)774M21.1026.2012.1525.90
UniLM(Dong et al.,2019)340M27.7029.7014.8530.20
BART(Lewis et al.,2020)406M26.3030.9013.9230.60
T5-base (Raffel et al.,2019)220M16.4023.009.1622.00
T5-large (Raffel et al., 2019)770M28.6030.1014.9631.60
KG-BART (Liu et al., 2020)406M30.9032.4016.8332.70
T5-base (our implementation)220M24.9031.2012.9932.40
CALM-base220M26.4031.4013.8833.00
CALM-large774M29.5031.9015.6133.20
+ +Table 3: Comparison between PTLMs on CommonGEN. Above baselines are reported number in the leaderboard. T5-base(our implementation) uses different hyperparmeter setting than that reported in the leaderboard. + +However, our model performs slightly worse compared to RoBERTa-large. We suspect this is because RoBERTa-large is optimized for more steps than T5-large and our CALM-large. This is also observed in many other tasks and datasets. + +# 4.2 PERFORMANCE ANALYSIS + +Analysis on Generative objective To investigate the contribution of each generative objective, we conduct an ablation study by continually pre-training three models from the same T5-base model with C2S, COR, and text infilling, which is the original objective for pre-training T5, as the objective for the intermediate task. We continually pre-train these models for the same number of steps and then evaluate their performance by fine-tuning on different target tasks. The result is shown in Table 4. + +
MethodsCSQA PIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
T5- Text Infilling61.9268.1925.1031.0013.1332.40
CALM-COR62.3668.7725.7031.2013.6532.60
CALM-C2S62.2468.7525.9031.4013.9432.80
+ +(a) Generative objectives + +
MethodsCSQAPIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
Multi-choice QA62.2168.8225.0031.2013.2832.60
True/False62.2467.8125.1031.2013.4132.60
Generative QA62.7370.6725.5031.2013.5832.60
+ +(b) Contrastive objectives + +Table 4: Analysis on Contrastive and Generative objectives. Left table shows the performance on downstream tasks by pre-training with different generative objective (COR, C2S, and original objective for pre-training T5). Right table shows the performance on downstream tasks by pre-training with different task formats of contrastive objective. + +We can see that both C2S and COR works better than the original masked span infilling objective on itself. This confirms the effectiveness of our proposed generative objectives on improving the commonsense reasoning ability of pre-trained text-to-text transformers. + +Task Formulation of the Contrastive objectives For contrastive objectives, we test three different task formats: Multi-choice QA, Generative QA, and True/False. Multi-choice QA and Generative QA takes the concatenation of the real sentence and the distractor. Then, Multi-choice QA output the index of the real sentence following other conventional Multi-choice QA tasks, and Generative QA output the real sentence respectively. True/False takes either the real sentence or the distractor and train the model to perform a binary classification problem of whether the input sentence makes sense. The result is shown in Table 4. We could find that the format of Generative QA performs the best. We suspect this is because the Generative QA format is closer to the format used during the original pre-training stage of the T5 model and the format used for fine-tuning. + +Performance with fewer training examples To investigate the effectiveness of our objective in the lowresource setting, we explore the performance of our model and baselines fine-tuning with different fractions of the training data. From Figure 4, we can see that the performance improvement yielded by our models upon the T5-base model is more significant in the low-resource regime. This shows that CALM may already pack some + +![](images/d53949681d0c53f39d987e7afd217298a347d7f0231c014211f25f5c381ab759.jpg) +Figure 4: Performance of compared models fine-tuned with different fraction of the datasets. + +commonsense knowledge in its parameters so that it does not require much data for fine-tuning before obtaining a good performance. In contrast, the original T5 model requires much data for fine-tuning, which suggests it may fail to encode much commonsense knowledge and must fit the correlation patterns in the downstream datasets to get a good performance. + +Comparison of Generated Data Table 5 shows the comparison of generated examples for the COMMONGEN test set between T5-base and CALM. We can see that the sentences generated by CALM are generally more acceptable in terms of commonsense plausibility while T5-base sometimes generates sentences that do not make sense. + +
Concept-setT5-baseCALM-base
Grass,Dog,Ball, Chasea dog is chased by a ball on the grass.dog chasing a ball in the grass.
Net, Cast,Boat,Waterfishing boat casts a net in the water.fisherman casts a net into the water from a fishing boat.
Hole,Tree,Plant, Diga man digs a hole in a tree to plant a new tree . he digs theman digging a hole to plant a tree.
Ingredient, Add,Pan, rya pan filled with ingredients adds a touch of spice to the fry .add the ingredients to a pan and fry.
Water,Hold,Hand,WalkA man holding a hand and walking in the water. A man is holding water.man holding a bottle of water in his hand as he walks down the street.
Place, Use,Metal toolA man uses a metal tool to make a piece of metal.woman uses a metal tool to make a piece of jewelry.
+ +# Table 5: Comparison of generated sentences with same concept-set. + +Knowledge Probing To investigate how much concept-centric knowledge our model pack, we conducted two probing methods with our model $:$ Language Model Analysis (LAMA) probe (Petroni et al., 2019), Knowledge Intensive Language Task (KILT) (Petroni et al., 2020). We summarize the results on Table 6 and Appendix A.4. We could find that our model outperforms the baseline. + +# 5 RELATED WORK + +Self-Supervised Language Representation Pre-Training. Motivated by the fact that words can have different meanings in different contexts, contextual language representation methods (McCann et al., 2017; Peters et al., 2018) have been developed and shown superior performance on downstream tasks compared with static word embeddings Mikolov et al. (2013); Pennington et al. (2014). More recently, large scale language models based on transformer architecture (Vaswani et al., 2017) pretrained with either mask language modeling objective (Devlin et al., 2018; Liu et al., 2019; Lan et al., 2019) or mask span infilling objective (Lewis et al., 2019; Raffel et al., 2019) have been explored further advanced the state-of-the-art on multiple NLU and NLG tasks. Our method is based on these techniques and we focus on improving the commonsense reasoning ability of pre-trained text-to-text transformers. More recently, Clark et al. (2020) propose a new self-supervised pre-training objective called Replaced Token Detection (RTD). RTD uses a mask language model like BERT to fill in the mask and train a discriminator to predict whether a token is generated or real. This pre-training paradigm is related to our proposed joint training framework. Some major differences include that (1) Our method employs sentence-level distractors that are in general grammatically correct but not in line with commonsense, thus require the model to perform relational commonsense reasoning while RTD is a token-level discrimination task and can often be solved with syntactic and shallow semantic knowledge (Rosset et al., 2020); (2) Our method unifies generative and contrastive objectives with one model, which can be applied to both NLU and NLG downstream tasks; and (3) The discriminator in our framework is “contrastive”, takes both the real sentence and the distractor as input simultaneously. + +Table 6: Experimental results on Knowledge Probing. Left table shows the mean precision on LAMA probing task of ConceptNET. Right table shows the performance on Fact checking and Entity linking, which are from KILT task. + +
MethodsMRRPrecision@50Precision @10Precision@1
T5-Base11.5338.5221.605.93
CALM (Mix-only)11.7738.9321.926.10
CALM12.0939.6922.536.46
+ +
MethodsFEVERAY2
T5-base76.6574.97
CALM (Mix-only)77.0576.27
CALM77.4477.24
+ +(a) LAMA probe (b) KILT task + +Knowledge-augmented PTLMs. As standard pre-trained language models usually do not explicitly model knowledge, a number of works have examined the problem of incorporating world knowledge with the PTLMs. Recent work Zhang et al. (2019); Peters et al. (2019); Wang et al. (2020); Liu et al. (2020) utilizes an external knowledge base to incorporate entity knowledge with PTLMs; however, these approaches require specialized resources like knowledge bases, which limits the domain they can be applied to. Xiong et al. (2020) proposes WikiLM that encodes world knowledge into the parameters of a BERT(Devlin et al., 2018)-like pre-trained model with a novel entity replacement detection objective that incorporates Wikipedia to form distractors. Their approach differs from ours because it requires an external knowledge base (i.e., Wikipedia) which limits the domain it can be applied, is limited to discriminative pre-training objectives and downstream tasks, and focuses on world knowledge instead of relational commonsense knowledge. More recently, (Rosset et al., 2020) propose KALM, an entity-aware language model with more world knowledge packed into its parameters. Their method is restricted to the training of language models instead of masked language models or text-to-text transformers which can be used for more downstream tasks. Also, all the aforementioned work mainly focuses on world knowledge of named entities. In contrast, our work mainly focuses on commonsense knowledge about quotidian concepts. + +# 6 CONCLUSION + +We propose novel self-supervised strategies that encourage the model to focus on concept-centric information that is related to commonsense understanding and reasoning instead of simple word coocurrence patterns so that the commonsense learning capability of pre-trained text-to-text transformers can be improved. Despite merely continually pre-trained on a small dataset with only around $2 0 \mathrm { k }$ steps, our CALM model consistently outperforms the T5-base model on all commonsense-related datasets, and even yields better performance compared with some larger size PTLMs on the COMMONGEN dataset. The performance gain is larger when we use fewer examples for fine-tuning on different downstream tasks, indicating that CALM effectively encodes more commonsense knowledge and rely less on fitting superficial patterns of datasets compared to traditional pre-trained language models. Our work suggests that text-to-text models can be pre-trained with better parameter and sample efficiency by carefully designed self-supervised objectives that focus more on the ability (e.g., commonsense reasoning ability) required by target tasks. + +# ACKNOWLEDGEMENT + +This research is supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via Contract No. 2019-19051600007, the DARPA MCS program under Contract No. N660011924033 with the United States Office Of Naval Research, the Defense Advanced Research Projects Agency with award W911NF-19-20271, and NSF SMA 18-29268. 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URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ BJlzm64tDH. + +Zhengyan Zhang, Xu Han, Zhiyuan Liu, Xin Jiang, Maosong Sun, and Qun Liu. Ernie: Enhanced language representation with informative entities. arXiv preprint arXiv:1905.07129, 2019. + +# A APPENDIX + +# A.1 PRE-TRAINING DETAILS + +The following details apply to both base architecture and joint-training architecture. We implement our pre-train models using Pytorch-lightning (Falcon, 2019) and Hugginface’s Pytorch Transformers (Wolf et al., 2019). For pre-training phase, we use the Adam optimizer with maximum sequence length 256, train batch size 8, gradient accumulation 8, warmup steps 10000, weight decay 0.01 and adam epsilon 1e-6. We train the models with 8 V100 GPUs and FP32 precision for 17 hours. The model is pre-trained for at most 3 epochs to prevent overfitting. We searched for the best learning rate for our model out of [1e-4, 2e-5, 2e-6, 5e-7]. + +# A.2 FINE-TUNING DETAILS + +For fine-tuning, we use 4 V100 GPUs and use FP32. For all discriminative tasks, we use the Adam optimizer with maximum sequence length 256, batch size 4 and gradient accumulation 16. For generative task, we use the Adam optimizer with maximum source length 32, maximum target length 32, batch size 8, gradient accumulation 16. For all tasks, we use warmup fraction 0.01. Learning rates and train epochs are listed in Table 7. + +Table 7: Fine-tuning hyperparameters. + +
HyperparameterCommonsenseQAOpenbookQAPIQAaNLICommonGEN
Learning rate[1e-4,2e-4,3e-4][5e-5,1e-4,2e-4,3e-4][1e-4,2e-4,3e-4][2e-5,3e-5][2e-5]
Train Epochs2020201020
+ +# A.3 DATASET PROPERTIES + +• CommonsenseQA (Talmor et al., 2018) is a multiple-choice question answering task, which picks the most appropriate answer on general commonsense questions. + +• OpenbookQA (Mihaylov et al., 2018) is a multiple-choice question answering task, which is modeled after open book exams on elementary-level core science questions. The task requires open book fact and additional commonsense which is not contained in the book. To test the commonsense reasoning ability, we do not use open book fact. + +• PIQA (Bisk et al., 2020) is multiple-choice question answering task, which chooses the most appropriate solution for physical commonsense questions. + +• aNLI (Bhagavatula et al., 2019) is a binary-classification task, which picks the most plausible explanatory hypothesis given two observations from narrative contexts. + +• CommonGEN (Lin et al., 2020) is a constrained text generation task, which generates a coherent sentence describing an everyday scenario using common concepts. + +Table 8: Properties of Commonsense benchmark datasets. + +
DatasetTrainDevelopmentTestSource ExampleTarget Example
CommonsenseQA9,7411,2211,140context:What homeentertainment equipment requirescable? options: 1: radio shack 2: substation 3:cabinet 4: television 5:desk4
OpenbookQA4,957500500context: You can make a telescope with options: 1: straw 2: glass 3: candle 4: mailing tube2
PIQA16,1131,8383.084context:When boiling butter,when it's ready,you can options:1: Pour it onto a plate 2:Pour it into a jar2
aNLI169,6541,5323.040context: It was my birthday.WhenI got home the party was set up for my brother. options: 1: Iwas so excited. 2: Iwas so mad.2
CommonGEN67,3894.0186.042generate a sentence with these concepts: Apple Grow TreeApple grows on the tree
+ +# A.4 KNOWLEDGE PROBING + +LAMA probe is consisting of a set of knowledge sources, each comprised of a set of fact. It defines that a pre-trained language model knows a fact (subject, relation, object) such as (Bird, CapableOf, Fly) if it can predict masked objects in cloze statement such as "Birds can [MASK]". For evaluation, we first filtered out examples that mask label is not in vocabulary list of T5. Then, we evaluate the model based on how highly it ranks the ground truth token against every other word in a fixed vocabulary list of T5, and get mean precision at k to check whether the object is ranked among the top k results. We summarize the results of ConceptNet (Speer & Havasi, 2012) in Table 6. Unlike other language models which are optimised to masked word anywhere in a given sequence, T5 is trained with different denoising method. It might cause low performance on such slot filling task, but compared to T5, our model shows better performance compared to base model. + +KILT task is a benchmark for assessing models that need to access specific knowledge in a defined snapshot of Wikipedia to solve tasks spanning five domains. The goal is to analyze the model whether it has task-agnostic representations of knowledge. We test our model on domain of fact checking, entity linking. Fact checking verifies textual claims against textual sources. For this task, we use FEVER (Thorne et al., 2018) which is a large dataset for claim veracity that requires evidence from multiple Wikipedia pages to determine whether the claim is supported or refuted. Entity Linking assigns Wikipedia page to entities mentioned in text. We use AIDA CoNLL-YAGO (AY2) (Hoffart et al., 2011) which supplements the CoNLL 2003 (Tjong Kim Sang & De Meulder, 2003) with Wikipedia URL annotations for all entities. + +# A.5 EXPERIMENTS WITH BART AS BACKBONE + +To show that our approach is versatile to different pre-trained models, we conduct experiments with BART as the backbone model. We can see that our approach consistently and significantly (with p-value $< 0 . 0 1$ ) improves BART-base on all datasets. This result shows that our method is versatile to different pre-trained models. + +Table 9: Experimental results with BART as backbone model. Best models are bold. + +
MethodsCSQAOBQAPIQAaNLICommonGEN
AccuracyBLEU-4METEORCIDErSPICE
BART-base (Mix-only)56.31(±0.28)58.30(±1.1)(official dev) 67.53(±1.01)59.85(±1.14)25.1029.5013.1630.20
CALM (BART-base)58.22(±0.21)59.10(±1.0)69.40(±1.23)61.28(±0.30)26.4029.9013.7131.10
+ +# A.6 EXPERIMENTS WITH NOUN/VERB AS CONCEPTS + +We also conducted an ablation study about the choice of using either nouns or verbs as concepts. We can see that using either nouns-only or verbs-only as concepts for our approach leads to substantial performance drop. This supports our choice about using both nouns and verbs as concepts. + +
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
CALM-nouns62.45(±0.42)59.40(±0.5)69.05(±0.70)61.55(±0.58)25.7031.2013.1732.60
CALM-verbs62.51(±0.47)59.10(±0.7)69.24(±0.65)61.40(±0.51)25.6031.2013.2432.60
+ +# Table 10: Experimental results with Noun/Verb as Concepts. Best models are bold. + +# A.7 HUMAN EVALUATION ON COMMONGEN GENERATIONS + +We conducted a human evaluation of CommonGEN predictions between T5 and CALM. We asked three annotators to choose the most reasonable sentence between T5-base and CALM-base predictions. The evaluation was conducted on 50 test sentences in binary selection by majority voting. Cohen’s Kappa score, which is a measurement of inter-annotator agreement, was 0.73. Annotators say that for $60 \%$ of test sentences, CALM-base generated better. \ No newline at end of file diff --git a/parse/train/3k20LAiHYL2/3k20LAiHYL2_content_list.json b/parse/train/3k20LAiHYL2/3k20LAiHYL2_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..29a3b39f0010dc5ea72ecac429e1fe1f8bae703f --- /dev/null +++ b/parse/train/3k20LAiHYL2/3k20LAiHYL2_content_list.json @@ -0,0 +1,1828 @@ +[ + { + "type": "text", + "text": "PRE-TRAINING TEXT-TO-TEXT TRANSFORMERS FOR CONCEPT-CENTRIC COMMON SENSE ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Wangchunshu Zhou1∗, Dong-Ho Lee2∗, Ravi Kiran Selvam2, Seyeon Lee2, Bill Yuchen $\\mathbf { L i n ^ { 2 } }$ , Xiang Ren2 \n1 Beihang University 2 University of Southern California \nzhouwangchunshu@buaa.edu.cn, {dongho.lee, xiangren}@usc.edu ", + "bbox": [ + 184, + 167, + 697, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 263, + 544, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Pre-trained language models (PTLM) have achieved impressive results in a range of natural language understanding (NLU) and generation (NLG) tasks. However, current pre-training objectives such as masked token prediction (for BERT-style PTLMs) and masked span infilling (for T5-style PTLMs) do not explicitly model the relational commonsense knowledge about everyday concepts, which is crucial to many downstream tasks that need common sense to understand or generate. To augment PTLMs with concept-centric commonsense knowledge, in this paper, we propose both generative and contrastive objectives for learning common sense from the text, and use them as intermediate self-supervised learning tasks for incrementally pre-training PTLMs (before task-specific fine-tuning on downstream datasets). Furthermore, we develop a joint pre-training framework to unify generative and contrastive objectives so that they can mutually reinforce each other. Extensive experimental results show that our method, concept-aware language model $( { \\bf C A L M } ) ^ { 1 }$ , can pack more commonsense knowledge into the parameters of a pre-trained text-to-text transformer without relying on external knowledge graphs, yielding better performance on both NLU and NLG tasks. We show that while only incrementally pre-trained on a relatively small corpus for a few steps, CALM outperforms baseline methods by a consistent margin and even comparable with some larger PTLMs, which suggests that CALM can serve as a general, “plug-and-play” method for improving the commonsense reasoning ability of a PTLM. ", + "bbox": [ + 233, + 295, + 766, + 573 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 599, + 336, + 614 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Pre-trained language models (PLTMs) such as BERT (Devlin et al., 2018) and T5 (Raffel et al., 2019) have revolutionized the field of NLP, yielding impressive performance on various conventional natural language understanding (NLU) and generation (NLG) tasks. BERT and its novel variants such as RoBERTa (Liu et al., 2019) and ALBERT (Lan et al., 2019) capture syntactical and semantic knowledge mainly from the pre-training task of masked language modeling, while T5-style models such as BART (Lewis et al., 2019) instead focus on masked span infilling tasks. Though yielding better performance on many downstream tasks, these pre-training objectives, however, do not explicitly guide the models to reason with concept-centric commonsense knowledge from language, including the relation and composition of daily concepts in our lives. This leaves room for equipping current PTLMs with richer commonsense reasoning ability. ", + "bbox": [ + 174, + 623, + 825, + 762 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For example, consider a multi-choice question “What do you fill with ink to write notes on a piece of copy paper? (A) fountain pen (B) pencil case (C) printer (D) notepad”. The current state-of-the-art question answering model, UnifiedQA (Khashabi et al., 2020), which was fine-tuned on T5-large with multiple datasets, still predicts ‘(C) printer’ as its answer. The model may be overly sensitive to the co-occurrence between phrases in question sentence like ‘ink’ and ‘copy paper’ and the answer choice ‘printer’, but fails to reason with the concept-centric knowledge that ‘fountain pen’ is a writing instrument that needs to be filled with ‘ink’. Such mistake in commonsense reasoning becomes a bottleneck for current PTLMs (Davis & Marcus, 2015). Towards augmenting PTLMs with more knowledge, prior works mainly focus on training larger models (Brown et al., 2020), adding specific architectures to exploit external knowledge (Peters et al., 2019), or incorporating knowledge bases for pre-training (Xiong et al., 2020). In this paper, we instead look to explicitly teach pre-trained models to write and reason with common concepts through novel pre-training strategies. ", + "bbox": [ + 174, + 770, + 825, + 881 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We present two kinds of self-supervised pre-training tasks: concept-to-sentence generation (C2S) and concept order recovering (COR). C2S trains the pre-trained model to compose (“write\") sentences given a set of concepts, and expects the generated sentences to be fluent and plausible in terms of commonsense. COR aims to teach models to detect and revise a corrupted sentence with incorrect ordering of concepts. As illustrated in Figure 1, both tasks require a pre-trained model to recall relevant commonsense facts about the concepts and to understand the underlying commonsense relations between them. Both of the proposed objectives can explicitly encourage the model to capture the relational concept-centric commonsense knowledge and perform compositional reasoning. ", + "bbox": [ + 174, + 166, + 825, + 279 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Specifically, we need a generative pre-training objective to encourage models to capture this generative commonsense reasoning ability, so that models can learn to generate sentences with commonsense knowledge for both C2S and COR. Also, to teach modes to distinguish truth sentences from less plausible ones, we need to teach models with discriminative commonsense through contrastive self-training. To unify both generative and contrastive objectives within a joint learning framework so that the model can learn both generative and discriminative commonsense knowledge at the same time, we propose to use the sentences generated by the model itself as the distractors and train the model to distinguish the generated sentences from real sentences. In this way, the model is forced to acquire new commonsense knowledge in order to distinguish the distractors generated by itself, which probably exploit the knowledge the model already possesses. Therefore, the model is trained to iteratively improve upon itself in a self-play fashion. We share all the parameters between the generator (trained with the generative objective) and the discriminator (trained with the contrastive objective), then train multiple objectives with different prefixes. Compared to previous works (Peters et al., 2019; Li et al., 2019; Xiong et al., 2020) that utilize external knowledge bases like Wikidata or ConceptNet, our approach can directly improve the generative and discriminative commonsense reasoning ability of PTLMs at the same time without relying on external knowledge bases. ", + "bbox": [ + 174, + 285, + 825, + 506 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To evaluate the effectiveness of our proposed method, we apply our method in an intermediate-task transfer learning setting (Pruksachatkun et al., 2020) based on the pre-trained T5-base model to train a Concept-Aware Language Model (CALM). While only continually pre-trained on a small dataset for a relatively fewer number of updates (compared to conventional pre-training), CALM consistently outperforms T5-base on four commonsense-related NLU datasets (i.e., COMMONSENSEQA, OPENBOOKQA, PIQA, and ANLI) and COMMONGEN, a commonsense-related NLG dataset. Our results and careful ablation studies demonstrate the potential of our method to serve as a “plug-and-play” method for any pre-trained text-to-text transformer before fine-tuning on commonsense-related tasks. To the best of our knowledge, our work is the first to investigate concept-centric self-supervised objectives that improve both generative and discriminative commonsense reasoning ability of a pre-trained language model. ", + "bbox": [ + 174, + 513, + 825, + 666 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 SELF-SUPERVISED OBJECTIVES FOR CONCEPT-CENTRIC LEARNING ", + "text_level": 1, + "bbox": [ + 173, + 698, + 766, + 713 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we first describe the proposed generative and contrastive objectives used for improving the commonsense reasoning ability of pre-trained text-to-text transformers. Then, we introduce the joint learning framework which unifies the proposed self-supervised objectives and learn a unified text-to-text transformer based on pre-trained models such as T5. ", + "bbox": [ + 174, + 729, + 823, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GENERATIVE OBJECTIVES ", + "text_level": 1, + "bbox": [ + 176, + 809, + 398, + 824 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Similar to many other pre-training tasks such as masked language modeling, we aim to teach models to recover original sentences from corrupted inputs, which is often regarded as a denoising process. We propose two generative self-supervised pre-training objectives: concept-to-sentence generation (C2S) and concept order recovering (COR). ", + "bbox": [ + 174, + 833, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Concept Extraction. Given an input $\\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } ]$ , we first conduct part-of-speech tagging with Spacy for the sentence and extract Verb, Noun, and Proper Nouns from the sentence to use as concepts2. Next, we form concept-sets $\\mathcal { C } = [ v _ { 1 } , v _ { 2 } , \\dotsc , v _ { p } , n _ { 1 } , n _ { 2 } \\dotsc , n _ { q } ]$ where $v _ { i }$ and $n _ { i }$ denotes the i-th verb or noun/proper noun concept (token) in $\\mathbf { x }$ . We denote $\\mathcal { C } _ { v }$ and $\\mathcal { C } _ { n }$ as the set of verb and noun/proper noun concepts respectively in $\\mathcal { C }$ . (i.e. $\\mathcal { C } _ { v } = [ v _ { 1 } , v _ { 2 } , \\ldots , v _ { p } ]$ and $\\mathcal { C } _ { n } = [ n _ { 1 } , n _ { 2 } , \\ldots , n _ { q } ] .$ ) ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/71120918b9f339ca61128329519fd7917e09ca81e0765b3859051f9b3cc76d5b.jpg", + "image_caption": [ + "Figure 1: Two self-supervised pre-training objectives that teach text-to-text transformers with generative common sense: (1) Concept-to-Sentence Generation (C2S) pre-trains the model to recover the original sentence with a shuffled concept set, e.g., {forward, Simpson, ignore, information, prosecutor $\\} $ “The information was forwarded to Simpson’s prosecutors, but it was ignored.” (2) Concept Order Recovering (COR), similarly, teaches the model to correct the mispositioned concepts in the original sentence. For example, the concepts (stops, fights, bar, drives, performance), are randomly reordered in the input, while the model should recover the original sentence. " + ], + "image_footnote": [], + "bbox": [ + 181, + 85, + 818, + 261 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 393, + 826, + 436 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Concept-to-Sentence Generation (C2S). The concept-to-sentence generation (C2S) objective requires the text-to-text transformer to recover the original sentence given only a few unordered keywords of the sentence. Specifically, given a sentence, we shuffle the extracted concept-set $\\mathcal { C }$ to create the perturbed source sequence and train the model to generate the original sentence with a prefix (denoted as ${ < } \\mathbf { C } { \\boldsymbol { 2 } } \\mathbf { s } >$ ) as described in Fig. 1. Formally, the C2S objective can be formulated as: ", + "bbox": [ + 174, + 443, + 825, + 513 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1112622d2622f70eade71ee380d1126109f9230f51f251b1db06b6a4e2ddb39f.jpg", + "text": "$$\nL _ { c 2 s } = \\mathbb { E } \\Big ( \\sum _ { i = 1 } ^ { n } - \\log p ( x _ { i } | < c 2 \\mathsf { s } > ; \\mathsf { P E R M U T E } ( \\mathcal { C } ) ; x _ { 1 : i - 1 } ) \\Big )\n$$", + "text_format": "latex", + "bbox": [ + 303, + 513, + 694, + 555 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the PERMUTE() function randomly shuffle the concepts in the concept-set. This objective requires the model to construct an acceptable commonsense sentence by adhering to and reasoning over the commonsense relations between the given concepts. Therefore, relational commonsense knowledge is implicitly injected into the parameters of the model. The C2S objective is motivated by the task proposed in Lin et al. (2020). Compared to their work, the concept-set used in C2S covers more concepts such as named entities, while the original task only includes the concepts appearing in ConceptNet. We apply the task in a general domain and as a pre-training objective, instead of merely serving as an evaluation task. ", + "bbox": [ + 173, + 555, + 825, + 666 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Concept Order Recovering (COR). As for the concept order recovering (COR) objective, we shuffle the order of concept in a sentence and train the model to recover the original sentence. As illustrated in Figure 1, given an input sentence “tree grows on the apple,”, the models would shuffle the concepts including “tree”, “grow”, and “apple” to recover the original sentence “apple grows on the tree.” The noise introduced by concept shuffling is different from that by traditional self-supervised objectives like mask language modeling and mask span prediction because the corrupted source sentences are in general complete (i.e., no tokens or spans are masked) and grammatically correct, while not acceptable in terms of commonsense because the order and relation between concepts are shuffled. By training the model to detect and correct the disorder of concepts in a sentence, the model is expected to acquire some relational commonsense knowledge like “apple generally grows on a tree” instead of “tree grows on an apple.” ", + "bbox": [ + 173, + 672, + 826, + 827 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Formally, the COR objective can be formulated as: ", + "bbox": [ + 174, + 833, + 508, + 848 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7dd4a0f8a8cfad0cb5270bf23de7a795b48115782b4a21fa664d041f275a67eb.jpg", + "text": "$$\nL _ { c o r } = \\mathbb { E } \\Big ( \\sum _ { i = 1 } ^ { n } - \\log p ( x _ { i } | < \\mathrm { c o r } > ; \\mathrm { C O N C E P T - P E R M U T E } ( \\mathbf { x } , \\mathcal { C } ) ; x _ { 1 : i - 1 } ) \\Big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 258, + 849, + 738, + 890 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where ${ < } \\mathsf { C O T } >$ is the prefix for the COR objective illustrated in Figure 1. The function CONCEPTPERMUTE() permutes the order between concepts in the same category (i.e. noun or verb) in the sentence, which can be formally defined as: ", + "bbox": [ + 174, + 103, + 828, + 146 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/670bdf95e62afb22f4a4a1733a3e6d7cbbda230c302ea47ec193f4373fc48869.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { \\scriptscriptstyle { N C E P T - P E R M U T E } } ( \\mathbf { x } , \\mathcal { C } ) = [ x _ { 1 } ^ { \\prime } , x _ { 2 } ^ { \\prime } , \\ldots , x _ { n } ^ { \\prime } ] \\mathrm { \\ w h e r e \\ } x _ { i } ^ { \\prime } = \\left\\{ \\begin{array} { l l } { x _ { i } } & { x _ { i } \\notin \\mathcal C } \\\\ { \\mathrm { P E R M U T E } ( \\mathcal { C } _ { v } ) [ j ] } & { x _ { i } = v _ { j } } \\\\ { \\mathrm { P E R M U T E } ( \\mathcal { C } _ { n } ) [ j ] } & { x _ { i } = n _ { j } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 154, + 785, + 207 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our proposed objectives require the model to capture the relational commonsense knowledge between concepts and perform relational (COR) and compositional (C2S) commonsense reasoning in order to successfully reconstruct the original sentence. Therefore, the model is encouraged to acquire conceptcentric commonsense knowledge more effectively. In contrast, conventional pre-training objectives like masked language modeling and masked span infilling mainly focus on general token-level co-occurrence patterns and thus are less effective for learning commonsense knowledge. ", + "bbox": [ + 174, + 220, + 826, + 305 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 CONTRASTIVE OBJECTIVE ", + "text_level": 1, + "bbox": [ + 174, + 315, + 400, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The contrastive objective encourages the pretrained model to distinguish the real sentence from a distractor sentence: a sentence that is similar to the real sentence, generally grammatically correct, but may not follow common sense. We expect it to improve the pre-trained model’s discriminative commonsense reasoning ability so that the model’s performance on commonsensereasoning-discriminative tasks, like CommonsenseQA, can be improved. We formulate the contrastive objective as a Generative QA task: we take the concatenation of a prefix $ (question / context), the real sentence $x$ (answer), and the distractor $x ^ { \\prime }$ (distractor) as the input and train the model to output the real sentence $x$ . Formally, we have the loss function of the contrastive objective defined as: ", + "bbox": [ + 174, + 334, + 465, + 597 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Generative QA ", + "text_level": 1, + "bbox": [ + 596, + 335, + 714, + 351 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4bcd87719b77670a28c639b496395502d69cf89639fb487bb622f6fbb338a228.jpg", + "image_caption": [ + "Figure 2: Overview of Contrastive self-supervised pre-training objectives. Generative QA style contrastive objective requires the model to distinguish truth sentences from less plausible ones. " + ], + "image_footnote": [], + "bbox": [ + 495, + 359, + 805, + 506 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ba38e2fe9b57a46a030f46b174e007526bfab9e72343dee93b5a2c7b6ee8196b.jpg", + "text": "$$\nL _ { c o n t } = \\mathbb { E } \\big ( - \\log p ( x | < \\mathrm { c o n t } > ; \\mathrm { P E R M U T E } ( x ; x ^ { \\prime } ) ) \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 325, + 595, + 671, + 616 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the prefix $ is described in Figure 2. The distractor $x ^ { \\prime }$ is either constructed by concept shuffling as described previously (i.e. $x ^ { \\prime } =$ CONCEPT-PERMUTE $( \\mathbf { x } , { \\mathcal { C } } ) ,$ ) when used independently, or generated by a generator trained with the aforementioned generative objectives when used in the joint training framework, which will be described in the next section. ", + "bbox": [ + 174, + 619, + 825, + 676 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 JOINT TRAINING WITH GENERATIVE AND CONTRASTIVE OBJECTIVES ", + "text_level": 1, + "bbox": [ + 171, + 689, + 789, + 707 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The aforementioned generative and contrastive selfsupervised objectives can be applied independently or simply combined in a multi-task learning fashion. We argue that these two objectives can mutually reinforce each other: the generated sentences from the generative objective can help the contrastive module learn to distinguish commonsense sentences from less plausible ones. ", + "bbox": [ + 174, + 714, + 517, + 825 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Therefore, we propose a joint training framework to unify generative objectives and contrastive objectives by using the generator to produce distractors for learning towards contrastive objective. ", + "bbox": [ + 173, + 833, + 517, + 888 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Specifically, we have a generator $G _ { \\theta }$ (trained with the generative objectives) and a discriminator $D _ { \\phi }$ (trained with the contrastive objective). Given an input sentence $\\mathbf { x }$ , we first use the method for either C2S or COR to produce the corrupted source sequence $\\mathbf { x } ^ { \\prime }$ . Then, we use the generator $G _ { \\theta }$ trained with the corresponding objective to generate the recovered sentence $\\mathbf { x } ^ { \\prime \\prime } = G _ { \\theta } ( \\bar { x } ^ { \\prime } )$ . We then take $\\mathbf { x } ^ { \\prime \\prime }$ as the distractor to train the discriminator $D _ { \\phi }$ with the contrastive objective. The loss function of the proposed joint training framework consists of two parts: the first part is the loss of generative objectives, which is identical to the loss described in Eq.(1) and Eq.(2) and is used to update the generator $G _ { \\theta }$ . The second part is the loss of the contrastive objective as described in Eq.(4), which can be formulated as: ", + "bbox": [ + 173, + 896, + 516, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/c37b1c3310815c04035b080f4fd359a2c163fdc8dbb947b44517dc90c0a736d8.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm Aware Language Model (CALM).1: Pre-training Concept-
Input:Text-to-Text Transformer Tθ,Text corpus X=[x1,x2,...,xn]. repeat
foreach xi∈Xdo Extract the concept-set Ci; Construct the distractor sentence
x'=CONCEPT-PERMUTE(Xi,Ci); Update Tθ with Eq.(1,2, 4);
until maximum iterations reached; repeat
for each xi ∈Xdo
Update Tθwith Eq.(7)
until maximum iterationsreached;
", + "bbox": [ + 529, + 717, + 823, + 917 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/f5d2db4a0787731254436813ae341d1d74b47389aca725862fbd62b722ed5e9f.jpg", + "image_caption": [ + "Figure 3: Proposed Joint Training Framework. Given an input sentence $x$ (“She was the first woman to hold the position.”), we extract concept-set $\\mathcal { C }$ (woman, hold, position). Given $x$ and $\\mathcal { C }$ , we produce corrupted source sequence $x ^ { \\prime }$ either for C2S and COR. The generator trained with the corresponding objective recovers sentences as distractors $x ^ { \\prime \\prime }$ to the discriminator. The discriminator is trained to distinguish truth sentences from randomly selected distractor among two objectives. Parameters between the generator and discriminator are shared. " + ], + "image_footnote": [], + "bbox": [ + 176, + 94, + 818, + 222 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 332, + 825, + 444 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1d84e1cf0c5e1acd245645b5b393afeb107f43679d05242008d90ce19fa32c1f.jpg", + "text": "$$\n\\begin{array} { r l } & { L _ { c o n t \\_ j o i n t \\_ c 2 s } = \\mathbb { E } \\big ( - \\log \\mathrm { D } _ { \\phi } ( y | < \\mathrm { c o n t } > ; x ; \\mathbf { G } _ { \\theta } ( < \\mathrm { c } 2 \\mathsf { s } > ; \\mathsf { P E R M U T E } ( \\mathcal { C } ) ) \\big ) } \\\\ & { L _ { c o n t \\_ j o i n t \\_ c o r r } = \\mathbb { E } \\big ( - \\log \\mathrm { D } _ { \\phi } ( y | < \\mathrm { c o n t } > ; x ; \\mathbf { G } _ { \\theta } ( < \\mathrm { c o r } > ; \\mathrm { C o n c E P T } \\cdot \\mathrm { P E R M U T E } ( \\mathbf { x } , \\mathcal { C } ) ) \\big ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 448, + 781, + 503 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $L _ { c o n t \\_ j o i n t \\_ c 2 s }$ and $L _ { c o n t \\_ j o i n t \\_ c o r }$ is the contrastive loss with the distractor generated with either the C2S or the COR objective and $y$ is the original sentence. We then have the overall objective for the joint training framework defined as : ", + "bbox": [ + 176, + 511, + 823, + 553 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/710d503fa012702f933e0a1eb28b7f2e23d7278fbe31b3c7dcdf3048975267af.jpg", + "text": "$$\nL _ { j o i n t } = ( L _ { c 2 s } + L _ { c o r } ) + \\beta ( L _ { c o n t \\_ j o i n t \\_ c 2 s } + L _ { c o n t \\_ j o i n t \\_ c o r } ) .\n$$", + "text_format": "latex", + "bbox": [ + 287, + 560, + 710, + 578 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$L _ { c 2 s }$ and $L _ { c o r }$ are defined in Eq.(1) and Eq.(2) respectively and $\\beta$ is a hyperparameter controlling the relative weight between the generative and contrastive objectives. Note that since we would like to inject both generative and discriminative commonsense reasoning ability into the parameters of a single text-to-text transformer, we share the parameters between the generator $G _ { \\theta }$ and the discriminator $D _ { \\phi }$ . ", + "bbox": [ + 174, + 584, + 825, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, we describe the overall procedure to apply the proposed self-supervised objectives and the joint training framework on a pre-trained text-to-text transformer. We apply a two-stage training strategy. During the first stage, we apply our proposed generative and contrastive objectives individually on the model in a multi-task learning fashion with different prefixes. This provides a good starting point for the second stage where the joint training framework is applied. We summarize the workflow of our method in Algorithm 1. ", + "bbox": [ + 173, + 661, + 825, + 746 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 766, + 326, + 782 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, motivated by the observation of Pruksachatkun et al. (2020) that tasks requiring commonsense reasoning ability generally serve as good intermediate task, we test our method in the intermediate task transfer setting. Specifically, we initialize our model with T5-base, a pre-trained text-to-text transformer model, and training the model with our proposed method as intermediate task before fine-tuning and target downstream tasks. Another reason for adopting this setting is because we expect our method to serve as a “plug-and-play” method that can be applied to any pre-trained text-to-text transformer by simply continually training for a few steps. ", + "bbox": [ + 174, + 790, + 825, + 890 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Details for Pre-training and Fine-tuning CALM is continually pre-trained with our proposed self-supervised objectives as intermediate tasks based on the pre-trained T5-base model following the setting in Pruksachatkun et al. (2020). We randomly sample 500K sentences from the English Wikipedia corpus3, which is used for pre-training BERT and its variants, as the source dataset for our proposed self-supervised objectives which serve as intermediate tasks. We then fine-tune the CALM on each downstream task individually and report the average performance of three runs with different random seeds for fine-tuning on each dataset since the performance is sensitive to different random seeds. Training details and hyperparameter settings are presented in Appendix A.1 and A.2. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets We consider five commonsense benchmark datasets as target tasks. We categorize these datasets into discriminative and generative tasks. Discriminative tasks are classification tasks while generative tasks are text generation tasks. We consider four datasets for discriminative task: CommonsenseQA (Talmor et al., 2018), OpenbookQA (Mihaylov et al., 2018), PIQA (Bisk et al., 2020), aNLI (Bhagavatula et al., 2019) and one dataset for generative task: CommonGEN (Lin et al., 2020). Details on datasets are discussed in Appendix A.3. ", + "bbox": [ + 174, + 194, + 826, + 277 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Compared Methods We compare our model with following models continually trained with different intermediate tasks based on the pre-trained T5-base model: (1) T5-base is the pre-trained T5-base model without continually training on any intermediate task. (2) T5-base w/ additional epochs is continually pre-trained using the original pre-training objective of T5 with additional training steps. The total number of additional training steps is equal to that of our final model. (3) T5-base $\\mathbf { \\Gamma } + \\mathbf { S S M }$ is continual pre-trained with a variant of the salient span masking objective (Guu et al., 2020; Roberts et al., 2020) objective that masks text spans of concepts extracted with POS tagging instead of named entities extracted by a pre-trained NER model, which makes it more focused on concepts. (4) CALM(Generative-Only) is continually pre-trained with the proposed generative objectives including concept-to-sentence generation(C2S) and concept order recovering(COR) as intermediate tasks. (5) CALM(Contrastive-Only) is continually pre-trained with the proposed contrastive objective as described in section 2.2 using the distractor generated by concept shuffling. (6) CALM(Mix-only) is continually pre-trained with both the generative objectives and the contrastive objective, combined with a multi-task learning fashion with identical weights for each objective as the intermediate task. (7) CALM (w/o Mix warmup) is continually pre-trained with the joint training objective described in Eq (7) directly from the pre-trained T5-base model. (8) CALM is our main model trained as described in Algorithm 1. The difference between CALM and CALM (Joint) is that the former is initialized by the CALM(Mix). We also include the performance of the BERT-base model and two knowledge enhanced PTLMs that have similar architecture to BERT-base. ", + "bbox": [ + 173, + 285, + 825, + 549 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Evaluation Metrics For discriminative tasks, we choose accuracy as our metric following other conventional question answering tasks. For generative tasks, we report automated metrics including BLEU (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), CIDEr (Vedantam et al., 2015), and SPICE (Anderson et al., 2016) following the leaderboard of COMMONGEN (Lin et al., 2020). Results for COMMONGEN are on the test set and others are on the official development set. We tune the hyperparameters based on the models’ performance on a in-house split dev set. ", + "bbox": [ + 174, + 555, + 825, + 638 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 656, + 392, + 670 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The result is presented in Table 1. First, we can see that our CALM model consistently and significantly (with p-value $< 0 . 0 1$ ) outperforms the backbone T5-base model on all five datasets by a margin range from 1.5 to 2.9 accuracy on discriminative tasks and $1 . 5 / 0 . 6$ BLEU/SPICE score on CommonGEN. This is an impressive result since we are only performing intermediate training on a relatively small dataset for only around $2 0 \\mathrm { k }$ updates. It demonstrates the potential of our method for serving as a “plug-and-play” method for packing more commonsense knowledge into a pre-trained text-to-text transformer. Table 3 also shows that CALM performs comparably with several large-size PTLMs like BART, T5-large, and GPT-2 on the COMMONGEN dataset. The performance is worse than KG-BART (Liu et al., 2020), the current state-of-the-art on COMMONGEN, which is a contemporary work that exploits external knowledge bases as additional information, and is based on a larger backbone(i.e., BART (Lewis et al., 2019)). ", + "bbox": [ + 174, + 681, + 825, + 834 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In addition, we can observe that both the proposed generative and contrastive objective outperforms the backbone T5-base model, as well as its variants that continually pre-trained with the original masked span prediction objective and the concept-specific salient span masking scheme, when applied independently. Note that we find the variant of salient span masking that focuses on concept is not ", + "bbox": [ + 176, + 842, + 823, + 897 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/ada5dfc381fff46fe7830056ce2b2c898ebaae42eb6894dc7d7184a39f973ad4.jpg", + "table_caption": [ + "Table 1: Experimental results on commonsense reasoning datasets. The first group of models are baselines. The models in the middle group and last group except the CALM model are trained with the proposed objectives independently and the final CALM model is trained by joint training. Best models are bold and second best ones are underlined within each metric. " + ], + "table_footnote": [], + "table_body": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-base53.08(±0.16)57.60(±0.8)64.86(±0.52)61.88(±0.56)11--
ERNIE54.06(±0.12)58.90(±0.9)66.47(±0.58)63.04(±0.46)1
KnowBERT53.88(±0.15)58.50(±0.8)66.61(±0.63)63.18(±0.52)1-1-
T5-base61.88(±0.08)58.20(±1.0)68.14(±0.73)61.10(±0.38)24.9031.2012.9932.40
T5-base + cont. pretraining61.92(±0.45)58.10(±0.9)68.19(±0.77)61.15(±0.52)25.1031.0013.1232.40
T5-base + SSM62.08(±0.41)58.30(±0.8)68.27(±0.71)61.25(±0.51)25.2031.2013.2832.40
CALM (Generative-Only)62.28(±0.36)58.90(±0.4)68.91(±0.88)60.95(±0.46)25.8031.2013.8132.60
CALM (Contrastive-Only)62.73(±0.41)59.30(±0.3)70.67(±0.98)61.35(±0.06)25.5031.2013.5832.60
CALM(w/o Mix warmup)62.18(±0.48)59.00(±0.5)69.21(±0.57)61.25(±0.55)25.8031.2013.7732.60
CALM (Mix-only)63.02(±0.47)60.40(±0.4)70.07(±0.98)62.79(±0.55)26.0031.2013.8232.80
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
", + "bbox": [ + 176, + 99, + 821, + 253 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/af13aa76908dabacc756a8a5173f368814aa4989a06334fe6971b3401c9c2a8d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-large57.06(±0.12)60.40(±0.6)67.08(±0.61)66.75(±0.61)==
T5-large69.81(±1.02)61.40(±1.0)72.19(±1.09)75.54(±1.22)28.6030.1014.9631.60
CALM-large (Mix-only)70.26(±0.23)62.50(±1.0)73.70(±1.09)75.99(±1.26)29.2031.3015.2433.10
CALM-large71.31(±0.04)66.00(±1.0)75.11(±1.65)77.12(±0.34)29.5031.9015.6133.20
RoBERTa-large471.81(±0.25)63.90(±0.8)76.90(±0.62)82.35(±0.54)111-
", + "bbox": [ + 176, + 327, + 823, + 421 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2: Experimental results on large model. Comparison between large models of other PTLMs and CALM. Best models are bold and second best ones are underlined within each metric. ", + "bbox": [ + 174, + 431, + 823, + 459 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "very effective. We suspect this is because the resulting training data would be somewhat similar to the original text infilling objective because concepts are very common in the corpus and we only train for a few steps. The combination of the generative and contrastive objectives (i.e., CALM(Mix-only)) yields further improvement upon the model trained independently with either generative or contrastive objectives. Also, we find that the CALM model consistently outperforms CALM(Mix), demonstrating the effectiveness of the proposed joint training framework. Applying joint training directly on top of a pre-trained model (i.e., CALM(w/o Mix warmup)) does not work very well, demonstrating the necessity of applying mixed training to initialize the model before starting joint training. ", + "bbox": [ + 173, + 469, + 825, + 582 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To further confirm the effectiveness of our approach, we also apply our method to continually pre-train T5-large with the same data and number of training steps. We then compare the performance of the resulting model with that of the original T5-large model in Table 10. We find that both the proposed training objectives and the joint training framework consistently and significantly (with p-value $< 0 . 0 1$ ) improve upon the original T5-large, showing our approach is effective for models with different sizes. Our model also outperforms BERT-large by a large margin. ", + "bbox": [ + 174, + 589, + 442, + 782 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/5ea7c0ba53a26539e64742930742dcca418fe1ee79b7f4ebcd1d75ee2c73cab2.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodsParamsCommonGEN
BLEU-4METEORCIDErSPICE
GPT-2 (Radford et al.,2019)774M21.1026.2012.1525.90
UniLM(Dong et al.,2019)340M27.7029.7014.8530.20
BART(Lewis et al.,2020)406M26.3030.9013.9230.60
T5-base (Raffel et al.,2019)220M16.4023.009.1622.00
T5-large (Raffel et al., 2019)770M28.6030.1014.9631.60
KG-BART (Liu et al., 2020)406M30.9032.4016.8332.70
T5-base (our implementation)220M24.9031.2012.9932.40
CALM-base220M26.4031.4013.8833.00
CALM-large774M29.5031.9015.6133.20
", + "bbox": [ + 457, + 592, + 820, + 705 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 3: Comparison between PTLMs on CommonGEN. Above baselines are reported number in the leaderboard. T5-base(our implementation) uses different hyperparmeter setting than that reported in the leaderboard. ", + "bbox": [ + 454, + 717, + 825, + 772 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "However, our model performs slightly worse compared to RoBERTa-large. We suspect this is because RoBERTa-large is optimized for more steps than T5-large and our CALM-large. This is also observed in many other tasks and datasets. ", + "bbox": [ + 176, + 784, + 825, + 824 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 PERFORMANCE ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 835, + 398, + 849 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Analysis on Generative objective To investigate the contribution of each generative objective, we conduct an ablation study by continually pre-training three models from the same T5-base model with C2S, COR, and text infilling, which is the original objective for pre-training T5, as the objective for the intermediate task. We continually pre-train these models for the same number of steps and then evaluate their performance by fine-tuning on different target tasks. The result is shown in Table 4. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/6688f0c8845fb48b0e8203a3f67f1738c4c65e7fe82a3b7f29851e7c736114fd.jpg", + "table_caption": [], + "table_footnote": [ + "(a) Generative objectives " + ], + "table_body": "
MethodsCSQA PIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
T5- Text Infilling61.9268.1925.1031.0013.1332.40
CALM-COR62.3668.7725.7031.2013.6532.60
CALM-C2S62.2468.7525.9031.4013.9432.80
", + "bbox": [ + 176, + 93, + 501, + 148 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/cfa2066a516a506e2ef24a54037274ccbb2c97f27eca1ee5ec96a0fb7cbd7522.jpg", + "table_caption": [], + "table_footnote": [ + "(b) Contrastive objectives " + ], + "table_body": "
MethodsCSQAPIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
Multi-choice QA62.2168.8225.0031.2013.2832.60
True/False62.2467.8125.1031.2013.4132.60
Generative QA62.7370.6725.5031.2013.5832.60
", + "bbox": [ + 509, + 94, + 830, + 148 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 4: Analysis on Contrastive and Generative objectives. Left table shows the performance on downstream tasks by pre-training with different generative objective (COR, C2S, and original objective for pre-training T5). Right table shows the performance on downstream tasks by pre-training with different task formats of contrastive objective. ", + "bbox": [ + 173, + 171, + 825, + 227 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We can see that both C2S and COR works better than the original masked span infilling objective on itself. This confirms the effectiveness of our proposed generative objectives on improving the commonsense reasoning ability of pre-trained text-to-text transformers. ", + "bbox": [ + 174, + 237, + 825, + 280 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Task Formulation of the Contrastive objectives For contrastive objectives, we test three different task formats: Multi-choice QA, Generative QA, and True/False. Multi-choice QA and Generative QA takes the concatenation of the real sentence and the distractor. Then, Multi-choice QA output the index of the real sentence following other conventional Multi-choice QA tasks, and Generative QA output the real sentence respectively. True/False takes either the real sentence or the distractor and train the model to perform a binary classification problem of whether the input sentence makes sense. The result is shown in Table 4. We could find that the format of Generative QA performs the best. We suspect this is because the Generative QA format is closer to the format used during the original pre-training stage of the T5 model and the format used for fine-tuning. ", + "bbox": [ + 173, + 286, + 825, + 411 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Performance with fewer training examples To investigate the effectiveness of our objective in the lowresource setting, we explore the performance of our model and baselines fine-tuning with different fractions of the training data. From Figure 4, we can see that the performance improvement yielded by our models upon the T5-base model is more significant in the low-resource regime. This shows that CALM may already pack some ", + "bbox": [ + 174, + 419, + 421, + 584 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/d53949681d0c53f39d987e7afd217298a347d7f0231c014211f25f5c381ab759.jpg", + "image_caption": [ + "Figure 4: Performance of compared models fine-tuned with different fraction of the datasets. " + ], + "image_footnote": [], + "bbox": [ + 447, + 417, + 794, + 539 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "commonsense knowledge in its parameters so that it does not require much data for fine-tuning before obtaining a good performance. In contrast, the original T5 model requires much data for fine-tuning, which suggests it may fail to encode much commonsense knowledge and must fit the correlation patterns in the downstream datasets to get a good performance. ", + "bbox": [ + 174, + 585, + 825, + 640 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Comparison of Generated Data Table 5 shows the comparison of generated examples for the COMMONGEN test set between T5-base and CALM. We can see that the sentences generated by CALM are generally more acceptable in terms of commonsense plausibility while T5-base sometimes generates sentences that do not make sense. ", + "bbox": [ + 173, + 647, + 825, + 703 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/813fd8d9cffcae2d37a01c6c6c4e1a587fbc25a7f21c3d791a6fda2c1a57630d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Concept-setT5-baseCALM-base
Grass,Dog,Ball, Chasea dog is chased by a ball on the grass.dog chasing a ball in the grass.
Net, Cast,Boat,Waterfishing boat casts a net in the water.fisherman casts a net into the water from a fishing boat.
Hole,Tree,Plant, Diga man digs a hole in a tree to plant a new tree . he digs theman digging a hole to plant a tree.
Ingredient, Add,Pan, rya pan filled with ingredients adds a touch of spice to the fry .add the ingredients to a pan and fry.
Water,Hold,Hand,WalkA man holding a hand and walking in the water. A man is holding water.man holding a bottle of water in his hand as he walks down the street.
Place, Use,Metal toolA man uses a metal tool to make a piece of metal.woman uses a metal tool to make a piece of jewelry.
", + "bbox": [ + 176, + 704, + 821, + 768 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 5: Comparison of generated sentences with same concept-set. ", + "text_level": 1, + "bbox": [ + 266, + 777, + 730, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Knowledge Probing To investigate how much concept-centric knowledge our model pack, we conducted two probing methods with our model $:$ Language Model Analysis (LAMA) probe (Petroni et al., 2019), Knowledge Intensive Language Task (KILT) (Petroni et al., 2020). We summarize the results on Table 6 and Appendix A.4. We could find that our model outperforms the baseline. ", + "bbox": [ + 173, + 800, + 825, + 856 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 871, + 344, + 887 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Self-Supervised Language Representation Pre-Training. Motivated by the fact that words can have different meanings in different contexts, contextual language representation methods (McCann et al., 2017; Peters et al., 2018) have been developed and shown superior performance on downstream tasks compared with static word embeddings Mikolov et al. (2013); Pennington et al. (2014). More recently, large scale language models based on transformer architecture (Vaswani et al., 2017) pretrained with either mask language modeling objective (Devlin et al., 2018; Liu et al., 2019; Lan et al., 2019) or mask span infilling objective (Lewis et al., 2019; Raffel et al., 2019) have been explored further advanced the state-of-the-art on multiple NLU and NLG tasks. Our method is based on these techniques and we focus on improving the commonsense reasoning ability of pre-trained text-to-text transformers. More recently, Clark et al. (2020) propose a new self-supervised pre-training objective called Replaced Token Detection (RTD). RTD uses a mask language model like BERT to fill in the mask and train a discriminator to predict whether a token is generated or real. This pre-training paradigm is related to our proposed joint training framework. Some major differences include that (1) Our method employs sentence-level distractors that are in general grammatically correct but not in line with commonsense, thus require the model to perform relational commonsense reasoning while RTD is a token-level discrimination task and can often be solved with syntactic and shallow semantic knowledge (Rosset et al., 2020); (2) Our method unifies generative and contrastive objectives with one model, which can be applied to both NLU and NLG downstream tasks; and (3) The discriminator in our framework is “contrastive”, takes both the real sentence and the distractor as input simultaneously. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/7bd245446154dd54aec795629e87543d0df56471c9f65630426eb781a2201741.jpg", + "table_caption": [ + "Table 6: Experimental results on Knowledge Probing. Left table shows the mean precision on LAMA probing task of ConceptNET. Right table shows the performance on Fact checking and Entity linking, which are from KILT task. " + ], + "table_footnote": [], + "table_body": "
MethodsMRRPrecision@50Precision @10Precision@1
T5-Base11.5338.5221.605.93
CALM (Mix-only)11.7738.9321.926.10
CALM12.0939.6922.536.46
", + "bbox": [ + 176, + 102, + 598, + 159 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/1a823416d371a55b53adb7bfc037dd5cca8eab03e2f0eeb4b1fb9d920daf18ac.jpg", + "table_caption": [], + "table_footnote": [ + "(a) LAMA probe (b) KILT task " + ], + "table_body": "
MethodsFEVERAY2
T5-base76.6574.97
CALM (Mix-only)77.0576.27
CALM77.4477.24
", + "bbox": [ + 607, + 102, + 828, + 160 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 246, + 825, + 482 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Knowledge-augmented PTLMs. As standard pre-trained language models usually do not explicitly model knowledge, a number of works have examined the problem of incorporating world knowledge with the PTLMs. Recent work Zhang et al. (2019); Peters et al. (2019); Wang et al. (2020); Liu et al. (2020) utilizes an external knowledge base to incorporate entity knowledge with PTLMs; however, these approaches require specialized resources like knowledge bases, which limits the domain they can be applied to. Xiong et al. (2020) proposes WikiLM that encodes world knowledge into the parameters of a BERT(Devlin et al., 2018)-like pre-trained model with a novel entity replacement detection objective that incorporates Wikipedia to form distractors. Their approach differs from ours because it requires an external knowledge base (i.e., Wikipedia) which limits the domain it can be applied, is limited to discriminative pre-training objectives and downstream tasks, and focuses on world knowledge instead of relational commonsense knowledge. More recently, (Rosset et al., 2020) propose KALM, an entity-aware language model with more world knowledge packed into its parameters. Their method is restricted to the training of language models instead of masked language models or text-to-text transformers which can be used for more downstream tasks. Also, all the aforementioned work mainly focuses on world knowledge of named entities. In contrast, our work mainly focuses on commonsense knowledge about quotidian concepts. ", + "bbox": [ + 174, + 489, + 825, + 712 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 733, + 318, + 750 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose novel self-supervised strategies that encourage the model to focus on concept-centric information that is related to commonsense understanding and reasoning instead of simple word coocurrence patterns so that the commonsense learning capability of pre-trained text-to-text transformers can be improved. Despite merely continually pre-trained on a small dataset with only around $2 0 \\mathrm { k }$ steps, our CALM model consistently outperforms the T5-base model on all commonsense-related datasets, and even yields better performance compared with some larger size PTLMs on the COMMONGEN dataset. The performance gain is larger when we use fewer examples for fine-tuning on different downstream tasks, indicating that CALM effectively encodes more commonsense knowledge and rely less on fitting superficial patterns of datasets compared to traditional pre-trained language models. Our work suggests that text-to-text models can be pre-trained with better parameter and sample efficiency by carefully designed self-supervised objectives that focus more on the ability (e.g., commonsense reasoning ability) required by target tasks. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 103, + 356, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This research is supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via Contract No. 2019-19051600007, the DARPA MCS program under Contract No. N660011924033 with the United States Office Of Naval Research, the Defense Advanced Research Projects Agency with award W911NF-19-20271, and NSF SMA 18-29268. We would like to thank all the collaborators in USC INK research lab for their constructive feedback on the work. 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", + "bbox": [ + 174, + 103, + 823, + 132 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 PRE-TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 136, + 390, + 150 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The following details apply to both base architecture and joint-training architecture. We implement our pre-train models using Pytorch-lightning (Falcon, 2019) and Hugginface’s Pytorch Transformers (Wolf et al., 2019). For pre-training phase, we use the Adam optimizer with maximum sequence length 256, train batch size 8, gradient accumulation 8, warmup steps 10000, weight decay 0.01 and adam epsilon 1e-6. We train the models with 8 V100 GPUs and FP32 precision for 17 hours. The model is pre-trained for at most 3 epochs to prevent overfitting. We searched for the best learning rate for our model out of [1e-4, 2e-5, 2e-6, 5e-7]. ", + "bbox": [ + 174, + 164, + 825, + 261 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 FINE-TUNING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 282, + 380, + 296 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For fine-tuning, we use 4 V100 GPUs and use FP32. For all discriminative tasks, we use the Adam optimizer with maximum sequence length 256, batch size 4 and gradient accumulation 16. For generative task, we use the Adam optimizer with maximum source length 32, maximum target length 32, batch size 8, gradient accumulation 16. For all tasks, we use warmup fraction 0.01. Learning rates and train epochs are listed in Table 7. ", + "bbox": [ + 174, + 309, + 825, + 380 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/7b3c51abb8508f9d0dc28b77986d343c8f3a653e19ec81c32ced03d339b0cdb1.jpg", + "table_caption": [ + "Table 7: Fine-tuning hyperparameters. " + ], + "table_footnote": [], + "table_body": "
HyperparameterCommonsenseQAOpenbookQAPIQAaNLICommonGEN
Learning rate[1e-4,2e-4,3e-4][5e-5,1e-4,2e-4,3e-4][1e-4,2e-4,3e-4][2e-5,3e-5][2e-5]
Train Epochs2020201020
", + "bbox": [ + 176, + 396, + 821, + 445 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 DATASET PROPERTIES", + "text_level": 1, + "bbox": [ + 176, + 503, + 372, + 518 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• CommonsenseQA (Talmor et al., 2018) is a multiple-choice question answering task, which picks the most appropriate answer on general commonsense questions. ", + "bbox": [ + 214, + 531, + 823, + 560 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• OpenbookQA (Mihaylov et al., 2018) is a multiple-choice question answering task, which is modeled after open book exams on elementary-level core science questions. The task requires open book fact and additional commonsense which is not contained in the book. To test the commonsense reasoning ability, we do not use open book fact. ", + "bbox": [ + 218, + 568, + 825, + 625 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• PIQA (Bisk et al., 2020) is multiple-choice question answering task, which chooses the most appropriate solution for physical commonsense questions. ", + "bbox": [ + 218, + 632, + 820, + 661 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• aNLI (Bhagavatula et al., 2019) is a binary-classification task, which picks the most plausible explanatory hypothesis given two observations from narrative contexts. ", + "bbox": [ + 212, + 669, + 823, + 696 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• CommonGEN (Lin et al., 2020) is a constrained text generation task, which generates a coherent sentence describing an everyday scenario using common concepts. ", + "bbox": [ + 215, + 705, + 821, + 733 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/f994caf7b382a3392ed7ae22b0543139ed655e5e056c920fef0648074112a2fc.jpg", + "table_caption": [ + "Table 8: Properties of Commonsense benchmark datasets. " + ], + "table_footnote": [], + "table_body": "
DatasetTrainDevelopmentTestSource ExampleTarget Example
CommonsenseQA9,7411,2211,140context:What homeentertainment equipment requirescable? options: 1: radio shack 2: substation 3:cabinet 4: television 5:desk4
OpenbookQA4,957500500context: You can make a telescope with options: 1: straw 2: glass 3: candle 4: mailing tube2
PIQA16,1131,8383.084context:When boiling butter,when it's ready,you can options:1: Pour it onto a plate 2:Pour it into a jar2
aNLI169,6541,5323.040context: It was my birthday.WhenI got home the party was set up for my brother. options: 1: Iwas so excited. 2: Iwas so mad.2
CommonGEN67,3894.0186.042generate a sentence with these concepts: Apple Grow TreeApple grows on the tree
", + "bbox": [ + 178, + 760, + 821, + 883 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.4 KNOWLEDGE PROBING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 380, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "LAMA probe is consisting of a set of knowledge sources, each comprised of a set of fact. It defines that a pre-trained language model knows a fact (subject, relation, object) such as (Bird, CapableOf, Fly) if it can predict masked objects in cloze statement such as \"Birds can [MASK]\". For evaluation, we first filtered out examples that mask label is not in vocabulary list of T5. Then, we evaluate the model based on how highly it ranks the ground truth token against every other word in a fixed vocabulary list of T5, and get mean precision at k to check whether the object is ranked among the top k results. We summarize the results of ConceptNet (Speer & Havasi, 2012) in Table 6. Unlike other language models which are optimised to masked word anywhere in a given sequence, T5 is trained with different denoising method. It might cause low performance on such slot filling task, but compared to T5, our model shows better performance compared to base model. ", + "bbox": [ + 173, + 130, + 825, + 268 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "KILT task is a benchmark for assessing models that need to access specific knowledge in a defined snapshot of Wikipedia to solve tasks spanning five domains. The goal is to analyze the model whether it has task-agnostic representations of knowledge. We test our model on domain of fact checking, entity linking. Fact checking verifies textual claims against textual sources. For this task, we use FEVER (Thorne et al., 2018) which is a large dataset for claim veracity that requires evidence from multiple Wikipedia pages to determine whether the claim is supported or refuted. Entity Linking assigns Wikipedia page to entities mentioned in text. We use AIDA CoNLL-YAGO (AY2) (Hoffart et al., 2011) which supplements the CoNLL 2003 (Tjong Kim Sang & De Meulder, 2003) with Wikipedia URL annotations for all entities. ", + "bbox": [ + 174, + 275, + 825, + 400 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.5 EXPERIMENTS WITH BART AS BACKBONE ", + "text_level": 1, + "bbox": [ + 176, + 417, + 514, + 431 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To show that our approach is versatile to different pre-trained models, we conduct experiments with BART as the backbone model. We can see that our approach consistently and significantly (with p-value $< 0 . 0 1$ ) improves BART-base on all datasets. This result shows that our method is versatile to different pre-trained models. ", + "bbox": [ + 174, + 443, + 825, + 500 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/da6ac46e761543eefb4c1ed5d757aa8e209ff3e467330e8d2f78887463ab566b.jpg", + "table_caption": [ + "Table 9: Experimental results with BART as backbone model. Best models are bold. " + ], + "table_footnote": [], + "table_body": "
MethodsCSQAOBQAPIQAaNLICommonGEN
AccuracyBLEU-4METEORCIDErSPICE
BART-base (Mix-only)56.31(±0.28)58.30(±1.1)(official dev) 67.53(±1.01)59.85(±1.14)25.1029.5013.1630.20
CALM (BART-base)58.22(±0.21)59.10(±1.0)69.40(±1.23)61.28(±0.30)26.4029.9013.7131.10
", + "bbox": [ + 176, + 512, + 821, + 570 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.6 EXPERIMENTS WITH NOUN/VERB AS CONCEPTS ", + "text_level": 1, + "bbox": [ + 174, + 621, + 555, + 636 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/7938293e05b0ce973b065c1134b0c22c0ec4f0826580fcbdccf5a1de51d548aa.jpg", + "table_caption": [ + "We also conducted an ablation study about the choice of using either nouns or verbs as concepts. We can see that using either nouns-only or verbs-only as concepts for our approach leads to substantial performance drop. This supports our choice about using both nouns and verbs as concepts. " + ], + "table_footnote": [], + "table_body": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
CALM-nouns62.45(±0.42)59.40(±0.5)69.05(±0.70)61.55(±0.58)25.7031.2013.1732.60
CALM-verbs62.51(±0.47)59.10(±0.7)69.24(±0.65)61.40(±0.51)25.6031.2013.2432.60
", + "bbox": [ + 176, + 702, + 821, + 775 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 10: Experimental results with Noun/Verb as Concepts. Best models are bold. ", + "text_level": 1, + "bbox": [ + 214, + 784, + 781, + 799 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.7 HUMAN EVALUATION ON COMMONGEN GENERATIONS ", + "text_level": 1, + "bbox": [ + 174, + 811, + 602, + 827 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We conducted a human evaluation of CommonGEN predictions between T5 and CALM. We asked three annotators to choose the most reasonable sentence between T5-base and CALM-base predictions. The evaluation was conducted on 50 test sentences in binary selection by majority voting. Cohen’s Kappa score, which is a measurement of inter-annotator agreement, was 0.73. Annotators say that for $60 \\%$ of test sentences, CALM-base generated better. 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However,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "score": 1.0, + "content": "current pre-training objectives such as masked token prediction (for BERT-style", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 281 + ], + "score": 1.0, + "content": "PTLMs) and masked span infilling (for T5-style PTLMs) do not explicitly model", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "score": 1.0, + "content": "the relational commonsense knowledge about everyday concepts, which is crucial", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "to many downstream tasks that need common sense to understand or generate. To", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "score": 1.0, + "content": "augment PTLMs with concept-centric commonsense knowledge, in this paper, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 312, + 470, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 324 + ], + "score": 1.0, + "content": "propose both generative and contrastive objectives for learning common sense", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "score": 1.0, + "content": "from the text, and use them as intermediate self-supervised learning tasks for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 469, + 345 + ], + "score": 1.0, + "content": "incrementally pre-training PTLMs (before task-specific fine-tuning on downstream", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 470, + 356 + ], + "score": 1.0, + "content": "datasets). Furthermore, we develop a joint pre-training framework to unify gen-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "erative and contrastive objectives so that they can mutually reinforce each other.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 365, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 470, + 379 + ], + "score": 1.0, + "content": "Extensive experimental results show that our method, concept-aware language", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 470, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 170, + 390 + ], + "score": 1.0, + "content": "model", + "type": "text" + }, + { + "bbox": [ + 170, + 376, + 211, + 388 + ], + "score": 0.54, + "content": "( { \\bf C A L M } ) ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 376, + 470, + 390 + ], + "score": 1.0, + "content": ", can pack more commonsense knowledge into the parameters of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "score": 1.0, + "content": "a pre-trained text-to-text transformer without relying on external knowledge graphs,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 470, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 470, + 412 + ], + "score": 1.0, + "content": "yielding better performance on both NLU and NLG tasks. We show that while only", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 470, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 422 + ], + "score": 1.0, + "content": "incrementally pre-trained on a relatively small corpus for a few steps, CALM out-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 421, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 470, + 433 + ], + "score": 1.0, + "content": "performs baseline methods by a consistent margin and even comparable with some", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 431, + 470, + 445 + ], + "spans": [ + { + "bbox": [ + 141, + 431, + 470, + 445 + ], + "score": 1.0, + "content": "larger PTLMs, which suggests that CALM can serve as a general, “plug-and-play”", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 443, + 425, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 425, + 455 + ], + "score": 1.0, + "content": "method for improving the commonsense reasoning ability of a PTLM.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 16.5, + "bbox_fs": [ + 141, + 234, + 470, + 455 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 475, + 206, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 208, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 208, + 491 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 506 + ], + "score": 1.0, + "content": "Pre-trained language models (PLTMs) such as BERT (Devlin et al., 2018) and T5 (Raffel et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "2019) have revolutionized the field of NLP, yielding impressive performance on various conventional", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "natural language understanding (NLU) and generation (NLG) tasks. 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Though yielding better", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "performance on many downstream tasks, these pre-training objectives, however, do not explicitly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "score": 1.0, + "content": "guide the models to reason with concept-centric commonsense knowledge from language, including", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "the relation and composition of daily concepts in our lives. This leaves room for equipping current", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 592, + 315, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 315, + 606 + ], + "score": 1.0, + "content": "PTLMs with richer commonsense reasoning ability.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 495, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "For example, consider a multi-choice question “What do you fill with ink to write notes on a piece of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 633 + ], + "score": 1.0, + "content": "copy paper? (A) fountain pen (B) pencil case (C) printer (D) notepad”. The current state-of-the-art", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "question answering model, UnifiedQA (Khashabi et al., 2020), which was fine-tuned on T5-large", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "with multiple datasets, still predicts ‘(C) printer’ as its answer. 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Towards augmenting PTLMs with more", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "knowledge, prior works mainly focus on training larger models (Brown et al., 2020), adding specific", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "architectures to exploit external knowledge (Peters et al., 2019), or incorporating knowledge bases", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "for pre-training (Xiong et al., 2020). 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In this paper, we instead look to explicitly teach pre-trained", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 460, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 460, + 128 + ], + "score": 1.0, + "content": "models to write and reason with common concepts through novel pre-training strategies.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "We present two kinds of self-supervised pre-training tasks: concept-to-sentence generation (C2S)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "and concept order recovering (COR). C2S trains the pre-trained model to compose (“write\")", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "sentences given a set of concepts, and expects the generated sentences to be fluent and plausible in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "terms of commonsense. COR aims to teach models to detect and revise a corrupted sentence with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 504, + 188 + ], + "score": 1.0, + "content": "incorrect ordering of concepts. As illustrated in Figure 1, both tasks require a pre-trained model to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "recall relevant commonsense facts about the concepts and to understand the underlying commonsense", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "relations between them. Both of the proposed objectives can explicitly encourage the model to capture", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 484, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 484, + 223 + ], + "score": 1.0, + "content": "the relational concept-centric commonsense knowledge and perform compositional reasoning.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 507, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 507, + 240 + ], + "score": 1.0, + "content": "Specifically, we need a generative pre-training objective to encourage models to capture this genera-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "tive commonsense reasoning ability, so that models can learn to generate sentences with common-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "sense knowledge for both C2S and COR. Also, to teach modes to distinguish truth sentences from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "less plausible ones, we need to teach models with discriminative commonsense through contrastive", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "self-training. To unify both generative and contrastive objectives within a joint learning framework", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "so that the model can learn both generative and discriminative commonsense knowledge at the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 303 + ], + "score": 1.0, + "content": "time, we propose to use the sentences generated by the model itself as the distractors and train the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "model to distinguish the generated sentences from real sentences. In this way, the model is forced", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "to acquire new commonsense knowledge in order to distinguish the distractors generated by itself,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "which probably exploit the knowledge the model already possesses. Therefore, the model is trained", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "to iteratively improve upon itself in a self-play fashion. We share all the parameters between the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "generator (trained with the generative objective) and the discriminator (trained with the contrastive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "objective), then train multiple objectives with different prefixes. Compared to previous works (Peters", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "et al., 2019; Li et al., 2019; Xiong et al., 2020) that utilize external knowledge bases like Wikidata", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "or ConceptNet, our approach can directly improve the generative and discriminative commonsense", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 391, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 470, + 403 + ], + "score": 1.0, + "content": "reasoning ability of PTLMs at the same time without relying on external knowledge bases.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "To evaluate the effectiveness of our proposed method, we apply our method in an intermediate-task", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "transfer learning setting (Pruksachatkun et al., 2020) based on the pre-trained T5-base model to train", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "a Concept-Aware Language Model (CALM). While only continually pre-trained on a small dataset", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "for a relatively fewer number of updates (compared to conventional pre-training), CALM consistently", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 452, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 463 + ], + "score": 1.0, + "content": "outperforms T5-base on four commonsense-related NLU datasets (i.e., COMMONSENSEQA, OPEN-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "BOOKQA, PIQA, and ANLI) and COMMONGEN, a commonsense-related NLG dataset. Our results", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "and careful ablation studies demonstrate the potential of our method to serve as a “plug-and-play”", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "method for any pre-trained text-to-text transformer before fine-tuning on commonsense-related tasks.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "To the best of our knowledge, our work is the first to investigate concept-centric self-supervised", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "objectives that improve both generative and discriminative commonsense reasoning ability of a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 518, + 221, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 221, + 529 + ], + "score": 1.0, + "content": "pre-trained language model.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 106, + 553, + 469, + 565 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 471, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 471, + 568 + ], + "score": 1.0, + "content": "2 SELF-SUPERVISED OBJECTIVES FOR CONCEPT-CENTRIC LEARNING", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "In this section, we first describe the proposed generative and contrastive objectives used for improving", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "the commonsense reasoning ability of pre-trained text-to-text transformers. Then, we introduce the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "joint learning framework which unifies the proposed self-supervised objectives and learn a unified", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 365, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 365, + 622 + ], + "score": 1.0, + "content": "text-to-text transformer based on pre-trained models such as T5.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 108, + 641, + 244, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 245, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 245, + 654 + ], + "score": 1.0, + "content": "2.1 GENERATIVE OBJECTIVES", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Similar to many other pre-training tasks such as masked language modeling, we aim to teach models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 671, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 507, + 684 + ], + "score": 1.0, + "content": "to recover original sentences from corrupted inputs, which is often regarded as a denoising process.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "We propose two generative self-supervised pre-training objectives: concept-to-sentence generation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 693, + 284, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 284, + 705 + ], + "score": 1.0, + "content": "(C2S) and concept order recovering (COR).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 257, + 723 + ], + "score": 1.0, + "content": "Concept Extraction. Given an input", + "type": "text" + }, + { + "bbox": [ + 258, + 710, + 341, + 722 + ], + "score": 0.91, + "content": "\\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 708, + 506, + 723 + ], + "score": 1.0, + "content": ", we first conduct part-of-speech tagging", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "with Spacy for the sentence and extract Verb, Noun, and Proper Nouns from the sentence to use as", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 146 + ], + "score": 1.0, + "content": "We present two kinds of self-supervised pre-training tasks: concept-to-sentence generation (C2S)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "and concept order recovering (COR). C2S trains the pre-trained model to compose (“write\")", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "sentences given a set of concepts, and expects the generated sentences to be fluent and plausible in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "terms of commonsense. COR aims to teach models to detect and revise a corrupted sentence with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 504, + 188 + ], + "score": 1.0, + "content": "incorrect ordering of concepts. As illustrated in Figure 1, both tasks require a pre-trained model to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "recall relevant commonsense facts about the concepts and to understand the underlying commonsense", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "relations between them. Both of the proposed objectives can explicitly encourage the model to capture", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 484, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 484, + 223 + ], + "score": 1.0, + "content": "the relational concept-centric commonsense knowledge and perform compositional reasoning.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 131, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 507, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 507, + 240 + ], + "score": 1.0, + "content": "Specifically, we need a generative pre-training objective to encourage models to capture this genera-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "tive commonsense reasoning ability, so that models can learn to generate sentences with common-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "sense knowledge for both C2S and COR. Also, to teach modes to distinguish truth sentences from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "less plausible ones, we need to teach models with discriminative commonsense through contrastive", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "self-training. To unify both generative and contrastive objectives within a joint learning framework", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "so that the model can learn both generative and discriminative commonsense knowledge at the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 303 + ], + "score": 1.0, + "content": "time, we propose to use the sentences generated by the model itself as the distractors and train the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "model to distinguish the generated sentences from real sentences. In this way, the model is forced", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "to acquire new commonsense knowledge in order to distinguish the distractors generated by itself,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "which probably exploit the knowledge the model already possesses. Therefore, the model is trained", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "to iteratively improve upon itself in a self-play fashion. We share all the parameters between the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "generator (trained with the generative objective) and the discriminator (trained with the contrastive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "objective), then train multiple objectives with different prefixes. Compared to previous works (Peters", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "et al., 2019; Li et al., 2019; Xiong et al., 2020) that utilize external knowledge bases like Wikidata", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "or ConceptNet, our approach can directly improve the generative and discriminative commonsense", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 391, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 470, + 403 + ], + "score": 1.0, + "content": "reasoning ability of PTLMs at the same time without relying on external knowledge bases.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 226, + 507, + 403 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "To evaluate the effectiveness of our proposed method, we apply our method in an intermediate-task", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "transfer learning setting (Pruksachatkun et al., 2020) based on the pre-trained T5-base model to train", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "a Concept-Aware Language Model (CALM). While only continually pre-trained on a small dataset", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "for a relatively fewer number of updates (compared to conventional pre-training), CALM consistently", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 452, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 463 + ], + "score": 1.0, + "content": "outperforms T5-base on four commonsense-related NLU datasets (i.e., COMMONSENSEQA, OPEN-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "BOOKQA, PIQA, and ANLI) and COMMONGEN, a commonsense-related NLG dataset. Our results", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "and careful ablation studies demonstrate the potential of our method to serve as a “plug-and-play”", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "method for any pre-trained text-to-text transformer before fine-tuning on commonsense-related tasks.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "To the best of our knowledge, our work is the first to investigate concept-centric self-supervised", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "objectives that improve both generative and discriminative commonsense reasoning ability of a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 518, + 221, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 221, + 529 + ], + "score": 1.0, + "content": "pre-trained language model.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 406, + 506, + 529 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 553, + 469, + 565 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 471, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 471, + 568 + ], + "score": 1.0, + "content": "2 SELF-SUPERVISED OBJECTIVES FOR CONCEPT-CENTRIC LEARNING", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "In this section, we first describe the proposed generative and contrastive objectives used for improving", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "the commonsense reasoning ability of pre-trained text-to-text transformers. Then, we introduce the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "joint learning framework which unifies the proposed self-supervised objectives and learn a unified", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 365, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 365, + 622 + ], + "score": 1.0, + "content": "text-to-text transformer based on pre-trained models such as T5.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 577, + 506, + 622 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 641, + 244, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 245, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 245, + 654 + ], + "score": 1.0, + "content": "2.1 GENERATIVE OBJECTIVES", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Similar to many other pre-training tasks such as masked language modeling, we aim to teach models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 671, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 507, + 684 + ], + "score": 1.0, + "content": "to recover original sentences from corrupted inputs, which is often regarded as a denoising process.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "We propose two generative self-supervised pre-training objectives: concept-to-sentence generation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 693, + 284, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 284, + 705 + ], + "score": 1.0, + "content": "(C2S) and concept order recovering (COR).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 660, + 507, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 257, + 723 + ], + "score": 1.0, + "content": "Concept Extraction. Given an input", + "type": "text" + }, + { + "bbox": [ + 258, + 710, + 341, + 722 + ], + "score": 0.91, + "content": "\\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 708, + 506, + 723 + ], + "score": 1.0, + "content": ", we first conduct part-of-speech tagging", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "with Spacy for the sentence and extract Verb, Noun, and Proper Nouns from the sentence to use as", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 264, + 326 + ], + "score": 1.0, + "content": "concepts2. 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(i.e.", + "type": "text" + }, + { + "bbox": [ + 308, + 334, + 392, + 347 + ], + "score": 0.92, + "content": "\\mathcal { C } _ { v } = [ v _ { 1 } , v _ { 2 } , \\ldots , v _ { p } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 335, + 410, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 410, + 334, + 501, + 347 + ], + "score": 0.93, + "content": "\\mathcal { C } _ { n } = [ n _ { 1 } , n _ { 2 } , \\ldots , n _ { q } ] .", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 335, + 506, + 347 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "Concept-to-Sentence Generation (C2S). The concept-to-sentence generation (C2S) objective re-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "quires the text-to-text transformer to recover the original sentence given only a few unordered", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 486, + 386 + ], + "score": 1.0, + "content": "keywords of the sentence. Specifically, given a sentence, we shuffle the extracted concept-set", + "type": "text" + }, + { + "bbox": [ + 486, + 374, + 493, + 383 + ], + "score": 0.8, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "create the perturbed source sequence and train the model to generate the original sentence with a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 395, + 507, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 180, + 408 + ], + "score": 1.0, + "content": "prefix (denoted as", + "type": "text" + }, + { + "bbox": [ + 180, + 396, + 211, + 406 + ], + "score": 0.86, + "content": "{ < } \\mathbf { C } { \\boldsymbol { 2 } } \\mathbf { s } >", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 395, + 507, + 408 + ], + "score": 1.0, + "content": ") as described in Fig. 1. Formally, the C2S objective can be formulated as:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 407, + 425, + 440 + ], + "lines": [ + { + "bbox": [ + 186, + 407, + 425, + 440 + ], + "spans": [ + { + "bbox": [ + 186, + 407, + 425, + 440 + ], + "score": 0.94, + "content": "L _ { c 2 s } = \\mathbb { E } \\Big ( \\sum _ { i = 1 } ^ { n } - \\log p ( x _ { i } | < c 2 \\mathsf { s } > ; \\mathsf { P E R M U T E } ( \\mathcal { C } ) ; x _ { 1 : i - 1 } ) \\Big )", + "type": "interline_equation", + "image_path": "1112622d2622f70eade71ee380d1126109f9230f51f251b1db06b6a4e2ddb39f.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 407, + 425, + 423.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 186, + 423.5, + 425, + 440.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "where the PERMUTE() function randomly shuffle the concepts in the concept-set. This objective", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 450, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 506, + 465 + ], + "score": 1.0, + "content": "requires the model to construct an acceptable commonsense sentence by adhering to and reasoning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "over the commonsense relations between the given concepts. Therefore, relational commonsense", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "knowledge is implicitly injected into the parameters of the model. The C2S objective is motivated by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "the task proposed in Lin et al. (2020). Compared to their work, the concept-set used in C2S covers", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "more concepts such as named entities, while the original task only includes the concepts appearing in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "ConceptNet. We apply the task in a general domain and as a pre-training objective, instead of merely", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 518, + 226, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 226, + 529 + ], + "score": 1.0, + "content": "serving as an evaluation task.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 506, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "Concept Order Recovering (COR). As for the concept order recovering (COR) objective, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "shuffle the order of concept in a sentence and train the model to recover the original sentence. As", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "illustrated in Figure 1, given an input sentence “tree grows on the apple,”, the models would shuffle the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "concepts including “tree”, “grow”, and “apple” to recover the original sentence “apple grows on the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "tree.” The noise introduced by concept shuffling is different from that by traditional self-supervised", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "objectives like mask language modeling and mask span prediction because the corrupted source", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "sentences are in general complete (i.e., no tokens or spans are masked) and grammatically correct,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "while not acceptable in terms of commonsense because the order and relation between concepts are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "shuffled. By training the model to detect and correct the disorder of concepts in a sentence, the model", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "is expected to acquire some relational commonsense knowledge like “apple generally grows on a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 272, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 272, + 656 + ], + "score": 1.0, + "content": "tree” instead of “tree grows on an apple.”", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 311, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 312, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 312, + 673 + ], + "score": 1.0, + "content": "Formally, the COR objective can be formulated as:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 673, + 452, + 705 + ], + "lines": [ + { + "bbox": [ + 158, + 673, + 452, + 705 + ], + "spans": [ + { + "bbox": [ + 158, + 673, + 452, + 705 + ], + "score": 0.93, + "content": "L _ { c o r } = \\mathbb { E } \\Big ( \\sum _ { i = 1 } ^ { n } - \\log p ( x _ { i } | < \\mathrm { c o r } > ; 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For example, the concepts (stops, fights, bar, drives, performance),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 282, + 468, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 468, + 295 + ], + "score": 1.0, + "content": "are randomly reordered in the input, while the model should recover the original sentence.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 506, + 346 + ], + "lines": [], + "index": 11, + "bbox_fs": [ + 105, + 311, + 506, + 347 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "Concept-to-Sentence Generation (C2S). The concept-to-sentence generation (C2S) objective re-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "quires the text-to-text transformer to recover the original sentence given only a few unordered", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 486, + 386 + ], + "score": 1.0, + "content": "keywords of the sentence. 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Formally, the C2S objective can be formulated as:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 350, + 507, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 407, + 425, + 440 + ], + "lines": [ + { + "bbox": [ + 186, + 407, + 425, + 440 + ], + "spans": [ + { + "bbox": [ + 186, + 407, + 425, + 440 + ], + "score": 0.94, + "content": "L _ { c 2 s } = \\mathbb { E } \\Big ( \\sum _ { i = 1 } ^ { n } - \\log p ( x _ { i } | < c 2 \\mathsf { s } > ; \\mathsf { P E R M U T E } ( \\mathcal { C } ) ; x _ { 1 : i - 1 } ) \\Big )", + "type": "interline_equation", + "image_path": "1112622d2622f70eade71ee380d1126109f9230f51f251b1db06b6a4e2ddb39f.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 407, + 425, + 423.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 186, + 423.5, + 425, + 440.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "where the PERMUTE() function randomly shuffle the concepts in the concept-set. This objective", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 450, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 506, + 465 + ], + "score": 1.0, + "content": "requires the model to construct an acceptable commonsense sentence by adhering to and reasoning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "over the commonsense relations between the given concepts. Therefore, relational commonsense", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "knowledge is implicitly injected into the parameters of the model. 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}, + { + "type": "text", + "bbox": [ + 107, + 566, + 317, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 318, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 318, + 578 + ], + "score": 1.0, + "content": "The aforementioned generative and contrastive self-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 577, + 317, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 317, + 590 + ], + "score": 1.0, + "content": "supervised objectives can be applied independently", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 588, + 318, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 318, + 600 + ], + "score": 1.0, + "content": "or simply combined in a multi-task learning fashion.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 599, + 318, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 318, + 611 + ], + "score": 1.0, + "content": "We argue that these two objectives can mutually re-", + "type": 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Algorithm Aware Language Model (CALM).1: Pre-training Concept-
Input:Text-to-Text Transformer Tθ,Text corpus X=[x1,x2,...,xn]. repeat
foreach xi∈Xdo Extract the concept-set Ci; Construct the distractor sentence
x'=CONCEPT-PERMUTE(Xi,Ci); Update Tθ with Eq.(1,2, 4);
until maximum iterations reached; repeat
for each xi ∈Xdo
Update Tθwith Eq.(7)
until maximum iterationsreached;
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The function CONCEPT-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "PERMUTE() permutes the order between concepts in the same category (i.e. noun or verb) in the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 283, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 283, + 117 + ], + "score": 1.0, + "content": "sentence, which can be formally defined as:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 507, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 122, + 481, + 164 + ], + "lines": [ + { + "bbox": [ + 131, + 122, + 481, + 164 + ], + "spans": [ + { + "bbox": [ + 131, + 122, + 481, + 164 + ], + "score": 0.73, + "content": "\\begin{array} { r } { \\mathrm { \\scriptscriptstyle { N C E P T - P E R M U T E } } ( \\mathbf { x } , \\mathcal { C } ) = [ x _ { 1 } ^ { \\prime } , x _ { 2 } ^ { \\prime } , \\ldots , x _ { n } ^ { \\prime } ] \\mathrm { \\ w h e r e \\ } x _ { i } ^ { \\prime } = \\left\\{ \\begin{array} { l l } { x _ { i } } & { x _ { i } \\notin \\mathcal C } \\\\ { \\mathrm { P E R M U T E } ( \\mathcal { C } _ { v } ) [ j ] } & { x _ { i } = v _ { j } } \\\\ { \\mathrm { P E R M U T E } ( \\mathcal { C } _ { n } ) [ j ] } & { x _ { i } = n _ { j } } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "670bdf95e62afb22f4a4a1733a3e6d7cbbda230c302ea47ec193f4373fc48869.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 131, + 122, + 481, + 136.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 131, + 136.0, + 481, + 150.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 131, + 150.0, + 481, + 164.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 175, + 506, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "Our proposed objectives require the model to capture the relational commonsense knowledge between", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "concepts and perform relational (COR) and compositional (C2S) commonsense reasoning in order to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 198, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 507, + 210 + ], + "score": 1.0, + "content": "successfully reconstruct the original sentence. 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The distractor", + "type": "text" + }, + { + "bbox": [ + 367, + 491, + 376, + 501 + ], + "score": 0.88, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "is either constructed by concept", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 255, + 515 + ], + "score": 1.0, + "content": "shuffling as described previously (i.e.", + "type": "text" + }, + { + "bbox": [ + 256, + 502, + 278, + 513 + ], + "score": 0.76, + "content": "x ^ { \\prime } =", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 502, + 362, + 515 + ], + "score": 1.0, + "content": "CONCEPT-PERMUTE", + "type": "text" + }, + { + "bbox": [ + 362, + 502, + 387, + 514 + ], + "score": 0.88, + "content": "( \\mathbf { x } , { \\mathcal { C } } ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 502, + 506, + 515 + ], + "score": 1.0, + "content": ") when used independently, or", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "generated by a generator trained with the aforementioned generative objectives when used in the joint", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 524, + 363, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 363, + 536 + ], + "score": 1.0, + "content": "training framework, which will be described in the next section.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 490, + 506, + 536 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 546, + 483, + 560 + ], + "lines": [ + { + "bbox": [ + 104, + 545, + 484, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 484, + 561 + ], + "score": 1.0, + "content": "3 JOINT TRAINING WITH GENERATIVE AND CONTRASTIVE OBJECTIVES", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 317, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 318, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 318, + 578 + ], + "score": 1.0, + "content": "The aforementioned generative and contrastive self-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 577, + 317, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 317, + 590 + ], + "score": 1.0, + "content": "supervised objectives can be applied independently", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 588, + 318, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 318, + 600 + ], + "score": 1.0, + "content": "or simply combined in a multi-task learning fashion.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 599, + 318, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 318, + 611 + ], + "score": 1.0, + "content": "We argue that 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The loss function of", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "the proposed joint training framework consists of two parts: the first part is the loss of generative", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "objectives, which is identical to the loss described in Eq.(1) and Eq.(2) and is used to update the", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 147, + 343 + ], + "score": 1.0, + "content": "generator", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 147, + 330, + 160, + 341 + ], + "score": 0.89, + "content": "G _ { \\theta }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 161, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ". The second part is the loss of the contrastive objective as described in Eq.(4), which", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 195, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 195, + 353 + ], + "score": 1.0, + "content": "can be formulated as:", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 64.5, + "bbox_fs": [ + 105, + 709, + 317, + 735 + ] + }, + { + "type": "table", + "bbox": [ + 324, + 568, + 504, + 727 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 324, + 568, + 504, + 727 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 324, + 568, + 504, + 727 + ], + "spans": [ + { + "bbox": [ + 324, + 568, + 504, + 727 + ], + "score": 0.648, + "html": "
Algorithm Aware Language Model (CALM).1: Pre-training Concept-
Input:Text-to-Text Transformer Tθ,Text corpus X=[x1,x2,...,xn]. repeat
foreach xi∈Xdo Extract the concept-set Ci; Construct the distractor sentence
x'=CONCEPT-PERMUTE(Xi,Ci); Update Tθ with Eq.(1,2, 4);
until maximum iterations reached; repeat
for each xi ∈Xdo
Update Tθwith Eq.(7)
until maximum iterationsreached;
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Given an input sentence", + "type": "text" + }, + { + "bbox": [ + 417, + 187, + 424, + 195 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "(“She was the first", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 196, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 328, + 208 + ], + "score": 1.0, + "content": "woman to hold the position.”), we extract concept-set", + "type": "text" + }, + { + "bbox": [ + 328, + 196, + 335, + 205 + ], + "score": 0.76, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 196, + 469, + 208 + ], + "score": 1.0, + "content": "(woman, hold, position). Given", + "type": "text" + }, + { + "bbox": [ + 469, + 198, + 476, + 205 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 196, + 495, + 208 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 496, + 196, + 502, + 205 + ], + "score": 0.75, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 196, + 506, + 208 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 206, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 265, + 218 + ], + "score": 1.0, + "content": "we produce corrupted source sequence", + "type": "text" + }, + { + "bbox": [ + 266, + 207, + 276, + 216 + ], + "score": 0.86, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 206, + 506, + 218 + ], + "score": 1.0, + "content": "either for C2S and COR. The generator trained with the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 335, + 230 + ], + "score": 1.0, + "content": "corresponding objective recovers sentences as distractors", + "type": "text" + }, + { + "bbox": [ + 335, + 217, + 348, + 227 + ], + "score": 0.88, + "content": "x ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "to the discriminator. The discriminator", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 228, + 507, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 507, + 241 + ], + "score": 1.0, + "content": "is trained to distinguish truth sentences from randomly selected distractor among two objectives.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 239, + 362, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 362, + 251 + ], + "score": 1.0, + "content": "Parameters between the generator and discriminator are shared.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 361, + 277 + ], + "score": 1.0, + "content": "(trained with the contrastive objective). Given an input sentence", + "type": "text" + }, + { + "bbox": [ + 361, + 267, + 368, + 274 + ], + "score": 0.34, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 264, + 506, + 277 + ], + "score": 1.0, + "content": ", we first use the method for either", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 331, + 288 + ], + "score": 1.0, + "content": "C2S or COR to produce the corrupted source sequence", + "type": "text" + }, + { + "bbox": [ + 332, + 276, + 342, + 286 + ], + "score": 0.85, + "content": "\\mathbf { x } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 275, + 459, + 288 + ], + "score": 1.0, + "content": ". Then, we use the generator", + "type": "text" + }, + { + "bbox": [ + 460, + 276, + 473, + 286 + ], + "score": 0.88, + "content": "G _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "trained", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 504, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 378, + 299 + ], + "score": 1.0, + "content": "with the corresponding objective to generate the recovered sentence", + "type": "text" + }, + { + "bbox": [ + 379, + 286, + 433, + 298 + ], + "score": 0.93, + "content": "\\mathbf { x } ^ { \\prime \\prime } = G _ { \\theta } ( \\bar { x } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 286, + 491, + 299 + ], + "score": 1.0, + "content": ". We then take", + "type": "text" + }, + { + "bbox": [ + 491, + 286, + 504, + 297 + ], + "score": 0.86, + "content": "\\mathbf { x } ^ { \\prime \\prime }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 278, + 309 + ], + "score": 1.0, + "content": "as the distractor to train the discriminator", + "type": "text" + }, + { + "bbox": [ + 278, + 298, + 293, + 309 + ], + "score": 0.9, + "content": "D _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "with the contrastive objective. The loss function of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "the proposed joint training framework consists of two parts: the first part is the loss of generative", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "objectives, which is identical to the loss described in Eq.(1) and Eq.(2) and is used to update the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 147, + 343 + ], + "score": 1.0, + "content": "generator", + "type": "text" + }, + { + "bbox": [ + 147, + 330, + 160, + 341 + ], + "score": 0.89, + "content": "G _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ". The second part is the loss of the contrastive objective as described in Eq.(4), which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 195, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 195, + 353 + ], + "score": 1.0, + "content": "can be formulated as:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 355, + 478, + 399 + ], + "lines": [ + { + "bbox": [ + 121, + 355, + 478, + 399 + ], + "spans": [ + { + "bbox": [ + 121, + 355, + 478, + 399 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { L _ { c o n t \\_ j o i n t \\_ c 2 s } = \\mathbb { E } \\big ( - \\log \\mathrm { D } _ { \\phi } ( y | < \\mathrm { c o n t } > ; x ; \\mathbf { G } _ { \\theta } ( < \\mathrm { c } 2 \\mathsf { s } > ; \\mathsf { P E R M U T E } ( \\mathcal { C } ) ) \\big ) } \\\\ & { L _ { c o n t \\_ j o i n t \\_ c o r r } = \\mathbb { E } \\big ( - \\log \\mathrm { D } _ { \\phi } ( y | < \\mathrm { c o n t } > ; x ; \\mathbf { G } _ { \\theta } ( < \\mathrm { c o r } > ; \\mathrm { C o n c E P T } \\cdot \\mathrm { P E R M U T E } ( \\mathbf { x } , \\mathcal { C } ) ) \\big ) } \\end{array}", + "type": "interline_equation", + "image_path": "1d84e1cf0c5e1acd245645b5b393afeb107f43679d05242008d90ce19fa32c1f.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 121, + 355, + 478, + 369.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 121, + 369.6666666666667, + 478, + 384.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 121, + 384.33333333333337, + 478, + 399.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 405, + 504, + 438 + ], + "lines": [ + { + "bbox": [ + 104, + 403, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 133, + 419 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 405, + 194, + 417 + ], + "score": 0.88, + "content": "L _ { c o n t \\_ j o i n t \\_ c 2 s }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 403, + 213, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 405, + 274, + 417 + ], + "score": 0.87, + "content": "L _ { c o n t \\_ j o i n t \\_ c o r }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 403, + 506, + 419 + ], + "score": 1.0, + "content": "is the contrastive loss with the distractor generated with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 266, + 428 + ], + "score": 1.0, + "content": "either the C2S or the COR objective and", + "type": "text" + }, + { + "bbox": [ + 266, + 418, + 273, + 428 + ], + "score": 0.82, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "is the original sentence. We then have the overall objective", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 426, + 284, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 284, + 439 + ], + "score": 1.0, + "content": "for the joint training framework defined as :", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 444, + 435, + 458 + ], + "lines": [ + { + "bbox": [ + 176, + 444, + 435, + 458 + ], + "spans": [ + { + "bbox": [ + 176, + 444, + 435, + 458 + ], + "score": 0.87, + "content": "L _ { j o i n t } = ( L _ { c 2 s } + L _ { c o r } ) + \\beta ( L _ { c o n t \\_ j o i n t \\_ c 2 s } + L _ { c o n t \\_ j o i n t \\_ c o r } ) .", + "type": "interline_equation", + "image_path": "710d503fa012702f933e0a1eb28b7f2e23d7278fbe31b3c7dcdf3048975267af.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 176, + 444, + 435, + 458 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 107, + 462, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 107, + 464, + 126, + 475 + ], + "score": 0.9, + "content": "L _ { c 2 s }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 462, + 145, + 477 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 145, + 464, + 164, + 475 + ], + "score": 0.92, + "content": "L _ { c o r }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 462, + 366, + 477 + ], + "score": 1.0, + "content": "are defined in Eq.(1) and Eq.(2) respectively and", + "type": "text" + }, + { + "bbox": [ + 366, + 464, + 374, + 475 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 462, + 505, + 477 + ], + "score": 1.0, + "content": "is a hyperparameter controlling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "the relative weight between the generative and contrastive objectives. Note that since we would", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "like to inject both generative and discriminative commonsense reasoning ability into the parameters", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 456, + 509 + ], + "score": 1.0, + "content": "of a single text-to-text transformer, we share the parameters between the generator", + "type": "text" + }, + { + "bbox": [ + 456, + 497, + 469, + 507 + ], + "score": 0.88, + "content": "G _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 505, + 181, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 162, + 521 + ], + "score": 1.0, + "content": "discriminator", + "type": "text" + }, + { + "bbox": [ + 162, + 508, + 177, + 520 + ], + "score": 0.9, + "content": "D _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 505, + 181, + 521 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "Finally, we describe the overall procedure to apply the proposed self-supervised objectives and the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 536, + 507, + 548 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 507, + 548 + ], + "score": 1.0, + "content": "joint training framework on a pre-trained text-to-text transformer. We apply a two-stage training strat-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "egy. During the first stage, we apply our proposed generative and contrastive objectives individually", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 556, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 505, + 572 + ], + "score": 1.0, + "content": "on the model in a multi-task learning fashion with different prefixes. This provides a good starting", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "point for the second stage where the joint training framework is applied. We summarize the workflow", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 580, + 230, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 230, + 591 + ], + "score": 1.0, + "content": "of our method in Algorithm 1.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 607, + 200, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 201, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 201, + 622 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 640 + ], + "score": 1.0, + "content": "In this section, motivated by the observation of Pruksachatkun et al. 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We then fine-tune the CALM", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "on each downstream task individually and report the average performance of three runs with different", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "random seeds for fine-tuning on each dataset since the performance is sensitive to different random", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 474, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 474, + 150 + ], + "score": 1.0, + "content": "seeds. Training details and hyperparameter settings are presented in Appendix A.1 and A.2.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 506, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "Datasets We consider five commonsense benchmark datasets as target tasks. We categorize these", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "datasets into discriminative and generative tasks. Discriminative tasks are classification tasks while", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "score": 1.0, + "content": "generative tasks are text generation tasks. We consider four datasets for discriminative task: Com-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "monsenseQA (Talmor et al., 2018), OpenbookQA (Mihaylov et al., 2018), PIQA (Bisk et al., 2020),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 506, + 210 + ], + "score": 1.0, + "content": "aNLI (Bhagavatula et al., 2019) and one dataset for generative task: CommonGEN (Lin et al., 2020).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 311, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 311, + 221 + ], + "score": 1.0, + "content": "Details on datasets are discussed in Appendix A.3.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "Compared Methods We compare our model with following models continually trained with different", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "intermediate tasks based on the pre-trained T5-base model: (1) T5-base is the pre-trained T5-base", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "model without continually training on any intermediate task. (2) T5-base w/ additional epochs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 258, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 506, + 273 + ], + "score": 1.0, + "content": "is continually pre-trained using the original pre-training objective of T5 with additional training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "steps. The total number of additional training steps is equal to that of our final model. (3) T5-base", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 136, + 291 + ], + "score": 0.57, + "content": "\\mathbf { \\Gamma } + \\mathbf { S S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "is continual pre-trained with a variant of the salient span masking objective (Guu et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 306 + ], + "score": 1.0, + "content": "2020; Roberts et al., 2020) objective that masks text spans of concepts extracted with POS tagging", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "instead of named entities extracted by a pre-trained NER model, which makes it more focused on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "concepts. (4) CALM(Generative-Only) is continually pre-trained with the proposed generative", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "objectives including concept-to-sentence generation(C2S) and concept order recovering(COR) as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "intermediate tasks. (5) CALM(Contrastive-Only) is continually pre-trained with the proposed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "contrastive objective as described in section 2.2 using the distractor generated by concept shuffling. (6)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "CALM(Mix-only) is continually pre-trained with both the generative objectives and the contrastive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "objective, combined with a multi-task learning fashion with identical weights for each objective as the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "score": 1.0, + "content": "intermediate task. (7) CALM (w/o Mix warmup) is continually pre-trained with the joint training", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "score": 1.0, + "content": "objective described in Eq (7) directly from the pre-trained T5-base model. (8) CALM is our main", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "model trained as described in Algorithm 1. The difference between CALM and CALM (Joint) is that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "the former is initialized by the CALM(Mix). We also include the performance of the BERT-base", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 424, + 466, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 466, + 436 + ], + "score": 1.0, + "content": "model and two knowledge enhanced PTLMs that have similar architecture to BERT-base.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "Evaluation Metrics For discriminative tasks, we choose accuracy as our metric following other", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "conventional question answering tasks. For generative tasks, we report automated metrics including", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 475 + ], + "score": 1.0, + "content": "BLEU (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), CIDEr (Vedantam et al., 2015),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 507, + 486 + ], + "score": 1.0, + "content": "and SPICE (Anderson et al., 2016) following the leaderboard of COMMONGEN (Lin et al., 2020).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Results for COMMONGEN are on the test set and others are on the official development set. We tune", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 438, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 438, + 508 + ], + "score": 1.0, + "content": "the hyperparameters based on the models’ performance on a in-house split dev set.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 240, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 241, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 241, + 532 + ], + "score": 1.0, + "content": "4.1 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "The result is presented in Table 1. First, we can see that our CALM model consistently and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 214, + 565 + ], + "score": 1.0, + "content": "significantly (with p-value", + "type": "text" + }, + { + "bbox": [ + 214, + 552, + 243, + 562 + ], + "score": 0.82, + "content": "< 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 551, + 505, + 565 + ], + "score": 1.0, + "content": ") outperforms the backbone T5-base model on all five datasets by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 390, + 576 + ], + "score": 1.0, + "content": "a margin range from 1.5 to 2.9 accuracy on discriminative tasks and", + "type": "text" + }, + { + "bbox": [ + 390, + 563, + 420, + 573 + ], + "score": 0.56, + "content": "1 . 5 / 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "BLEU/SPICE score", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 572, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 588 + ], + "score": 1.0, + "content": "on CommonGEN. This is an impressive result since we are only performing intermediate training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 296, + 597 + ], + "score": 1.0, + "content": "on a relatively small dataset for only around", + "type": "text" + }, + { + "bbox": [ + 296, + 585, + 313, + 595 + ], + "score": 0.53, + "content": "2 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "updates. It demonstrates the potential of our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "method for serving as a “plug-and-play” method for packing more commonsense knowledge into a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "pre-trained text-to-text transformer. Table 3 also shows that CALM performs comparably with several", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "large-size PTLMs like BART, T5-large, and GPT-2 on the COMMONGEN dataset. The performance", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "is worse than KG-BART (Liu et al., 2020), the current state-of-the-art on COMMONGEN, which is a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "contemporary work that exploits external knowledge bases as additional information, and is based on", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 650, + 311, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 311, + 663 + ], + "score": 1.0, + "content": "a larger backbone(i.e., BART (Lewis et al., 2019)).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "In addition, we can observe that both the proposed generative and contrastive objective outperforms", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "the backbone T5-base model, as well as its variants that continually pre-trained with the original", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "masked span prediction objective and the concept-specific salient span masking scheme, when applied", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "independently. 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We categorize these", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "datasets into discriminative and generative tasks. Discriminative tasks are classification tasks while", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "score": 1.0, + "content": "generative tasks are text generation tasks. We consider four datasets for discriminative task: Com-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "monsenseQA (Talmor et al., 2018), OpenbookQA (Mihaylov et al., 2018), PIQA (Bisk et al., 2020),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 506, + 210 + ], + "score": 1.0, + "content": "aNLI (Bhagavatula et al., 2019) and one dataset for generative task: CommonGEN (Lin et al., 2020).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 311, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 311, + 221 + ], + "score": 1.0, + "content": "Details on datasets are discussed in Appendix A.3.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 153, + 507, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "Compared Methods We compare our model with following models continually trained with different", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "intermediate tasks based on the pre-trained T5-base model: (1) T5-base is the pre-trained T5-base", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "model without continually training on any intermediate task. (2) T5-base w/ additional epochs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 258, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 506, + 273 + ], + "score": 1.0, + "content": "is continually pre-trained using the original pre-training objective of T5 with additional training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "steps. The total number of additional training steps is equal to that of our final model. (3) T5-base", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 136, + 291 + ], + "score": 0.57, + "content": "\\mathbf { \\Gamma } + \\mathbf { S S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "is continual pre-trained with a variant of the salient span masking objective (Guu et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 306 + ], + "score": 1.0, + "content": "2020; Roberts et al., 2020) objective that masks text spans of concepts extracted with POS tagging", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "instead of named entities extracted by a pre-trained NER model, which makes it more focused on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "concepts. (4) CALM(Generative-Only) is continually pre-trained with the proposed generative", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "objectives including concept-to-sentence generation(C2S) and concept order recovering(COR) as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "intermediate tasks. (5) CALM(Contrastive-Only) is continually pre-trained with the proposed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "contrastive objective as described in section 2.2 using the distractor generated by concept shuffling. (6)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "CALM(Mix-only) is continually pre-trained with both the generative objectives and the contrastive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "objective, combined with a multi-task learning fashion with identical weights for each objective as the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 393 + ], + "score": 1.0, + "content": "intermediate task. (7) CALM (w/o Mix warmup) is continually pre-trained with the joint training", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "score": 1.0, + "content": "objective described in Eq (7) directly from the pre-trained T5-base model. (8) CALM is our main", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "model trained as described in Algorithm 1. The difference between CALM and CALM (Joint) is that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "the former is initialized by the CALM(Mix). We also include the performance of the BERT-base", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 424, + 466, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 466, + 436 + ], + "score": 1.0, + "content": "model and two knowledge enhanced PTLMs that have similar architecture to BERT-base.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 226, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "Evaluation Metrics For discriminative tasks, we choose accuracy as our metric following other", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "conventional question answering tasks. For generative tasks, we report automated metrics including", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 507, + 475 + ], + "score": 1.0, + "content": "BLEU (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), CIDEr (Vedantam et al., 2015),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 507, + 486 + ], + "score": 1.0, + "content": "and SPICE (Anderson et al., 2016) following the leaderboard of COMMONGEN (Lin et al., 2020).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Results for COMMONGEN are on the test set and others are on the official development set. We tune", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 495, + 438, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 438, + 508 + ], + "score": 1.0, + "content": "the hyperparameters based on the models’ performance on a in-house split dev set.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 439, + 507, + 508 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 240, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 241, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 241, + 532 + ], + "score": 1.0, + "content": "4.1 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "The result is presented in Table 1. First, we can see that our CALM model consistently and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 214, + 565 + ], + "score": 1.0, + "content": "significantly (with p-value", + "type": "text" + }, + { + "bbox": [ + 214, + 552, + 243, + 562 + ], + "score": 0.82, + "content": "< 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 551, + 505, + 565 + ], + "score": 1.0, + "content": ") outperforms the backbone T5-base model on all five datasets by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 390, + 576 + ], + "score": 1.0, + "content": "a margin range from 1.5 to 2.9 accuracy on discriminative tasks and", + "type": "text" + }, + { + "bbox": [ + 390, + 563, + 420, + 573 + ], + "score": 0.56, + "content": "1 . 5 / 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "BLEU/SPICE score", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 572, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 588 + ], + "score": 1.0, + "content": "on CommonGEN. This is an impressive result since we are only performing intermediate training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 296, + 597 + ], + "score": 1.0, + "content": "on a relatively small dataset for only around", + "type": "text" + }, + { + "bbox": [ + 296, + 585, + 313, + 595 + ], + "score": 0.53, + "content": "2 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "updates. It demonstrates the potential of our", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "method for serving as a “plug-and-play” method for packing more commonsense knowledge into a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "pre-trained text-to-text transformer. Table 3 also shows that CALM performs comparably with several", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "large-size PTLMs like BART, T5-large, and GPT-2 on the COMMONGEN dataset. The performance", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "is worse than KG-BART (Liu et al., 2020), the current state-of-the-art on COMMONGEN, which is a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "contemporary work that exploits external knowledge bases as additional information, and is based on", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 650, + 311, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 311, + 663 + ], + "score": 1.0, + "content": "a larger backbone(i.e., BART (Lewis et al., 2019)).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 540, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "In addition, we can observe that both the proposed generative and contrastive objective outperforms", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "the backbone T5-base model, as well as its variants that continually pre-trained with the original", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "masked span prediction objective and the concept-specific salient span masking scheme, when applied", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "independently. Note that we find the variant of salient span masking that focuses on concept is not", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 667, + 506, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 79, + 503, + 201 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 79, + 503, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 503, + 201 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 503, + 201 + ], + "score": 0.982, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-base53.08(±0.16)57.60(±0.8)64.86(±0.52)61.88(±0.56)11--
ERNIE54.06(±0.12)58.90(±0.9)66.47(±0.58)63.04(±0.46)1
KnowBERT53.88(±0.15)58.50(±0.8)66.61(±0.63)63.18(±0.52)1-1-
T5-base61.88(±0.08)58.20(±1.0)68.14(±0.73)61.10(±0.38)24.9031.2012.9932.40
T5-base + cont. pretraining61.92(±0.45)58.10(±0.9)68.19(±0.77)61.15(±0.52)25.1031.0013.1232.40
T5-base + SSM62.08(±0.41)58.30(±0.8)68.27(±0.71)61.25(±0.51)25.2031.2013.2832.40
CALM (Generative-Only)62.28(±0.36)58.90(±0.4)68.91(±0.88)60.95(±0.46)25.8031.2013.8132.60
CALM (Contrastive-Only)62.73(±0.41)59.30(±0.3)70.67(±0.98)61.35(±0.06)25.5031.2013.5832.60
CALM(w/o Mix warmup)62.18(±0.48)59.00(±0.5)69.21(±0.57)61.25(±0.55)25.8031.2013.7732.60
CALM (Mix-only)63.02(±0.47)60.40(±0.4)70.07(±0.98)62.79(±0.55)26.0031.2013.8232.80
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
", + "type": "table", + "image_path": "ada5dfc381fff46fe7830056ce2b2c898ebaae42eb6894dc7d7184a39f973ad4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 503, + 119.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 119.66666666666666, + 503, + 160.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 160.33333333333331, + 503, + 200.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 209, + 505, + 254 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "Table 1: Experimental results on commonsense reasoning datasets. The first group of models are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "baselines. The models in the middle group and last group except the CALM model are trained with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "the proposed objectives independently and the final CALM model is trained by joint training. Best", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 242, + 398, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 398, + 254 + ], + "score": 1.0, + "content": "models are bold and second best ones are underlined within each metric.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "table", + "bbox": [ + 108, + 259, + 504, + 334 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 259, + 504, + 334 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 259, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 108, + 259, + 504, + 334 + ], + "score": 0.98, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-large57.06(±0.12)60.40(±0.6)67.08(±0.61)66.75(±0.61)==
T5-large69.81(±1.02)61.40(±1.0)72.19(±1.09)75.54(±1.22)28.6030.1014.9631.60
CALM-large (Mix-only)70.26(±0.23)62.50(±1.0)73.70(±1.09)75.99(±1.26)29.2031.3015.2433.10
CALM-large71.31(±0.04)66.00(±1.0)75.11(±1.65)77.12(±0.34)29.5031.9015.6133.20
RoBERTa-large471.81(±0.25)63.90(±0.8)76.90(±0.62)82.35(±0.54)111-
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Best models are bold and second best ones are underlined within each metric.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "very effective. We suspect this is because the resulting training data would be somewhat similar to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "the original text infilling objective because concepts are very common in the corpus and we only train", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "for a few steps. The combination of the generative and contrastive objectives (i.e., CALM(Mix-only))", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "score": 1.0, + "content": "yields further improvement upon the model trained independently with either generative or contrastive", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 415, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 431 + ], + "score": 1.0, + "content": "objectives. Also, we find that the CALM model consistently outperforms CALM(Mix), demonstrating", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "the effectiveness of the proposed joint training framework. Applying joint training directly on top", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "of a pre-trained model (i.e., CALM(w/o Mix warmup)) does not work very well, demonstrating the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 449, + 461, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 461, + 463 + ], + "score": 1.0, + "content": "necessity of applying mixed training to initialize the model before starting joint training.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 271, + 620 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 272, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 272, + 478 + ], + "score": 1.0, + "content": "To further confirm the effectiveness of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 271, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 271, + 489 + ], + "score": 1.0, + "content": "our approach, we also apply our method", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 272, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 272, + 501 + ], + "score": 1.0, + "content": "to continually pre-train T5-large with the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 498, + 273, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 273, + 512 + ], + "score": 1.0, + "content": "same data and number of training steps.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 510, + 272, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 272, + 523 + ], + "score": 1.0, + "content": "We then compare the performance of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 271, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 271, + 533 + ], + "score": 1.0, + "content": "resulting model with that of the original", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 532, + 272, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 272, + 545 + ], + "score": 1.0, + "content": "T5-large model in Table 10. We find that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 543, + 272, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 272, + 555 + ], + "score": 1.0, + "content": "both the proposed training objectives and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 272, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 272, + 567 + ], + "score": 1.0, + "content": "the joint training framework consistently", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 565, + 272, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 236, + 578 + ], + "score": 1.0, + "content": "and significantly (with p-value", + "type": "text" + }, + { + "bbox": [ + 236, + 565, + 267, + 576 + ], + "score": 0.81, + "content": "< 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 565, + 272, + 578 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 577, + 273, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 273, + 587 + ], + "score": 1.0, + "content": "improve upon the original T5-large, show-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 588, + 272, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 272, + 598 + ], + "score": 1.0, + "content": "ing our approach is effective for models", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 598, + 273, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 273, + 609 + ], + "score": 1.0, + "content": "with different sizes. Our model also out-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 609, + 273, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 273, + 622 + ], + "score": 1.0, + "content": "performs BERT-large by a large margin.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26.5 + }, + { + "type": "table", + "bbox": [ + 280, + 469, + 502, + 559 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 280, + 469, + 502, + 559 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 280, + 469, + 502, + 559 + ], + "spans": [ + { + "bbox": [ + 280, + 469, + 502, + 559 + ], + "score": 0.977, + "html": "
MethodsParamsCommonGEN
BLEU-4METEORCIDErSPICE
GPT-2 (Radford et al.,2019)774M21.1026.2012.1525.90
UniLM(Dong et al.,2019)340M27.7029.7014.8530.20
BART(Lewis et al.,2020)406M26.3030.9013.9230.60
T5-base (Raffel et al.,2019)220M16.4023.009.1622.00
T5-large (Raffel et al., 2019)770M28.6030.1014.9631.60
KG-BART (Liu et al., 2020)406M30.9032.4016.8332.70
T5-base (our implementation)220M24.9031.2012.9932.40
CALM-base220M26.4031.4013.8833.00
CALM-large774M29.5031.9015.6133.20
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We suspect this is because", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "RoBERTa-large is optimized for more steps than T5-large and our CALM-large. This is also observed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 642, + 240, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 240, + 654 + ], + "score": 1.0, + "content": "in many other tasks and datasets.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 244, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 245, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 245, + 675 + ], + "score": 1.0, + "content": "4.2 PERFORMANCE ANALYSIS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Analysis on Generative objective To investigate the contribution of each generative objective, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "conduct an ablation study by continually pre-training three models from the same T5-base model with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "C2S, COR, and text infilling, which is the original objective for pre-training T5, as the objective for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the intermediate task. We continually pre-train these models for the same number of steps and then", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "evaluate their performance by fine-tuning on different target tasks. The result is shown in Table 4.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 79, + 503, + 201 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 79, + 503, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 503, + 201 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 503, + 201 + ], + "score": 0.982, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-base53.08(±0.16)57.60(±0.8)64.86(±0.52)61.88(±0.56)11--
ERNIE54.06(±0.12)58.90(±0.9)66.47(±0.58)63.04(±0.46)1
KnowBERT53.88(±0.15)58.50(±0.8)66.61(±0.63)63.18(±0.52)1-1-
T5-base61.88(±0.08)58.20(±1.0)68.14(±0.73)61.10(±0.38)24.9031.2012.9932.40
T5-base + cont. pretraining61.92(±0.45)58.10(±0.9)68.19(±0.77)61.15(±0.52)25.1031.0013.1232.40
T5-base + SSM62.08(±0.41)58.30(±0.8)68.27(±0.71)61.25(±0.51)25.2031.2013.2832.40
CALM (Generative-Only)62.28(±0.36)58.90(±0.4)68.91(±0.88)60.95(±0.46)25.8031.2013.8132.60
CALM (Contrastive-Only)62.73(±0.41)59.30(±0.3)70.67(±0.98)61.35(±0.06)25.5031.2013.5832.60
CALM(w/o Mix warmup)62.18(±0.48)59.00(±0.5)69.21(±0.57)61.25(±0.55)25.8031.2013.7732.60
CALM (Mix-only)63.02(±0.47)60.40(±0.4)70.07(±0.98)62.79(±0.55)26.0031.2013.8232.80
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
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MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-large57.06(±0.12)60.40(±0.6)67.08(±0.61)66.75(±0.61)==
T5-large69.81(±1.02)61.40(±1.0)72.19(±1.09)75.54(±1.22)28.6030.1014.9631.60
CALM-large (Mix-only)70.26(±0.23)62.50(±1.0)73.70(±1.09)75.99(±1.26)29.2031.3015.2433.10
CALM-large71.31(±0.04)66.00(±1.0)75.11(±1.65)77.12(±0.34)29.5031.9015.6133.20
RoBERTa-large471.81(±0.25)63.90(±0.8)76.90(±0.62)82.35(±0.54)111-
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Best models are bold and second best ones are underlined within each metric.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 341, + 505, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 372, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "very effective. We suspect this is because the resulting training data would be somewhat similar to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "the original text infilling objective because concepts are very common in the corpus and we only train", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "for a few steps. The combination of the generative and contrastive objectives (i.e., CALM(Mix-only))", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "score": 1.0, + "content": "yields further improvement upon the model trained independently with either generative or contrastive", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 415, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 431 + ], + "score": 1.0, + "content": "objectives. Also, we find that the CALM model consistently outperforms CALM(Mix), demonstrating", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 442 + ], + "score": 1.0, + "content": "the effectiveness of the proposed joint training framework. Applying joint training directly on top", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "of a pre-trained model (i.e., CALM(w/o Mix warmup)) does not work very well, demonstrating the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 449, + 461, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 461, + 463 + ], + "score": 1.0, + "content": "necessity of applying mixed training to initialize the model before starting joint training.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 373, + 506, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 271, + 620 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 272, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 272, + 478 + ], + "score": 1.0, + "content": "To further confirm the effectiveness of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 271, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 271, + 489 + ], + "score": 1.0, + "content": "our approach, we also apply our method", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 272, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 272, + 501 + ], + "score": 1.0, + "content": "to continually pre-train T5-large with the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 498, + 273, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 273, + 512 + ], + "score": 1.0, + "content": "same data and number of training steps.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 510, + 272, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 272, + 523 + ], + "score": 1.0, + "content": "We then compare the performance of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 271, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 271, + 533 + ], + "score": 1.0, + "content": "resulting model with that of the original", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 532, + 272, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 272, + 545 + ], + "score": 1.0, + "content": "T5-large model in Table 10. We find that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 543, + 272, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 272, + 555 + ], + "score": 1.0, + "content": "both the proposed training objectives and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 272, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 272, + 567 + ], + "score": 1.0, + "content": "the joint training framework consistently", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 565, + 272, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 236, + 578 + ], + "score": 1.0, + "content": "and significantly (with p-value", + "type": "text" + }, + { + "bbox": [ + 236, + 565, + 267, + 576 + ], + "score": 0.81, + "content": "< 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 565, + 272, + 578 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 577, + 273, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 273, + 587 + ], + "score": 1.0, + "content": "improve upon the original T5-large, show-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 588, + 272, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 272, + 598 + ], + "score": 1.0, + "content": "ing our approach is effective for models", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 598, + 273, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 273, + 609 + ], + "score": 1.0, + "content": "with different sizes. Our model also out-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 609, + 273, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 273, + 622 + ], + "score": 1.0, + "content": "performs BERT-large by a large margin.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 466, + 273, + 622 + ] + }, + { + "type": "table", + "bbox": [ + 280, + 469, + 502, + 559 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 280, + 469, + 502, + 559 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 280, + 469, + 502, + 559 + ], + "spans": [ + { + "bbox": [ + 280, + 469, + 502, + 559 + ], + "score": 0.977, + "html": "
MethodsParamsCommonGEN
BLEU-4METEORCIDErSPICE
GPT-2 (Radford et al.,2019)774M21.1026.2012.1525.90
UniLM(Dong et al.,2019)340M27.7029.7014.8530.20
BART(Lewis et al.,2020)406M26.3030.9013.9230.60
T5-base (Raffel et al.,2019)220M16.4023.009.1622.00
T5-large (Raffel et al., 2019)770M28.6030.1014.9631.60
KG-BART (Liu et al., 2020)406M30.9032.4016.8332.70
T5-base (our implementation)220M24.9031.2012.9932.40
CALM-base220M26.4031.4013.8833.00
CALM-large774M29.5031.9015.6133.20
", + "type": "table", + "image_path": "5ea7c0ba53a26539e64742930742dcca418fe1ee79b7f4ebcd1d75ee2c73cab2.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 280, + 469, + 502, + 484.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 280, + 484.0, + 502, + 499.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 280, + 499.0, + 502, + 514.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 280, + 514.0, + 502, + 529.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 280, + 529.0, + 502, + 544.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 280, + 544.0, + 502, + 559.0 + ], + "spans": [], + "index": 39 + } + ] + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 278, + 568, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 278, + 568, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 278, + 568, + 506, + 579 + ], + "score": 1.0, + "content": "Table 3: Comparison between PTLMs on CommonGEN.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 278, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 278, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "Above baselines are reported number in the leaderboard.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 277, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 277, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "T5-base(our implementation) uses different hyperparme-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 277, + 601, + 470, + 612 + ], + "spans": [ + { + "bbox": [ + 277, + 601, + 470, + 612 + ], + "score": 1.0, + "content": "ter setting than that reported in the leaderboard.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 277, + 568, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 621, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "However, our model performs slightly worse compared to RoBERTa-large. We suspect this is because", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "RoBERTa-large is optimized for more steps than T5-large and our CALM-large. This is also observed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 642, + 240, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 240, + 654 + ], + "score": 1.0, + "content": "in many other tasks and datasets.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 619, + 505, + 654 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 662, + 244, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 245, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 245, + 675 + ], + "score": 1.0, + "content": "4.2 PERFORMANCE ANALYSIS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Analysis on Generative objective To investigate the contribution of each generative objective, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "conduct an ablation study by continually pre-training three models from the same T5-base model with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "C2S, COR, and text infilling, which is the original objective for pre-training T5, as the objective for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the intermediate task. We continually pre-train these models for the same number of steps and then", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "evaluate their performance by fine-tuning on different target tasks. The result is shown in Table 4.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 676, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 74, + 307, + 118 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 74, + 307, + 118 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 74, + 307, + 118 + ], + "spans": [ + { + "bbox": [ + 108, + 74, + 307, + 118 + ], + "score": 0.959, + "html": "
MethodsCSQA PIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
T5- Text Infilling61.9268.1925.1031.0013.1332.40
CALM-COR62.3668.7725.7031.2013.6532.60
CALM-C2S62.2468.7525.9031.4013.9432.80
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MethodsCSQAPIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
Multi-choice QA62.2168.8225.0031.2013.2832.60
True/False62.2467.8125.1031.2013.4132.60
Generative QA62.7370.6725.5031.2013.5832.60
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Left table shows the performance", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "on downstream tasks by pre-training with different generative objective (COR, C2S, and original", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "objective for pre-training T5). Right table shows the performance on downstream tasks by pre-training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 168, + 312, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 312, + 181 + ], + "score": 1.0, + "content": "with different task formats of contrastive objective.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "We can see that both C2S and COR works better than the original masked span infilling objective", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "on itself. This confirms the effectiveness of our proposed generative objectives on improving the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 211, + 394, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 394, + 222 + ], + "score": 1.0, + "content": "commonsense reasoning ability of pre-trained text-to-text transformers.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "Task Formulation of the Contrastive objectives For contrastive objectives, we test three different", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 238, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 250 + ], + "score": 1.0, + "content": "task formats: Multi-choice QA, Generative QA, and True/False. Multi-choice QA and Generative", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "score": 1.0, + "content": "QA takes the concatenation of the real sentence and the distractor. Then, Multi-choice QA output the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "index of the real sentence following other conventional Multi-choice QA tasks, and Generative QA", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "output the real sentence respectively. True/False takes either the real sentence or the distractor and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "score": 1.0, + "content": "train the model to perform a binary classification problem of whether the input sentence makes sense.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "The result is shown in Table 4. We could find that the format of Generative QA performs the best.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "We suspect this is because the Generative QA format is closer to the format used during the original", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 315, + 390, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 390, + 329 + ], + "score": 1.0, + "content": "pre-training stage of the T5 model and the format used for fine-tuning.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 258, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 258, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 258, + 345 + ], + "score": 1.0, + "content": "Performance with fewer training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 343, + 259, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 259, + 355 + ], + "score": 1.0, + "content": "examples To investigate the effec-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 354, + 258, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 258, + 366 + ], + "score": 1.0, + "content": "tiveness of our objective in the low-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 366, + 258, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 258, + 376 + ], + "score": 1.0, + "content": "resource setting, we explore the per-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 376, + 257, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 257, + 387 + ], + "score": 1.0, + "content": "formance of our model and baselines", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 387, + 258, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 258, + 398 + ], + "score": 1.0, + "content": "fine-tuning with different fractions of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 398, + 258, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 258, + 409 + ], + "score": 1.0, + "content": "the training data. From Figure 4, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 409, + 258, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 258, + 421 + ], + "score": 1.0, + "content": "can see that the performance improve-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 420, + 258, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 258, + 431 + ], + "score": 1.0, + "content": "ment yielded by our models upon the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 430, + 257, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 257, + 442 + ], + "score": 1.0, + "content": "T5-base model is more significant in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 442, + 258, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 258, + 453 + ], + "score": 1.0, + "content": "the low-resource regime. This shows", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 451, + 257, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 257, + 466 + ], + "score": 1.0, + "content": "that CALM may already pack some", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 30.5 + }, + { + "type": "image", + "bbox": [ + 274, + 331, + 486, + 427 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 274, + 331, + 486, + 427 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 274, + 331, + 486, + 427 + ], + "spans": [ + { + "bbox": [ + 274, + 331, + 486, + 427 + ], + "score": 0.963, + "type": "image", + "image_path": "d53949681d0c53f39d987e7afd217298a347d7f0231c014211f25f5c381ab759.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 274, + 331, + 486, + 344.7142857142857 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 274, + 344.7142857142857, + 486, + 358.42857142857144 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 274, + 358.42857142857144, + 486, + 372.14285714285717 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 274, + 372.14285714285717, + 486, + 385.8571428571429 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 274, + 385.8571428571429, + 486, + 399.5714285714286 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 274, + 399.5714285714286, + 486, + 413.28571428571433 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 274, + 413.28571428571433, + 486, + 427.00000000000006 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 264, + 429, + 504, + 451 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 264, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 264, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "Figure 4: Performance of compared models fine-tuned with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 264, + 439, + 397, + 452 + ], + "spans": [ + { + "bbox": [ + 264, + 439, + 397, + 452 + ], + "score": 1.0, + "content": "different fraction of the datasets.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "index": 39.75 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 504, + 476 + ], + "score": 1.0, + "content": "commonsense knowledge in its parameters so that it does not require much data for fine-tuning before", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 472, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 489 + ], + "score": 1.0, + "content": "obtaining a good performance. In contrast, the original T5 model requires much data for fine-tuning,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "which suggests it may fail to encode much commonsense knowledge and must fit the correlation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 496, + 360, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 360, + 509 + ], + "score": 1.0, + "content": "patterns in the downstream datasets to get a good performance.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Comparison of Generated Data Table 5 shows the comparison of generated examples for the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "COMMONGEN test set between T5-base and CALM. We can see that the sentences generated by", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "CALM are generally more acceptable in terms of commonsense plausibility while T5-base sometimes", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 547, + 283, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 283, + 558 + ], + "score": 1.0, + "content": "generates sentences that do not make sense.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5 + }, + { + "type": "table", + "bbox": [ + 108, + 558, + 503, + 609 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 558, + 503, + 609 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 108, + 558, + 503, + 609 + ], + "spans": [ + { + "bbox": [ + 108, + 558, + 503, + 609 + ], + "score": 0.97, + "html": "
Concept-setT5-baseCALM-base
Grass,Dog,Ball, Chasea dog is chased by a ball on the grass.dog chasing a ball in the grass.
Net, Cast,Boat,Waterfishing boat casts a net in the water.fisherman casts a net into the water from a fishing boat.
Hole,Tree,Plant, Diga man digs a hole in a tree to plant a new tree . he digs theman digging a hole to plant a tree.
Ingredient, Add,Pan, rya pan filled with ingredients adds a touch of spice to the fry .add the ingredients to a pan and fry.
Water,Hold,Hand,WalkA man holding a hand and walking in the water. A man is holding water.man holding a bottle of water in his hand as he walks down the street.
Place, Use,Metal toolA man uses a metal tool to make a piece of metal.woman uses a metal tool to make a piece of jewelry.
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MethodsCSQA PIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
T5- Text Infilling61.9268.1925.1031.0013.1332.40
CALM-COR62.3668.7725.7031.2013.6532.60
CALM-C2S62.2468.7525.9031.4013.9432.80
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MethodsCSQAPIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
Multi-choice QA62.2168.8225.0031.2013.2832.60
True/False62.2467.8125.1031.2013.4132.60
Generative QA62.7370.6725.5031.2013.5832.60
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Left table shows the performance", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "on downstream tasks by pre-training with different generative objective (COR, C2S, and original", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "objective for pre-training T5). Right table shows the performance on downstream tasks by pre-training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 168, + 312, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 312, + 181 + ], + "score": 1.0, + "content": "with different task formats of contrastive objective.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 135, + 506, + 181 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "We can see that both C2S and COR works better than the original masked span infilling objective", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "on itself. This confirms the effectiveness of our proposed generative objectives on improving the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 211, + 394, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 394, + 222 + ], + "score": 1.0, + "content": "commonsense reasoning ability of pre-trained text-to-text transformers.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 187, + 505, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "Task Formulation of the Contrastive objectives For contrastive objectives, we test three different", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 238, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 250 + ], + "score": 1.0, + "content": "task formats: Multi-choice QA, Generative QA, and True/False. Multi-choice QA and Generative", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 504, + 261 + ], + "score": 1.0, + "content": "QA takes the concatenation of the real sentence and the distractor. Then, Multi-choice QA output the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "index of the real sentence following other conventional Multi-choice QA tasks, and Generative QA", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "output the real sentence respectively. True/False takes either the real sentence or the distractor and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "score": 1.0, + "content": "train the model to perform a binary classification problem of whether the input sentence makes sense.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "The result is shown in Table 4. We could find that the format of Generative QA performs the best.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "We suspect this is because the Generative QA format is closer to the format used during the original", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 315, + 390, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 390, + 329 + ], + "score": 1.0, + "content": "pre-training stage of the T5 model and the format used for fine-tuning.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 227, + 507, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 258, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 258, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 258, + 345 + ], + "score": 1.0, + "content": "Performance with fewer training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 343, + 259, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 259, + 355 + ], + "score": 1.0, + "content": "examples To investigate the effec-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 354, + 258, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 258, + 366 + ], + "score": 1.0, + "content": "tiveness of our objective in the low-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 366, + 258, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 258, + 376 + ], + "score": 1.0, + "content": "resource setting, we explore the per-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 376, + 257, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 257, + 387 + ], + "score": 1.0, + "content": "formance of our model and baselines", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 387, + 258, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 258, + 398 + ], + "score": 1.0, + "content": "fine-tuning with different fractions of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 398, + 258, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 258, + 409 + ], + "score": 1.0, + "content": "the training data. From Figure 4, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 409, + 258, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 258, + 421 + ], + "score": 1.0, + "content": "can see that the performance improve-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 420, + 258, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 258, + 431 + ], + "score": 1.0, + "content": "ment yielded by our models upon the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 430, + 257, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 257, + 442 + ], + "score": 1.0, + "content": "T5-base model is more significant in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 442, + 258, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 258, + 453 + ], + "score": 1.0, + "content": "the low-resource regime. This shows", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 451, + 257, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 257, + 466 + ], + "score": 1.0, + "content": "that CALM may already pack some", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 330, + 259, + 466 + ] + }, + { + "type": "image", + "bbox": [ + 274, + 331, + 486, + 427 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 274, + 331, + 486, + 427 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 274, + 331, + 486, + 427 + ], + "spans": [ + { + "bbox": [ + 274, + 331, + 486, + 427 + ], + "score": 0.963, + "type": "image", + "image_path": "d53949681d0c53f39d987e7afd217298a347d7f0231c014211f25f5c381ab759.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 274, + 331, + 486, + 344.7142857142857 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 274, + 344.7142857142857, + 486, + 358.42857142857144 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 274, + 358.42857142857144, + 486, + 372.14285714285717 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 274, + 372.14285714285717, + 486, + 385.8571428571429 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 274, + 385.8571428571429, + 486, + 399.5714285714286 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 274, + 399.5714285714286, + 486, + 413.28571428571433 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 274, + 413.28571428571433, + 486, + 427.00000000000006 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 264, + 429, + 504, + 451 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 264, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 264, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "Figure 4: Performance of compared models fine-tuned with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 264, + 439, + 397, + 452 + ], + "spans": [ + { + "bbox": [ + 264, + 439, + 397, + 452 + ], + "score": 1.0, + "content": "different fraction of the datasets.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "index": 39.75 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 504, + 476 + ], + "score": 1.0, + "content": "commonsense knowledge in its parameters so that it does not require much data for fine-tuning before", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 472, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 489 + ], + "score": 1.0, + "content": "obtaining a good performance. In contrast, the original T5 model requires much data for fine-tuning,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "which suggests it may fail to encode much commonsense knowledge and must fit the correlation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 496, + 360, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 360, + 509 + ], + "score": 1.0, + "content": "patterns in the downstream datasets to get a good performance.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 463, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Comparison of Generated Data Table 5 shows the comparison of generated examples for the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "COMMONGEN test set between T5-base and CALM. We can see that the sentences generated by", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "CALM are generally more acceptable in terms of commonsense plausibility while T5-base sometimes", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 547, + 283, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 283, + 558 + ], + "score": 1.0, + "content": "generates sentences that do not make sense.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 513, + 506, + 558 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 558, + 503, + 609 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 558, + 503, + 609 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 108, + 558, + 503, + 609 + ], + "spans": [ + { + "bbox": [ + 108, + 558, + 503, + 609 + ], + "score": 0.97, + "html": "
Concept-setT5-baseCALM-base
Grass,Dog,Ball, Chasea dog is chased by a ball on the grass.dog chasing a ball in the grass.
Net, Cast,Boat,Waterfishing boat casts a net in the water.fisherman casts a net into the water from a fishing boat.
Hole,Tree,Plant, Diga man digs a hole in a tree to plant a new tree . he digs theman digging a hole to plant a tree.
Ingredient, Add,Pan, rya pan filled with ingredients adds a touch of spice to the fry .add the ingredients to a pan and fry.
Water,Hold,Hand,WalkA man holding a hand and walking in the water. A man is holding water.man holding a bottle of water in his hand as he walks down the street.
Place, Use,Metal toolA man uses a metal tool to make a piece of metal.woman uses a metal tool to make a piece of jewelry.
", + "type": "table", + "image_path": "813fd8d9cffcae2d37a01c6c6c4e1a587fbc25a7f21c3d791a6fda2c1a57630d.jpg" + } + ] + } + ], + "index": 54, + "virtual_lines": [ + { + "bbox": [ + 108, + 558, + 503, + 575.0 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 108, + 575.0, + 503, + 592.0 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 108, + 592.0, + 503, + 609.0 + ], + "spans": [], + "index": 55 + } + ] + } + ], + "index": 54 + }, + { + "type": "title", + "bbox": [ + 163, + 616, + 447, + 627 + ], + "lines": [ + { + "bbox": [ + 162, + 614, + 449, + 630 + ], + "spans": [ + { + "bbox": [ + 162, + 614, + 449, + 630 + ], + "score": 1.0, + "content": "Table 5: Comparison of generated sentences with same concept-set.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 56 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 647 + ], + "score": 1.0, + "content": "Knowledge Probing To investigate how much concept-centric knowledge our model pack, we", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 295, + 657 + ], + "score": 1.0, + "content": "conducted two probing methods with our model", + "type": "text" + }, + { + "bbox": [ + 296, + 647, + 302, + 654 + ], + "score": 0.28, + "content": ":", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "Language Model Analysis (LAMA) probe (Petroni", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "et al., 2019), Knowledge Intensive Language Task (KILT) (Petroni et al., 2020). We summarize the", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 667, + 483, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 483, + 679 + ], + "score": 1.0, + "content": "results on Table 6 and Appendix A.4. We could find that our model outperforms the baseline.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 58.5, + "bbox_fs": [ + 105, + 632, + 506, + 679 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 690, + 211, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 689, + 213, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 213, + 705 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 61 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Self-Supervised Language Representation Pre-Training. Motivated by the fact that words can", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "have different meanings in different contexts, contextual language representation methods (McCann", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "et al., 2017; Peters et al., 2018) have been developed and shown superior performance on downstream", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "tasks compared with static word embeddings Mikolov et al. (2013); Pennington et al. (2014). More", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "recently, large scale language models based on transformer architecture (Vaswani et al., 2017) pre-", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "trained with either mask language modeling objective (Devlin et al., 2018; Liu et al., 2019; Lan et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "2019) or mask span infilling objective (Lewis et al., 2019; Raffel et al., 2019) have been explored", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "further advanced the state-of-the-art on multiple NLU and NLG tasks. Our method is based on these", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "techniques and we focus on improving the commonsense reasoning ability of pre-trained text-to-text", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "transformers. More recently, Clark et al. (2020) propose a new self-supervised pre-training objective", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "score": 1.0, + "content": "called Replaced Token Detection (RTD). RTD uses a mask language model like BERT to fill in the", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "mask and train a discriminator to predict whether a token is generated or real. This pre-training", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "paradigm is related to our proposed joint training framework. Some major differences include that (1)", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "Our method employs sentence-level distractors that are in general grammatically correct but not in", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "line with commonsense, thus require the model to perform relational commonsense reasoning while", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "RTD is a token-level discrimination task and can often be solved with syntactic and shallow semantic", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "knowledge (Rosset et al., 2020); (2) Our method unifies generative and contrastive objectives with one", + "type": "text", + "cross_page": true + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "model, which can be applied to both NLU and NLG downstream tasks; and (3) The discriminator in", + "type": "text", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "our framework is “contrastive”, takes both the real sentence and the distractor as input simultaneously.", + "type": "text", + "cross_page": true + } + ], + "index": 25 + } + ], + "index": 62.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 81, + 366, + 126 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 81, + 366, + 126 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 81, + 366, + 126 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 366, + 126 + ], + "score": 0.95, + "html": "
MethodsMRRPrecision@50Precision @10Precision@1
T5-Base11.5338.5221.605.93
CALM (Mix-only)11.7738.9321.926.10
CALM12.0939.6922.536.46
", + "type": "table", + "image_path": "7bd245446154dd54aec795629e87543d0df56471c9f65630426eb781a2201741.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 81, + 366, + 96.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 96.0, + 366, + 111.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 111.0, + 366, + 126.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 107, + 142, + 504, + 175 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 141, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 155 + ], + "score": 1.0, + "content": "Table 6: Experimental results on Knowledge Probing. Left table shows the mean precision on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 152, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 506, + 167 + ], + "score": 1.0, + "content": "LAMA probing task of ConceptNET. Right table shows the performance on Fact checking and Entity", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 164, + 249, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 249, + 176 + ], + "score": 1.0, + "content": "linking, which are from KILT task.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + } + ], + "index": 4.0 + }, + { + "type": "table", + "bbox": [ + 372, + 81, + 507, + 127 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 372, + 81, + 507, + 127 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 372, + 81, + 507, + 127 + ], + "spans": [ + { + "bbox": [ + 372, + 81, + 507, + 127 + ], + "score": 0.574, + "html": "
MethodsFEVERAY2
T5-base76.6574.97
CALM (Mix-only)77.0576.27
CALM77.4477.24
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MethodsMRRPrecision@50Precision @10Precision@1
T5-Base11.5338.5221.605.93
CALM (Mix-only)11.7738.9321.926.10
CALM12.0939.6922.536.46
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MethodsFEVERAY2
T5-base76.6574.97
CALM (Mix-only)77.0576.27
CALM77.4477.24
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Learning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 280, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 280, + 302 + ], + "score": 1.0, + "content": "rates and train epochs are listed in Table 7.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "table", + "bbox": [ + 108, + 314, + 503, + 353 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 314, + 503, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 314, + 503, + 353 + ], + "spans": [ + { + "bbox": [ + 108, + 314, + 503, + 353 + ], + "score": 0.973, + "html": "
HyperparameterCommonsenseQAOpenbookQAPIQAaNLICommonGEN
Learning rate[1e-4,2e-4,3e-4][5e-5,1e-4,2e-4,3e-4][1e-4,2e-4,3e-4][2e-5,3e-5][2e-5]
Train Epochs2020201020
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The task", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 473, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 141, + 473, + 506, + 484 + ], + "score": 1.0, + "content": "requires open book fact and additional commonsense which is not contained in the book. To", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 483, + 424, + 496 + ], + "spans": [ + { + "bbox": [ + 141, + 483, + 424, + 496 + ], + "score": 1.0, + "content": "test the commonsense reasoning ability, we do not use open book fact.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 134, + 501, + 502, + 524 + ], + "lines": [ + { + "bbox": [ + 133, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 133, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "• PIQA (Bisk et al., 2020) is multiple-choice question answering task, which chooses the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 513, + 398, + 524 + ], + "spans": [ + { + "bbox": [ + 142, + 513, + 398, + 524 + ], + "score": 1.0, + "content": "most appropriate solution for physical commonsense questions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 130, + 530, + 504, + 552 + ], + "lines": [ + { + "bbox": [ + 134, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 134, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "• aNLI (Bhagavatula et al., 2019) is a binary-classification task, which picks the most plausible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 541, + 429, + 553 + ], + "spans": [ + { + "bbox": [ + 141, + 541, + 429, + 553 + ], + "score": 1.0, + "content": "explanatory hypothesis given two observations from narrative contexts.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 132, + 559, + 503, + 581 + ], + "lines": [ + { + "bbox": [ + 135, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 135, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "• CommonGEN (Lin et al., 2020) is a constrained text generation task, which generates a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 569, + 447, + 583 + ], + "spans": [ + { + "bbox": [ + 141, + 569, + 447, + 583 + ], + "score": 1.0, + "content": "coherent sentence describing an everyday scenario using common concepts.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "table", + "bbox": [ + 109, + 602, + 503, + 700 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 602, + 503, + 700 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 602, + 503, + 700 + ], + "spans": [ + { + "bbox": [ + 109, + 602, + 503, + 700 + ], + "score": 0.977, + "html": "
DatasetTrainDevelopmentTestSource ExampleTarget Example
CommonsenseQA9,7411,2211,140context:What homeentertainment equipment requirescable? options: 1: radio shack 2: substation 3:cabinet 4: television 5:desk4
OpenbookQA4,957500500context: You can make a telescope with options: 1: straw 2: glass 3: candle 4: mailing tube2
PIQA16,1131,8383.084context:When boiling butter,when it's ready,you can options:1: Pour it onto a plate 2:Pour it into a jar2
aNLI169,6541,5323.040context: It was my birthday.WhenI got home the party was set up for my brother. options: 1: Iwas so excited. 2: Iwas so mad.2
CommonGEN67,3894.0186.042generate a sentence with these concepts: Apple Grow TreeApple grows on the tree
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We implement", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 141, + 507, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 507, + 154 + ], + "score": 1.0, + "content": "our pre-train models using Pytorch-lightning (Falcon, 2019) and Hugginface’s Pytorch Transform-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 165 + ], + "score": 1.0, + "content": "ers (Wolf et al., 2019). 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HyperparameterCommonsenseQAOpenbookQAPIQAaNLICommonGEN
Learning rate[1e-4,2e-4,3e-4][5e-5,1e-4,2e-4,3e-4][1e-4,2e-4,3e-4][2e-5,3e-5][2e-5]
Train Epochs2020201020
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The task", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 473, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 141, + 473, + 506, + 484 + ], + "score": 1.0, + "content": "requires open book fact and additional commonsense which is not contained in the book. 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DatasetTrainDevelopmentTestSource ExampleTarget Example
CommonsenseQA9,7411,2211,140context:What homeentertainment equipment requirescable? options: 1: radio shack 2: substation 3:cabinet 4: television 5:desk4
OpenbookQA4,957500500context: You can make a telescope with options: 1: straw 2: glass 3: candle 4: mailing tube2
PIQA16,1131,8383.084context:When boiling butter,when it's ready,you can options:1: Pour it onto a plate 2:Pour it into a jar2
aNLI169,6541,5323.040context: It was my birthday.WhenI got home the party was set up for my brother. options: 1: Iwas so excited. 2: Iwas so mad.2
CommonGEN67,3894.0186.042generate a sentence with these concepts: Apple Grow TreeApple grows on the tree
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It might cause low performance on such slot filling task, but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 425, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 425, + 214 + ], + "score": 1.0, + "content": "compared to T5, our model shows better performance compared to base model.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "KILT task is a benchmark for assessing models that need to access specific knowledge in a defined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 242 + ], + "score": 1.0, + "content": "snapshot of Wikipedia to solve tasks spanning five domains. The goal is to analyze the model whether", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 239, + 507, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 507, + 254 + ], + "score": 1.0, + "content": "it has task-agnostic representations of knowledge. We test our model on domain of fact checking,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "entity linking. Fact checking verifies textual claims against textual sources. 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We use AIDA CoNLL-YAGO (AY2) (Hoffart", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "et al., 2011) which supplements the CoNLL 2003 (Tjong Kim Sang & De Meulder, 2003) with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 281, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 281, + 319 + ], + "score": 1.0, + "content": "Wikipedia URL annotations for all entities.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 315, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 317, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 317, + 343 + ], + "score": 1.0, + "content": "A.5 EXPERIMENTS WITH BART AS BACKBONE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "To show that our approach is versatile to different pre-trained models, we conduct experiments with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "BART as the backbone model. We can see that our approach consistently and significantly (with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 137, + 386 + ], + "score": 1.0, + "content": "p-value", + "type": "text" + }, + { + "bbox": [ + 137, + 374, + 164, + 385 + ], + "score": 0.78, + "content": "< 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 374, + 506, + 386 + ], + "score": 1.0, + "content": ") improves BART-base on all datasets. This result shows that our method is versatile to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 222, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 222, + 396 + ], + "score": 1.0, + "content": "different pre-trained models.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "table", + "bbox": [ + 108, + 406, + 503, + 452 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 406, + 503, + 452 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 406, + 503, + 452 + ], + "spans": [ + { + "bbox": [ + 108, + 406, + 503, + 452 + ], + "score": 0.973, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
AccuracyBLEU-4METEORCIDErSPICE
BART-base (Mix-only)56.31(±0.28)58.30(±1.1)(official dev) 67.53(±1.01)59.85(±1.14)25.1029.5013.1630.20
CALM (BART-base)58.22(±0.21)59.10(±1.0)69.40(±1.23)61.28(±0.30)26.4029.9013.7131.10
", + "type": "table", + "image_path": "da6ac46e761543eefb4c1ed5d757aa8e209ff3e467330e8d2f78887463ab566b.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 108, + 406, + 503, + 421.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 108, + 421.3333333333333, + 503, + 436.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 108, + 436.66666666666663, + 503, + 451.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 129, + 460, + 481, + 472 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 459, + 482, + 472 + ], + "spans": [ + { + "bbox": [ + 128, + 459, + 482, + 472 + ], + "score": 1.0, + "content": "Table 9: Experimental results with BART as backbone model. Best models are bold.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 27.0 + }, + { + "type": "title", + "bbox": [ + 107, + 492, + 340, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 340, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 340, + 504 + ], + "score": 1.0, + "content": "A.6 EXPERIMENTS WITH NOUN/VERB AS CONCEPTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "table", + "bbox": [ + 108, + 556, + 503, + 614 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 512, + 505, + 546 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "We also conducted an ablation study about the choice of using either nouns or verbs as concepts. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "can see that using either nouns-only or verbs-only as concepts for our approach leads to substantial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 470, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 470, + 548 + ], + "score": 1.0, + "content": "performance drop. This supports our choice about using both nouns and verbs as concepts.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "table_body", + "bbox": [ + 108, + 556, + 503, + 614 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 556, + 503, + 614 + ], + "spans": [ + { + "bbox": [ + 108, + 556, + 503, + 614 + ], + "score": 0.973, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
CALM-nouns62.45(±0.42)59.40(±0.5)69.05(±0.70)61.55(±0.58)25.7031.2013.1732.60
CALM-verbs62.51(±0.47)59.10(±0.7)69.24(±0.65)61.40(±0.51)25.6031.2013.2432.60
", + "type": "table", + "image_path": "7938293e05b0ce973b065c1134b0c22c0ec4f0826580fcbdccf5a1de51d548aa.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 108, + 556, + 503, + 575.3333333333334 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 108, + 575.3333333333334, + 503, + 594.6666666666667 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 108, + 594.6666666666667, + 503, + 614.0000000000001 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 131, + 621, + 478, + 633 + ], + "lines": [ + { + "bbox": [ + 131, + 621, + 479, + 635 + ], + "spans": [ + { + "bbox": [ + 131, + 621, + 479, + 635 + ], + "score": 1.0, + "content": "Table 10: Experimental results with Noun/Verb as Concepts. Best models are bold.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 643, + 369, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 370, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 370, + 656 + ], + "score": 1.0, + "content": "A.7 HUMAN EVALUATION ON COMMONGEN GENERATIONS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "We conducted a human evaluation of CommonGEN predictions between T5 and CALM. We asked", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 674, + 507, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 507, + 688 + ], + "score": 1.0, + "content": "three annotators to choose the most reasonable sentence between T5-base and CALM-base predictions.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "score": 1.0, + "content": "The evaluation was conducted on 50 test sentences in binary selection by majority voting. Cohen’s", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "score": 1.0, + "content": "Kappa score, which is a measurement of inter-annotator agreement, was 0.73. Annotators say that for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 707, + 318, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 126, + 719 + ], + "score": 0.84, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 707, + 318, + 721 + ], + "score": 1.0, + "content": "of test sentences, CALM-base generated better.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 233, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 234, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 234, + 95 + ], + "score": 1.0, + "content": "A.4 KNOWLEDGE PROBING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "LAMA probe is consisting of a set of knowledge sources, each comprised of a set of fact. It defines", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "that a pre-trained language model knows a fact (subject, relation, object) such as (Bird, CapableOf,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 507, + 138 + ], + "score": 1.0, + "content": "Fly) if it can predict masked objects in cloze statement such as \"Birds can [MASK]\". For evaluation,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "we first filtered out examples that mask label is not in vocabulary list of T5. Then, we evaluate", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 160 + ], + "score": 1.0, + "content": "the model based on how highly it ranks the ground truth token against every other word in a fixed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 170 + ], + "score": 1.0, + "content": "vocabulary list of T5, and get mean precision at k to check whether the object is ranked among the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "top k results. We summarize the results of ConceptNet (Speer & Havasi, 2012) in Table 6. Unlike", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "other language models which are optimised to masked word anywhere in a given sequence, T5 is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "score": 1.0, + "content": "trained with different denoising method. It might cause low performance on such slot filling task, but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 425, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 425, + 214 + ], + "score": 1.0, + "content": "compared to T5, our model shows better performance compared to base model.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 102, + 507, + 214 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "KILT task is a benchmark for assessing models that need to access specific knowledge in a defined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 230, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 242 + ], + "score": 1.0, + "content": "snapshot of Wikipedia to solve tasks spanning five domains. The goal is to analyze the model whether", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 239, + 507, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 507, + 254 + ], + "score": 1.0, + "content": "it has task-agnostic representations of knowledge. We test our model on domain of fact checking,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "entity linking. Fact checking verifies textual claims against textual sources. For this task, we use", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "FEVER (Thorne et al., 2018) which is a large dataset for claim veracity that requires evidence from", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "score": 1.0, + "content": "multiple Wikipedia pages to determine whether the claim is supported or refuted. Entity Linking", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "assigns Wikipedia page to entities mentioned in text. We use AIDA CoNLL-YAGO (AY2) (Hoffart", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "et al., 2011) which supplements the CoNLL 2003 (Tjong Kim Sang & De Meulder, 2003) with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 281, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 281, + 319 + ], + "score": 1.0, + "content": "Wikipedia URL annotations for all entities.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 219, + 507, + 319 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 315, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 317, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 317, + 343 + ], + "score": 1.0, + "content": "A.5 EXPERIMENTS WITH BART AS BACKBONE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "To show that our approach is versatile to different pre-trained models, we conduct experiments with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "BART as the backbone model. We can see that our approach consistently and significantly (with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 137, + 386 + ], + "score": 1.0, + "content": "p-value", + "type": "text" + }, + { + "bbox": [ + 137, + 374, + 164, + 385 + ], + "score": 0.78, + "content": "< 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 374, + 506, + 386 + ], + "score": 1.0, + "content": ") improves BART-base on all datasets. This result shows that our method is versatile to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 222, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 222, + 396 + ], + "score": 1.0, + "content": "different pre-trained models.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 352, + 506, + 396 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 406, + 503, + 452 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 406, + 503, + 452 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 406, + 503, + 452 + ], + "spans": [ + { + "bbox": [ + 108, + 406, + 503, + 452 + ], + "score": 0.973, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
AccuracyBLEU-4METEORCIDErSPICE
BART-base (Mix-only)56.31(±0.28)58.30(±1.1)(official dev) 67.53(±1.01)59.85(±1.14)25.1029.5013.1630.20
CALM (BART-base)58.22(±0.21)59.10(±1.0)69.40(±1.23)61.28(±0.30)26.4029.9013.7131.10
", + "type": "table", + "image_path": "da6ac46e761543eefb4c1ed5d757aa8e209ff3e467330e8d2f78887463ab566b.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 108, + 406, + 503, + 421.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 108, + 421.3333333333333, + 503, + 436.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 108, + 436.66666666666663, + 503, + 451.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 129, + 460, + 481, + 472 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 459, + 482, + 472 + ], + "spans": [ + { + "bbox": [ + 128, + 459, + 482, + 472 + ], + "score": 1.0, + "content": "Table 9: Experimental results with BART as backbone model. Best models are bold.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 27.0 + }, + { + "type": "title", + "bbox": [ + 107, + 492, + 340, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 340, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 340, + 504 + ], + "score": 1.0, + "content": "A.6 EXPERIMENTS WITH NOUN/VERB AS CONCEPTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "table", + "bbox": [ + 108, + 556, + 503, + 614 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 512, + 505, + 546 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "We also conducted an ablation study about the choice of using either nouns or verbs as concepts. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "can see that using either nouns-only or verbs-only as concepts for our approach leads to substantial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 470, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 470, + 548 + ], + "score": 1.0, + "content": "performance drop. This supports our choice about using both nouns and verbs as concepts.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "table_body", + "bbox": [ + 108, + 556, + 503, + 614 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 556, + 503, + 614 + ], + "spans": [ + { + "bbox": [ + 108, + 556, + 503, + 614 + ], + "score": 0.973, + "html": "
MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
CALM-nouns62.45(±0.42)59.40(±0.5)69.05(±0.70)61.55(±0.58)25.7031.2013.1732.60
CALM-verbs62.51(±0.47)59.10(±0.7)69.24(±0.65)61.40(±0.51)25.6031.2013.2432.60
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Best models are bold.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 643, + 369, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 370, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 370, + 656 + ], + "score": 1.0, + "content": "A.7 HUMAN EVALUATION ON COMMONGEN GENERATIONS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "We conducted a human evaluation of CommonGEN predictions between T5 and CALM. We asked", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 674, + 507, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 507, + 688 + ], + "score": 1.0, + "content": "three annotators to choose the most reasonable sentence between T5-base and CALM-base predictions.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "score": 1.0, + "content": "The evaluation was conducted on 50 test sentences in binary selection by majority voting. Cohen’s", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "score": 1.0, + "content": "Kappa score, which is a measurement of inter-annotator agreement, was 0.73. Annotators say that for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 707, + 318, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 126, + 719 + ], + "score": 0.84, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 707, + 318, + 721 + ], + "score": 1.0, + "content": "of test sentences, CALM-base generated better.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 663, + 507, + 721 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/3k20LAiHYL2/3k20LAiHYL2_model.json b/parse/train/3k20LAiHYL2/3k20LAiHYL2_model.json new file mode 100644 index 0000000000000000000000000000000000000000..efb1680950a1ab8ab3ff046ff29464f0a469b933 --- /dev/null +++ b/parse/train/3k20LAiHYL2/3k20LAiHYL2_model.json @@ -0,0 +1,17476 @@ +[ + { + "layout_dets": [ + { + "category_id": 1, + "poly": [ + 298, + 1373, + 1404, + 1373, + 1404, + 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Algorithm Aware Language Model (CALM).1: Pre-training Concept-
Input:Text-to-Text Transformer Tθ,Text corpus X=[x1,x2,...,xn]. repeat
foreach xi∈Xdo Extract the concept-set Ci; Construct the distractor sentence
x'=CONCEPT-PERMUTE(Xi,Ci); Update Tθ with Eq.(1,2, 4);
until maximum iterations reached; repeat
for each xi ∈Xdo
Update Tθwith Eq.(7)
until maximum iterationsreached;
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MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-base53.08(±0.16)57.60(±0.8)64.86(±0.52)61.88(±0.56)11--
ERNIE54.06(±0.12)58.90(±0.9)66.47(±0.58)63.04(±0.46)1
KnowBERT53.88(±0.15)58.50(±0.8)66.61(±0.63)63.18(±0.52)1-1-
T5-base61.88(±0.08)58.20(±1.0)68.14(±0.73)61.10(±0.38)24.9031.2012.9932.40
T5-base + cont. pretraining61.92(±0.45)58.10(±0.9)68.19(±0.77)61.15(±0.52)25.1031.0013.1232.40
T5-base + SSM62.08(±0.41)58.30(±0.8)68.27(±0.71)61.25(±0.51)25.2031.2013.2832.40
CALM (Generative-Only)62.28(±0.36)58.90(±0.4)68.91(±0.88)60.95(±0.46)25.8031.2013.8132.60
CALM (Contrastive-Only)62.73(±0.41)59.30(±0.3)70.67(±0.98)61.35(±0.06)25.5031.2013.5832.60
CALM(w/o Mix warmup)62.18(±0.48)59.00(±0.5)69.21(±0.57)61.25(±0.55)25.8031.2013.7732.60
CALM (Mix-only)63.02(±0.47)60.40(±0.4)70.07(±0.98)62.79(±0.55)26.0031.2013.8232.80
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
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MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
BERT-large57.06(±0.12)60.40(±0.6)67.08(±0.61)66.75(±0.61)==
T5-large69.81(±1.02)61.40(±1.0)72.19(±1.09)75.54(±1.22)28.6030.1014.9631.60
CALM-large (Mix-only)70.26(±0.23)62.50(±1.0)73.70(±1.09)75.99(±1.26)29.2031.3015.2433.10
CALM-large71.31(±0.04)66.00(±1.0)75.11(±1.65)77.12(±0.34)29.5031.9015.6133.20
RoBERTa-large471.81(±0.25)63.90(±0.8)76.90(±0.62)82.35(±0.54)111-
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MethodsParamsCommonGEN
BLEU-4METEORCIDErSPICE
GPT-2 (Radford et al.,2019)774M21.1026.2012.1525.90
UniLM(Dong et al.,2019)340M27.7029.7014.8530.20
BART(Lewis et al.,2020)406M26.3030.9013.9230.60
T5-base (Raffel et al.,2019)220M16.4023.009.1622.00
T5-large (Raffel et al., 2019)770M28.6030.1014.9631.60
KG-BART (Liu et al., 2020)406M30.9032.4016.8332.70
T5-base (our implementation)220M24.9031.2012.9932.40
CALM-base220M26.4031.4013.8833.00
CALM-large774M29.5031.9015.6133.20
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Concept-setT5-baseCALM-base
Grass,Dog,Ball, Chasea dog is chased by a ball on the grass.dog chasing a ball in the grass.
Net, Cast,Boat,Waterfishing boat casts a net in the water.fisherman casts a net into the water from a fishing boat.
Hole,Tree,Plant, Diga man digs a hole in a tree to plant a new tree . he digs theman digging a hole to plant a tree.
Ingredient, Add,Pan, rya pan filled with ingredients adds a touch of spice to the fry .add the ingredients to a pan and fry.
Water,Hold,Hand,WalkA man holding a hand and walking in the water. A man is holding water.man holding a bottle of water in his hand as he walks down the street.
Place, Use,Metal toolA man uses a metal tool to make a piece of metal.woman uses a metal tool to make a piece of jewelry.
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MethodsCSQA PIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
T5- Text Infilling61.9268.1925.1031.0013.1332.40
CALM-COR62.3668.7725.7031.2013.6532.60
CALM-C2S62.2468.7525.9031.4013.9432.80
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MethodsCSQAPIQACommonGEN
AccuracyBLEU-4METEORCIDErSPICE
Multi-choice QA62.2168.8225.0031.2013.2832.60
True/False62.2467.8125.1031.2013.4132.60
Generative QA62.7370.6725.5031.2013.5832.60
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DatasetTrainDevelopmentTestSource ExampleTarget Example
CommonsenseQA9,7411,2211,140context:What homeentertainment equipment requirescable? options: 1: radio shack 2: substation 3:cabinet 4: television 5:desk4
OpenbookQA4,957500500context: You can make a telescope with options: 1: straw 2: glass 3: candle 4: mailing tube2
PIQA16,1131,8383.084context:When boiling butter,when it's ready,you can options:1: Pour it onto a plate 2:Pour it into a jar2
aNLI169,6541,5323.040context: It was my birthday.WhenI got home the party was set up for my brother. options: 1: Iwas so excited. 2: Iwas so mad.2
CommonGEN67,3894.0186.042generate a sentence with these concepts: Apple Grow TreeApple grows on the tree
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HyperparameterCommonsenseQAOpenbookQAPIQAaNLICommonGEN
Learning rate[1e-4,2e-4,3e-4][5e-5,1e-4,2e-4,3e-4][1e-4,2e-4,3e-4][2e-5,3e-5][2e-5]
Train Epochs2020201020
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MethodsCSQAOBQAPIQAaNLICommonGEN
Accuracy (official dev)BLEU-4METEORCIDErSPICE
CALM63.32(±0.35)60.90(±0.4)71.01(±0.61)63.20(±0.52)26.4031.4013.8833.00
CALM-nouns62.45(±0.42)59.40(±0.5)69.05(±0.70)61.55(±0.58)25.7031.2013.1732.60
CALM-verbs62.51(±0.47)59.10(±0.7)69.24(±0.65)61.40(±0.51)25.6031.2013.2432.60
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MethodsCSQAOBQAPIQAaNLICommonGEN
AccuracyBLEU-4METEORCIDErSPICE
BART-base (Mix-only)56.31(±0.28)58.30(±1.1)(official dev) 67.53(±1.01)59.85(±1.14)25.1029.5013.1630.20
CALM (BART-base)58.22(±0.21)59.10(±1.0)69.40(±1.23)61.28(±0.30)26.4029.9013.7131.10
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/dev/null +++ b/parse/train/SJzRZ-WCZ/images/fdd3120429a9c3fe73ff39747859481cbeef7c5310cb76872713fd46a5d54bd6.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9153d0dcd8c2942c1c9f69df92554b5bdfd1b7d05c1071b6927d80d526ce8546 +size 11642 diff --git a/parse/train/SkNksoRctQ/SkNksoRctQ.md b/parse/train/SkNksoRctQ/SkNksoRctQ.md new file mode 100644 index 0000000000000000000000000000000000000000..23ba9c00d3b9c8d87765bf1c3e84086742e3f31e --- /dev/null +++ b/parse/train/SkNksoRctQ/SkNksoRctQ.md @@ -0,0 +1,417 @@ +# FLUCTUATION-DISSIPATION RELATIONS FOR STOCHASTIC GRADIENT DESCENT + +# Sho Yaida + +Facebook AI Research +Facebook Inc. +Menlo Park, California 94025, USA shoyaida@fb.com + +# ABSTRACT + +The notion of the stationary equilibrium ensemble has played a central role in statistical mechanics. In machine learning as well, training serves as generalized equilibration that drives the probability distribution of model parameters toward stationarity. Here, we derive stationary fluctuation-dissipation relations that link measurable quantities and hyperparameters in the stochastic gradient descent algorithm. These relations hold exactly for any stationary state and can in particular be used to adaptively set training schedule. We can further use the relations to efficiently extract information pertaining to a loss-function landscape such as the magnitudes of its Hessian and anharmonicity. Our claims are empirically verified. + +# 1 INTRODUCTION + +Equilibration rules the long-term fate of many macroscopic dynamical systems. For instance, as we pour water into a glass and let it be, the stationary state of tranquility is eventually attained. Zooming into the tranquil water with a microscope would reveal, however, a turmoil of stochastic fluctuations that maintain the apparent stationarity in balance. This is vividly exemplified by the Brownian motion (Brown, 1828): a pollen immersed in water is constantly bombarded by jittery molecular movements, resulting in the macroscopically observable diffusive motion of the solute. Out of the effort in bridging microscopic and macroscopic realms through the Brownian movement came a prototype of fluctuation-dissipation relations (Einstein, 1905; Von Smoluchowski, 1906). These relations quantitatively link degrees of noisy microscopic fluctuations to smooth macroscopic dissipative phenomena and have since been codified in the linear response theory for physical systems (Onsager, 1931; Green, 1954; Kubo, 1957), a cornerstone of statistical mechanics. + +Machine learning begets another form of equilibration. As a model learns patterns in data, its performance first improves and then plateaus, again reaching apparent stationarity. This dynamical process naturally comes equipped with stochastic fluctuations as well: often given data too gigantic to consume at once, training proceeds in small batches and random selections of these mini-batches consequently give rise to the noisy dynamical excursion of the model parameters in the loss-function landscape, reminiscent of the Brownian motion. It is thus natural to wonder if there exist analogous fluctuation-dissipation relations that quantitatively link the noise in mini-batched data to the observable evolution of the model performance and that in turn facilitate the learning process. + +Here, we derive such fluctuation-dissipation relations for the stochastic gradient descent algorithm. The only assumption made is stationarity of the probability distribution that governs the model parameters at sufficiently long time. Our results thus apply to generic cases with non-Gaussian mini-batch noises and nonconvex loss-function landscapes. Practically, the first relation (FDR1) offers the metric for assessing equilibration and yields an adaptive algorithm that sets learning-rate schedule on the fly. The second relation (FDR2) further helps us determine the properties of the lossfunction landscape, including the strength of its Hessian and the degree of anharmonicity, i.e., the deviation from the idealized harmonic limit of a quadratic loss surface and a constant noise matrix. + +Our approach should be contrasted with recent attempts to import the machinery of stochastic differential calculus into the study of the stochastic gradient descent algorithm (Mandt et al., 2015; Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2017; + +Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). This line of work all assumes Gaussian noises and sometimes additionally employs the quadratic harmonic approximation for loss-function landscapes. The more severe drawback, however, is the usage of the analogy with continuous-time stochastic differential equations, which is inconsistent in general (see Section 2.3.3). Instead, the stochastic gradient descent algorithm can be properly treated within the framework of the KramersMoyal expansion (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen & Moody, 1993). + +The paper is organized as follows. In Section 2, after setting up notations and deriving a stationary fluctuation-dissipation theorem (FDT), we derive two specific fluctuation-dissipation relations. The first relation (FDR1) can be used to check stationarity and the second relation (FDR2) to delineate the shape of the loss-function landscape, as empirically borne out in Section 3. An adaptive scheduling method is proposed and tested in Section 3.3. We conclude in Section 4 with future outlooks. + +# 2 FLUCTUATION-DISSIPATION RELATIONS + +A model is parametrized by a weight coordinate, $\pmb { \theta } = \{ \theta _ { i } \} _ { i = 1 , \dots , P }$ . The training set of $N _ { \mathrm { s } }$ examples is utilized by the model to learby a full-batch loss function, $\begin{array} { r } { \dot { f } \left( \pmb { \theta } \right) \equiv \frac { 1 } { N _ { \mathrm { s } } } \sum _ { \alpha = 1 } ^ { N _ { \mathrm { s } } } { f _ { \alpha } \left( \pmb { \theta } \right) } } \end{array}$ e mod, with $f _ { \alpha } \left( \pmb { \theta } \right)$ erall performance is evaluatedquantifying the performance of the model on a particular sample $\alpha$ : the smaller the loss is, the better the model is expected to perform. The learning process can thus be cast as an optimization problem of minimizing the loss function. One of the most commonly used optimization schemes is the stochastic gradient descent (SGD) algorithm (Robbins $\&$ Monro, 1951) in which a mini-batch $B \subset \{ 1 , 2 , \ldots , N _ { \mathrm { s } } \}$ of size $| B |$ is stochastically chosen for training at each time step. Specifically, the update equation is given by + +$$ +\pmb \theta ( t + 1 ) = \pmb \theta ( t ) - \eta \pmb \nabla f ^ { \mathcal { B } } \left[ \pmb \theta ( t ) \right] , +$$ + +where $\eta > 0$ is a learning rate and a mini-batch loss $\begin{array} { r } { f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \equiv \frac { 1 } { \left| \mathcal { B } \right| } \sum _ { \alpha \in \mathcal { B } } f _ { \alpha } \left( \pmb { \theta } \right) } \end{array}$ . Note that + +$$ +\left[ \nabla f ^ { \boldsymbol { B } } \left( \pmb { \theta } \right) \right] _ { \mathrm { m . b . } } = \nabla f \left( \pmb { \theta } \right) , +$$ + +with $\mathbb { I } . . . \mathbb { J } _ { \mathrm { m . b . } }$ . denoting the average over mini-batch realizations. For later purposes, it is convenient to define a full two-point noise matrix $\widetilde { C }$ through1 + +$$ +\widetilde { C } _ { i , j } ( \pmb { \theta } ) \equiv [ [ \partial _ { i } f ^ { \mathcal { B } } ( \pmb { \theta } ) ] [ \partial _ { j } f ^ { \mathcal { B } } ( \pmb { \theta } ) ] ] ] _ { \mathrm { m . b . } } +$$ + +and, more generally, higher-point noise tensors + +$$ +\tilde { C } _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } } \left( \pmb { \theta } \right) \equiv \left[ \left[ \partial _ { i _ { 1 } } f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \right] \left[ \partial _ { i _ { 2 } } f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \right] \cdots \left[ \partial _ { i _ { k } } f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \right] \right] _ { \mathrm { m . b . } } . +$$ + +Below, we shall not make any assumptions on the distribution of the noise vector $\nabla f ^ { B }$ – other than that a mini-batch is independent and identically distributed from the $N _ { \mathrm { s } }$ training samples at each time step – and the noise distribution is therefore allowed to have nontrivial higher connected moments indicative of non-Gaussianity. + +It is empirically often observed that the performance of the model plateaus after some training through SGD. It is thus natural to hypothesize the existence of a stationary-state distribution, $p _ { \mathrm { s s } } \left( \pmb { \theta } \right)$ , that dictates the SGD sampling at long time (see Section 2.3.4 for discussion on this assumption). For any observable quantity, $\mathcal { O } \left( \pmb { \theta } \right)$ , – something that can be measured during training such as $\pmb { \theta } ^ { 2 }$ and $f \left( \pmb { \theta } \right) -$ its stationary-state average is then defined as + +$$ +\left. \mathcal O \left( \pmb { \theta } \right) \right. \equiv \int \mathrm { d } \pmb { \theta } p _ { \mathrm { s s } } \left( \pmb { \theta } \right) \mathcal O \left( \pmb { \theta } \right) . +$$ + +In general the probability distribution of the model parameters evolves as $p ( \pmb { \theta } , t + 1 ) =$ $\begin{array} { r } { \left[ \int \bar { \mathrm { d } } \pmb { \theta } ^ { \prime } p ( \pmb { \theta } ^ { \prime } , t ) \delta \bar { \left\{ \pmb { \theta } - \left[ \pmb { \theta } ^ { \prime } - \eta \nabla f ^ { \mathcal { B } } \left( \pmb { \theta } ^ { \prime } \right) \right] \right\} } \right] _ { \mathrm { m . b . } } } \end{array}$ and in particular for the stationary state + +$$ +\begin{array} { r c l } { \displaystyle \int \mathrm { d } \theta p _ { \mathrm { s s } } \left( \theta , t \right) \mathcal { O } \left( \theta \right) } & { = } & { \displaystyle \int \mathrm { d } \theta \ d p _ { \mathrm { s s } } \left( \theta , t + 1 \right) \mathcal { O } \left( \theta \right) } \\ & { = } & { \displaystyle \left[ \int \mathrm { d } \theta \int \mathrm { d } \theta ^ { \prime } p _ { \mathrm { s s } } ( \theta ^ { \prime } , t ) \delta \left\{ \theta - \left[ \theta ^ { \prime } - \eta \nabla f ^ { \mathcal { B } } \left( \theta ^ { \prime } \right) \right] \right\} \mathcal { O } \left( \theta \right) \right] _ { \mathrm { m . b . } } } \\ & { = } & { \displaystyle \int \mathrm { d } \theta ^ { \prime } p _ { \mathrm { s s } } \left( \theta ^ { \prime } \right) \left[ \mathcal { O } \left[ \theta ^ { \prime } - \eta \nabla f ^ { \mathcal { B } } \left( \theta ^ { \prime } \right) \right] \right] _ { \mathrm { m . b . } } . } \end{array} +$$ + +Thus follows the master equation + +$$ + { \mathcal O ( { \pmb \theta } ) } = { [ { \mathcal O [ { \pmb \theta } - \eta { \pmb \nabla } f ^ { \beta } ( { \pmb \theta } ) } ] } ] _ { \mathrm { m . b . } } . +$$ + +In the next two subsections, we apply this general formula to simple observables in order to derive various stationary fluctuation-dissipation relations. Incidentally, the discrete version of the FokkerPlanck equation can be derived through the Kramers-Moyal expansion, considering the more general nonstationary version of the above equation and performing the Taylor expansion in $\eta$ and repeated integrations by parts (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen & Moody, 1993). + +# 2.1 FIRST FLUCTUATION-DISSIPATION RELATION + +Applying the master equation (FDT) to the linear observable, + +$$ +\left. \pmb { \theta } \right. = \left. \left[ \pmb { \theta } - \eta \pmb { \nabla } f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \right] _ { \mathrm { m . b . } } \right. = \left. \pmb { \theta } \right. - \eta \left. \pmb { \nabla } f \left( \pmb { \theta } \right) \right. . +$$ + +We thus have + +$$ +\langle \nabla f \rangle = 0 . +$$ + +This is natural because there is no particular direction that the gradient picks on average as the model parameter stochastically bounces around the local minimum or, more generally, wanders around the loss-function landscape according to the stationary distribution. + +Performing similar algebra for the quadratic observable $\langle \theta _ { i } \theta _ { j } \rangle$ yields + +$$ +\langle \theta _ { i } \left( \partial _ { j } f \right) \rangle + \langle \left( \partial _ { i } f \right) \theta _ { j } \rangle = \eta \left. \widetilde { C } _ { i , j } \right. . +$$ + +In particular, taking the trace of this matrix-form relation, we obtain + +$$ +\langle \pmb { \theta } \cdot ( \pmb { \nabla } f ) \rangle = \frac { 1 } { 2 } \eta \left. \mathrm { T r } \widetilde { C } \right. . +$$ + +More generally, in the case of SGD with momentum $\mu$ and dampening $\nu$ , whose update equation is given by + +$$ +\begin{array} { r c l } { { \mathbf v ( t + 1 ) } } & { { = } } & { { \mu \mathbf v ( t ) - ( 1 - \nu ) \nabla f ^ { \mathcal { B } } \left[ \pmb { \theta } ( t ) \right] , } } \\ { { \pmb \theta ( t + 1 ) } } & { { = } } & { { \pmb \theta ( t ) + \eta \mathbf v ( t + 1 ) , } } \end{array} +$$ + +a similar derivation yields (see Appendix A) + +$$ +\langle \pmb \theta \cdot ( \pmb \nabla f ) \rangle = \frac { ( 1 + \mu ) } { 2 ( 1 - \nu ) } \eta \left. \mathbf { v } ^ { 2 } \right. . +$$ + +The last equation reduces to the equation (FDR1) when $\mu = \nu = 0$ with $\mathbf { v } = - \nabla f ^ { B }$ . Also note that $\left. \pmb { \theta } \cdot ( \pmb { \nabla } f ) \right. = \left. ( \pmb { \theta } - \pmb { \theta } _ { \mathrm { c } } ) \cdot ( \pmb { \nabla } f ) \right.$ for an arbitrary constant vector $\theta _ { \mathrm { c } }$ because of the equation (8). + +This first fluctuation-dissipation relation is easy to evaluate on the fly during training, exactly holds without any approximation if sampled well from the stationary distribution, and can thus be used as the standard metric to check if learning has plateaued, just as similar relations can be used to check equilibration in Monte Carlo simulations of physical systems (Santen & Krauth, 2000). [It should be cautioned, however, that the fluctuation-dissipation relations are necessary but not sufficient to ensure stationarity (Odriozola & Berthier, 2011).] Such a metric can in turn be used to schedule changes in hyperparameters, as shall be demonstrated in Section 3.3. + +# 2.2 SECOND FLUCTUATION-DISSIPATION RELATION + +Applying the master equation (FDT) on the full-batch loss function and Taylor-expanding it in the learning rate $\eta$ yields the closed-form expression + +$$ +\begin{array} { r c l } { f \left( \pmb { \theta } \right) \rangle } & { = } & { \displaystyle \left. \left[ f \left[ \pmb { \theta } - \eta \boldsymbol { \nabla } f ^ { B } \left( \pmb { \theta } \right) \right] \right] _ { \mathrm { m . b . } } \right. } \\ & { = } & { \displaystyle \left. f + \sum _ { k = 1 } ^ { \infty } \frac { \left( - \eta \right) ^ { k } } { k ! } \sum _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } = 1 } ^ { P } \left( \partial _ { i _ { 1 } } \partial _ { i _ { 2 } } \cdots \partial _ { i _ { k } } f \right) \left[ \left( \partial _ { i _ { 1 } } f ^ { B } \right) \left( \partial _ { i _ { 2 } } f ^ { B } \right) \cdots \left( \partial _ { i _ { k } } f ^ { B } \right) \right] _ { \mathrm { m . b . } } \right. } \\ & { = } & { \displaystyle \left. f \right. - \eta \left. \left( \pmb { \nabla } f \right) ^ { 2 } \right. + \sum _ { k = 2 } ^ { \infty } \frac { \left( - \eta \right) ^ { k } } { k ! } \left. \sum _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } } \widetilde { C } _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } } \right. } \end{array} +$$ + +where we recalled the equation (4) and introduced + +$$ +F _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } } \left( \pmb { \theta } \right) \equiv \partial _ { i _ { 1 } } \partial _ { i _ { 2 } } \cdot \cdot \cdot \partial _ { i _ { k } } f \left( \pmb { \theta } \right) . +$$ + +In particular, ${ { H } _ { i , j } } \left( \pmb { \theta } \right) \equiv { { F } _ { i , j } } \left( \pmb { \theta } \right)$ is the Hessian matrix. Reorganizing terms, we obtain + +$$ +\left. \left( \nabla f \right) ^ { 2 } \right. = \frac { \eta } { 2 } \left. \mathrm { T r } \left( H \tilde { C } \right) \right. - \eta ^ { 2 } \left[ \sum _ { k = 3 } ^ { \infty } \frac { \left( - \eta \right) ^ { k - 3 } } { k ! } \left. \sum _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } } \widetilde { C } _ { i _ { 1 } , i _ { 2 } , \ldots , i _ { k } } \right. \right] . +$$ + +In the case of SGD with momentum and dampening, the left-hand side is replaced by $( 1 ~ -$ $\nu ) \left. \left( \nabla f \right) ^ { 2 } \right. - \mu \left. \mathbf { v } \cdot \pmb { \nabla } f \right.$ and $\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \dots , i _ { k } }$ by more hideous expressions (see Appendix A). + +We can extract at least two types of information on the loss-function landscape by evaluating the dependence of the left-hand side, $G ( \eta ) \equiv \left. \left( \nabla f \right) ^ { 2 } \right.$ , on the learning rate $\eta$ . First, in the small learning rate regime, the value of $2 G ( \eta ) / \eta$ approximates $\mathrm { T r } \left( H { \tilde { C } } \right)$ around a local ravine. Second, nonlinearity of $G ( \eta )$ at higher $\eta$ indicates discernible effects of anharmonicity. In such a regime, the Hessian matrix $\pmb { H }$ cannot be approximated as constant (which also implies that $\{ F _ { i _ { 1 } , i _ { 2 } , . . . , i _ { k } } \} _ { k > 2 }$ are nontrivial) and/or the noise two-point matrix $\widetilde { C }$ cannot be regarded as constant. Such nonlinearity especially indicates the breakdown of the harmonic approximation, that is, the quadratic truncation of the loss-function landscape, often used to analyze the regime explored at small learning rates. + +# 2.3 REMARKS + +# 2.3.1 INTUITION WITHIN THE HARMONIC APPROXIMATION + +In order to gain some intuition about the fluctuation-dissipation relations, let us momentarily employ the harmonic approximation, i.e., assume that there is a local minimum of the loss function at $\theta \ : = \ : \theta ^ { \star }$ and retain only up to quadratic terms of the Taylor expansions around it: $f ( \pmb \theta ) ~ \approx$ $\begin{array} { r } { f _ { 0 } { + } \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ { P } h _ { i , j } ( \theta _ { i } { - } \theta _ { i } ^ { \star } ) ( \theta _ { j } { - } \theta _ { j } ^ { \star } ) } \end{array}$ . Within this approximation, $\langle \pmb \theta \cdot ( \pmb \nabla f ) \rangle = \langle ( \pmb \theta - \pmb \theta ^ { \star } ) \cdot ( \pmb \nabla f ) \rangle \approx$ $2 \left. f - f _ { 0 } \right.$ . The relation (FDR1) then becomes $\begin{array} { r } { \langle f - f _ { 0 } \rangle \approx \frac { 1 } { 4 } \eta \left. \mathrm { T r } \widetilde { C } \right. } \end{array}$ , linking the height of the noise ball to the noise amplitude. This is in line with, for instance, the theorem 4.6 of the reference Bottou et al. (2018) and substantiates the analogy between SGD and simulated annealing, with the learning rate $\eta$ – multiplied by $\mathrm { T r } \widetilde { C }$ – playing the role of temperature (Bottou, 1991). + +# 2.3.2 HIGHER-ORDER RELATIONS + +Additional relations can be derived by repeating similar calculations for higher-order observables. For example, at the cubic order, + +$$ +\left. \theta _ { i } \theta _ { j } \left( \partial _ { k } f \right) + \theta _ { i } \left( \partial _ { j } f \right) \theta _ { k } + \left( \partial _ { i } f \right) \theta _ { j } \theta _ { k } \right. = \eta \left. \theta _ { i } \widetilde { C } _ { j , k } + \theta _ { j } \widetilde { C } _ { k , i } + \theta _ { k } \widetilde { C } _ { i , j } \right. - \eta ^ { 2 } \left. \widetilde { C } _ { i , j , k } \right. . +$$ + +The systematic investigation of higher-order relations is relegated to future work. + +# 2.3.3 SGD6=SDE + +There is no limit in which SGD asymptotically reduces to the stochastic differential equation (SDE). In order to take such a limit with continuous time differential $\mathrm d t \to 0 ^ { + }$ , each SGD update must become infinitesimal. One may thus try $\mathrm { d } t \equiv \eta 0 ^ { + }$ , as in recent work adapting the view that SGD $=$ SDE (Mandt et al., 2015; Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2017; Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). But this in turn forces the noise vector with zero mean, √ $\nabla f ^ { B } - \nabla f$ , to be multiplied by $\mathrm { d } t$ . This is in contrast to the scaling $\sqrt { \mathrm { d } t }$ needed for the standard machinery of SDE – Ito-Stratonovich calculus and all ˆ that – to apply; the additional factor of $\mathrm { d } t ^ { 1 / 2 }$ makes the effective noise covariance be suppressed by $\mathrm { d } t$ and the resulting equation in the continuous-time limit, if anything, would just be an ordinary differential equation without noise2 [unless noise with the proper scaling is explicitly added as in stochastic gradient Langevin dynamics (Welling & Teh, 2011; Teh et al., 2016) and natural Langevin dynamics (Marceau-Caron & Ollivier, 2017; Nado et al., 2018)]. + +In short, the recent work views $\eta = { \sqrt { \eta } } { \sqrt { \mathrm { d } t } }$ and sends $\mathrm d t \to 0 ^ { + }$ while pretending that $\eta$ is finite, which is inconsistent. This is not just a technical subtlety. When unjustifiably passing onto the continuous-time Fokker-Planck equation, the diffusive term is incorrectly governed by the connected two-point noise matrix $C _ { i , j } \left( \pmb { \theta } \right) \equiv \widetilde { C } _ { i , j } \left( \pmb { \theta } \right) - \left[ \partial _ { i } f \left( \pmb { \theta } \right) \right] \left[ \partial _ { j } f \left( \pmb { \theta } \right) \right]$ rather than the full two-point noise matrix $\widetilde { C } _ { i , j } \left( \pmb { \theta } \right)$ that appears herein.3 We must instead employ the discrete-time version of the Fokker-Planck equation derived in references Van Kampen (1992); Gardiner (2009); Risken (1984); Radons et al. (1990); Leen & Moody (1993), as has been followed in the equation (6). + +# 2.3.4 ON STATIONARITY + +In contrast to statistical mechanics where an equilibrium state is dictated by a handful of thermodynamic variables, in machine learning a stationary state generically depends not only on hyperparameters but also on a part of its learning history. The stationarity assumption made herein, which is codified in the equation (6), is weaker than the typicality assumption underlying statistical mechanics and can hold even in the presence of lingering memory. In the full-batch limit $| B | = N _ { \mathrm { s } }$ , for instance, any distribution delta-peaked at a local minimum is stationary. For sufficiently small learning rates $\eta$ as well, it is natural to expect multiple stationary distributions that form disconnected ponds around these minima, which merge upon increasing $\eta$ and fragment upon decreasing $\eta$ . + +It is beyond the scope of the present paper to formulate conditions under which stationary distributions exist. Indeed, if the formulation were too generic, there could be counterexamples to such a putative existence statement. A case in point is a model with the unregularized cross entropy loss, whose model parameters keep cascading toward infinity in order to sharpen its softmax output (Neyshabur et al., 2014; 2017) with logarithmically diverging $\pmb { \theta } ^ { 2 }$ (Soudry et al., 2018). It would be interesting to see if there are any other nontrivial caveats. + +# 3 EMPIRICAL TESTS + +In this section we empirically bear out our theoretical claims in the last section. To this end, two simple models of supervised learning are used (see Appendix B for full specifications): a multilayer perceptron (MLP) learning patterns in the MNIST training data (LeCun et al., 1998) through SGD without momentum and a convolutional neural network (CNN) learning patterns in the CIFAR10 training data (Krizhevsky & Hinton, 2009) through SGD with momentum $\mu = 0 . 9$ . For both models, the mini-batch size is set to be $| B | = 1 0 0$ , and the training data are shuffled at each epoch $\begin{array} { r } { t = \frac { N _ { \mathrm { s } } } { | B | } \hat { t } _ { \mathrm { e p o c h } } } \end{array}$ with $\hat { t } _ { \mathrm { e p o c h } } \in \mathbb { N }$ . In order to avoid the overfitting cascade mentioned in Section 2.3.4, the $L ^ { 2 }$ -regularization term $\scriptstyle { \frac { 1 } { 2 } } \lambda \theta ^ { 2 }$ with the weight decay $\lambda = 0 . 0 1$ is included in the loss function $f$ . + +Before proceeding further, let us define the half-running average of an observable $\mathcal { O }$ as + +$$ +\overline { { \mathcal { O } } } ( t ) \equiv \frac { 1 } { t - t _ { 0 } } \sum _ { t ^ { \prime } = t _ { 0 } + 1 } ^ { t } \mathcal { O } ( t ^ { \prime } ) \mathrm { w i t h } t _ { 0 } = \lfloor t / 2 \rfloor . +$$ + +This is the average of the observable up to the time step $t$ , with the initial half discarded as containing transient. If SGD drives the distribution of the model parameters to stationarity at long time, then + +$$ +\operatorname * { l i m } _ { t \infty } \overline { { \mathcal { O } } } ( t ) = \langle \mathcal { O } \rangle \ . +$$ + +# 3.1 FIRST FLUCTUATION-DISSIPATION RELATION AND EQUILIBRATION + +In order to assess the proximity to stationarity, define + +$$ +{ \mathcal { O } } _ { \mathrm { L } } \equiv \pmb { \theta } \cdot \left( \pmb { \nabla } f ^ { \mathcal { B } } \right) \quad \mathrm { a n d } \quad { \mathcal { O } } _ { \mathrm { R } } \equiv \frac { ( 1 + \mu ) } { 2 ( 1 - \nu ) } \eta \mathbf { v } ^ { 2 } +$$ + +(with $\mathbf { v }$ replaced by $- \nabla f ^ { B }$ for SGD without momentum).4 Both of these observables can easily be measured on the fly at each time step during training and, according to the relation (FDR1’), the running averages of these two observables should converge to each other upon equilibration. + +![](images/dd6e095e05fa6ef441af1144be34ad1c2a5f82267adbadf967e8e05ab9904b6d.jpg) +Figure 1: Approaches toward stationarity during the initial trainings for the MLP on the MNIST data (a) and for the CNN on the CIFAR-10 data (b). Top panels depict the half-running average ${ \overline { { f ^ { B } } } } ( t )$ (dark green) and the instantaneous value $f ^ { B } ( t )$ (light green) of the mini-batch loss. Bottom panels depict the convergence of the half-running averages of the observables $O _ { \mathrm { { L } } } = \pmb { \theta } \cdot \pmb { \nabla } f ^ { B }$ and $\begin{array} { r } { \dot { \mathcal { O } } _ { \mathrm { R } } = \frac { ( 1 + \mu ) } { 2 ( 1 - \nu ) } \eta \mathbf { v } ^ { 2 } } \end{array}$ , whose stationary-state averages should agree according to the relation (FDR1’). + +In order to verify this claim, we first train the model with the learning rate $\eta = 0 . 1$ for $\hat { t } _ { \mathrm { e p o c h } } ^ { \mathrm { t o t a l } } = 1 0 0$ epochs, that is, for ttotal $\begin{array} { r } { t ^ { \mathrm { t o t a l } } = \frac { N _ { \mathrm { s } } } { | \mathcal { B } | } \hat { t } _ { \mathrm { e p o c h } } ^ { \mathrm { t o t a l } } = 1 0 0 \frac { N _ { \mathrm { s } } } { | \mathcal { B } | } } \end{array}$ time steps. As shown in the figure 1, the observables $\overline { { \mathcal { O } } } _ { \mathrm { L } } ( t )$ and $\overline { { \mathcal { O } } } _ { \mathrm { R } } ( t )$ converge to each other. We then take the model at the end of the initial 100- epoch training and sequentially train it further at various learning rates $\eta$ (see Appendix B). The observables $\breve { \mathcal { O } } _ { \mathrm { L } } ( t )$ and $\overline { { \mathcal { O } } } _ { \mathrm { R } } ( t )$ again converge to each other, as plotted in the figure 2. Note that the smaller the learning rate is, the longer it takes to equilibrate. + +# 3.2 SECOND FLUCTUATION-DISSIPATION RELATION AND SHAPE OF LOSS-FUNCTION LANDSCAPE + +In order to assess the loss-function landscape information from the relation (FDR2), define + +$$ +{ \mathcal { O } } _ { \mathrm { F B } } \equiv \left( 1 - \nu \right) \left( \nabla f \right) ^ { 2 } - \mu \mathbf { v } \cdot \nabla f ^ { \mathcal { B } } +$$ + +![](images/c0374e79625d8da67e29747429a8245babd8d8942f04402779473459843e631c.jpg) +Figure 2: Approaches toward stationarity during the sequential runs for various learning rates $\eta$ , seen through the half-running averages of the observables $O _ { \mathrm { L } } = \pmb { \theta } \cdot \pmb { \nabla } f ^ { B }$ (solid) and $\begin{array} { r } { \mathcal { O } _ { \mathrm { R } } = \frac { ( 1 + \mu ) } { 2 ( 1 - \nu ) } \eta \mathbf { v } ^ { 2 } } \end{array}$ (dotted light-colored). They agree at sufficiently long times but the relaxation time to reach such a stationary regime increases as the learning rate $\eta$ decreases. + +(with the second term nonexistent for SGD without momentum).5 Note that $\left( \nabla f \right) ^ { 2 }$ is a full-batch – not mini-batch – quantity. Given its computational cost, here we measure this first term only at the end of each epoch and take the half-running average over these sparse sample points, discarding the initial half of the run. + +The half-running average of the full-batch observable $\overline { { \mathcal { O } } } _ { \mathrm { F B } }$ at the end of sufficiently long training, which is a good proxy for $\langle \mathcal { O } _ { \mathrm { F B } } \rangle$ , is plotted in the figure 3 as a function of the learning rate $\eta$ . As predicted by the relation (FDR2), at small learning rates $\eta$ , the observable $\langle \mathcal { O } _ { \mathrm { F B } } \rangle$ approaches zero; its slope – divided by $\left. \operatorname { T r } \tilde { C } \right.$ if preferred – measures the magnitude of the Hessian matrix, component-wise averaged over directions in which the noise preferentially fluctuates. Meanwhile, nonlinearity at higher learning rates $\eta$ measures the degree of anharmonicity experienced over the distribution $p _ { \mathrm { s s } } \left( \bar { \theta } \right)$ . We see that anharmonic effects are pronounced especially for the CNN on the CIFAR-10 data even at moderately small learning rates. This invalidates the use of the quadratic harmonic approximation for the loss-function landscape and/or the assumption of the constant noise matrix for this model except at very small learning rates. + +# 3.3 FIRST FLUCTUATION-DISSIPATION RELATION AND LEARNING-RATE SCHEDULES + +Saturation of the relation (FDR1) suggests the learning stationarity, at which point it might be wise to decrease the learning rate $\eta$ . Such scheduling is often carried out in an ad hoc manner but we can now algorithmize this procedure as follows: + +1. Evaluate the half-running averages $\overline { { \mathcal { O } } } _ { \mathrm { L } } ( t )$ and $\overline { { \mathcal { O } } } _ { \mathrm { R } } ( t )$ at the end of each epoch. 2. If $\left| \frac { \overline { { \mathcal { O } } } _ { \mathrm { L } } ( t ) } { \overline { { \mathcal { O } } } _ { \mathrm { R } } ( t ) } - 1 \right| < X$ , then decrease the learning rate as $\eta ( 1 - Y ) \eta$ and also set $t = 0$ for the purpose of evaluating half-running averages. + +Here, two scheduling hyperparameters $X$ and $Y$ are introduced, which control the threshold for saturation of the relation (FDR1) and the amount of decrease in the learning rate, respectively. + +Plotted in the figure 4 are results for SGD without momentum, with the Xavier initialization (Glorot & Bengio, 2010) and training through (i) preset training schedule with decrease of the learning rate by a factor of 10 for each 100 epochs, (ii) an adaptive scheduler with $X = 0 . 0 1$ $1 \%$ threshold) and + +![](images/103dc6de5ea9ad9ef08c5e066700c85d465ad811619762e61470c6bd0105ac25.jpg) +Figure 3: The stationary-state average of the full-batch observable $\mathcal { O } _ { \mathrm { F B } }$ as a function of the learning rate $\eta$ , estimated through half-running averages. Dots and error bars denote mean values and $9 5 \%$ confidence intervals over several distinct runs, respectively. The straight red line connects the origin and the point with the smallest $\eta$ explored. (a) For the MLP on the MNIST data, linear dependence on $\eta$ for $\eta \lesssim 0 . 0 1$ supports the validity of the harmonic approximation there. (b) For the CNN on the CIFAR-10 data, anharmonicity is pronounced even down to $\eta \sim 0 . 0 0 1$ . + +$Y = 0 . 1$ $1 0 \%$ decrease), and (iii) the AMSGrad algorithm (J. Reddi et al., 2018) with the default hyperparameters. The adaptive scheduler attains comparable accuracies with the preset scheduling at long time and outperforms AMSGrad (see Appendix C for additional simulations). + +![](images/f01849030d6cbc9c9569481343254672d25b6a856d2e487ca17f88423fe9d67c.jpg) +Figure 4: Comparison of preset training schedule (black) and adaptive training schedule (blue), employing SGD without momentum both for the MLP on the MNIST data (a) and the CNN on the CIFAR-10 data (b), along with the AMSGrad algorithm (green). From top to bottom, plotted are the learning rate $\eta$ , the full-batch training loss $f$ , and prediction accuracies on the training-set images (solid) and the 10000 test-set images (dashed). + +These two scheduling methods span different subspaces of all the possible schedules. The adaptive scheduling method proposed herein has a theoretical grounding and in practice much less dimensionality for tuning of scheduling hyperparameters than the presetting method, thus ameliorating the optimization of scheduling hyperparameters. The systematic comparison between the two scheduling methods for state-of-the-arts architectures, and also the comparison with the AMSGrad algorithm for natural language processing tasks, could be a worthwhile avenue to pursue in the future. + +# 4 CONCLUSION + +In this paper, we have derived the fluctuation-dissipation relations with no assumptions other than stationarity of the probability distribution. These relations hold exactly even when the noise is nonGaussian and the loss function is nonconvex. The relations have been empirically verified and used to probe the properties of the loss-function landscapes for the simple models. The relations further have resulted in the algorithm to adaptively set learning-rate schedule on the fly rather than presetting it in an ad hoc manner. In addition to systematically testing the performance of this adaptive scheduling algorithm, it would be interesting to investigate non-Gaussianity and noncovexity in more details through higher-point observables, both analytically and numerically. It would also be interesting to further elucidate the physics of machine learning by extending our formalism to incorporate nonstationary dynamics, linearly away from stationarity (Onsager, 1931; Green, 1954; Kubo, 1957) and beyond (Jarzynski, 1997; Crooks, 1999), so that it can in particular properly treat overfitting cascading dynamics and time-dependent sample distributions. + +# ACKNOWLEDGMENTS + +The author thanks Ludovic Berthier, Leon Bottou, Guy Gur-Ari, Kunihiko Kaneko, Ari Morcos, ´ Dheevatsa Mudigere, Yann Ollivier, Yuandong Tian, and Mark Tygert for discussions. 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In Proceedings of the 28th International Conference on Machine Learning, pp. 681–688, 2011. +Zhanxing Zhu, Jingfeng Wu, Bing Yu, Lei Wu, and Jinwen Ma. The anisotropic noise in stochastic gradient descent: Its behavior of escaping from minima and regularization effects. arXiv preprint arXiv:1803.00195, 2018. + +# A SGD WITH MOMENTUM AND DAMPENING + +For SGD with momentum $\mu$ and dampening $\nu$ , the update equation is given by + +$$ +\begin{array} { r c l } { { \mathbf v ( t + 1 ) } } & { { = } } & { { \mu \mathbf v ( t ) - ( 1 - \nu ) \nabla f ^ { \mathcal { B } } \left[ \pmb { \theta } ( t ) \right] , } } \\ { { \pmb \theta ( t + 1 ) } } & { { = } } & { { \pmb \theta ( t ) + \eta \mathbf v ( t + 1 ) . } } \end{array} +$$ + +Here $\mathbf { v } = \{ v _ { i } \} _ { i = 1 , \ldots , P }$ is the velocity and $\eta > 0$ the learning rate; SGD without momentum is the special case with $\dot { \mu } = 0$ . Again hypothesizing the existence of a stationary-state distribution $p _ { \mathrm { s s } } \left( \bar { \pmb { \theta } } , \mathbf { v } \right)$ , the stationary-state average of an observable $\mathcal { O } \left( \theta , \mathbf { v } \right)$ is defined as + +$$ +\left. \mathcal { O } \left( \pmb { \theta } , \mathbf { v } \right) \right. \equiv \int \mathrm { d } \pmb { \theta } \mathrm { d } \mathbf { v } p _ { \mathrm { s s } } \left( \pmb { \theta } , \mathbf { v } \right) \mathcal { O } \left( \pmb { \theta } , \mathbf { v } \right) . +$$ + +Just as in the main text, from the assumed stationarity follows the master equation for SGD with momentum and dampening + +$$ +\begin{array} { r } { \langle \mathcal { O } \left( \pmb { \theta } , \mathbf { v } \right) \rangle = \big \langle \left[ \mathcal { O } \left\{ \pmb { \theta } + \eta \left[ \mu \mathbf { v } - ( 1 - \nu ) \pmb { \nabla } f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \right] , \mu \mathbf { v } - ( 1 - \nu ) \pmb { \nabla } f ^ { \mathcal { B } } \left( \pmb { \theta } \right) \right\} \right] _ { \mathrm { m . b . } } \big \rangle . } \end{array} +$$ + +For the linear observables, + +$$ +\left. { { \bf { v } } } \right. = \mu \left. { { \bf { v } } } \right. - \left( { 1 - \nu } \right) \left. { \nabla f \left( { \pmb { \theta } } \right) } \right. +$$ + +and + +$$ +\left. \pmb { \theta } \right. = \left. \pmb { \theta } \right. + \eta \left[ \mu \left. \mathbf { v } \right. - \left( 1 - \nu \right) \left. \pmb { \nabla } f \left( \pmb { \theta } \right) \right. \right] = \left. \pmb { \theta } \right. + \eta \left. \mathbf { v } \right. , +$$ + +thus + +$$ +\langle { \bf v } \rangle = 0 \mathrm { a n d } \langle \nabla f \rangle = 0 . +$$ + +For the quadratic observables + +$$ +\left. v _ { i } v _ { j } \right. = \mu ^ { 2 } \left. v _ { i } v _ { j } \right. + ( 1 - \nu ) ^ { 2 } \left. \widetilde { C } _ { i , j } \right. - ( 1 - \nu ) \mu \left[ \left. v _ { i } \left( \partial _ { j } f \right) \right. + \left. \left( \partial _ { i } f \right) v _ { j } \right. \right] , +$$ + +$$ +\left. v _ { i } \theta _ { j } \right. - \eta \left. v _ { i } v _ { j } \right. = \mu \left. v _ { i } \theta _ { j } \right. - \left( 1 - \nu \right) \left. \left( \partial _ { i } f \right) \theta _ { j } \right. , +$$ + +and + +$$ +( 1 - \nu ) \left[ \langle \theta _ { i } \left( \partial _ { j } f \right) \rangle + \langle \left( \partial _ { i } f \right) \theta _ { j } \rangle \right] - \mu \left( \langle \theta _ { i } v _ { j } \rangle + \langle v _ { i } \theta _ { j } \rangle \right) = \eta \left. v _ { i } v _ { j } \right. . +$$ + +Note that the relations (26) and (27) are trivially satisfied at each time step if the left-hand side observables are evaluated at one step ahead and thus their being satisfied for running averages has nothing to do with equilibration [the same can be said about the relation (23)]; the only nontrivial relation is the equation (28), which is a consequence of setting $\langle \theta _ { i } \theta _ { j } \rangle$ constant of time. After taking traces and some rearrangement, we obtain the relation (FDR1’) in the main text. + +For the full-batch loss function, the algebra similar to the one in the main text yields + +$$ +\begin{array} { r l r } { { \Big [ ( 1 - \nu ) ( \nabla f ) ^ { 2 } - \mu \mathbf { v } \cdot \nabla f \Big ] } } \\ & { = } & { \eta \sum _ { i , j = 1 } ^ { P } H _ { i , j } \{ ( 1 - \nu ) ^ { 2 } \widetilde { C } _ { i , j } - \mu ( 1 - \nu ) [ v _ { i } ( \partial _ { j } f ) + ( \partial _ { i } f ) v _ { j } ] + \mu ^ { 2 } v _ { i } v _ { j } \} + O ( \eta ^ { 2 } ) . } \end{array} +$$ + +# B MODELS AND SIMULATION PROTOCOLS + +B.1 MLP ON MNIST THROUGH SGD WITHOUT MOMENTUM + +The MNIST training data consist of $N _ { \mathrm { s } } = 6 0 0 0 0$ black-white images of hand-written digits with 28-by-28 pixels (LeCun et al., 1998). We preprocess the data through an affine transformation such that their mean and variance (over both the training data and pixels) are zero and one, respectively. + +Our multilayer perceptron (MLP) consists of a 784-dimensional input layer followed by a hidden layer of 200 neurons with ReLU activations, another hidden layer of 200 neurons with ReLU activations, and a 10-dimensional output layer with the softmax activation. The model performance is evaluated by the cross-entropy loss supplemented by the $L ^ { 2 }$ -regularization term $\scriptstyle { \frac { 1 } { 2 } } \lambda \theta ^ { 2 }$ with the weight decay $\lambda = 0 . 0 1$ . + +Throughout the paper, the MLP is trained on the MNIST data through SGD without momentum. +The data are shuffled at each epoch with the mini-batch size $| B | = 1 0 { \bar { 0 } }$ . + +The MLP is initialized through the Xavier method (Glorot & Bengio, 2010) and trained for $\hat { t } _ { \mathrm { e p o c h } } ^ { \mathrm { t o t a l } } ~ = ~ 1 0 0$ $\eta \mathit { \Pi } = \ 0 . 1$ t with. This $( \eta , \hat { t } _ { \mathrm { e p o c h } } ^ { \mathrm { t o t a l } } ) = ( 0 . 0 5 , 5 0 0 ) \to ( 0 . 0 2 , 5 0 0 ) \to ( 0 . 0 1 , 5 0 0 ) \to ( 0 . 0 0 5 , 1 0 0 0 ) \to ( 0 . 0 0 3 , 1 0 0 0 )$ data shuffling, all starting from the common model parameter attained at the end of the initial 100- epoch run. The figure 2 depicts trajectories for one particular seed, while the figure 3 plots means and error bars over these distinct seeds. + +# B.2 CNN ON CIFAR-10 THROUGH SGD WITH MOMENTUM + +The CIFAR-10 training data consist of $N _ { \mathrm { s } } ~ = ~ 5 0 0 0 0$ color images of objects – divided into ten categories – with 32-by-32 pixels in each of 3 color channels, each pixel ranging in $[ 0 , 1 ]$ (Krizhevsky & Hinton, 2009). We preprocess the data through uniformly subtracting 0.5 and multiplying by 2 so that each pixel ranges in $[ - 1 , 1 ]$ . + +In order to describe the architecture of our convolutional neural network (CNN) in detail, let us associate a tuple $[ F , C , S , P ; M ]$ to a convolutional layer with filter width $F$ , a number of channels $C$ , stride $S$ , and padding $P$ , followed by ReLU activations and a max-pooling layer of width $M$ . Then, as in the demo at Karpathy (2014), our CNN consists of a (32, 32, 3) input layer followed by a convolutional layer with $[ 5 , 1 6 , 1 , 2 ; 2 ]$ , another convolutional layer with $[ 5 , 2 0 , 1 , 2 ; 2 ]$ , yet another convolutional layer with $[ 5 , 2 0 , 1 , 2 ; 2 ]$ , and finally a fully-connected 10-dimensional output layer with the softmax activation. The model performance is evaluated by the cross-entropy loss supplemented by the $L ^ { 2 }$ -regularization term $\scriptstyle { \frac { 1 } { 2 } } \lambda ^ { \setminus } \lambda ^ { 2 }$ with the weight decay $\lambda = 0 . 0 1$ . + +Throughout the paper (except in Section 3.3 where the adaptive scheduling method is tested for SGD without momentum), the CNN is trained on the CIFAR-10 data through SGD with momentum $\mu = 0 . 9$ and dampening $\nu = 0$ . The data are shuffled at each epoch with the mini-batch size $| B | = 1 0 0$ . + +$\hat { t } _ { \mathrm { e p o c h } } ^ { \mathrm { t o t a l } } = 1 0 0$ initialized through the Xavepochs with the learning rate $\eta = 0 . 1$ hod (Glorot & Bengio, 2010) an. We then sequentially train it with $( \eta , \hat { t } _ { \mathrm { e p o c h } } ^ { \mathrm { t o t a l } } ) =$ ( $0 . 0 5 , 2 0 0 ) ( 0 . 0 2 , 2 0 0 ) ( 0 . 0 1 , 2 0 0 ) ( 0 . 0 0 5 , 4 0 0 ) ( 0 . 0 0 3 , 4 0 0 ) ( 0 . 0 0 2 , 4 0 0 ) ( 0 . 0 0 2 , 4 0 0 ) ( 0 . 0 0 3 , 4 0 0 )$ $( 0 . 0 0 1 5 , 4 0 0 ) ( 0 . 0 0 1 , 4 0 0 ) ( 0 . 0 0 0 5 , 8 0 0 ) \nonumber$ $( 0 . 0 0 0 5 , 8 0 0 ) \to ( 0 . 0 0 0 2 5 , 8 0 0 ) \to ( 0 . 0 0 0 1 , 8 0 0 )$ . At each junction of the sequence, the velocity $\mathbf { v }$ is zeroed. This sequential-run protocol is carried out with 16 distinct seeds for the random-number generator used in data shuffling, all starting from the common model parameter attained at the end of the initial 100-epoch run. The figure 2 depicts trajectories for one particular seed, while the figure 3 plots means and error bars over these distinct seeds. + +# C ADDITIONAL SIMULATIONS + +# C.1 ADAM VERSUS AMSGRAD + +Plotted in the figure S1 are the comparisons between Adam (Kingma & Ba, 2014) and AMSGrad (J. Reddi et al., 2018) algorithms with the default hyperparameters $\alpha = 1 0 ^ { - 3 }$ , $( \beta _ { 1 } , \beta _ { 2 } ) =$ (0.9, 0.999), and $\epsilon = 1 0 ^ { - 8 }$ . The AMSGrad algorithm marginally outperforms the Adam algorithm for the tasks at hand and thus the results with the AMSGrad are presented in the main text. + +![](images/8def667daebc2aa4b197f16b14c7b40436d97606a86f6209006025629d06c79c.jpg) +Figure S1: Comparison of AMSGrad (green) and Adam (orange) algorithms for the MLP on the MNIST data (a) and the CNN on the CIFAR-10 data (b). Top rows plot the full-batch training loss $f$ while bottom rows plot prediction accuracies on the training-set images (solid) and the 10000 test-set images (dashed). + +# C.2 INITIAL ACCURACY GAIN WITH DIFFERENT SCHEDULING HYPERPARAMETERS + +In the figure 4(a) for the MNIST classification task with the MLP, the proposed adaptive method with the scheduling hyperparameters $X ~ = ~ 0 . 0 1$ and $Y ~ = ~ 0 . 1$ outperforms the AMSGrad algorithm in terms of accuracy attained at long time and also exhibits a quick initial convergence. In the figure 4(b) for the CIFAR-10 classification task with the CNN, however, while the proposed adaptive method attains better accuracy at long time, its initial accuracy gain is visibly slower than the AMSGrad algorithm. This lag in initial accuracy gain can be ameliorated by choosing another combination of the scheduling hyperparameters, e.g., $X \ = \ 0 . 1$ and $Y = 0 . 3$ , at the expense of degradation in generalization accuracy with respect to the original choice $X = 0 . 0 1$ and $Y = 0 . 1$ . See the figure S2. + +![](images/f5057f4b1b2f6d6801470a5a515afb41274608c952c91a877d63ba45a92a3a55.jpg) +Figure S2: Comparison of preset training schedule (black) and adaptive training schedule (purple) – now with the scheduling hyperparameters $X ~ = ~ 0 . 1$ and $Y \ = \ 0 . 3$ – employing SGD without momentum, and the AMSGrad algorithm (green), for the CNN on the CIFAR-10 data with the same initial seed as in the main text (a) and three different initial seeds (b-d). From top to bottom, plotted are the learning rate $\eta$ , the full-batch training loss $f$ , and prediction accuracies on the training-set images (solid) and the 10000 test-set images (dashed). \ No newline at end of file diff --git a/parse/train/SkNksoRctQ/SkNksoRctQ_content_list.json b/parse/train/SkNksoRctQ/SkNksoRctQ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2a04905d3d82fcd7571e579fdda528f1e328d88a --- /dev/null +++ b/parse/train/SkNksoRctQ/SkNksoRctQ_content_list.json @@ -0,0 +1,1983 @@ +[ + { + "type": "text", + "text": "FLUCTUATION-DISSIPATION RELATIONS FOR STOCHASTIC GRADIENT DESCENT ", + "text_level": 1, + "bbox": [ + 174, + 101, + 705, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sho Yaida ", + "text_level": 1, + "bbox": [ + 184, + 170, + 256, + 183 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Facebook AI Research \nFacebook Inc. \nMenlo Park, California 94025, USA shoyaida@fb.com ", + "bbox": [ + 184, + 184, + 421, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The notion of the stationary equilibrium ensemble has played a central role in statistical mechanics. In machine learning as well, training serves as generalized equilibration that drives the probability distribution of model parameters toward stationarity. Here, we derive stationary fluctuation-dissipation relations that link measurable quantities and hyperparameters in the stochastic gradient descent algorithm. These relations hold exactly for any stationary state and can in particular be used to adaptively set training schedule. We can further use the relations to efficiently extract information pertaining to a loss-function landscape such as the magnitudes of its Hessian and anharmonicity. Our claims are empirically verified. ", + "bbox": [ + 233, + 306, + 764, + 431 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 454, + 336, + 470 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Equilibration rules the long-term fate of many macroscopic dynamical systems. For instance, as we pour water into a glass and let it be, the stationary state of tranquility is eventually attained. Zooming into the tranquil water with a microscope would reveal, however, a turmoil of stochastic fluctuations that maintain the apparent stationarity in balance. This is vividly exemplified by the Brownian motion (Brown, 1828): a pollen immersed in water is constantly bombarded by jittery molecular movements, resulting in the macroscopically observable diffusive motion of the solute. Out of the effort in bridging microscopic and macroscopic realms through the Brownian movement came a prototype of fluctuation-dissipation relations (Einstein, 1905; Von Smoluchowski, 1906). These relations quantitatively link degrees of noisy microscopic fluctuations to smooth macroscopic dissipative phenomena and have since been codified in the linear response theory for physical systems (Onsager, 1931; Green, 1954; Kubo, 1957), a cornerstone of statistical mechanics. ", + "bbox": [ + 174, + 486, + 825, + 637 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Machine learning begets another form of equilibration. As a model learns patterns in data, its performance first improves and then plateaus, again reaching apparent stationarity. This dynamical process naturally comes equipped with stochastic fluctuations as well: often given data too gigantic to consume at once, training proceeds in small batches and random selections of these mini-batches consequently give rise to the noisy dynamical excursion of the model parameters in the loss-function landscape, reminiscent of the Brownian motion. It is thus natural to wonder if there exist analogous fluctuation-dissipation relations that quantitatively link the noise in mini-batched data to the observable evolution of the model performance and that in turn facilitate the learning process. ", + "bbox": [ + 174, + 645, + 823, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Here, we derive such fluctuation-dissipation relations for the stochastic gradient descent algorithm. The only assumption made is stationarity of the probability distribution that governs the model parameters at sufficiently long time. Our results thus apply to generic cases with non-Gaussian mini-batch noises and nonconvex loss-function landscapes. Practically, the first relation (FDR1) offers the metric for assessing equilibration and yields an adaptive algorithm that sets learning-rate schedule on the fly. The second relation (FDR2) further helps us determine the properties of the lossfunction landscape, including the strength of its Hessian and the degree of anharmonicity, i.e., the deviation from the idealized harmonic limit of a quadratic loss surface and a constant noise matrix. ", + "bbox": [ + 174, + 763, + 823, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our approach should be contrasted with recent attempts to import the machinery of stochastic differential calculus into the study of the stochastic gradient descent algorithm (Mandt et al., 2015; Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2017; ", + "bbox": [ + 176, + 882, + 823, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). This line of work all assumes Gaussian noises and sometimes additionally employs the quadratic harmonic approximation for loss-function landscapes. The more severe drawback, however, is the usage of the analogy with continuous-time stochastic differential equations, which is inconsistent in general (see Section 2.3.3). Instead, the stochastic gradient descent algorithm can be properly treated within the framework of the KramersMoyal expansion (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen & Moody, 1993). ", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The paper is organized as follows. In Section 2, after setting up notations and deriving a stationary fluctuation-dissipation theorem (FDT), we derive two specific fluctuation-dissipation relations. The first relation (FDR1) can be used to check stationarity and the second relation (FDR2) to delineate the shape of the loss-function landscape, as empirically borne out in Section 3. An adaptive scheduling method is proposed and tested in Section 3.3. We conclude in Section 4 with future outlooks. ", + "bbox": [ + 174, + 208, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 FLUCTUATION-DISSIPATION RELATIONS ", + "text_level": 1, + "bbox": [ + 174, + 315, + 537, + 332 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A model is parametrized by a weight coordinate, $\\pmb { \\theta } = \\{ \\theta _ { i } \\} _ { i = 1 , \\dots , P }$ . The training set of $N _ { \\mathrm { s } }$ examples is utilized by the model to learby a full-batch loss function, $\\begin{array} { r } { \\dot { f } \\left( \\pmb { \\theta } \\right) \\equiv \\frac { 1 } { N _ { \\mathrm { s } } } \\sum _ { \\alpha = 1 } ^ { N _ { \\mathrm { s } } } { f _ { \\alpha } \\left( \\pmb { \\theta } \\right) } } \\end{array}$ e mod, with $f _ { \\alpha } \\left( \\pmb { \\theta } \\right)$ erall performance is evaluatedquantifying the performance of the model on a particular sample $\\alpha$ : the smaller the loss is, the better the model is expected to perform. The learning process can thus be cast as an optimization problem of minimizing the loss function. One of the most commonly used optimization schemes is the stochastic gradient descent (SGD) algorithm (Robbins $\\&$ Monro, 1951) in which a mini-batch $B \\subset \\{ 1 , 2 , \\ldots , N _ { \\mathrm { s } } \\}$ of size $| B |$ is stochastically chosen for training at each time step. Specifically, the update equation is given by ", + "bbox": [ + 173, + 347, + 825, + 465 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/faf41f795c68a01f1e78f3a95b39a1c2b42c3584ecc871188a222772ea43c102.jpg", + "text": "$$\n\\pmb \\theta ( t + 1 ) = \\pmb \\theta ( t ) - \\eta \\pmb \\nabla f ^ { \\mathcal { B } } \\left[ \\pmb \\theta ( t ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 474, + 611, + 493 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\eta > 0$ is a learning rate and a mini-batch loss $\\begin{array} { r } { f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\equiv \\frac { 1 } { \\left| \\mathcal { B } \\right| } \\sum _ { \\alpha \\in \\mathcal { B } } f _ { \\alpha } \\left( \\pmb { \\theta } \\right) } \\end{array}$ . Note that ", + "bbox": [ + 173, + 500, + 767, + 520 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7c882868a8ebbd570e66251f4b62da090e71348c129c2ccdee6e64c4f8720ed6.jpg", + "text": "$$\n\\left[ \\nabla f ^ { \\boldsymbol { B } } \\left( \\pmb { \\theta } \\right) \\right] _ { \\mathrm { m . b . } } = \\nabla f \\left( \\pmb { \\theta } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 405, + 530, + 594, + 551 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "with $\\mathbb { I } . . . \\mathbb { J } _ { \\mathrm { m . b . } }$ . denoting the average over mini-batch realizations. For later purposes, it is convenient to define a full two-point noise matrix $\\widetilde { C }$ through1 ", + "bbox": [ + 174, + 559, + 825, + 592 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/93a92da04daa633144c1cb552f51528761213fa11716fee794e4e5bf9785db7a.jpg", + "text": "$$\n\\widetilde { C } _ { i , j } ( \\pmb { \\theta } ) \\equiv [ [ \\partial _ { i } f ^ { \\mathcal { B } } ( \\pmb { \\theta } ) ] [ \\partial _ { j } f ^ { \\mathcal { B } } ( \\pmb { \\theta } ) ] ] ] _ { \\mathrm { m . b . } }\n$$", + "text_format": "latex", + "bbox": [ + 362, + 599, + 632, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "and, more generally, higher-point noise tensors ", + "bbox": [ + 174, + 630, + 482, + 645 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/24043a79c7e4b2bb0efb643e8d5ba58cb66c94630e7f2c1ceaa8770401b8cb72.jpg", + "text": "$$\n\\tilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\left( \\pmb { \\theta } \\right) \\equiv \\left[ \\left[ \\partial _ { i _ { 1 } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { i _ { 2 } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\cdots \\left[ \\partial _ { i _ { k } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\right] _ { \\mathrm { m . b . } } .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 654, + 717, + 676 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Below, we shall not make any assumptions on the distribution of the noise vector $\\nabla f ^ { B }$ – other than that a mini-batch is independent and identically distributed from the $N _ { \\mathrm { s } }$ training samples at each time step – and the noise distribution is therefore allowed to have nontrivial higher connected moments indicative of non-Gaussianity. ", + "bbox": [ + 174, + 685, + 825, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "It is empirically often observed that the performance of the model plateaus after some training through SGD. It is thus natural to hypothesize the existence of a stationary-state distribution, $p _ { \\mathrm { s s } } \\left( \\pmb { \\theta } \\right)$ , that dictates the SGD sampling at long time (see Section 2.3.4 for discussion on this assumption). For any observable quantity, $\\mathcal { O } \\left( \\pmb { \\theta } \\right)$ , – something that can be measured during training such as $\\pmb { \\theta } ^ { 2 }$ and $f \\left( \\pmb { \\theta } \\right) -$ its stationary-state average is then defined as ", + "bbox": [ + 173, + 748, + 825, + 819 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/94278d1546a8f41e6bf0f2088fc70e9333694a23a8d62b20fe5d1c579bd74329.jpg", + "text": "$$\n\\left. \\mathcal O \\left( \\pmb { \\theta } \\right) \\right. \\equiv \\int \\mathrm { d } \\pmb { \\theta } p _ { \\mathrm { s s } } \\left( \\pmb { \\theta } \\right) \\mathcal O \\left( \\pmb { \\theta } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 828, + 607, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In general the probability distribution of the model parameters evolves as $p ( \\pmb { \\theta } , t + 1 ) =$ $\\begin{array} { r } { \\left[ \\int \\bar { \\mathrm { d } } \\pmb { \\theta } ^ { \\prime } p ( \\pmb { \\theta } ^ { \\prime } , t ) \\delta \\bar { \\left\\{ \\pmb { \\theta } - \\left[ \\pmb { \\theta } ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } ^ { \\prime } \\right) \\right] \\right\\} } \\right] _ { \\mathrm { m . b . } } } \\end{array}$ and in particular for the stationary state ", + "bbox": [ + 174, + 103, + 825, + 136 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6951a2b906a707d090b5bf3ad90ac8bc037c4c6f0ef2d8f7caab4d3a1c2626e1.jpg", + "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\int \\mathrm { d } \\theta p _ { \\mathrm { s s } } \\left( \\theta , t \\right) \\mathcal { O } \\left( \\theta \\right) } & { = } & { \\displaystyle \\int \\mathrm { d } \\theta \\ d p _ { \\mathrm { s s } } \\left( \\theta , t + 1 \\right) \\mathcal { O } \\left( \\theta \\right) } \\\\ & { = } & { \\displaystyle \\left[ \\int \\mathrm { d } \\theta \\int \\mathrm { d } \\theta ^ { \\prime } p _ { \\mathrm { s s } } ( \\theta ^ { \\prime } , t ) \\delta \\left\\{ \\theta - \\left[ \\theta ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\theta ^ { \\prime } \\right) \\right] \\right\\} \\mathcal { O } \\left( \\theta \\right) \\right] _ { \\mathrm { m . b . } } } \\\\ & { = } & { \\displaystyle \\int \\mathrm { d } \\theta ^ { \\prime } p _ { \\mathrm { s s } } \\left( \\theta ^ { \\prime } \\right) \\left[ \\mathcal { O } \\left[ \\theta ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\theta ^ { \\prime } \\right) \\right] \\right] _ { \\mathrm { m . b . } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 141, + 802, + 244 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Thus follows the master equation ", + "bbox": [ + 174, + 248, + 395, + 262 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0ffb2372834b8ec8232844d9b1568e34ee777129622569610c2743ab3c46b4ff.jpg", + "text": "$$\n { \\mathcal O ( { \\pmb \\theta } ) } = { [ { \\mathcal O [ { \\pmb \\theta } - \\eta { \\pmb \\nabla } f ^ { \\beta } ( { \\pmb \\theta } ) } ] } ] _ { \\mathrm { m . b . } } .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 267, + 637, + 289 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the next two subsections, we apply this general formula to simple observables in order to derive various stationary fluctuation-dissipation relations. Incidentally, the discrete version of the FokkerPlanck equation can be derived through the Kramers-Moyal expansion, considering the more general nonstationary version of the above equation and performing the Taylor expansion in $\\eta$ and repeated integrations by parts (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen & Moody, 1993). ", + "bbox": [ + 173, + 291, + 825, + 376 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 FIRST FLUCTUATION-DISSIPATION RELATION ", + "text_level": 1, + "bbox": [ + 174, + 393, + 526, + 407 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Applying the master equation (FDT) to the linear observable, ", + "bbox": [ + 174, + 419, + 575, + 434 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c662baacc287502c21297d8daaeb0b59a355a40fc8b6a19d1af9deb8f39d6ed7.jpg", + "text": "$$\n\\left. \\pmb { \\theta } \\right. = \\left. \\left[ \\pmb { \\theta } - \\eta \\pmb { \\nabla } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] _ { \\mathrm { m . b . } } \\right. = \\left. \\pmb { \\theta } \\right. - \\eta \\left. \\pmb { \\nabla } f \\left( \\pmb { \\theta } \\right) \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 439, + 678, + 462 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We thus have ", + "bbox": [ + 174, + 464, + 264, + 478 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b92ca98e5c611ff5d19c9a6589d474eca7d209515b0446f84cc0fa42effb4eb6.jpg", + "text": "$$\n\\langle \\nabla f \\rangle = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 460, + 477, + 539, + 493 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This is natural because there is no particular direction that the gradient picks on average as the model parameter stochastically bounces around the local minimum or, more generally, wanders around the loss-function landscape according to the stationary distribution. ", + "bbox": [ + 174, + 496, + 825, + 539 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Performing similar algebra for the quadratic observable $\\langle \\theta _ { i } \\theta _ { j } \\rangle$ yields ", + "bbox": [ + 173, + 545, + 625, + 560 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f56879e724da5f7ba7c32189a48d9bf6a61690c6702057e2b51af38c9b36c4af.jpg", + "text": "$$\n\\langle \\theta _ { i } \\left( \\partial _ { j } f \\right) \\rangle + \\langle \\left( \\partial _ { i } f \\right) \\theta _ { j } \\rangle = \\eta \\left. \\widetilde { C } _ { i , j } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 566, + 624, + 594 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In particular, taking the trace of this matrix-form relation, we obtain ", + "bbox": [ + 171, + 598, + 620, + 613 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2df2418efc4668835bfeab7c81005e8a8b573e712db1b26873f25126d112abe7.jpg", + "text": "$$\n\\langle \\pmb { \\theta } \\cdot ( \\pmb { \\nabla } f ) \\rangle = \\frac { 1 } { 2 } \\eta \\left. \\mathrm { T r } \\widetilde { C } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 618, + 593, + 648 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "More generally, in the case of SGD with momentum $\\mu$ and dampening $\\nu$ , whose update equation is given by ", + "bbox": [ + 171, + 652, + 823, + 683 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/901070c31094cea11f7bb6b45e3cc9942cb97e067647b6bf93bb7a999d88b685.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\mathbf v ( t + 1 ) } } & { { = } } & { { \\mu \\mathbf v ( t ) - ( 1 - \\nu ) \\nabla f ^ { \\mathcal { B } } \\left[ \\pmb { \\theta } ( t ) \\right] , } } \\\\ { { \\pmb \\theta ( t + 1 ) } } & { { = } } & { { \\pmb \\theta ( t ) + \\eta \\mathbf v ( t + 1 ) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 686, + 647, + 727 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "a similar derivation yields (see Appendix A) ", + "bbox": [ + 174, + 729, + 467, + 744 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/74874fcbc349287a32ae42d19afb08c61b36c9ac0b853cd64b5fd9bf0d5cedac.jpg", + "text": "$$\n\\langle \\pmb \\theta \\cdot ( \\pmb \\nabla f ) \\rangle = \\frac { ( 1 + \\mu ) } { 2 ( 1 - \\nu ) } \\eta \\left. \\mathbf { v } ^ { 2 } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 750, + 607, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The last equation reduces to the equation (FDR1) when $\\mu = \\nu = 0$ with $\\mathbf { v } = - \\nabla f ^ { B }$ . Also note that $\\left. \\pmb { \\theta } \\cdot ( \\pmb { \\nabla } f ) \\right. = \\left. ( \\pmb { \\theta } - \\pmb { \\theta } _ { \\mathrm { c } } ) \\cdot ( \\pmb { \\nabla } f ) \\right.$ for an arbitrary constant vector $\\theta _ { \\mathrm { c } }$ because of the equation (8). ", + "bbox": [ + 171, + 790, + 823, + 820 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This first fluctuation-dissipation relation is easy to evaluate on the fly during training, exactly holds without any approximation if sampled well from the stationary distribution, and can thus be used as the standard metric to check if learning has plateaued, just as similar relations can be used to check equilibration in Monte Carlo simulations of physical systems (Santen & Krauth, 2000). [It should be cautioned, however, that the fluctuation-dissipation relations are necessary but not sufficient to ensure stationarity (Odriozola & Berthier, 2011).] Such a metric can in turn be used to schedule changes in hyperparameters, as shall be demonstrated in Section 3.3. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 SECOND FLUCTUATION-DISSIPATION RELATION ", + "text_level": 1, + "bbox": [ + 173, + 103, + 544, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Applying the master equation (FDT) on the full-batch loss function and Taylor-expanding it in the learning rate $\\eta$ yields the closed-form expression ", + "bbox": [ + 171, + 128, + 825, + 159 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/99aa08971b83899a98ed4879bea52b257f7ee185412ed4678f2a5cf135692e8a.jpg", + "text": "$$\n\\begin{array} { r c l } { f \\left( \\pmb { \\theta } \\right) \\rangle } & { = } & { \\displaystyle \\left. \\left[ f \\left[ \\pmb { \\theta } - \\eta \\boldsymbol { \\nabla } f ^ { B } \\left( \\pmb { \\theta } \\right) \\right] \\right] _ { \\mathrm { m . b . } } \\right. } \\\\ & { = } & { \\displaystyle \\left. f + \\sum _ { k = 1 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k } } { k ! } \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } \\left( \\partial _ { i _ { 1 } } \\partial _ { i _ { 2 } } \\cdots \\partial _ { i _ { k } } f \\right) \\left[ \\left( \\partial _ { i _ { 1 } } f ^ { B } \\right) \\left( \\partial _ { i _ { 2 } } f ^ { B } \\right) \\cdots \\left( \\partial _ { i _ { k } } f ^ { B } \\right) \\right] _ { \\mathrm { m . b . } } \\right. } \\\\ & { = } & { \\displaystyle \\left. f \\right. - \\eta \\left. \\left( \\pmb { \\nabla } f \\right) ^ { 2 } \\right. + \\sum _ { k = 2 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k } } { k ! } \\left. \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 164, + 828, + 280 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where we recalled the equation (4) and introduced ", + "bbox": [ + 173, + 286, + 504, + 301 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4211d605971ec4f1f22304d5ffbb6e3f62ed1cd919eeede218f43e63390af1e7.jpg", + "text": "$$\nF _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\left( \\pmb { \\theta } \\right) \\equiv \\partial _ { i _ { 1 } } \\partial _ { i _ { 2 } } \\cdot \\cdot \\cdot \\partial _ { i _ { k } } f \\left( \\pmb { \\theta } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 374, + 309, + 624, + 327 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In particular, ${ { H } _ { i , j } } \\left( \\pmb { \\theta } \\right) \\equiv { { F } _ { i , j } } \\left( \\pmb { \\theta } \\right)$ is the Hessian matrix. Reorganizing terms, we obtain ", + "bbox": [ + 171, + 333, + 740, + 349 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2e3980171ceb1fc599439671eea53d32c0fc531f468c2b77fe0b9198e57cb333.jpg", + "text": "$$\n\\left. \\left( \\nabla f \\right) ^ { 2 } \\right. = \\frac { \\eta } { 2 } \\left. \\mathrm { T r } \\left( H \\tilde { C } \\right) \\right. - \\eta ^ { 2 } \\left[ \\sum _ { k = 3 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k - 3 } } { k ! } \\left. \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\right. \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 199, + 357, + 800, + 409 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 771, + 405, + 823, + 417 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the case of SGD with momentum and dampening, the left-hand side is replaced by $( 1 ~ -$ $\\nu ) \\left. \\left( \\nabla f \\right) ^ { 2 } \\right. - \\mu \\left. \\mathbf { v } \\cdot \\pmb { \\nabla } f \\right.$ and $\\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\dots , i _ { k } }$ by more hideous expressions (see Appendix A). ", + "bbox": [ + 173, + 417, + 825, + 454 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can extract at least two types of information on the loss-function landscape by evaluating the dependence of the left-hand side, $G ( \\eta ) \\equiv \\left. \\left( \\nabla f \\right) ^ { 2 } \\right.$ , on the learning rate $\\eta$ . First, in the small learning rate regime, the value of $2 G ( \\eta ) / \\eta$ approximates $\\mathrm { T r } \\left( H { \\tilde { C } } \\right)$ around a local ravine. Second, nonlinearity of $G ( \\eta )$ at higher $\\eta$ indicates discernible effects of anharmonicity. In such a regime, the Hessian matrix $\\pmb { H }$ cannot be approximated as constant (which also implies that $\\{ F _ { i _ { 1 } , i _ { 2 } , . . . , i _ { k } } \\} _ { k > 2 }$ are nontrivial) and/or the noise two-point matrix $\\widetilde { C }$ cannot be regarded as constant. Such nonlinearity especially indicates the breakdown of the harmonic approximation, that is, the quadratic truncation of the loss-function landscape, often used to analyze the regime explored at small learning rates. ", + "bbox": [ + 173, + 462, + 826, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 REMARKS ", + "text_level": 1, + "bbox": [ + 174, + 617, + 287, + 632 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3.1 INTUITION WITHIN THE HARMONIC APPROXIMATION ", + "text_level": 1, + "bbox": [ + 173, + 643, + 593, + 659 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to gain some intuition about the fluctuation-dissipation relations, let us momentarily employ the harmonic approximation, i.e., assume that there is a local minimum of the loss function at $\\theta \\ : = \\ : \\theta ^ { \\star }$ and retain only up to quadratic terms of the Taylor expansions around it: $f ( \\pmb \\theta ) ~ \\approx$ $\\begin{array} { r } { f _ { 0 } { + } \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { P } h _ { i , j } ( \\theta _ { i } { - } \\theta _ { i } ^ { \\star } ) ( \\theta _ { j } { - } \\theta _ { j } ^ { \\star } ) } \\end{array}$ . Within this approximation, $\\langle \\pmb \\theta \\cdot ( \\pmb \\nabla f ) \\rangle = \\langle ( \\pmb \\theta - \\pmb \\theta ^ { \\star } ) \\cdot ( \\pmb \\nabla f ) \\rangle \\approx$ $2 \\left. f - f _ { 0 } \\right.$ . The relation (FDR1) then becomes $\\begin{array} { r } { \\langle f - f _ { 0 } \\rangle \\approx \\frac { 1 } { 4 } \\eta \\left. \\mathrm { T r } \\widetilde { C } \\right. } \\end{array}$ , linking the height of the noise ball to the noise amplitude. This is in line with, for instance, the theorem 4.6 of the reference Bottou et al. (2018) and substantiates the analogy between SGD and simulated annealing, with the learning rate $\\eta$ – multiplied by $\\mathrm { T r } \\widetilde { C }$ – playing the role of temperature (Bottou, 1991). ", + "bbox": [ + 171, + 667, + 826, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3.2 HIGHER-ORDER RELATIONS ", + "text_level": 1, + "bbox": [ + 176, + 813, + 421, + 828 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Additional relations can be derived by repeating similar calculations for higher-order observables. For example, at the cubic order, ", + "bbox": [ + 173, + 838, + 823, + 867 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/253dabe82cea34cd22e4882e49e1217e38877dd6c19c2772e321d653b7dea368.jpg", + "text": "$$\n\\left. \\theta _ { i } \\theta _ { j } \\left( \\partial _ { k } f \\right) + \\theta _ { i } \\left( \\partial _ { j } f \\right) \\theta _ { k } + \\left( \\partial _ { i } f \\right) \\theta _ { j } \\theta _ { k } \\right. = \\eta \\left. \\theta _ { i } \\widetilde { C } _ { j , k } + \\theta _ { j } \\widetilde { C } _ { k , i } + \\theta _ { k } \\widetilde { C } _ { i , j } \\right. - \\eta ^ { 2 } \\left. \\widetilde { C } _ { i , j , k } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 875, + 784, + 902 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The systematic investigation of higher-order relations is relegated to future work. ", + "bbox": [ + 173, + 909, + 705, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3.3 SGD6=SDE ", + "text_level": 1, + "bbox": [ + 174, + 103, + 310, + 118 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "There is no limit in which SGD asymptotically reduces to the stochastic differential equation (SDE). In order to take such a limit with continuous time differential $\\mathrm d t \\to 0 ^ { + }$ , each SGD update must become infinitesimal. One may thus try $\\mathrm { d } t \\equiv \\eta 0 ^ { + }$ , as in recent work adapting the view that SGD $=$ SDE (Mandt et al., 2015; Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2017; Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). But this in turn forces the noise vector with zero mean, √ $\\nabla f ^ { B } - \\nabla f$ , to be multiplied by $\\mathrm { d } t$ . This is in contrast to the scaling $\\sqrt { \\mathrm { d } t }$ needed for the standard machinery of SDE – Ito-Stratonovich calculus and all ˆ that – to apply; the additional factor of $\\mathrm { d } t ^ { 1 / 2 }$ makes the effective noise covariance be suppressed by $\\mathrm { d } t$ and the resulting equation in the continuous-time limit, if anything, would just be an ordinary differential equation without noise2 [unless noise with the proper scaling is explicitly added as in stochastic gradient Langevin dynamics (Welling & Teh, 2011; Teh et al., 2016) and natural Langevin dynamics (Marceau-Caron & Ollivier, 2017; Nado et al., 2018)]. ", + "bbox": [ + 173, + 128, + 825, + 299 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In short, the recent work views $\\eta = { \\sqrt { \\eta } } { \\sqrt { \\mathrm { d } t } }$ and sends $\\mathrm d t \\to 0 ^ { + }$ while pretending that $\\eta$ is finite, which is inconsistent. This is not just a technical subtlety. When unjustifiably passing onto the continuous-time Fokker-Planck equation, the diffusive term is incorrectly governed by the connected two-point noise matrix $C _ { i , j } \\left( \\pmb { \\theta } \\right) \\equiv \\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right) - \\left[ \\partial _ { i } f \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { j } f \\left( \\pmb { \\theta } \\right) \\right]$ rather than the full two-point noise matrix $\\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right)$ that appears herein.3 We must instead employ the discrete-time version of the Fokker-Planck equation derived in references Van Kampen (1992); Gardiner (2009); Risken (1984); Radons et al. (1990); Leen & Moody (1993), as has been followed in the equation (6). ", + "bbox": [ + 174, + 306, + 825, + 410 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3.4 ON STATIONARITY ", + "text_level": 1, + "bbox": [ + 176, + 428, + 356, + 441 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In contrast to statistical mechanics where an equilibrium state is dictated by a handful of thermodynamic variables, in machine learning a stationary state generically depends not only on hyperparameters but also on a part of its learning history. The stationarity assumption made herein, which is codified in the equation (6), is weaker than the typicality assumption underlying statistical mechanics and can hold even in the presence of lingering memory. In the full-batch limit $| B | = N _ { \\mathrm { s } }$ , for instance, any distribution delta-peaked at a local minimum is stationary. For sufficiently small learning rates $\\eta$ as well, it is natural to expect multiple stationary distributions that form disconnected ponds around these minima, which merge upon increasing $\\eta$ and fragment upon decreasing $\\eta$ . ", + "bbox": [ + 174, + 452, + 825, + 564 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "It is beyond the scope of the present paper to formulate conditions under which stationary distributions exist. Indeed, if the formulation were too generic, there could be counterexamples to such a putative existence statement. A case in point is a model with the unregularized cross entropy loss, whose model parameters keep cascading toward infinity in order to sharpen its softmax output (Neyshabur et al., 2014; 2017) with logarithmically diverging $\\pmb { \\theta } ^ { 2 }$ (Soudry et al., 2018). It would be interesting to see if there are any other nontrivial caveats. ", + "bbox": [ + 173, + 570, + 825, + 654 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EMPIRICAL TESTS ", + "text_level": 1, + "bbox": [ + 176, + 676, + 356, + 691 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section we empirically bear out our theoretical claims in the last section. To this end, two simple models of supervised learning are used (see Appendix B for full specifications): a multilayer perceptron (MLP) learning patterns in the MNIST training data (LeCun et al., 1998) through SGD without momentum and a convolutional neural network (CNN) learning patterns in the CIFAR10 training data (Krizhevsky & Hinton, 2009) through SGD with momentum $\\mu = 0 . 9$ . For both models, the mini-batch size is set to be $| B | = 1 0 0$ , and the training data are shuffled at each epoch $\\begin{array} { r } { t = \\frac { N _ { \\mathrm { s } } } { | B | } \\hat { t } _ { \\mathrm { e p o c h } } } \\end{array}$ with $\\hat { t } _ { \\mathrm { e p o c h } } \\in \\mathbb { N }$ . In order to avoid the overfitting cascade mentioned in Section 2.3.4, the $L ^ { 2 }$ -regularization term $\\scriptstyle { \\frac { 1 } { 2 } } \\lambda \\theta ^ { 2 }$ with the weight decay $\\lambda = 0 . 0 1$ is included in the loss function $f$ . ", + "bbox": [ + 173, + 708, + 825, + 828 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Before proceeding further, let us define the half-running average of an observable $\\mathcal { O }$ as ", + "bbox": [ + 174, + 103, + 741, + 119 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6f4b20d9ef2037c1fadba026b8e477f3149b00f7c865c0613bc2eaf652f2024a.jpg", + "text": "$$\n\\overline { { \\mathcal { O } } } ( t ) \\equiv \\frac { 1 } { t - t _ { 0 } } \\sum _ { t ^ { \\prime } = t _ { 0 } + 1 } ^ { t } \\mathcal { O } ( t ^ { \\prime } ) \\mathrm { w i t h } t _ { 0 } = \\lfloor t / 2 \\rfloor .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 118, + 661, + 162 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This is the average of the observable up to the time step $t$ , with the initial half discarded as containing transient. If SGD drives the distribution of the model parameters to stationarity at long time, then ", + "bbox": [ + 176, + 164, + 821, + 193 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/3a8ffa5554f0fc893082c5151a11b8cea44c4f8bad2b0db390f536c798d65045.jpg", + "text": "$$\n\\operatorname * { l i m } _ { t \\infty } \\overline { { \\mathcal { O } } } ( t ) = \\langle \\mathcal { O } \\rangle \\ .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 193, + 562, + 217 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 FIRST FLUCTUATION-DISSIPATION RELATION AND EQUILIBRATION ", + "text_level": 1, + "bbox": [ + 176, + 229, + 673, + 244 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In order to assess the proximity to stationarity, define ", + "bbox": [ + 173, + 256, + 524, + 270 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6c82645f48c7891c82f54b900d39af6861097d2a3f3932044e91a5c35e51ca31.jpg", + "text": "$$\n{ \\mathcal { O } } _ { \\mathrm { L } } \\equiv \\pmb { \\theta } \\cdot \\left( \\pmb { \\nabla } f ^ { \\mathcal { B } } \\right) \\quad \\mathrm { a n d } \\quad { \\mathcal { O } } _ { \\mathrm { R } } \\equiv \\frac { ( 1 + \\mu ) } { 2 ( 1 - \\nu ) } \\eta \\mathbf { v } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 346, + 271, + 651, + 305 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(with $\\mathbf { v }$ replaced by $- \\nabla f ^ { B }$ for SGD without momentum).4 Both of these observables can easily be measured on the fly at each time step during training and, according to the relation (FDR1’), the running averages of these two observables should converge to each other upon equilibration. ", + "bbox": [ + 174, + 306, + 825, + 351 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/dd6e095e05fa6ef441af1144be34ad1c2a5f82267adbadf967e8e05ab9904b6d.jpg", + "image_caption": [ + "Figure 1: Approaches toward stationarity during the initial trainings for the MLP on the MNIST data (a) and for the CNN on the CIFAR-10 data (b). Top panels depict the half-running average ${ \\overline { { f ^ { B } } } } ( t )$ (dark green) and the instantaneous value $f ^ { B } ( t )$ (light green) of the mini-batch loss. Bottom panels depict the convergence of the half-running averages of the observables $O _ { \\mathrm { { L } } } = \\pmb { \\theta } \\cdot \\pmb { \\nabla } f ^ { B }$ and $\\begin{array} { r } { \\dot { \\mathcal { O } } _ { \\mathrm { R } } = \\frac { ( 1 + \\mu ) } { 2 ( 1 - \\nu ) } \\eta \\mathbf { v } ^ { 2 } } \\end{array}$ , whose stationary-state averages should agree according to the relation (FDR1’). " + ], + "image_footnote": [], + "bbox": [ + 179, + 373, + 825, + 584 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In order to verify this claim, we first train the model with the learning rate $\\eta = 0 . 1$ for $\\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } = 1 0 0$ epochs, that is, for ttotal $\\begin{array} { r } { t ^ { \\mathrm { t o t a l } } = \\frac { N _ { \\mathrm { s } } } { | \\mathcal { B } | } \\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } = 1 0 0 \\frac { N _ { \\mathrm { s } } } { | \\mathcal { B } | } } \\end{array}$ time steps. As shown in the figure 1, the observables $\\overline { { \\mathcal { O } } } _ { \\mathrm { L } } ( t )$ and $\\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t )$ converge to each other. We then take the model at the end of the initial 100- epoch training and sequentially train it further at various learning rates $\\eta$ (see Appendix B). The observables $\\breve { \\mathcal { O } } _ { \\mathrm { L } } ( t )$ and $\\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t )$ again converge to each other, as plotted in the figure 2. Note that the smaller the learning rate is, the longer it takes to equilibrate. ", + "bbox": [ + 173, + 694, + 825, + 787 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 SECOND FLUCTUATION-DISSIPATION RELATION AND SHAPE OF LOSS-FUNCTION LANDSCAPE ", + "text_level": 1, + "bbox": [ + 171, + 804, + 766, + 830 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In order to assess the loss-function landscape information from the relation (FDR2), define ", + "bbox": [ + 173, + 842, + 767, + 857 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/cb68ce98139ae94de2389569f949c4cffaa92deee446491caed18365a16ef533.jpg", + "text": "$$\n{ \\mathcal { O } } _ { \\mathrm { F B } } \\equiv \\left( 1 - \\nu \\right) \\left( \\nabla f \\right) ^ { 2 } - \\mu \\mathbf { v } \\cdot \\nabla f ^ { \\mathcal { B } }\n$$", + "text_format": "latex", + "bbox": [ + 379, + 858, + 619, + 878 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/c0374e79625d8da67e29747429a8245babd8d8942f04402779473459843e631c.jpg", + "image_caption": [ + "Figure 2: Approaches toward stationarity during the sequential runs for various learning rates $\\eta$ , seen through the half-running averages of the observables $O _ { \\mathrm { L } } = \\pmb { \\theta } \\cdot \\pmb { \\nabla } f ^ { B }$ (solid) and $\\begin{array} { r } { \\mathcal { O } _ { \\mathrm { R } } = \\frac { ( 1 + \\mu ) } { 2 ( 1 - \\nu ) } \\eta \\mathbf { v } ^ { 2 } } \\end{array}$ (dotted light-colored). They agree at sufficiently long times but the relaxation time to reach such a stationary regime increases as the learning rate $\\eta$ decreases. " + ], + "image_footnote": [], + "bbox": [ + 169, + 114, + 825, + 324 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(with the second term nonexistent for SGD without momentum).5 Note that $\\left( \\nabla f \\right) ^ { 2 }$ is a full-batch – not mini-batch – quantity. Given its computational cost, here we measure this first term only at the end of each epoch and take the half-running average over these sparse sample points, discarding the initial half of the run. ", + "bbox": [ + 173, + 422, + 825, + 479 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The half-running average of the full-batch observable $\\overline { { \\mathcal { O } } } _ { \\mathrm { F B } }$ at the end of sufficiently long training, which is a good proxy for $\\langle \\mathcal { O } _ { \\mathrm { F B } } \\rangle$ , is plotted in the figure 3 as a function of the learning rate $\\eta$ . As predicted by the relation (FDR2), at small learning rates $\\eta$ , the observable $\\langle \\mathcal { O } _ { \\mathrm { F B } } \\rangle$ approaches zero; its slope – divided by $\\left. \\operatorname { T r } \\tilde { C } \\right.$ if preferred – measures the magnitude of the Hessian matrix, component-wise averaged over directions in which the noise preferentially fluctuates. Meanwhile, nonlinearity at higher learning rates $\\eta$ measures the degree of anharmonicity experienced over the distribution $p _ { \\mathrm { s s } } \\left( \\bar { \\theta } \\right)$ . We see that anharmonic effects are pronounced especially for the CNN on the CIFAR-10 data even at moderately small learning rates. This invalidates the use of the quadratic harmonic approximation for the loss-function landscape and/or the assumption of the constant noise matrix for this model except at very small learning rates. ", + "bbox": [ + 173, + 486, + 825, + 636 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 FIRST FLUCTUATION-DISSIPATION RELATION AND LEARNING-RATE SCHEDULES ", + "text_level": 1, + "bbox": [ + 176, + 655, + 766, + 669 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Saturation of the relation (FDR1) suggests the learning stationarity, at which point it might be wise to decrease the learning rate $\\eta$ . Such scheduling is often carried out in an ad hoc manner but we can now algorithmize this procedure as follows: ", + "bbox": [ + 176, + 680, + 825, + 723 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. Evaluate the half-running averages $\\overline { { \\mathcal { O } } } _ { \\mathrm { L } } ( t )$ and $\\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t )$ at the end of each epoch. 2. If $\\left| \\frac { \\overline { { \\mathcal { O } } } _ { \\mathrm { L } } ( t ) } { \\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t ) } - 1 \\right| < X$ , then decrease the learning rate as $\\eta ( 1 - Y ) \\eta$ and also set $t = 0$ for the purpose of evaluating half-running averages. ", + "bbox": [ + 210, + 733, + 826, + 794 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here, two scheduling hyperparameters $X$ and $Y$ are introduced, which control the threshold for saturation of the relation (FDR1) and the amount of decrease in the learning rate, respectively. ", + "bbox": [ + 171, + 804, + 823, + 834 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Plotted in the figure 4 are results for SGD without momentum, with the Xavier initialization (Glorot & Bengio, 2010) and training through (i) preset training schedule with decrease of the learning rate by a factor of 10 for each 100 epochs, (ii) an adaptive scheduler with $X = 0 . 0 1$ $1 \\%$ threshold) and ", + "bbox": [ + 176, + 840, + 823, + 883 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/103dc6de5ea9ad9ef08c5e066700c85d465ad811619762e61470c6bd0105ac25.jpg", + "image_caption": [ + "Figure 3: The stationary-state average of the full-batch observable $\\mathcal { O } _ { \\mathrm { F B } }$ as a function of the learning rate $\\eta$ , estimated through half-running averages. Dots and error bars denote mean values and $9 5 \\%$ confidence intervals over several distinct runs, respectively. The straight red line connects the origin and the point with the smallest $\\eta$ explored. (a) For the MLP on the MNIST data, linear dependence on $\\eta$ for $\\eta \\lesssim 0 . 0 1$ supports the validity of the harmonic approximation there. (b) For the CNN on the CIFAR-10 data, anharmonicity is pronounced even down to $\\eta \\sim 0 . 0 0 1$ . " + ], + "image_footnote": [], + "bbox": [ + 171, + 111, + 826, + 324 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$Y = 0 . 1$ $1 0 \\%$ decrease), and (iii) the AMSGrad algorithm (J. Reddi et al., 2018) with the default hyperparameters. The adaptive scheduler attains comparable accuracies with the preset scheduling at long time and outperforms AMSGrad (see Appendix C for additional simulations). ", + "bbox": [ + 173, + 449, + 826, + 492 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/f01849030d6cbc9c9569481343254672d25b6a856d2e487ca17f88423fe9d67c.jpg", + "image_caption": [ + "Figure 4: Comparison of preset training schedule (black) and adaptive training schedule (blue), employing SGD without momentum both for the MLP on the MNIST data (a) and the CNN on the CIFAR-10 data (b), along with the AMSGrad algorithm (green). From top to bottom, plotted are the learning rate $\\eta$ , the full-batch training loss $f$ , and prediction accuracies on the training-set images (solid) and the 10000 test-set images (dashed). " + ], + "image_footnote": [], + "bbox": [ + 179, + 532, + 825, + 732 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "These two scheduling methods span different subspaces of all the possible schedules. The adaptive scheduling method proposed herein has a theoretical grounding and in practice much less dimensionality for tuning of scheduling hyperparameters than the presetting method, thus ameliorating the optimization of scheduling hyperparameters. The systematic comparison between the two scheduling methods for state-of-the-arts architectures, and also the comparison with the AMSGrad algorithm for natural language processing tasks, could be a worthwhile avenue to pursue in the future. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 318, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we have derived the fluctuation-dissipation relations with no assumptions other than stationarity of the probability distribution. These relations hold exactly even when the noise is nonGaussian and the loss function is nonconvex. The relations have been empirically verified and used to probe the properties of the loss-function landscapes for the simple models. The relations further have resulted in the algorithm to adaptively set learning-rate schedule on the fly rather than presetting it in an ad hoc manner. In addition to systematically testing the performance of this adaptive scheduling algorithm, it would be interesting to investigate non-Gaussianity and noncovexity in more details through higher-point observables, both analytically and numerically. It would also be interesting to further elucidate the physics of machine learning by extending our formalism to incorporate nonstationary dynamics, linearly away from stationarity (Onsager, 1931; Green, 1954; Kubo, 1957) and beyond (Jarzynski, 1997; Crooks, 1999), so that it can in particular properly treat overfitting cascading dynamics and time-dependent sample distributions. ", + "bbox": [ + 174, + 133, + 825, + 300 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 319, + 326, + 330 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The author thanks Ludovic Berthier, Leon Bottou, Guy Gur-Ari, Kunihiko Kaneko, Ari Morcos, ´ Dheevatsa Mudigere, Yann Ollivier, Yuandong Tian, and Mark Tygert for discussions. Special thanks go to Daniel Adam Roberts who prompted the practical application of the fluctuationdissipation relations, leading to the adaptive method in Section 3.3. ", + "bbox": [ + 174, + 342, + 825, + 397 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 420, + 285, + 435 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Jing An, Jianfeng Lu, and Lexing Ying. Stochastic modified equations for the asynchronous stochastic gradient descent. arXiv preprint arXiv:1805.08244, 2018. 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The anisotropic noise in stochastic gradient descent: Its behavior of escaping from minima and regularization effects. arXiv preprint arXiv:1803.00195, 2018. ", + "bbox": [ + 166, + 101, + 826, + 488 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A SGD WITH MOMENTUM AND DAMPENING ", + "text_level": 1, + "bbox": [ + 174, + 102, + 560, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For SGD with momentum $\\mu$ and dampening $\\nu$ , the update equation is given by ", + "bbox": [ + 173, + 133, + 687, + 148 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/bce382bece204cdd9d7c5b86c3402bb5fae32955db2144c7583e3ab5d116f9bf.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\mathbf v ( t + 1 ) } } & { { = } } & { { \\mu \\mathbf v ( t ) - ( 1 - \\nu ) \\nabla f ^ { \\mathcal { B } } \\left[ \\pmb { \\theta } ( t ) \\right] , } } \\\\ { { \\pmb \\theta ( t + 1 ) } } & { { = } } & { { \\pmb \\theta ( t ) + \\eta \\mathbf v ( t + 1 ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 154, + 647, + 194 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Here $\\mathbf { v } = \\{ v _ { i } \\} _ { i = 1 , \\ldots , P }$ is the velocity and $\\eta > 0$ the learning rate; SGD without momentum is the special case with $\\dot { \\mu } = 0$ . Again hypothesizing the existence of a stationary-state distribution $p _ { \\mathrm { s s } } \\left( \\bar { \\pmb { \\theta } } , \\mathbf { v } \\right)$ , the stationary-state average of an observable $\\mathcal { O } \\left( \\theta , \\mathbf { v } \\right)$ is defined as ", + "bbox": [ + 173, + 196, + 825, + 242 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/096c94b6875c9cd98ad072ad86ac58c745c1a3022b8fafaf1428930c3580fb16.jpg", + "text": "$$\n\\left. \\mathcal { O } \\left( \\pmb { \\theta } , \\mathbf { v } \\right) \\right. \\equiv \\int \\mathrm { d } \\pmb { \\theta } \\mathrm { d } \\mathbf { v } p _ { \\mathrm { s s } } \\left( \\pmb { \\theta } , \\mathbf { v } \\right) \\mathcal { O } \\left( \\pmb { \\theta } , \\mathbf { v } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 356, + 247, + 643, + 281 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Just as in the main text, from the assumed stationarity follows the master equation for SGD with momentum and dampening ", + "bbox": [ + 171, + 286, + 825, + 315 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/1b6cab6c929fb0d342bca7cac2ee8529cfabf6a7b24e6ae9ae724d3a5459ebef.jpg", + "text": "$$\n\\begin{array} { r } { \\langle \\mathcal { O } \\left( \\pmb { \\theta } , \\mathbf { v } \\right) \\rangle = \\big \\langle \\left[ \\mathcal { O } \\left\\{ \\pmb { \\theta } + \\eta \\left[ \\mu \\mathbf { v } - ( 1 - \\nu ) \\pmb { \\nabla } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] , \\mu \\mathbf { v } - ( 1 - \\nu ) \\pmb { \\nabla } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right\\} \\right] _ { \\mathrm { m . b . } } \\big \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 320, + 769, + 343 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the linear observables, ", + "bbox": [ + 174, + 354, + 349, + 369 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/56e722199f8b386ad0ab7bccc45f4124d6b3e24e787aaaf63c3c70250d6100a3.jpg", + "text": "$$\n\\left. { { \\bf { v } } } \\right. = \\mu \\left. { { \\bf { v } } } \\right. - \\left( { 1 - \\nu } \\right) \\left. { \\nabla f \\left( { \\pmb { \\theta } } \\right) } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 388, + 367, + 609, + 385 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 388, + 200, + 401 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/966f700669b92c7a55ec0ebf7081f6f281b9a069ff5407861d2e2ee605ef17ef.jpg", + "text": "$$\n\\left. \\pmb { \\theta } \\right. = \\left. \\pmb { \\theta } \\right. + \\eta \\left[ \\mu \\left. \\mathbf { v } \\right. - \\left( 1 - \\nu \\right) \\left. \\pmb { \\nabla } f \\left( \\pmb { \\theta } \\right) \\right. \\right] = \\left. \\pmb { \\theta } \\right. + \\eta \\left. \\mathbf { v } \\right. ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 400, + 692, + 417 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "thus ", + "bbox": [ + 173, + 421, + 204, + 434 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/f860935c0373d6778d1296b93a39dc4e97674c50ac1cf5c257474e650a1952bc.jpg", + "text": "$$\n\\langle { \\bf v } \\rangle = 0 \\mathrm { a n d } \\langle \\nabla f \\rangle = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 433, + 593, + 450 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the quadratic observables ", + "bbox": [ + 174, + 460, + 369, + 474 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/bf1809f07960777ec419b5ed6595461b2c405b81a849572412aa3d4fe6fe7077.jpg", + "text": "$$\n\\left. v _ { i } v _ { j } \\right. = \\mu ^ { 2 } \\left. v _ { i } v _ { j } \\right. + ( 1 - \\nu ) ^ { 2 } \\left. \\widetilde { C } _ { i , j } \\right. - ( 1 - \\nu ) \\mu \\left[ \\left. v _ { i } \\left( \\partial _ { j } f \\right) \\right. + \\left. \\left( \\partial _ { i } f \\right) v _ { j } \\right. \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 245, + 481, + 753, + 510 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/2d313fe80cf86d262a907e4a1e7c6609d46dcb47ceb8e49d73d1d1f25b98c319.jpg", + "text": "$$\n\\left. v _ { i } \\theta _ { j } \\right. - \\eta \\left. v _ { i } v _ { j } \\right. = \\mu \\left. v _ { i } \\theta _ { j } \\right. - \\left( 1 - \\nu \\right) \\left. \\left( \\partial _ { i } f \\right) \\theta _ { j } \\right. ,\n$$", + "text_format": "latex", + "bbox": [ + 290, + 513, + 738, + 532 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 535, + 200, + 547 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/3d555849a75c7dbbb35299f2cb6eeb4bbe8b7bbf9da25c879de79f26e887ff7d.jpg", + "text": "$$\n( 1 - \\nu ) \\left[ \\langle \\theta _ { i } \\left( \\partial _ { j } f \\right) \\rangle + \\langle \\left( \\partial _ { i } f \\right) \\theta _ { j } \\rangle \\right] - \\mu \\left( \\langle \\theta _ { i } v _ { j } \\rangle + \\langle v _ { i } \\theta _ { j } \\rangle \\right) = \\eta \\left. v _ { i } v _ { j } \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 546, + 723, + 564 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Note that the relations (26) and (27) are trivially satisfied at each time step if the left-hand side observables are evaluated at one step ahead and thus their being satisfied for running averages has nothing to do with equilibration [the same can be said about the relation (23)]; the only nontrivial relation is the equation (28), which is a consequence of setting $\\langle \\theta _ { i } \\theta _ { j } \\rangle$ constant of time. After taking traces and some rearrangement, we obtain the relation (FDR1’) in the main text. ", + "bbox": [ + 173, + 566, + 825, + 637 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the full-batch loss function, the algebra similar to the one in the main text yields ", + "bbox": [ + 173, + 643, + 723, + 659 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/6d7836cd5404f3e5e2e65671cda667be3e3ef0fa65626270a18399f58fd87bb4.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\Big [ ( 1 - \\nu ) ( \\nabla f ) ^ { 2 } - \\mu \\mathbf { v } \\cdot \\nabla f \\Big ] } } \\\\ & { = } & { \\eta \\sum _ { i , j = 1 } ^ { P } H _ { i , j } \\{ ( 1 - \\nu ) ^ { 2 } \\widetilde { C } _ { i , j } - \\mu ( 1 - \\nu ) [ v _ { i } ( \\partial _ { j } f ) + ( \\partial _ { i } f ) v _ { j } ] + \\mu ^ { 2 } v _ { i } v _ { j } \\} + O ( \\eta ^ { 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 662, + 805, + 738 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B MODELS AND SIMULATION PROTOCOLS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 539, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 MLP ON MNIST THROUGH SGD WITHOUT MOMENTUM ", + "bbox": [ + 174, + 132, + 607, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The MNIST training data consist of $N _ { \\mathrm { s } } = 6 0 0 0 0$ black-white images of hand-written digits with 28-by-28 pixels (LeCun et al., 1998). We preprocess the data through an affine transformation such that their mean and variance (over both the training data and pixels) are zero and one, respectively. ", + "bbox": [ + 174, + 159, + 825, + 202 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Our multilayer perceptron (MLP) consists of a 784-dimensional input layer followed by a hidden layer of 200 neurons with ReLU activations, another hidden layer of 200 neurons with ReLU activations, and a 10-dimensional output layer with the softmax activation. The model performance is evaluated by the cross-entropy loss supplemented by the $L ^ { 2 }$ -regularization term $\\scriptstyle { \\frac { 1 } { 2 } } \\lambda \\theta ^ { 2 }$ with the weight decay $\\lambda = 0 . 0 1$ . ", + "bbox": [ + 174, + 208, + 825, + 279 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Throughout the paper, the MLP is trained on the MNIST data through SGD without momentum. \nThe data are shuffled at each epoch with the mini-batch size $| B | = 1 0 { \\bar { 0 } }$ . ", + "bbox": [ + 176, + 285, + 821, + 314 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The MLP is initialized through the Xavier method (Glorot & Bengio, 2010) and trained for $\\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } ~ = ~ 1 0 0$ $\\eta \\mathit { \\Pi } = \\ 0 . 1$ t with. This $( \\eta , \\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } ) = ( 0 . 0 5 , 5 0 0 ) \\to ( 0 . 0 2 , 5 0 0 ) \\to ( 0 . 0 1 , 5 0 0 ) \\to ( 0 . 0 0 5 , 1 0 0 0 ) \\to ( 0 . 0 0 3 , 1 0 0 0 )$ data shuffling, all starting from the common model parameter attained at the end of the initial 100- epoch run. The figure 2 depicts trajectories for one particular seed, while the figure 3 plots means and error bars over these distinct seeds. ", + "bbox": [ + 174, + 320, + 825, + 422 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.2 CNN ON CIFAR-10 THROUGH SGD WITH MOMENTUM", + "text_level": 1, + "bbox": [ + 174, + 439, + 601, + 454 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The CIFAR-10 training data consist of $N _ { \\mathrm { s } } ~ = ~ 5 0 0 0 0$ color images of objects – divided into ten categories – with 32-by-32 pixels in each of 3 color channels, each pixel ranging in $[ 0 , 1 ]$ (Krizhevsky & Hinton, 2009). We preprocess the data through uniformly subtracting 0.5 and multiplying by 2 so that each pixel ranges in $[ - 1 , 1 ]$ . ", + "bbox": [ + 174, + 465, + 825, + 522 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In order to describe the architecture of our convolutional neural network (CNN) in detail, let us associate a tuple $[ F , C , S , P ; M ]$ to a convolutional layer with filter width $F$ , a number of channels $C$ , stride $S$ , and padding $P$ , followed by ReLU activations and a max-pooling layer of width $M$ . Then, as in the demo at Karpathy (2014), our CNN consists of a (32, 32, 3) input layer followed by a convolutional layer with $[ 5 , 1 6 , 1 , 2 ; 2 ]$ , another convolutional layer with $[ 5 , 2 0 , 1 , 2 ; 2 ]$ , yet another convolutional layer with $[ 5 , 2 0 , 1 , 2 ; 2 ]$ , and finally a fully-connected 10-dimensional output layer with the softmax activation. The model performance is evaluated by the cross-entropy loss supplemented by the $L ^ { 2 }$ -regularization term $\\scriptstyle { \\frac { 1 } { 2 } } \\lambda ^ { \\setminus } \\lambda ^ { 2 }$ with the weight decay $\\lambda = 0 . 0 1$ . ", + "bbox": [ + 173, + 529, + 825, + 641 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Throughout the paper (except in Section 3.3 where the adaptive scheduling method is tested for SGD without momentum), the CNN is trained on the CIFAR-10 data through SGD with momentum $\\mu = 0 . 9$ and dampening $\\nu = 0$ . The data are shuffled at each epoch with the mini-batch size $| B | = 1 0 0$ . ", + "bbox": [ + 174, + 647, + 825, + 703 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$\\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } = 1 0 0$ initialized through the Xavepochs with the learning rate $\\eta = 0 . 1$ hod (Glorot & Bengio, 2010) an. We then sequentially train it with $( \\eta , \\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } ) =$ ( $0 . 0 5 , 2 0 0 ) ( 0 . 0 2 , 2 0 0 ) ( 0 . 0 1 , 2 0 0 ) ( 0 . 0 0 5 , 4 0 0 ) ( 0 . 0 0 3 , 4 0 0 ) ( 0 . 0 0 2 , 4 0 0 ) ( 0 . 0 0 2 , 4 0 0 ) ( 0 . 0 0 3 , 4 0 0 )$ $( 0 . 0 0 1 5 , 4 0 0 ) ( 0 . 0 0 1 , 4 0 0 ) ( 0 . 0 0 0 5 , 8 0 0 ) \\nonumber$ $( 0 . 0 0 0 5 , 8 0 0 ) \\to ( 0 . 0 0 0 2 5 , 8 0 0 ) \\to ( 0 . 0 0 0 1 , 8 0 0 )$ . At each junction of the sequence, the velocity $\\mathbf { v }$ is zeroed. This sequential-run protocol is carried out with 16 distinct seeds for the random-number generator used in data shuffling, all starting from the common model parameter attained at the end of the initial 100-epoch run. The figure 2 depicts trajectories for one particular seed, while the figure 3 plots means and error bars over these distinct seeds. ", + "bbox": [ + 174, + 709, + 825, + 824 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C ADDITIONAL SIMULATIONS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 437, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.1 ADAM VERSUS AMSGRAD ", + "text_level": 1, + "bbox": [ + 176, + 133, + 405, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Plotted in the figure S1 are the comparisons between Adam (Kingma & Ba, 2014) and AMSGrad (J. Reddi et al., 2018) algorithms with the default hyperparameters $\\alpha = 1 0 ^ { - 3 }$ , $( \\beta _ { 1 } , \\beta _ { 2 } ) =$ (0.9, 0.999), and $\\epsilon = 1 0 ^ { - 8 }$ . The AMSGrad algorithm marginally outperforms the Adam algorithm for the tasks at hand and thus the results with the AMSGrad are presented in the main text. ", + "bbox": [ + 173, + 160, + 825, + 215 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/8def667daebc2aa4b197f16b14c7b40436d97606a86f6209006025629d06c79c.jpg", + "image_caption": [ + "Figure S1: Comparison of AMSGrad (green) and Adam (orange) algorithms for the MLP on the MNIST data (a) and the CNN on the CIFAR-10 data (b). Top rows plot the full-batch training loss $f$ while bottom rows plot prediction accuracies on the training-set images (solid) and the 10000 test-set images (dashed). " + ], + "image_footnote": [], + "bbox": [ + 179, + 242, + 825, + 450 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.2 INITIAL ACCURACY GAIN WITH DIFFERENT SCHEDULING HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 173, + 553, + 754, + 566 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In the figure 4(a) for the MNIST classification task with the MLP, the proposed adaptive method with the scheduling hyperparameters $X ~ = ~ 0 . 0 1$ and $Y ~ = ~ 0 . 1$ outperforms the AMSGrad algorithm in terms of accuracy attained at long time and also exhibits a quick initial convergence. In the figure 4(b) for the CIFAR-10 classification task with the CNN, however, while the proposed adaptive method attains better accuracy at long time, its initial accuracy gain is visibly slower than the AMSGrad algorithm. This lag in initial accuracy gain can be ameliorated by choosing another combination of the scheduling hyperparameters, e.g., $X \\ = \\ 0 . 1$ and $Y = 0 . 3$ , at the expense of degradation in generalization accuracy with respect to the original choice $X = 0 . 0 1$ and $Y = 0 . 1$ . See the figure S2. ", + "bbox": [ + 173, + 577, + 825, + 704 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/f5057f4b1b2f6d6801470a5a515afb41274608c952c91a877d63ba45a92a3a55.jpg", + "image_caption": [ + "Figure S2: Comparison of preset training schedule (black) and adaptive training schedule (purple) – now with the scheduling hyperparameters $X ~ = ~ 0 . 1$ and $Y \\ = \\ 0 . 3$ – employing SGD without momentum, and the AMSGrad algorithm (green), for the CNN on the CIFAR-10 data with the same initial seed as in the main text (a) and three different initial seeds (b-d). From top to bottom, plotted are the learning rate $\\eta$ , the full-batch training loss $f$ , and prediction accuracies on the training-set images (solid) and the 10000 test-set images (dashed). 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Here, we derive stationary fluctuation-dissipation relations that link", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 469, + 299 + ], + "score": 1.0, + "content": "measurable quantities and hyperparameters in the stochastic gradient descent al-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 297, + 470, + 309 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 470, + 309 + ], + "score": 1.0, + "content": "gorithm. These relations hold exactly for any stationary state and can in particular", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 308, + 469, + 320 + ], + "spans": [ + { + "bbox": [ + 142, + 308, + 469, + 320 + ], + "score": 1.0, + "content": "be used to adaptively set training schedule. We can further use the relations to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 320, + 469, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 320, + 469, + 331 + ], + "score": 1.0, + "content": "efficiently extract information pertaining to a loss-function landscape such as the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 469, + 343 + ], + "score": 1.0, + "content": "magnitudes of its Hessian and anharmonicity. Our claims are empirically verified.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 360, + 206, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 208, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 208, + 376 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "Equilibration rules the long-term fate of many macroscopic dynamical systems. For instance, as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "score": 1.0, + "content": "we pour water into a glass and let it be, the stationary state of tranquility is eventually attained.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "Zooming into the tranquil water with a microscope would reveal, however, a turmoil of stochastic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 431 + ], + "score": 1.0, + "content": "fluctuations that maintain the apparent stationarity in balance. This is vividly exemplified by the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "Brownian motion (Brown, 1828): a pollen immersed in water is constantly bombarded by jittery", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "molecular movements, resulting in the macroscopically observable diffusive motion of the solute.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "Out of the effort in bridging microscopic and macroscopic realms through the Brownian movement", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "came a prototype of fluctuation-dissipation relations (Einstein, 1905; Von Smoluchowski, 1906).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "These relations quantitatively link degrees of noisy microscopic fluctuations to smooth macroscopic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "dissipative phenomena and have since been codified in the linear response theory for physical sys-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 458, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 458, + 507 + ], + "score": 1.0, + "content": "tems (Onsager, 1931; Green, 1954; Kubo, 1957), a cornerstone of statistical mechanics.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Machine learning begets another form of equilibration. As a model learns patterns in data, its per-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "formance first improves and then plateaus, again reaching apparent stationarity. This dynamical", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "process naturally comes equipped with stochastic fluctuations as well: often given data too gigantic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "to consume at once, training proceeds in small batches and random selections of these mini-batches", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "score": 1.0, + "content": "consequently give rise to the noisy dynamical excursion of the model parameters in the loss-function", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "landscape, reminiscent of the Brownian motion. It is thus natural to wonder if there exist analogous", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "fluctuation-dissipation relations that quantitatively link the noise in mini-batched data to the observ-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 456, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 456, + 601 + ], + "score": 1.0, + "content": "able evolution of the model performance and that in turn facilitate the learning process.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "Here, we derive such fluctuation-dissipation relations for the stochastic gradient descent algorithm.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "The only assumption made is stationarity of the probability distribution that governs the model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "parameters at sufficiently long time. Our results thus apply to generic cases with non-Gaussian", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "mini-batch noises and nonconvex loss-function landscapes. Practically, the first relation (FDR1)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "offers the metric for assessing equilibration and yields an adaptive algorithm that sets learning-rate", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "schedule on the fly. The second relation (FDR2) further helps us determine the properties of the loss-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "function landscape, including the strength of its Hessian and the degree of anharmonicity, i.e., the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 682, + 503, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 503, + 694 + ], + "score": 1.0, + "content": "deviation from the idealized harmonic limit of a quadratic loss surface and a constant noise matrix.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Our approach should be contrasted with recent attempts to import the machinery of stochastic dif-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "ferential calculus into the study of the stochastic gradient descent algorithm (Mandt et al., 2015;", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "Li et al., 2015; 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We can further use the relations to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 320, + 469, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 320, + 469, + 331 + ], + "score": 1.0, + "content": "efficiently extract information pertaining to a loss-function landscape such as the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 469, + 343 + ], + "score": 1.0, + "content": "magnitudes of its Hessian and anharmonicity. Our claims are empirically verified.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 243, + 470, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 360, + 206, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 208, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 208, + 376 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "Equilibration rules the long-term fate of many macroscopic dynamical systems. For instance, as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 504, + 408 + ], + "score": 1.0, + "content": "we pour water into a glass and let it be, the stationary state of tranquility is eventually attained.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "Zooming into the tranquil water with a microscope would reveal, however, a turmoil of stochastic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 431 + ], + "score": 1.0, + "content": "fluctuations that maintain the apparent stationarity in balance. This is vividly exemplified by the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "Brownian motion (Brown, 1828): a pollen immersed in water is constantly bombarded by jittery", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "molecular movements, resulting in the macroscopically observable diffusive motion of the solute.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "Out of the effort in bridging microscopic and macroscopic realms through the Brownian movement", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "came a prototype of fluctuation-dissipation relations (Einstein, 1905; Von Smoluchowski, 1906).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "These relations quantitatively link degrees of noisy microscopic fluctuations to smooth macroscopic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "dissipative phenomena and have since been codified in the linear response theory for physical sys-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 458, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 458, + 507 + ], + "score": 1.0, + "content": "tems (Onsager, 1931; Green, 1954; Kubo, 1957), a cornerstone of statistical mechanics.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 385, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 504, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "Machine learning begets another form of equilibration. As a model learns patterns in data, its per-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "formance first improves and then plateaus, again reaching apparent stationarity. This dynamical", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "process naturally comes equipped with stochastic fluctuations as well: often given data too gigantic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 556 + ], + "score": 1.0, + "content": "to consume at once, training proceeds in small batches and random selections of these mini-batches", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "score": 1.0, + "content": "consequently give rise to the noisy dynamical excursion of the model parameters in the loss-function", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "landscape, reminiscent of the Brownian motion. It is thus natural to wonder if there exist analogous", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "fluctuation-dissipation relations that quantitatively link the noise in mini-batched data to the observ-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 456, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 456, + 601 + ], + "score": 1.0, + "content": "able evolution of the model performance and that in turn facilitate the learning process.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 511, + 506, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "Here, we derive such fluctuation-dissipation relations for the stochastic gradient descent algorithm.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "The only assumption made is stationarity of the probability distribution that governs the model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "parameters at sufficiently long time. Our results thus apply to generic cases with non-Gaussian", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "mini-batch noises and nonconvex loss-function landscapes. Practically, the first relation (FDR1)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "offers the metric for assessing equilibration and yields an adaptive algorithm that sets learning-rate", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "schedule on the fly. The second relation (FDR2) further helps us determine the properties of the loss-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "function landscape, including the strength of its Hessian and the degree of anharmonicity, i.e., the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 682, + 503, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 503, + 694 + ], + "score": 1.0, + "content": "deviation from the idealized harmonic limit of a quadratic loss surface and a constant noise matrix.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 605, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Our approach should be contrasted with recent attempts to import the machinery of stochastic dif-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "ferential calculus into the study of the stochastic gradient descent algorithm (Mandt et al., 2015;", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2017;", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 699, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). This line of work all assumes Gaussian", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "noises and sometimes additionally employs the quadratic harmonic approximation for loss-function", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "landscapes. The more severe drawback, however, is the usage of the analogy with continuous-time", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "stochastic differential equations, which is inconsistent in general (see Section 2.3.3). Instead, the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "stochastic gradient descent algorithm can be properly treated within the framework of the Kramers-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Moyal expansion (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen &", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 169, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 169, + 162 + ], + "score": 1.0, + "content": "Moody, 1993).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "The paper is organized as follows. In Section 2, after setting up notations and deriving a station-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "ary fluctuation-dissipation theorem (FDT), we derive two specific fluctuation-dissipation relations.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 504, + 199 + ], + "score": 1.0, + "content": "The first relation (FDR1) can be used to check stationarity and the second relation (FDR2) to de-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "lineate the shape of the loss-function landscape, as empirically borne out in Section 3. An adaptive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "scheduling method is proposed and tested in Section 3.3. We conclude in Section 4 with future", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 147, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 147, + 232 + ], + "score": 1.0, + "content": "outlooks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 250, + 329, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 330, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 330, + 264 + ], + "score": 1.0, + "content": "2 FLUCTUATION-DISSIPATION RELATIONS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 302, + 291 + ], + "score": 1.0, + "content": "A model is parametrized by a weight coordinate,", + "type": "text" + }, + { + "bbox": [ + 302, + 276, + 372, + 290 + ], + "score": 0.93, + "content": "\\pmb { \\theta } = \\{ \\theta _ { i } \\} _ { i = 1 , \\dots , P }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 275, + 451, + 291 + ], + "score": 1.0, + "content": ". The training set of", + "type": "text" + }, + { + "bbox": [ + 451, + 277, + 464, + 287 + ], + "score": 0.89, + "content": "N _ { \\mathrm { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 275, + 506, + 291 + ], + "score": 1.0, + "content": "examples", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 511, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 227, + 322 + ], + "score": 1.0, + "content": "is utilized by the model to learby a full-batch loss function,", + "type": "text" + }, + { + "bbox": [ + 227, + 299, + 333, + 315 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\dot { f } \\left( \\pmb { \\theta } \\right) \\equiv \\frac { 1 } { N _ { \\mathrm { s } } } \\sum _ { \\alpha = 1 } ^ { N _ { \\mathrm { s } } } { f _ { \\alpha } \\left( \\pmb { \\theta } \\right) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 288, + 358, + 322 + ], + "score": 1.0, + "content": "e mod, with", + "type": "text" + }, + { + "bbox": [ + 359, + 301, + 385, + 313 + ], + "score": 0.93, + "content": "f _ { \\alpha } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 288, + 511, + 322 + ], + "score": 1.0, + "content": "erall performance is evaluatedquantifying the performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 254, + 326 + ], + "score": 1.0, + "content": "of the model on a particular sample", + "type": "text" + }, + { + "bbox": [ + 254, + 316, + 262, + 324 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 313, + 505, + 326 + ], + "score": 1.0, + "content": ": the smaller the loss is, the better the model is expected to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "perform. The learning process can thus be cast as an optimization problem of minimizing the loss", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "function. One of the most commonly used optimization schemes is the stochastic gradient descent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 215, + 360 + ], + "score": 1.0, + "content": "(SGD) algorithm (Robbins", + "type": "text" + }, + { + "bbox": [ + 216, + 347, + 225, + 357 + ], + "score": 0.26, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 345, + 372, + 360 + ], + "score": 1.0, + "content": "Monro, 1951) in which a mini-batch", + "type": "text" + }, + { + "bbox": [ + 372, + 346, + 451, + 358 + ], + "score": 0.92, + "content": "B \\subset \\{ 1 , 2 , \\ldots , N _ { \\mathrm { s } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 345, + 481, + 360 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 482, + 347, + 495, + 358 + ], + "score": 0.88, + "content": "| B |", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 491, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 491, + 370 + ], + "score": 1.0, + "content": "stochastically chosen for training at each time step. Specifically, the update equation is given by", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 376, + 374, + 391 + ], + "lines": [ + { + "bbox": [ + 237, + 376, + 374, + 391 + ], + "spans": [ + { + "bbox": [ + 237, + 376, + 374, + 391 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t + 1 ) = \\pmb \\theta ( t ) - \\eta \\pmb \\nabla f ^ { \\mathcal { B } } \\left[ \\pmb \\theta ( t ) \\right] ,", + "type": "interline_equation", + "image_path": "faf41f795c68a01f1e78f3a95b39a1c2b42c3584ecc871188a222772ea43c102.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 237, + 376, + 374, + 391 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 470, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 471, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 133, + 414 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 399, + 158, + 410 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 396, + 316, + 414 + ], + "score": 1.0, + "content": "is a learning rate and a mini-batch loss", + "type": "text" + }, + { + "bbox": [ + 316, + 397, + 426, + 413 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\equiv \\frac { 1 } { \\left| \\mathcal { B } \\right| } \\sum _ { \\alpha \\in \\mathcal { B } } f _ { \\alpha } \\left( \\pmb { \\theta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 396, + 471, + 414 + ], + "score": 1.0, + "content": ". 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For later purposes, it is convenient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 308, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 260, + 471 + ], + "score": 1.0, + "content": "to define a full two-point noise matrix", + "type": "text" + }, + { + "bbox": [ + 261, + 456, + 270, + 468 + ], + "score": 0.86, + "content": "\\widetilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 456, + 308, + 471 + ], + "score": 1.0, + "content": "through1", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 475, + 387, + 493 + ], + "lines": [ + { + "bbox": [ + 222, + 475, + 387, + 493 + ], + "spans": [ + { + "bbox": [ + 222, + 475, + 387, + 493 + ], + "score": 0.92, + "content": "\\widetilde { C } _ { i , j } ( \\pmb { \\theta } ) \\equiv [ [ \\partial _ { i } f ^ { \\mathcal { B } } ( \\pmb { \\theta } ) ] [ \\partial _ { j } f ^ { \\mathcal { B } } ( \\pmb { \\theta } ) ] ] ] _ { \\mathrm { m . b . } }", + "type": "interline_equation", + "image_path": "93a92da04daa633144c1cb552f51528761213fa11716fee794e4e5bf9785db7a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 222, + 475, + 387, + 493 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 499, + 295, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 297, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 297, + 513 + ], + "score": 1.0, + "content": "and, more generally, higher-point noise tensors", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 518, + 439, + 536 + ], + "lines": [ + { + "bbox": [ + 172, + 518, + 439, + 536 + ], + "spans": [ + { + "bbox": [ + 172, + 518, + 439, + 536 + ], + "score": 0.9, + "content": "\\tilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\left( \\pmb { \\theta } \\right) \\equiv \\left[ \\left[ \\partial _ { i _ { 1 } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { i _ { 2 } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\cdots \\left[ \\partial _ { i _ { k } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\right] _ { \\mathrm { m . b . } } .", + "type": "interline_equation", + "image_path": "24043a79c7e4b2bb0efb643e8d5ba58cb66c94630e7f2c1ceaa8770401b8cb72.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 172, + 518, + 439, + 536 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 431, + 556 + ], + "score": 1.0, + "content": "Below, we shall not make any assumptions on the distribution of the noise vector", + "type": "text" + }, + { + "bbox": [ + 432, + 543, + 454, + 556 + ], + "score": 0.69, + "content": "\\nabla f ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "– other than", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 375, + 567 + ], + "score": 1.0, + "content": "that a mini-batch is independent and identically distributed from the", + "type": "text" + }, + { + "bbox": [ + 375, + 556, + 388, + 566 + ], + "score": 0.88, + "content": "N _ { \\mathrm { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "training samples at each time", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "step – and the noise distribution is therefore allowed to have nontrivial higher connected moments", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 228, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 228, + 589 + ], + "score": 1.0, + "content": "indicative of non-Gaussianity.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "score": 1.0, + "content": "It is empirically often observed that the performance of the model plateaus after some training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 473, + 617 + ], + "score": 1.0, + "content": "through SGD. It is thus natural to hypothesize the existence of a stationary-state distribution,", + "type": "text" + }, + { + "bbox": [ + 473, + 605, + 501, + 617 + ], + "score": 0.89, + "content": "p _ { \\mathrm { s s } } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 604, + 505, + 617 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "that dictates the SGD sampling at long time (see Section 2.3.4 for discussion on this assumption).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 625, + 504, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 625, + 225, + 640 + ], + "score": 1.0, + "content": "For any observable quantity,", + "type": "text" + }, + { + "bbox": [ + 225, + 627, + 250, + 639 + ], + "score": 0.9, + "content": "\\mathcal { O } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 625, + 492, + 640 + ], + "score": 1.0, + "content": ", – something that can be measured during training such as", + "type": "text" + }, + { + "bbox": [ + 492, + 627, + 504, + 637 + ], + "score": 0.85, + "content": "\\pmb { \\theta } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 335, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 123, + 651 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 638, + 154, + 650 + ], + "score": 0.77, + "content": "f \\left( \\pmb { \\theta } \\right) -", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 637, + 335, + 651 + ], + "score": 1.0, + "content": "its stationary-state average is then defined as", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 656, + 372, + 682 + ], + "lines": [ + { + "bbox": [ + 240, + 656, + 372, + 682 + ], + "spans": [ + { + "bbox": [ + 240, + 656, + 372, + 682 + ], + "score": 0.94, + "content": "\\left. \\mathcal O \\left( \\pmb { \\theta } \\right) \\right. \\equiv \\int \\mathrm { d } \\pmb { \\theta } p _ { \\mathrm { s s } } \\left( \\pmb { \\theta } \\right) \\mathcal O \\left( \\pmb { \\theta } \\right) .", + "type": "interline_equation", + "image_path": "94278d1546a8f41e6bf0f2088fc70e9333694a23a8d62b20fe5d1c579bd74329.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 240, + 656, + 372, + 682 + ], + "spans": [], + "index": 38 + } + ] + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 693, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 693, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 119, + 693, + 265, + 707 + ], + "score": 1.0, + "content": "1A connected noise covariant matrix,", + "type": "text" + }, + { + "bbox": [ + 265, + 693, + 428, + 706 + ], + "score": 0.91, + "content": "C _ { i , j } \\left( \\pmb { \\theta } \\right) \\ \\equiv \\ \\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right) - \\left[ \\partial _ { i } f \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { j } f \\left( \\pmb { \\theta } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 693, + 506, + 707 + ], + "score": 1.0, + "content": ", will not appear in", + "type": "text" + } + ] + }, + { + "bbox": [ + 102, + 701, + 506, + 726 + ], + "spans": [ + { + "bbox": [ + 102, + 701, + 402, + 726 + ], + "score": 1.0, + "content": "fluctuation-dissipation relations below but scales nicely with mini-batch sizes as", + "type": "text" + }, + { + "bbox": [ + 403, + 705, + 466, + 723 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\propto \\frac { 1 } { \\left| B \\right| } \\left( 1 - \\frac { \\left| B \\right| } { N _ { \\mathrm { s } } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 705, + 506, + 721 + ], + "score": 1.0, + "content": "(Li et al.,", + "type": "text" + } + ] + }, + { + "bbox": [ + 104, + 718, + 134, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 718, + 134, + 734 + ], + "score": 1.0, + "content": "2017).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). This line of work all assumes Gaussian", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "noises and sometimes additionally employs the quadratic harmonic approximation for loss-function", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "landscapes. The more severe drawback, however, is the usage of the analogy with continuous-time", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "stochastic differential equations, which is inconsistent in general (see Section 2.3.3). Instead, the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "stochastic gradient descent algorithm can be properly treated within the framework of the Kramers-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Moyal expansion (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen &", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 169, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 169, + 162 + ], + "score": 1.0, + "content": "Moody, 1993).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 82, + 505, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "The paper is organized as follows. In Section 2, after setting up notations and deriving a station-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "ary fluctuation-dissipation theorem (FDT), we derive two specific fluctuation-dissipation relations.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 504, + 199 + ], + "score": 1.0, + "content": "The first relation (FDR1) can be used to check stationarity and the second relation (FDR2) to de-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "lineate the shape of the loss-function landscape, as empirically borne out in Section 3. An adaptive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "scheduling method is proposed and tested in Section 3.3. We conclude in Section 4 with future", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 147, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 147, + 232 + ], + "score": 1.0, + "content": "outlooks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 165, + 505, + 232 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 250, + 329, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 330, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 330, + 264 + ], + "score": 1.0, + "content": "2 FLUCTUATION-DISSIPATION RELATIONS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 302, + 291 + ], + "score": 1.0, + "content": "A model is parametrized by a weight coordinate,", + "type": "text" + }, + { + "bbox": [ + 302, + 276, + 372, + 290 + ], + "score": 0.93, + "content": "\\pmb { \\theta } = \\{ \\theta _ { i } \\} _ { i = 1 , \\dots , P }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 275, + 451, + 291 + ], + "score": 1.0, + "content": ". The training set of", + "type": "text" + }, + { + "bbox": [ + 451, + 277, + 464, + 287 + ], + "score": 0.89, + "content": "N _ { \\mathrm { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 275, + 506, + 291 + ], + "score": 1.0, + "content": "examples", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 511, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 227, + 322 + ], + "score": 1.0, + "content": "is utilized by the model to learby a full-batch loss function,", + "type": "text" + }, + { + "bbox": [ + 227, + 299, + 333, + 315 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\dot { f } \\left( \\pmb { \\theta } \\right) \\equiv \\frac { 1 } { N _ { \\mathrm { s } } } \\sum _ { \\alpha = 1 } ^ { N _ { \\mathrm { s } } } { f _ { \\alpha } \\left( \\pmb { \\theta } \\right) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 288, + 358, + 322 + ], + "score": 1.0, + "content": "e mod, with", + "type": "text" + }, + { + "bbox": [ + 359, + 301, + 385, + 313 + ], + "score": 0.93, + "content": "f _ { \\alpha } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 288, + 511, + 322 + ], + "score": 1.0, + "content": "erall performance is evaluatedquantifying the performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 254, + 326 + ], + "score": 1.0, + "content": "of the model on a particular sample", + "type": "text" + }, + { + "bbox": [ + 254, + 316, + 262, + 324 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 313, + 505, + 326 + ], + "score": 1.0, + "content": ": the smaller the loss is, the better the model is expected to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "perform. The learning process can thus be cast as an optimization problem of minimizing the loss", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "function. One of the most commonly used optimization schemes is the stochastic gradient descent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 215, + 360 + ], + "score": 1.0, + "content": "(SGD) algorithm (Robbins", + "type": "text" + }, + { + "bbox": [ + 216, + 347, + 225, + 357 + ], + "score": 0.26, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 345, + 372, + 360 + ], + "score": 1.0, + "content": "Monro, 1951) in which a mini-batch", + "type": "text" + }, + { + "bbox": [ + 372, + 346, + 451, + 358 + ], + "score": 0.92, + "content": "B \\subset \\{ 1 , 2 , \\ldots , N _ { \\mathrm { s } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 345, + 481, + 360 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 482, + 347, + 495, + 358 + ], + "score": 0.88, + "content": "| B |", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 491, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 491, + 370 + ], + "score": 1.0, + "content": "stochastically chosen for training at each time step. Specifically, the update equation is given by", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 275, + 511, + 370 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 376, + 374, + 391 + ], + "lines": [ + { + "bbox": [ + 237, + 376, + 374, + 391 + ], + "spans": [ + { + "bbox": [ + 237, + 376, + 374, + 391 + ], + "score": 0.92, + "content": "\\pmb \\theta ( t + 1 ) = \\pmb \\theta ( t ) - \\eta \\pmb \\nabla f ^ { \\mathcal { B } } \\left[ \\pmb \\theta ( t ) \\right] ,", + "type": "interline_equation", + "image_path": "faf41f795c68a01f1e78f3a95b39a1c2b42c3584ecc871188a222772ea43c102.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 237, + 376, + 374, + 391 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 470, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 471, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 133, + 414 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 399, + 158, + 410 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 396, + 316, + 414 + ], + "score": 1.0, + "content": "is a learning rate and a mini-batch loss", + "type": "text" + }, + { + "bbox": [ + 316, + 397, + 426, + 413 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\equiv \\frac { 1 } { \\left| \\mathcal { B } \\right| } \\sum _ { \\alpha \\in \\mathcal { B } } f _ { \\alpha } \\left( \\pmb { \\theta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 396, + 471, + 414 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 396, + 471, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 420, + 364, + 437 + ], + "lines": [ + { + "bbox": [ + 248, + 420, + 364, + 437 + ], + "spans": [ + { + "bbox": [ + 248, + 420, + 364, + 437 + ], + "score": 0.92, + "content": "\\left[ \\nabla f ^ { \\boldsymbol { B } } \\left( \\pmb { \\theta } \\right) \\right] _ { \\mathrm { m . b . } } = \\nabla f \\left( \\pmb { \\theta } \\right) ,", + "type": "interline_equation", + "image_path": "7c882868a8ebbd570e66251f4b62da090e71348c129c2ccdee6e64c4f8720ed6.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 248, + 420, + 364, + 437 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 127, + 458 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 444, + 162, + 457 + ], + "score": 0.89, + "content": "\\mathbb { I } . . . \\mathbb { J } _ { \\mathrm { m . b . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 443, + 505, + 458 + ], + "score": 1.0, + "content": ". denoting the average over mini-batch realizations. For later purposes, it is convenient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 308, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 260, + 471 + ], + "score": 1.0, + "content": "to define a full two-point noise matrix", + "type": "text" + }, + { + "bbox": [ + 261, + 456, + 270, + 468 + ], + "score": 0.86, + "content": "\\widetilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 456, + 308, + 471 + ], + "score": 1.0, + "content": "through1", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 443, + 505, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 475, + 387, + 493 + ], + "lines": [ + { + "bbox": [ + 222, + 475, + 387, + 493 + ], + "spans": [ + { + "bbox": [ + 222, + 475, + 387, + 493 + ], + "score": 0.92, + "content": "\\widetilde { C } _ { i , j } ( \\pmb { \\theta } ) \\equiv [ [ \\partial _ { i } f ^ { \\mathcal { B } } ( \\pmb { \\theta } ) ] [ \\partial _ { j } f ^ { \\mathcal { B } } ( \\pmb { \\theta } ) ] ] ] _ { \\mathrm { m . b . } }", + "type": "interline_equation", + "image_path": "93a92da04daa633144c1cb552f51528761213fa11716fee794e4e5bf9785db7a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 222, + 475, + 387, + 493 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 499, + 295, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 297, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 297, + 513 + ], + "score": 1.0, + "content": "and, more generally, higher-point noise tensors", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 498, + 297, + 513 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 518, + 439, + 536 + ], + "lines": [ + { + "bbox": [ + 172, + 518, + 439, + 536 + ], + "spans": [ + { + "bbox": [ + 172, + 518, + 439, + 536 + ], + "score": 0.9, + "content": "\\tilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\left( \\pmb { \\theta } \\right) \\equiv \\left[ \\left[ \\partial _ { i _ { 1 } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { i _ { 2 } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\cdots \\left[ \\partial _ { i _ { k } } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] \\right] _ { \\mathrm { m . b . } } .", + "type": "interline_equation", + "image_path": "24043a79c7e4b2bb0efb643e8d5ba58cb66c94630e7f2c1ceaa8770401b8cb72.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 172, + 518, + 439, + 536 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 431, + 556 + ], + "score": 1.0, + "content": "Below, we shall not make any assumptions on the distribution of the noise vector", + "type": "text" + }, + { + "bbox": [ + 432, + 543, + 454, + 556 + ], + "score": 0.69, + "content": "\\nabla f ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "– other than", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 375, + 567 + ], + "score": 1.0, + "content": "that a mini-batch is independent and identically distributed from the", + "type": "text" + }, + { + "bbox": [ + 375, + 556, + 388, + 566 + ], + "score": 0.88, + "content": "N _ { \\mathrm { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "training samples at each time", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "step – and the noise distribution is therefore allowed to have nontrivial higher connected moments", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 228, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 228, + 589 + ], + "score": 1.0, + "content": "indicative of non-Gaussianity.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 543, + 505, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "score": 1.0, + "content": "It is empirically often observed that the performance of the model plateaus after some training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 473, + 617 + ], + "score": 1.0, + "content": "through SGD. 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\\left[ \\pmb { \\theta } ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } ^ { \\prime } \\right) \\right] \\right\\} } \\right] _ { \\mathrm { m . b . } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 91, + 462, + 110 + ], + "score": 1.0, + "content": "and in particular for the stationary state", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 112, + 491, + 194 + ], + "lines": [ + { + "bbox": [ + 115, + 112, + 491, + 194 + ], + "spans": [ + { + "bbox": [ + 115, + 112, + 491, + 194 + ], + "score": 0.94, + "content": "\\begin{array} { r c l } { \\displaystyle \\int \\mathrm { d } \\theta p _ { \\mathrm { s s } } \\left( \\theta , t \\right) \\mathcal { O } \\left( \\theta \\right) } & { = } & { \\displaystyle \\int \\mathrm { d } \\theta \\ d p _ { \\mathrm { s s } } \\left( \\theta , t + 1 \\right) \\mathcal { O } \\left( \\theta \\right) } \\\\ & { = } & { \\displaystyle \\left[ \\int \\mathrm { d } \\theta \\int \\mathrm { d } \\theta ^ { \\prime } p _ { \\mathrm { s s } } ( \\theta ^ { \\prime } , t ) \\delta \\left\\{ \\theta - \\left[ \\theta ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\theta ^ { \\prime } \\right) \\right] \\right\\} \\mathcal { O } \\left( \\theta \\right) \\right] _ { \\mathrm { m . b . } } } \\\\ & { = } & { \\displaystyle \\int \\mathrm { d } \\theta ^ { \\prime } p _ { \\mathrm { s s } } \\left( \\theta ^ { \\prime } \\right) \\left[ \\mathcal { O } \\left[ \\theta ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\theta ^ { \\prime } \\right) \\right] \\right] _ { \\mathrm { m . b . } } . } \\end{array}", + "type": "interline_equation", + "image_path": "6951a2b906a707d090b5bf3ad90ac8bc037c4c6f0ef2d8f7caab4d3a1c2626e1.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 115, + 112, + 491, + 139.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 115, + 139.33333333333334, + 491, + 166.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 115, + 166.66666666666669, + 491, + 194.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 197, + 242, + 208 + ], + "lines": [ + { + "bbox": [ + 106, + 195, + 242, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 242, + 210 + ], + "score": 1.0, + "content": "Thus follows the master equation", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 212, + 390, + 229 + ], + "lines": [ + { + "bbox": [ + 222, + 212, + 390, + 229 + ], + "spans": [ + { + "bbox": [ + 222, + 212, + 390, + 229 + ], + "score": 0.92, + "content": " { \\mathcal O ( { \\pmb \\theta } ) } = { [ { \\mathcal O [ { \\pmb \\theta } - \\eta { \\pmb \\nabla } f ^ { \\beta } ( { \\pmb \\theta } ) } ] } ] _ { \\mathrm { m . b . } } .", + "type": "interline_equation", + "image_path": "0ffb2372834b8ec8232844d9b1568e34ee777129622569610c2743ab3c46b4ff.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 222, + 212, + 390, + 229 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "In the next two subsections, we apply this general formula to simple observables in order to derive", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "various stationary fluctuation-dissipation relations. Incidentally, the discrete version of the Fokker-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "Planck equation can be derived through the Kramers-Moyal expansion, considering the more general", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 443, + 277 + ], + "score": 1.0, + "content": "nonstationary version of the above equation and performing the Taylor expansion in", + "type": "text" + }, + { + "bbox": [ + 444, + 267, + 451, + 276 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "and repeated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "integrations by parts (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 286, + 179, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 179, + 300 + ], + "score": 1.0, + "content": "& Moody, 1993).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 312, + 322, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 322, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 322, + 325 + ], + "score": 1.0, + "content": "2.1 FIRST FLUCTUATION-DISSIPATION RELATION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 352, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 354, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 354, + 346 + ], + "score": 1.0, + "content": "Applying the master equation (FDT) to the linear observable,", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 348, + 415, + 366 + ], + "lines": [ + { + "bbox": [ + 198, + 348, + 415, + 366 + ], + "spans": [ + { + "bbox": [ + 198, + 348, + 415, + 366 + ], + "score": 0.89, + "content": "\\left. \\pmb { \\theta } \\right. = \\left. \\left[ \\pmb { \\theta } - \\eta \\pmb { \\nabla } f ^ { \\mathcal { B } } \\left( \\pmb { \\theta } \\right) \\right] _ { \\mathrm { m . b . } } \\right. = \\left. \\pmb { \\theta } \\right. - \\eta \\left. \\pmb { \\nabla } f \\left( \\pmb { \\theta } \\right) \\right. .", + "type": "interline_equation", + "image_path": "c662baacc287502c21297d8daaeb0b59a355a40fc8b6a19d1af9deb8f39d6ed7.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 198, + 348, + 415, + 366 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 162, + 379 + ], + "lines": [ + { + "bbox": [ + 106, 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direction that the gradient picks on average as the model", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "parameter stochastically bounces around the local minimum or, more generally, wanders around the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 415, + 361, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 361, + 428 + ], + "score": 1.0, + "content": "loss-function landscape according to the stationary distribution.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 383, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 384, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 330, + 445 + ], + "score": 1.0, + "content": "Performing similar algebra for the quadratic observable", + "type": "text" + }, + { + "bbox": [ + 331, + 432, + 356, + 444 + ], + "score": 0.92, + "content": "\\langle \\theta _ { i } \\theta _ { j } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 431, + 384, + 445 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 449, + 382, + 471 + ], + "lines": [ + { + "bbox": [ + 231, + 449, + 382, + 471 + ], + "spans": [ + { + "bbox": [ + 231, + 449, + 382, + 471 + ], + "score": 0.93, + "content": "\\langle \\theta _ { i } \\left( \\partial _ { j } f \\right) \\rangle + \\langle \\left( \\partial _ { i } f \\right) \\theta _ { j } \\rangle = \\eta \\left. \\widetilde { C } _ { i , j } \\right. .", + "type": "interline_equation", + "image_path": "f56879e724da5f7ba7c32189a48d9bf6a61690c6702057e2b51af38c9b36c4af.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 231, + 449, + 382, + 471 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", 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Also note that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 637, + 490, + 651 + ], + "spans": [ + { + "bbox": [ + 107, + 637, + 239, + 650 + ], + "score": 0.91, + "content": "\\left. \\pmb { \\theta } \\cdot ( \\pmb { \\nabla } f ) \\right. = \\left. ( \\pmb { \\theta } - \\pmb { \\theta } _ { \\mathrm { c } } ) \\cdot ( \\pmb { \\nabla } f ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 637, + 366, + 651 + ], + "score": 1.0, + "content": "for an arbitrary constant vector", + "type": "text" + }, + { + "bbox": [ + 366, + 639, + 377, + 649 + ], + "score": 0.88, + "content": "\\theta _ { \\mathrm { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 637, + 490, + 651 + ], + "score": 1.0, + "content": "because of the equation (8).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "This first fluctuation-dissipation relation is easy to evaluate on the fly during training, exactly holds", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "without any approximation if sampled well from the stationary distribution, and can thus be used as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "the standard metric to check if learning has plateaued, just as similar relations can be used to check", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "equilibration in Monte Carlo simulations of physical systems (Santen & Krauth, 2000). 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\\left[ \\theta ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\theta ^ { \\prime } \\right) \\right] \\right\\} \\mathcal { O } \\left( \\theta \\right) \\right] _ { \\mathrm { m . b . } } } \\\\ & { = } & { \\displaystyle \\int \\mathrm { d } \\theta ^ { \\prime } p _ { \\mathrm { s s } } \\left( \\theta ^ { \\prime } \\right) \\left[ \\mathcal { O } \\left[ \\theta ^ { \\prime } - \\eta \\nabla f ^ { \\mathcal { B } } \\left( \\theta ^ { \\prime } \\right) \\right] \\right] _ { \\mathrm { m . b . } } . } \\end{array}", + "type": "interline_equation", + "image_path": "6951a2b906a707d090b5bf3ad90ac8bc037c4c6f0ef2d8f7caab4d3a1c2626e1.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 115, + 112, + 491, + 139.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 115, + 139.33333333333334, + 491, + 166.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 115, + 166.66666666666669, + 491, + 194.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 197, + 242, + 208 + ], + "lines": [ + { + "bbox": [ + 106, + 195, + 242, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 242, + 210 + ], + "score": 1.0, + "content": "Thus follows the master equation", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 195, + 242, + 210 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 212, + 390, + 229 + ], + "lines": [ + { + "bbox": [ + 222, + 212, + 390, + 229 + ], + "spans": [ + { + "bbox": [ + 222, + 212, + 390, + 229 + ], + "score": 0.92, + "content": " { \\mathcal O ( { \\pmb \\theta } ) } = { [ { \\mathcal O [ { \\pmb \\theta } - \\eta { \\pmb \\nabla } f ^ { \\beta } ( { \\pmb \\theta } ) } ] } ] _ { \\mathrm { m . b . } } .", + "type": "interline_equation", + "image_path": "0ffb2372834b8ec8232844d9b1568e34ee777129622569610c2743ab3c46b4ff.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 222, + 212, + 390, + 229 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "score": 1.0, + "content": "In the next two subsections, we apply this general formula to simple observables in order to derive", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "various stationary fluctuation-dissipation relations. Incidentally, the discrete version of the Fokker-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "Planck equation can be derived through the Kramers-Moyal expansion, considering the more general", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 266, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 443, + 277 + ], + "score": 1.0, + "content": "nonstationary version of the above equation and performing the Taylor expansion in", + "type": "text" + }, + { + "bbox": [ + 444, + 267, + 451, + 276 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 266, + 505, + 277 + ], + "score": 1.0, + "content": "and repeated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "integrations by parts (Van Kampen, 1992; Gardiner, 2009; Risken, 1984; Radons et al., 1990; Leen", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 286, + 179, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 179, + 300 + ], + "score": 1.0, + "content": "& Moody, 1993).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 232, + 506, + 300 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 312, + 322, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 322, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 322, + 325 + ], + "score": 1.0, + "content": "2.1 FIRST FLUCTUATION-DISSIPATION RELATION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 352, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 354, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 354, + 346 + ], + "score": 1.0, + "content": "Applying the 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"f56879e724da5f7ba7c32189a48d9bf6a61690c6702057e2b51af38c9b36c4af.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 231, + 449, + 382, + 471 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 474, + 380, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 380, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 380, + 487 + ], + "score": 1.0, + "content": "In particular, taking the trace of this matrix-form relation, we obtain", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 473, + 380, + 487 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 490, + 363, + 514 + ], + "lines": [ + { + "bbox": [ + 249, + 490, + 363, + 514 + ], + "spans": [ + { + "bbox": [ + 249, + 490, + 363, + 514 + ], + "score": 0.94, + "content": "\\langle \\pmb { \\theta } \\cdot ( \\pmb { \\nabla } f ) \\rangle = \\frac { 1 } { 2 } \\eta \\left. \\mathrm { T r } \\widetilde { C } \\right. .", + "type": "interline_equation", + "image_path": "2df2418efc4668835bfeab7c81005e8a8b573e712db1b26873f25126d112abe7.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 249, + 490, + 363, + 514 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 504, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 318, + 531 + ], + "score": 1.0, + "content": "More generally, in the case of SGD with momentum", + "type": "text" + }, + { + "bbox": [ + 319, + 519, + 326, + 529 + ], + "score": 0.83, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 516, + 391, + 531 + ], + "score": 1.0, + "content": "and dampening", + "type": "text" + }, + { + "bbox": [ + 391, + 520, + 398, + 528 + ], + "score": 0.77, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 516, + 505, + 531 + ], + "score": 1.0, + "content": ", whose 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Also note that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 637, + 490, + 651 + ], + "spans": [ + { + "bbox": [ + 107, + 637, + 239, + 650 + ], + "score": 0.91, + "content": "\\left. \\pmb { \\theta } \\cdot ( \\pmb { \\nabla } f ) \\right. = \\left. ( \\pmb { \\theta } - \\pmb { \\theta } _ { \\mathrm { c } } ) \\cdot ( \\pmb { \\nabla } f ) \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 637, + 366, + 651 + ], + "score": 1.0, + "content": "for an arbitrary constant vector", + "type": "text" + }, + { + "bbox": [ + 366, + 639, + 377, + 649 + ], + "score": 0.88, + "content": "\\theta _ { \\mathrm { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 637, + 490, + 651 + ], + "score": 1.0, + "content": "because of the equation (8).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 625, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "This first fluctuation-dissipation relation is easy to evaluate on the fly during training, exactly holds", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 107, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "without any approximation if sampled well from the stationary distribution, and can thus be used as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "the standard metric to check if learning has plateaued, just as similar relations can be used to check", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "equilibration in Monte Carlo simulations of physical systems (Santen & Krauth, 2000). 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} \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } \\left( \\partial _ { i _ { 1 } } \\partial _ { i _ { 2 } } \\cdots \\partial _ { i _ { k } } f \\right) \\left[ \\left( \\partial _ { i _ { 1 } } f ^ { B } \\right) \\left( \\partial _ { i _ { 2 } } f ^ { B } \\right) \\cdots \\left( \\partial _ { i _ { k } } f ^ { B } \\right) \\right] _ { \\mathrm { m . b . } } \\right. } \\\\ & { = } & { \\displaystyle \\left. f \\right. - \\eta \\left. \\left( \\pmb { \\nabla } f \\right) ^ { 2 } \\right. + \\sum _ { k = 2 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k } } { k ! } \\left. \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "99aa08971b83899a98ed4879bea52b257f7ee185412ed4678f2a5cf135692e8a.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 130, + 507, + 160.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 160.66666666666666, + 507, + 191.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 191.33333333333331, + 507, + 221.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 309, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 309, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 309, + 240 + ], + "score": 1.0, + "content": "where we recalled the equation (4) and introduced", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 245, + 382, + 259 + ], + "lines": [ + { + "bbox": [ + 229, + 245, + 382, + 259 + ], + "spans": [ + { + "bbox": [ + 229, + 245, + 382, + 259 + ], + "score": 0.91, + "content": "F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\left( \\pmb { \\theta } \\right) \\equiv \\partial _ { i _ { 1 } } \\partial _ { i _ { 2 } } \\cdot \\cdot \\cdot \\partial _ { i _ { k } } f \\left( \\pmb { \\theta } \\right) .", + "type": "interline_equation", + "image_path": "4211d605971ec4f1f22304d5ffbb6e3f62ed1cd919eeede218f43e63390af1e7.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 229, + 245, + 382, + 259 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 264, + 453, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 454, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 160, + 279 + ], + "score": 1.0, + "content": "In particular,", + "type": "text" + }, + { + "bbox": [ + 160, + 264, + 238, + 278 + ], + "score": 0.93, + "content": "{ { H } _ { i , j } } \\left( \\pmb { \\theta } \\right) \\equiv { { F } _ { i , j } } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 263, + 454, + 279 + ], + "score": 1.0, + "content": "is the Hessian matrix. Reorganizing terms, we obtain", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 283, + 490, + 324 + ], + "lines": [ + { + "bbox": [ + 122, + 283, + 490, + 324 + ], + "spans": [ + { + "bbox": [ + 122, + 283, + 490, + 324 + ], + "score": 0.93, + "content": "\\left. \\left( \\nabla f \\right) ^ { 2 } \\right. = \\frac { \\eta } { 2 } \\left. \\mathrm { T r } \\left( H \\tilde { C } \\right) \\right. - \\eta ^ { 2 } \\left[ \\sum _ { k = 3 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k - 3 } } { k ! } \\left. \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\right. \\right] .", + "type": "interline_equation", + "image_path": "2e3980171ceb1fc599439671eea53d32c0fc531f468c2b77fe0b9198e57cb333.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 122, + 283, + 490, + 296.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 122, + 296.6666666666667, + 490, + 310.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 122, + 310.33333333333337, + 490, + 324.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 472, + 321, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 472, + 321, + 504, + 331 + ], + "spans": [], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 483, + 345 + ], + "score": 1.0, + "content": "In the case of SGD with momentum and dampening, the left-hand side is replaced by", + "type": "text" + }, + { + "bbox": [ + 483, + 332, + 505, + 344 + ], + "score": 0.76, + "content": "( 1 ~ -", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 341, + 473, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 215, + 363 + ], + "score": 0.92, + "content": "\\nu ) \\left. \\left( \\nabla f \\right) ^ { 2 } \\right. - \\mu \\left. \\mathbf { v } \\cdot \\pmb { \\nabla } f \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 341, + 234, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 344, + 276, + 359 + ], + "score": 0.93, + "content": "\\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\dots , i _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 341, + 473, + 363 + ], + "score": 1.0, + "content": "by more hideous expressions (see Appendix A).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 506, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "We can extract at least two types of information on the loss-function landscape by evaluating the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 378, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 246, + 397 + ], + "score": 1.0, + "content": "dependence of the left-hand side,", + "type": "text" + }, + { + "bbox": [ + 247, + 378, + 326, + 398 + ], + "score": 0.94, + "content": "G ( \\eta ) \\equiv \\left. \\left( \\nabla f \\right) ^ { 2 } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 378, + 415, + 397 + ], + "score": 1.0, + "content": ", on the learning rate", + "type": "text" + }, + { + "bbox": [ + 415, + 384, + 421, + 393 + ], + "score": 0.77, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 378, + 506, + 397 + ], + "score": 1.0, + "content": ". First, in the small", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 396, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 103, + 398, + 241, + 415 + ], + "score": 1.0, + "content": "learning rate regime, the value of", + "type": "text" + }, + { + "bbox": [ + 241, + 400, + 279, + 413 + ], + "score": 0.93, + "content": "2 G ( \\eta ) / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 398, + 335, + 415 + ], + "score": 1.0, + "content": "approximates", + "type": "text" + }, + { + "bbox": [ + 336, + 396, + 380, + 417 + ], + "score": 0.94, + "content": "\\mathrm { T r } \\left( H { \\tilde { C } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 398, + 506, + 415 + ], + "score": 1.0, + "content": "around a local ravine. Second,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 167, + 428 + ], + "score": 1.0, + "content": "nonlinearity of", + "type": "text" + }, + { + "bbox": [ + 168, + 415, + 189, + 427 + ], + "score": 0.9, + "content": "G ( \\eta )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 415, + 227, + 428 + ], + "score": 1.0, + "content": "at higher", + "type": "text" + }, + { + "bbox": [ + 228, + 417, + 234, + 427 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "indicates discernible effects of anharmonicity. 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Such nonlinearity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "especially indicates the breakdown of the harmonic approximation, that is, the quadratic truncation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 462, + 492, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 492, + 476 + ], + "score": 1.0, + "content": "of the loss-function landscape, often used to analyze the regime explored at small learning rates.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 489, + 176, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 177, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 177, + 502 + ], + "score": 1.0, + "content": "2.3 REMARKS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 363, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 365, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 365, + 522 + ], + "score": 1.0, + "content": "2.3.1 INTUITION WITHIN THE HARMONIC APPROXIMATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 105, + 529, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "In order to gain some intuition about the fluctuation-dissipation relations, let us momentarily em-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "ploy the harmonic approximation, i.e., assume that there is a local minimum of the loss function", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 551, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 118, + 565 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 118, + 552, + 154, + 562 + ], + "score": 0.89, + "content": "\\theta \\ : = \\ : \\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 551, + 469, + 565 + ], + "score": 1.0, + "content": "and retain only up to quadratic terms of the Taylor expansions around it:", + "type": "text" + }, + { + "bbox": [ + 470, + 552, + 504, + 564 + ], + "score": 0.9, + "content": "f ( \\pmb \\theta ) ~ \\approx", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 563, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 248, + 579 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f _ { 0 } { + } \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { P } h _ { i , j } ( \\theta _ { i } { - } \\theta _ { i } ^ { \\star } ) ( \\theta _ { j } { - } \\theta _ { j } ^ { \\star } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 563, + 360, + 580 + ], + "score": 1.0, + "content": ". Within this approximation,", + "type": "text" + }, + { + "bbox": [ + 361, + 564, + 505, + 577 + ], + "score": 0.91, + "content": "\\langle \\pmb \\theta \\cdot ( \\pmb \\nabla f ) \\rangle = \\langle ( \\pmb \\theta - \\pmb \\theta ^ { \\star } ) \\cdot ( \\pmb \\nabla f ) \\rangle \\approx", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 578, + 507, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 149, + 594 + ], + "score": 0.91, + "content": "2 \\left. f - f _ { 0 } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 578, + 302, + 596 + ], + "score": 1.0, + "content": ". The relation (FDR1) then becomes", + "type": "text" + }, + { + "bbox": [ + 302, + 578, + 399, + 598 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\langle f - f _ { 0 } \\rangle \\approx \\frac { 1 } { 4 } \\eta \\left. \\mathrm { T r } \\widetilde { C } \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 578, + 507, + 596 + ], + "score": 1.0, + "content": ", linking the height of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "noise ball to the noise amplitude. This is in line with, for instance, the theorem 4.6 of the refer-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "ence Bottou et al. (2018) and substantiates the analogy between SGD and simulated annealing, with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 618, + 463, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 173, + 633 + ], + "score": 1.0, + "content": "the learning rate", + "type": "text" + }, + { + "bbox": [ + 174, + 622, + 181, + 632 + ], + "score": 0.49, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 619, + 245, + 633 + ], + "score": 1.0, + "content": "– multiplied by", + "type": "text" + }, + { + "bbox": [ + 245, + 618, + 266, + 631 + ], + "score": 0.76, + "content": "\\mathrm { T r } \\widetilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 619, + 463, + 633 + ], + "score": 1.0, + "content": "– playing the role of temperature (Bottou, 1991).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 644, + 258, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 259, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 259, + 657 + ], + "score": 1.0, + "content": "2.3.2 HIGHER-ORDER RELATIONS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "Additional relations can be derived by repeating similar calculations for higher-order observables.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 675, + 235, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 235, + 687 + ], + "score": 1.0, + "content": "For example, at the cubic order,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 693, + 480, + 715 + ], + "lines": [ + { + "bbox": [ + 117, + 693, + 480, + 715 + ], + "spans": [ + { + "bbox": [ + 117, + 693, + 480, + 715 + ], + "score": 0.93, + "content": "\\left. \\theta _ { i } \\theta _ { j } \\left( \\partial _ { k } f \\right) + \\theta _ { i } \\left( \\partial _ { j } f \\right) \\theta _ { k } + \\left( \\partial _ { i } f \\right) \\theta _ { j } \\theta _ { k } \\right. = \\eta \\left. \\theta _ { i } \\widetilde { C } _ { j , k } + \\theta _ { j } \\widetilde { C } _ { k , i } + \\theta _ { k } \\widetilde { C } _ { i , j } \\right. - \\eta ^ { 2 } \\left. \\widetilde { C } _ { i , j , k } \\right. .", + "type": "interline_equation", + "image_path": "253dabe82cea34cd22e4882e49e1217e38877dd6c19c2772e321d653b7dea368.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 117, + 693, + 480, + 715 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 432, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 720, + 432, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 432, + 734 + ], + "score": 1.0, + "content": "The systematic investigation of higher-order relations is relegated to future work.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + } + ], + "page_idx": 3, + "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 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 333, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 333, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 333, + 95 + ], + "score": 1.0, + "content": "2.2 SECOND FLUCTUATION-DISSIPATION RELATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 117 + ], + "score": 1.0, + "content": "Applying the master equation (FDT) on the full-batch loss function and Taylor-expanding it in the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 303, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 159, + 127 + ], + "score": 1.0, + "content": "learning rate", + "type": "text" + }, + { + "bbox": [ + 159, + 117, + 166, + 126 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 114, + 303, + 127 + ], + "score": 1.0, + "content": "yields the closed-form expression", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 102, + 505, + 127 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 130, + 507, + 222 + ], + "lines": [ + { + "bbox": [ + 111, + 130, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 111, + 130, + 507, + 222 + ], + "score": 0.92, + "content": "\\begin{array} { r c l } { f \\left( \\pmb { \\theta } \\right) \\rangle } & { = } & { \\displaystyle \\left. \\left[ f \\left[ \\pmb { \\theta } - \\eta \\boldsymbol { \\nabla } f ^ { B } \\left( \\pmb { \\theta } \\right) \\right] \\right] _ { \\mathrm { m . b . } } \\right. } \\\\ & { = } & { \\displaystyle \\left. f + \\sum _ { k = 1 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k } } { k ! } \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } \\left( \\partial _ { i _ { 1 } } \\partial _ { i _ { 2 } } \\cdots \\partial _ { i _ { k } } f \\right) \\left[ \\left( \\partial _ { i _ { 1 } } f ^ { B } \\right) \\left( \\partial _ { i _ { 2 } } f ^ { B } \\right) \\cdots \\left( \\partial _ { i _ { k } } f ^ { B } \\right) \\right] _ { \\mathrm { m . b . } } \\right. } \\\\ & { = } & { \\displaystyle \\left. f \\right. - \\eta \\left. \\left( \\pmb { \\nabla } f \\right) ^ { 2 } \\right. + \\sum _ { k = 2 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k } } { k ! } \\left. \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "99aa08971b83899a98ed4879bea52b257f7ee185412ed4678f2a5cf135692e8a.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 130, + 507, + 160.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 160.66666666666666, + 507, + 191.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 191.33333333333331, + 507, + 221.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 309, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 309, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 309, + 240 + ], + "score": 1.0, + "content": "where we recalled the equation (4) and introduced", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 227, + 309, + 240 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 245, + 382, + 259 + ], + "lines": [ + { + "bbox": [ + 229, + 245, + 382, + 259 + ], + "spans": [ + { + "bbox": [ + 229, + 245, + 382, + 259 + ], + "score": 0.91, + "content": "F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\left( \\pmb { \\theta } \\right) \\equiv \\partial _ { i _ { 1 } } \\partial _ { i _ { 2 } } \\cdot \\cdot \\cdot \\partial _ { i _ { k } } f \\left( \\pmb { \\theta } \\right) .", + "type": "interline_equation", + "image_path": "4211d605971ec4f1f22304d5ffbb6e3f62ed1cd919eeede218f43e63390af1e7.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 229, + 245, + 382, + 259 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 264, + 453, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 454, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 160, + 279 + ], + "score": 1.0, + "content": "In particular,", + "type": "text" + }, + { + "bbox": [ + 160, + 264, + 238, + 278 + ], + "score": 0.93, + "content": "{ { H } _ { i , j } } \\left( \\pmb { \\theta } \\right) \\equiv { { F } _ { i , j } } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 263, + 454, + 279 + ], + "score": 1.0, + "content": "is the Hessian matrix. Reorganizing terms, we obtain", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 263, + 454, + 279 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 283, + 490, + 324 + ], + "lines": [ + { + "bbox": [ + 122, + 283, + 490, + 324 + ], + "spans": [ + { + "bbox": [ + 122, + 283, + 490, + 324 + ], + "score": 0.93, + "content": "\\left. \\left( \\nabla f \\right) ^ { 2 } \\right. = \\frac { \\eta } { 2 } \\left. \\mathrm { T r } \\left( H \\tilde { C } \\right) \\right. - \\eta ^ { 2 } \\left[ \\sum _ { k = 3 } ^ { \\infty } \\frac { \\left( - \\eta \\right) ^ { k - 3 } } { k ! } \\left. \\sum _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } = 1 } ^ { P } F _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\ldots , i _ { k } } \\right. \\right] .", + "type": "interline_equation", + "image_path": "2e3980171ceb1fc599439671eea53d32c0fc531f468c2b77fe0b9198e57cb333.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 122, + 283, + 490, + 296.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 122, + 296.6666666666667, + 490, + 310.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 122, + 310.33333333333337, + 490, + 324.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 472, + 321, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 472, + 321, + 504, + 331 + ], + "spans": [], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 472, + 321, + 504, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 483, + 345 + ], + "score": 1.0, + "content": "In the case of SGD with momentum and dampening, the left-hand side is replaced by", + "type": "text" + }, + { + "bbox": [ + 483, + 332, + 505, + 344 + ], + "score": 0.76, + "content": "( 1 ~ -", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 341, + 473, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 215, + 363 + ], + "score": 0.92, + "content": "\\nu ) \\left. \\left( \\nabla f \\right) ^ { 2 } \\right. - \\mu \\left. \\mathbf { v } \\cdot \\pmb { \\nabla } f \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 341, + 234, + 363 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 344, + 276, + 359 + ], + "score": 0.93, + "content": "\\widetilde { C } _ { i _ { 1 } , i _ { 2 } , \\dots , i _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 341, + 473, + 363 + ], + "score": 1.0, + "content": "by more hideous expressions (see Appendix A).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 329, + 505, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 506, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "We can extract at least two types of information on the loss-function landscape by evaluating the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 378, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 246, + 397 + ], + "score": 1.0, + "content": "dependence of the left-hand side,", + "type": "text" + }, + { + "bbox": [ + 247, + 378, + 326, + 398 + ], + "score": 0.94, + "content": "G ( \\eta ) \\equiv \\left. \\left( \\nabla f \\right) ^ { 2 } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 378, + 415, + 397 + ], + "score": 1.0, + "content": ", on the learning rate", + "type": "text" + }, + { + "bbox": [ + 415, + 384, + 421, + 393 + ], + "score": 0.77, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 378, + 506, + 397 + ], + "score": 1.0, + "content": ". First, in the small", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 396, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 103, + 398, + 241, + 415 + ], + "score": 1.0, + "content": "learning rate regime, the value of", + "type": "text" + }, + { + "bbox": [ + 241, + 400, + 279, + 413 + ], + "score": 0.93, + "content": "2 G ( \\eta ) / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 398, + 335, + 415 + ], + "score": 1.0, + "content": "approximates", + "type": "text" + }, + { + "bbox": [ + 336, + 396, + 380, + 417 + ], + "score": 0.94, + "content": "\\mathrm { T r } \\left( H { \\tilde { C } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 398, + 506, + 415 + ], + "score": 1.0, + "content": "around a local ravine. Second,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 167, + 428 + ], + "score": 1.0, + "content": "nonlinearity of", + "type": "text" + }, + { + "bbox": [ + 168, + 415, + 189, + 427 + ], + "score": 0.9, + "content": "G ( \\eta )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 415, + 227, + 428 + ], + "score": 1.0, + "content": "at higher", + "type": "text" + }, + { + "bbox": [ + 228, + 417, + 234, + 427 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "indicates discernible effects of anharmonicity. 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Such nonlinearity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "especially indicates the breakdown of the harmonic approximation, that is, the quadratic truncation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 462, + 492, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 492, + 476 + ], + "score": 1.0, + "content": "of the loss-function landscape, often used to analyze the regime explored at small learning rates.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 103, + 366, + 506, + 476 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 489, + 176, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 177, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 177, + 502 + ], + "score": 1.0, + "content": "2.3 REMARKS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 363, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 365, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 365, + 522 + ], + "score": 1.0, + "content": "2.3.1 INTUITION WITHIN THE HARMONIC APPROXIMATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 105, + 529, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "In order to gain some intuition about the fluctuation-dissipation relations, let us momentarily em-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "ploy the harmonic approximation, i.e., assume that there is a local minimum of the loss function", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 551, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 118, + 565 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 118, + 552, + 154, + 562 + ], + "score": 0.89, + "content": "\\theta \\ : = \\ : \\theta ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 551, + 469, + 565 + ], + "score": 1.0, + "content": "and retain only up to quadratic terms of the Taylor expansions around it:", + "type": "text" + }, + { + "bbox": [ + 470, + 552, + 504, + 564 + ], + "score": 0.9, + "content": "f ( \\pmb \\theta ) ~ \\approx", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 563, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 248, + 579 + ], + "score": 0.92, + "content": "\\begin{array} { r } { f _ { 0 } { + } \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { P } h _ { i , j } ( \\theta _ { i } { - } \\theta _ { i } ^ { \\star } ) ( \\theta _ { j } { - } \\theta _ { j } ^ { \\star } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 563, + 360, + 580 + ], + "score": 1.0, + "content": ". Within this approximation,", + "type": "text" + }, + { + "bbox": [ + 361, + 564, + 505, + 577 + ], + "score": 0.91, + "content": "\\langle \\pmb \\theta \\cdot ( \\pmb \\nabla f ) \\rangle = \\langle ( \\pmb \\theta - \\pmb \\theta ^ { \\star } ) \\cdot ( \\pmb \\nabla f ) \\rangle \\approx", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 578, + 507, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 149, + 594 + ], + "score": 0.91, + "content": "2 \\left. f - f _ { 0 } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 578, + 302, + 596 + ], + "score": 1.0, + "content": ". The relation (FDR1) then becomes", + "type": "text" + }, + { + "bbox": [ + 302, + 578, + 399, + 598 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\langle f - f _ { 0 } \\rangle \\approx \\frac { 1 } { 4 } \\eta \\left. \\mathrm { T r } \\widetilde { C } \\right. } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 578, + 507, + 596 + ], + "score": 1.0, + "content": ", linking the height of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "noise ball to the noise amplitude. This is in line with, for instance, the theorem 4.6 of the refer-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "ence Bottou et al. (2018) and substantiates the analogy between SGD and simulated annealing, with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 618, + 463, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 173, + 633 + ], + "score": 1.0, + "content": "the learning rate", + "type": "text" + }, + { + "bbox": [ + 174, + 622, + 181, + 632 + ], + "score": 0.49, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 619, + 245, + 633 + ], + "score": 1.0, + "content": "– multiplied by", + "type": "text" + }, + { + "bbox": [ + 245, + 618, + 266, + 631 + ], + "score": 0.76, + "content": "\\mathrm { T r } \\widetilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 619, + 463, + 633 + ], + "score": 1.0, + "content": "– playing the role of temperature (Bottou, 1991).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 531, + 507, + 633 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 644, + 258, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 259, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 259, + 657 + ], + "score": 1.0, + "content": "2.3.2 HIGHER-ORDER RELATIONS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 504, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "Additional relations can be derived by repeating similar calculations for higher-order observables.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 675, + 235, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 235, + 687 + ], + "score": 1.0, + "content": "For example, at the cubic order,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 663, + 505, + 687 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 693, + 480, + 715 + ], + "lines": [ + { + "bbox": [ + 117, + 693, + 480, + 715 + ], + "spans": [ + { + "bbox": [ + 117, + 693, + 480, + 715 + ], + "score": 0.93, + "content": "\\left. \\theta _ { i } \\theta _ { j } \\left( \\partial _ { k } f \\right) + \\theta _ { i } \\left( \\partial _ { j } f \\right) \\theta _ { k } + \\left( \\partial _ { i } f \\right) \\theta _ { j } \\theta _ { k } \\right. = \\eta \\left. \\theta _ { i } \\widetilde { C } _ { j , k } + \\theta _ { j } \\widetilde { C } _ { k , i } + \\theta _ { k } \\widetilde { C } _ { i , j } \\right. - \\eta ^ { 2 } \\left. \\widetilde { C } _ { i , j , k } \\right. .", + "type": "interline_equation", + "image_path": "253dabe82cea34cd22e4882e49e1217e38877dd6c19c2772e321d653b7dea368.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 117, + 693, + 480, + 715 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 432, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 720, + 432, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 432, + 734 + ], + "score": 1.0, + "content": "The systematic investigation of higher-order relations is relegated to future work.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 720, + 432, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 190, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 191, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 191, + 97 + ], + "score": 1.0, + "content": "2.3.3 SGD6=SDE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "There is no limit in which SGD asymptotically reduces to the stochastic differential equation (SDE).", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 362, + 126 + ], + "score": 1.0, + "content": "In order to take such a limit with continuous time differential", + "type": "text" + }, + { + "bbox": [ + 362, + 113, + 403, + 124 + ], + "score": 0.92, + "content": "\\mathrm d t \\to 0 ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 112, + 506, + 126 + ], + "score": 1.0, + "content": ", each SGD update must", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 272, + 137 + ], + "score": 1.0, + "content": "become infinitesimal. One may thus try", + "type": "text" + }, + { + "bbox": [ + 273, + 124, + 334, + 136 + ], + "score": 0.92, + "content": "\\mathrm { d } t \\equiv \\eta 0 ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 123, + 506, + 137 + ], + "score": 1.0, + "content": ", as in recent work adapting the view that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 127, + 148 + ], + "score": 1.0, + "content": "SGD", + "type": "text" + }, + { + "bbox": [ + 127, + 136, + 135, + 145 + ], + "score": 0.42, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 133, + 506, + 148 + ], + "score": 1.0, + "content": "SDE (Mandt et al., 2015; Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "2018; Chaudhari & Soatto, 2017; Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). But this", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 155, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 291, + 170 + ], + "score": 1.0, + "content": "in turn forces the noise vector with zero mean, √", + "type": "text" + }, + { + "bbox": [ + 291, + 156, + 338, + 169 + ], + "score": 0.93, + "content": "\\nabla f ^ { B } - \\nabla f", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 155, + 417, + 170 + ], + "score": 1.0, + "content": ", to be multiplied by", + "type": "text" + }, + { + "bbox": [ + 418, + 158, + 428, + 167 + ], + "score": 0.32, + "content": "\\mathrm { d } t", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 155, + 506, + 170 + ], + "score": 1.0, + "content": ". This is in contrast", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 164, + 182 + ], + "score": 1.0, + "content": "to the scaling", + "type": "text" + }, + { + "bbox": [ + 164, + 168, + 183, + 181 + ], + "score": 0.91, + "content": "\\sqrt { \\mathrm { d } t }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "needed for the standard machinery of SDE – Ito-Stratonovich calculus and all ˆ", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 180, + 269, + 194 + ], + "score": 1.0, + "content": "that – to apply; the additional factor of", + "type": "text" + }, + { + "bbox": [ + 269, + 180, + 292, + 192 + ], + "score": 0.9, + "content": "\\mathrm { d } t ^ { 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "makes the effective noise covariance be suppressed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 191, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 118, + 206 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 193, + 130, + 203 + ], + "score": 0.43, + "content": "\\mathrm { d } t", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 191, + 505, + 206 + ], + "score": 1.0, + "content": "and the resulting equation in the continuous-time limit, if anything, would just be an ordinary", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "differential equation without noise2 [unless noise with the proper scaling is explicitly added as in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "stochastic gradient Langevin dynamics (Welling & Teh, 2011; Teh et al., 2016) and natural Langevin", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 367, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 367, + 238 + ], + "score": 1.0, + "content": "dynamics (Marceau-Caron & Ollivier, 2017; Nado et al., 2018)].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 239, + 257 + ], + "score": 1.0, + "content": "In short, the recent work views", + "type": "text" + }, + { + "bbox": [ + 239, + 242, + 294, + 257 + ], + "score": 0.93, + "content": "\\eta = { \\sqrt { \\eta } } { \\sqrt { \\mathrm { d } t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 243, + 340, + 257 + ], + "score": 1.0, + "content": "and sends", + "type": "text" + }, + { + "bbox": [ + 340, + 243, + 381, + 254 + ], + "score": 0.91, + "content": "\\mathrm d t \\to 0 ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 243, + 474, + 257 + ], + "score": 1.0, + "content": "while pretending that", + "type": "text" + }, + { + "bbox": [ + 474, + 246, + 482, + 255 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "is fi-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "nite, which is inconsistent. This is not just a technical subtlety. When unjustifiably passing onto", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "the continuous-time Fokker-Planck equation, the diffusive term is incorrectly governed by the con-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 227, + 292 + ], + "score": 1.0, + "content": "nected two-point noise matrix", + "type": "text" + }, + { + "bbox": [ + 227, + 276, + 388, + 291 + ], + "score": 0.93, + "content": "C _ { i , j } \\left( \\pmb { \\theta } \\right) \\equiv \\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right) - \\left[ \\partial _ { i } f \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { j } f \\left( \\pmb { \\theta } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 277, + 506, + 292 + ], + "score": 1.0, + "content": "rather than the full two-point", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 159, + 305 + ], + "score": 1.0, + "content": "noise matrix", + "type": "text" + }, + { + "bbox": [ + 159, + 290, + 192, + 304 + ], + "score": 0.73, + "content": "\\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "that appears herein.3 We must instead employ the discrete-time version of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "Fokker-Planck equation derived in references Van Kampen (1992); Gardiner (2009); Risken (1984);", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 452, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 452, + 327 + ], + "score": 1.0, + "content": "Radons et al. (1990); Leen & Moody (1993), as has been followed in the equation (6).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 339, + 218, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 220, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 220, + 352 + ], + "score": 1.0, + "content": "2.3.4 ON STATIONARITY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "In contrast to statistical mechanics where an equilibrium state is dictated by a handful of thermo-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "dynamic variables, in machine learning a stationary state generically depends not only on hyperpa-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "rameters but also on a part of its learning history. The stationarity assumption made herein, which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "is codified in the equation (6), is weaker than the typicality assumption underlying statistical me-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 449, + 416 + ], + "score": 1.0, + "content": "chanics and can hold even in the presence of lingering memory. In the full-batch limit", + "type": "text" + }, + { + "bbox": [ + 449, + 402, + 487, + 415 + ], + "score": 0.9, + "content": "| B | = N _ { \\mathrm { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 402, + 506, + 416 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "instance, any distribution delta-peaked at a local minimum is stationary. For sufficiently small learn-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 144, + 436 + ], + "score": 1.0, + "content": "ing rates", + "type": "text" + }, + { + "bbox": [ + 145, + 426, + 151, + 436 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "as well, it is natural to expect multiple stationary distributions that form disconnected", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 482, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 341, + 449 + ], + "score": 1.0, + "content": "ponds around these minima, which merge upon increasing", + "type": "text" + }, + { + "bbox": [ + 341, + 437, + 348, + 447 + ], + "score": 0.77, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 434, + 472, + 449 + ], + "score": 1.0, + "content": "and fragment upon decreasing", + "type": "text" + }, + { + "bbox": [ + 472, + 439, + 478, + 447 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 434, + 482, + 449 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "It is beyond the scope of the present paper to formulate conditions under which stationary distri-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "butions exist. Indeed, if the formulation were too generic, there could be counterexamples to such", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 473, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 505, + 488 + ], + "score": 1.0, + "content": "a putative existence statement. A case in point is a model with the unregularized cross entropy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "loss, whose model parameters keep cascading toward infinity in order to sharpen its softmax out-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 369, + 509 + ], + "score": 1.0, + "content": "put (Neyshabur et al., 2014; 2017) with logarithmically diverging", + "type": "text" + }, + { + "bbox": [ + 369, + 496, + 380, + 506 + ], + "score": 0.85, + "content": "\\pmb { \\theta } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "(Soudry et al., 2018). It would", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 348, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 348, + 520 + ], + "score": 1.0, + "content": "be interesting to see if there are any other nontrivial caveats.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 536, + 218, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 534, + 220, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 220, + 551 + ], + "score": 1.0, + "content": "3 EMPIRICAL TESTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "In this section we empirically bear out our theoretical claims in the last section. To this end, two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "simple models of supervised learning are used (see Appendix B for full specifications): a multilayer", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "perceptron (MLP) learning patterns in the MNIST training data (LeCun et al., 1998) through SGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "without momentum and a convolutional neural network (CNN) learning patterns in the CIFAR-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 424, + 618 + ], + "score": 1.0, + "content": "10 training data (Krizhevsky & Hinton, 2009) through SGD with momentum", + "type": "text" + }, + { + "bbox": [ + 425, + 606, + 461, + 617 + ], + "score": 0.9, + "content": "\\mu = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 605, + 506, + 618 + ], + "score": 1.0, + "content": ". For both", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 266, + 628 + ], + "score": 1.0, + "content": "models, the mini-batch size is set to be", + "type": "text" + }, + { + "bbox": [ + 266, + 617, + 309, + 628 + ], + "score": 0.93, + "content": "| B | = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 617, + 505, + 628 + ], + "score": 1.0, + "content": ", and the training data are shuffled at each epoch", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 627, + 507, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 627, + 161, + 643 + ], + "score": 0.94, + "content": "\\begin{array} { r } { t = \\frac { N _ { \\mathrm { s } } } { | B | } \\hat { t } _ { \\mathrm { e p o c h } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 627, + 182, + 643 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 183, + 627, + 228, + 641 + ], + "score": 0.91, + "content": "\\hat { t } _ { \\mathrm { e p o c h } } \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 627, + 507, + 643 + ], + "score": 1.0, + "content": ". In order to avoid the overfitting cascade mentioned in Section 2.3.4,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 642, + 504, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 121, + 655 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 643, + 133, + 653 + ], + "score": 0.88, + "content": "L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 643, + 214, + 655 + ], + "score": 1.0, + "content": "-regularization term", + "type": "text" + }, + { + "bbox": [ + 215, + 642, + 238, + 656 + ], + "score": 0.92, + "content": "\\scriptstyle { \\frac { 1 } { 2 } } \\lambda \\theta ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 643, + 329, + 655 + ], + "score": 1.0, + "content": "with the weight decay", + "type": "text" + }, + { + "bbox": [ + 329, + 643, + 367, + 654 + ], + "score": 0.89, + "content": "\\lambda = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 643, + 493, 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1.0, + "content": "due to the relation FDR2, and one may thus neglect the dif-", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 168, + 712 + ], + "score": 1.0, + "content": "ference between", + "type": "text" + }, + { + "bbox": [ + 169, + 699, + 178, + 710 + ], + "score": 0.85, + "content": "\\widetilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 699, + 195, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 700, + 204, + 710 + ], + "score": 0.8, + "content": "_ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 699, + 379, + 712 + ], + "score": 1.0, + "content": ", and hence justify the naive use of SDE, when", + "type": "text" + }, + { + "bbox": [ + 379, + 700, + 416, + 711 + ], + "score": 0.92, + "content": "\\eta H \\ll 1", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "and the Gaussian-noise", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "assumption holds. In the similar vein, the reference Li et al. (2015) proves faster convergence between SGD", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 720, + 381, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 250, + 732 + ], + "score": 1.0, + "content": "and SDE when the term proportional to", + "type": "text" + }, + { + "bbox": [ + 250, + 720, + 292, + 732 + ], + "score": 0.93, + "content": "\\eta \\nabla \\left( \\nabla f \\right) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 721, + 381, + 732 + ], + "score": 1.0, + "content": "is added to the gradient.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 190, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 191, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 191, + 97 + ], + "score": 1.0, + "content": "2.3.3 SGD6=SDE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "There is no limit in which SGD asymptotically reduces to the stochastic differential equation (SDE).", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 362, + 126 + ], + "score": 1.0, + "content": "In order to take such a limit with continuous time differential", + "type": "text" + }, + { + "bbox": [ + 362, + 113, + 403, + 124 + ], + "score": 0.92, + "content": "\\mathrm d t \\to 0 ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 112, + 506, + 126 + ], + "score": 1.0, + "content": ", each SGD update must", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 272, + 137 + ], + "score": 1.0, + "content": "become infinitesimal. One may thus try", + "type": "text" + }, + { + "bbox": [ + 273, + 124, + 334, + 136 + ], + "score": 0.92, + "content": "\\mathrm { d } t \\equiv \\eta 0 ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 123, + 506, + 137 + ], + "score": 1.0, + "content": ", as in recent work adapting the view that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 127, + 148 + ], + "score": 1.0, + "content": "SGD", + "type": "text" + }, + { + "bbox": [ + 127, + 136, + 135, + 145 + ], + "score": 0.42, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 133, + 506, + 148 + ], + "score": 1.0, + "content": "SDE (Mandt et al., 2015; Li et al., 2015; Mandt et al., 2017; Li et al., 2017; Smith & Le,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "2018; Chaudhari & Soatto, 2017; Jastrzebski et al., 2017; Zhu et al., 2018; An et al., 2018). But this", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 155, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 291, + 170 + ], + "score": 1.0, + "content": "in turn forces the noise vector with zero mean, √", + "type": "text" + }, + { + "bbox": [ + 291, + 156, + 338, + 169 + ], + "score": 0.93, + "content": "\\nabla f ^ { B } - \\nabla f", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 155, + 417, + 170 + ], + "score": 1.0, + "content": ", to be multiplied by", + "type": "text" + }, + { + "bbox": [ + 418, + 158, + 428, + 167 + ], + "score": 0.32, + "content": "\\mathrm { d } t", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 155, + 506, + 170 + ], + "score": 1.0, + "content": ". This is in contrast", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 164, + 182 + ], + "score": 1.0, + "content": "to the scaling", + "type": "text" + }, + { + "bbox": [ + 164, + 168, + 183, + 181 + ], + "score": 0.91, + "content": "\\sqrt { \\mathrm { d } t }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "needed for the standard machinery of SDE – Ito-Stratonovich calculus and all ˆ", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 180, + 269, + 194 + ], + "score": 1.0, + "content": "that – to apply; the additional factor of", + "type": "text" + }, + { + "bbox": [ + 269, + 180, + 292, + 192 + ], + "score": 0.9, + "content": "\\mathrm { d } t ^ { 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "makes the effective noise covariance be suppressed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 191, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 118, + 206 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 193, + 130, + 203 + ], + "score": 0.43, + "content": "\\mathrm { d } t", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 191, + 505, + 206 + ], + "score": 1.0, + "content": "and the resulting equation in the continuous-time limit, if anything, would just be an ordinary", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "differential equation without noise2 [unless noise with the proper scaling is explicitly added as in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "stochastic gradient Langevin dynamics (Welling & Teh, 2011; Teh et al., 2016) and natural Langevin", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 367, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 367, + 238 + ], + "score": 1.0, + "content": "dynamics (Marceau-Caron & Ollivier, 2017; Nado et al., 2018)].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 102, + 506, + 238 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 239, + 257 + ], + "score": 1.0, + "content": "In short, the recent work views", + "type": "text" + }, + { + "bbox": [ + 239, + 242, + 294, + 257 + ], + "score": 0.93, + "content": "\\eta = { \\sqrt { \\eta } } { \\sqrt { \\mathrm { d } t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 243, + 340, + 257 + ], + "score": 1.0, + "content": "and sends", + "type": "text" + }, + { + "bbox": [ + 340, + 243, + 381, + 254 + ], + "score": 0.91, + "content": "\\mathrm d t \\to 0 ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 243, + 474, + 257 + ], + "score": 1.0, + "content": "while pretending that", + "type": "text" + }, + { + "bbox": [ + 474, + 246, + 482, + 255 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "is fi-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "nite, which is inconsistent. This is not just a technical subtlety. When unjustifiably passing onto", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "the continuous-time Fokker-Planck equation, the diffusive term is incorrectly governed by the con-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 227, + 292 + ], + "score": 1.0, + "content": "nected two-point noise matrix", + "type": "text" + }, + { + "bbox": [ + 227, + 276, + 388, + 291 + ], + "score": 0.93, + "content": "C _ { i , j } \\left( \\pmb { \\theta } \\right) \\equiv \\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right) - \\left[ \\partial _ { i } f \\left( \\pmb { \\theta } \\right) \\right] \\left[ \\partial _ { j } f \\left( \\pmb { \\theta } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 277, + 506, + 292 + ], + "score": 1.0, + "content": "rather than the full two-point", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 159, + 305 + ], + "score": 1.0, + "content": "noise matrix", + "type": "text" + }, + { + "bbox": [ + 159, + 290, + 192, + 304 + ], + "score": 0.73, + "content": "\\widetilde { C } _ { i , j } \\left( \\pmb { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "that appears herein.3 We must instead employ the discrete-time version of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "Fokker-Planck equation derived in references Van Kampen (1992); Gardiner (2009); Risken (1984);", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 452, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 452, + 327 + ], + "score": 1.0, + "content": "Radons et al. (1990); Leen & Moody (1993), as has been followed in the equation (6).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 242, + 506, + 327 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 339, + 218, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 220, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 220, + 352 + ], + "score": 1.0, + "content": "2.3.4 ON STATIONARITY", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "In contrast to statistical mechanics where an equilibrium state is dictated by a handful of thermo-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "dynamic variables, in machine learning a stationary state generically depends not only on hyperpa-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "rameters but also on a part of its learning history. The stationarity assumption made herein, which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "is codified in the equation (6), is weaker than the typicality assumption underlying statistical me-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 449, + 416 + ], + "score": 1.0, + "content": "chanics and can hold even in the presence of lingering memory. In the full-batch limit", + "type": "text" + }, + { + "bbox": [ + 449, + 402, + 487, + 415 + ], + "score": 0.9, + "content": "| B | = N _ { \\mathrm { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 402, + 506, + 416 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "instance, any distribution delta-peaked at a local minimum is stationary. For sufficiently small learn-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 144, + 436 + ], + "score": 1.0, + "content": "ing rates", + "type": "text" + }, + { + "bbox": [ + 145, + 426, + 151, + 436 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "as well, it is natural to expect multiple stationary distributions that form disconnected", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 482, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 341, + 449 + ], + "score": 1.0, + "content": "ponds around these minima, which merge upon increasing", + "type": "text" + }, + { + "bbox": [ + 341, + 437, + 348, + 447 + ], + "score": 0.77, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 434, + 472, + 449 + ], + "score": 1.0, + "content": "and fragment upon decreasing", + "type": "text" + }, + { + "bbox": [ + 472, + 439, + 478, + 447 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 434, + 482, + 449 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 359, + 506, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "It is beyond the scope of the present paper to formulate conditions under which stationary distri-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "butions exist. Indeed, if the formulation were too generic, there could be counterexamples to such", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 473, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 505, + 488 + ], + "score": 1.0, + "content": "a putative existence statement. A case in point is a model with the unregularized cross entropy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "loss, whose model parameters keep cascading toward infinity in order to sharpen its softmax out-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 369, + 509 + ], + "score": 1.0, + "content": "put (Neyshabur et al., 2014; 2017) with logarithmically diverging", + "type": "text" + }, + { + "bbox": [ + 369, + 496, + 380, + 506 + ], + "score": 0.85, + "content": "\\pmb { \\theta } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "(Soudry et al., 2018). It would", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 348, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 348, + 520 + ], + "score": 1.0, + "content": "be interesting to see if there are any other nontrivial caveats.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 452, + 505, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 536, + 218, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 534, + 220, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 220, + 551 + ], + "score": 1.0, + "content": "3 EMPIRICAL TESTS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "In this section we empirically bear out our theoretical claims in the last section. To this end, two", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "simple models of supervised learning are used (see Appendix B for full specifications): a multilayer", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "perceptron (MLP) learning patterns in the MNIST training data (LeCun et al., 1998) through SGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "without momentum and a convolutional neural network (CNN) learning patterns in the CIFAR-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 424, + 618 + ], + "score": 1.0, + "content": "10 training data (Krizhevsky & Hinton, 2009) through SGD with momentum", + "type": "text" + }, + { + "bbox": [ + 425, + 606, + 461, + 617 + ], + "score": 0.9, + "content": "\\mu = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 605, + 506, + 618 + ], + "score": 1.0, + "content": ". For both", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 266, + 628 + ], + "score": 1.0, + "content": "models, the mini-batch size is set to be", + "type": "text" + }, + { + "bbox": [ + 266, + 617, + 309, + 628 + ], + "score": 0.93, + "content": "| B | = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 617, + 505, + 628 + ], + "score": 1.0, + "content": ", and the training data are shuffled at each epoch", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 627, + 507, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 627, + 161, + 643 + ], + "score": 0.94, + "content": "\\begin{array} { r } { t = \\frac { N _ { \\mathrm { s } } } { | B | } \\hat { t } _ { \\mathrm { e p o c h } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 627, + 182, + 643 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 183, + 627, + 228, + 641 + ], + "score": 0.91, + "content": "\\hat { t } _ { \\mathrm { e p o c h } } \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 627, + 507, + 643 + ], + "score": 1.0, + "content": ". 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If SGD drives the distribution of the model parameters to stationarity at long time, then", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 153, + 344, + 172 + ], + "lines": [ + { + "bbox": [ + 266, + 153, + 344, + 172 + ], + "spans": [ + { + "bbox": [ + 266, + 153, + 344, + 172 + ], + "score": 0.91, + "content": "\\operatorname * { l i m } _ { t \\infty } \\overline { { \\mathcal { O } } } ( t ) = \\langle \\mathcal { O } \\rangle \\ .", + "type": "interline_equation", + "image_path": "3a8ffa5554f0fc893082c5151a11b8cea44c4f8bad2b0db390f536c798d65045.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 266, + 153, + 344, + 172 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 182, + 412, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 414, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 414, + 195 + ], + "score": 1.0, + "content": "3.1 FIRST FLUCTUATION-DISSIPATION RELATION AND EQUILIBRATION", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 321, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 321, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 321, + 216 + ], + "score": 1.0, + "content": "In order to assess the proximity to stationarity, define", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 215, + 399, + 242 + ], + "lines": [ + { + "bbox": [ + 212, + 215, + 399, + 242 + ], + "spans": [ + { + "bbox": [ + 212, + 215, + 399, + 242 + ], + "score": 0.93, + "content": "{ \\mathcal { O } } _ { \\mathrm { L } } \\equiv \\pmb { \\theta } \\cdot \\left( \\pmb { \\nabla } f ^ { \\mathcal { B } } \\right) \\quad \\mathrm { a n d } \\quad { \\mathcal { O } } _ { \\mathrm { R } } \\equiv \\frac { ( 1 + \\mu ) } { 2 ( 1 - 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Top panels depict the half-running average", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 493, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 494, + 131, + 508 + ], + "score": 0.92, + "content": "{ \\overline { { f ^ { B } } } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 493, + 299, + 511 + ], + "score": 1.0, + "content": "(dark green) and the instantaneous value", + "type": "text" + }, + { + "bbox": [ + 299, + 495, + 323, + 508 + ], + "score": 0.92, + "content": "f ^ { B } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 493, + 506, + 511 + ], + "score": 1.0, + "content": "(light green) of the mini-batch loss. 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As shown in the figure 1, the observables", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 577, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 132, + 591 + ], + "score": 0.92, + "content": "\\overline { { \\mathcal { O } } } _ { \\mathrm { L } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 577, + 153, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 578, + 180, + 591 + ], + "score": 0.92, + "content": "\\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 577, + 505, + 594 + ], + "score": 1.0, + "content": "converge to each other. 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t _ { 0 } } \\sum _ { t ^ { \\prime } = t _ { 0 } + 1 } ^ { t } \\mathcal { O } ( t ^ { \\prime } ) \\mathrm { w i t h } t _ { 0 } = \\lfloor t / 2 \\rfloor .", + "type": "interline_equation", + "image_path": "6f4b20d9ef2037c1fadba026b8e477f3149b00f7c865c0613bc2eaf652f2024a.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 94, + 405, + 111.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 206, + 111.5, + 405, + 129.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 130, + 503, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 325, + 143 + ], + "score": 1.0, + "content": "This is the average of the observable up to the time step", + "type": "text" + }, + { + "bbox": [ + 325, + 131, + 330, + 140 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 129, + 505, + 143 + ], + "score": 1.0, + "content": ", with the initial half discarded as containing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 496, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 496, + 154 + ], + "score": 1.0, + "content": "transient. If SGD drives the distribution of the model parameters to stationarity at long time, then", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 129, + 505, + 154 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 153, + 344, + 172 + ], + "lines": [ + { + "bbox": [ + 266, + 153, + 344, + 172 + ], + "spans": [ + { + "bbox": [ + 266, + 153, + 344, + 172 + ], + "score": 0.91, + "content": "\\operatorname * { l i m } _ { t \\infty } \\overline { { \\mathcal { O } } } ( t ) = \\langle \\mathcal { O } \\rangle \\ .", + "type": "interline_equation", + "image_path": "3a8ffa5554f0fc893082c5151a11b8cea44c4f8bad2b0db390f536c798d65045.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 266, + 153, + 344, + 172 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 182, + 412, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 414, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 414, + 195 + ], + "score": 1.0, + "content": "3.1 FIRST FLUCTUATION-DISSIPATION RELATION AND EQUILIBRATION", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 321, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 321, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 321, + 216 + ], + "score": 1.0, + "content": "In order to assess the proximity to stationarity, define", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 202, + 321, + 216 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 215, + 399, + 242 + ], + "lines": [ + { + "bbox": [ + 212, + 215, + 399, + 242 + ], + "spans": [ + { + "bbox": [ + 212, + 215, + 399, + 242 + ], + "score": 0.93, + "content": "{ \\mathcal { O } } _ { \\mathrm { L } } \\equiv \\pmb { \\theta } \\cdot \\left( \\pmb { \\nabla } f ^ { \\mathcal { B } } \\right) \\quad \\mathrm { a n d } \\quad { \\mathcal { O } } _ { \\mathrm { R } } \\equiv \\frac { ( 1 + \\mu ) } { 2 ( 1 - 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Top panels depict the half-running average", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 493, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 494, + 131, + 508 + ], + "score": 0.92, + "content": "{ \\overline { { f ^ { B } } } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 493, + 299, + 511 + ], + "score": 1.0, + "content": "(dark green) and the instantaneous value", + "type": "text" + }, + { + "bbox": [ + 299, + 495, + 323, + 508 + ], + "score": 0.92, + "content": "f ^ { B } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 493, + 506, + 511 + ], + "score": 1.0, + "content": "(light green) of the mini-batch loss. Bottom", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 422, + 519 + ], + "score": 1.0, + "content": "panels depict the convergence of the half-running averages of the observables", + "type": "text" + }, + { + "bbox": [ + 422, + 506, + 487, + 519 + ], + "score": 0.93, + "content": "O _ { \\mathrm { { L } } } = \\pmb { \\theta } \\cdot \\pmb { \\nabla } f ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 517, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 107, + 518, + 178, + 535 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\dot { \\mathcal { O } } _ { \\mathrm { R } } = \\frac { ( 1 + \\mu ) } { 2 ( 1 - \\nu ) } \\eta \\mathbf { v } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 517, + 505, + 535 + ], + "score": 1.0, + "content": ", whose stationary-state averages should agree according to the relation (FDR1’).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + } + ], + "index": 15.0 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 104, + 548, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 402, + 564 + ], + "score": 1.0, + "content": "In order to verify this claim, we first train the model with the learning rate", + "type": "text" + }, + { + "bbox": [ + 402, + 550, + 435, + 562 + ], + "score": 0.9, + "content": "\\eta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 548, + 450, + 564 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 450, + 549, + 505, + 564 + ], + "score": 0.9, + "content": "\\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } = 1 0 0", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 561, + 508, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 209, + 577 + ], + "score": 1.0, + "content": "epochs, that is, for ttotal", + "type": "text" + }, + { + "bbox": [ + 182, + 563, + 294, + 579 + ], + "score": 0.93, + "content": "\\begin{array} { r } { t ^ { \\mathrm { t o t a l } } = \\frac { N _ { \\mathrm { s } } } { | \\mathcal { B } | } \\hat { t } _ { \\mathrm { e p o c h } } ^ { \\mathrm { t o t a l } } = 1 0 0 \\frac { N _ { \\mathrm { s } } } { | \\mathcal { B } | } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 561, + 508, + 579 + ], + "score": 1.0, + "content": "time steps. As shown in the figure 1, the observables", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 577, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 132, + 591 + ], + "score": 0.92, + "content": "\\overline { { \\mathcal { O } } } _ { \\mathrm { L } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 577, + 153, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 578, + 180, + 591 + ], + "score": 0.92, + "content": "\\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 577, + 505, + 594 + ], + "score": 1.0, + "content": "converge to each other. 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The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 156, + 614 + ], + "score": 1.0, + "content": "observables", + "type": "text" + }, + { + "bbox": [ + 156, + 601, + 182, + 613 + ], + "score": 0.92, + "content": "\\breve { \\mathcal { O } } _ { \\mathrm { L } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 600, + 200, + 614 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 601, + 227, + 613 + ], + "score": 0.93, + "content": "\\overline { { \\mathcal { O } } } _ { \\mathrm { R } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "again converge to each other, as plotted in the figure 2. 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The systematic comparison between the two scheduling", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 709, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 721 + ], + "score": 1.0, + "content": "methods for state-of-the-arts architectures, and also the comparison with the AMSGrad algorithm", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 721, + 474, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 474, + 733 + ], + "score": 1.0, + "content": "for natural language processing tasks, could be a worthwhile avenue to pursue in the future.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 666, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 195, + 93 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 197, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 197, + 97 + ], + "score": 1.0, + "content": "4 CONCLUSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 105, + 107, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 505, + 119 + ], + "score": 1.0, + "content": "In this paper, we have derived the fluctuation-dissipation relations with no assumptions other than", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 119, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 129 + ], + "score": 1.0, + "content": "stationarity of the probability distribution. 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Special", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "thanks go to Daniel Adam Roberts who prompted the practical application of the fluctuation-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 377, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 377, + 317 + ], + "score": 1.0, + "content": "dissipation relations, leading to the adaptive method in Section 3.3.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 333, + 175, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 176, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 176, + 346 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "Jing An, Jianfeng Lu, and Lexing Ying. 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It would also", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "be interesting to further elucidate the physics of machine learning by extending our formalism to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "incorporate nonstationary dynamics, linearly away from stationarity (Onsager, 1931; Green, 1954;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "Kubo, 1957) and beyond (Jarzynski, 1997; Crooks, 1999), so that it can in particular properly treat", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 400, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 400, + 240 + ], + "score": 1.0, + "content": "overfitting cascading dynamics and time-dependent sample distributions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 107, + 505, + 240 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 253, + 200, + 262 + ], + "lines": [ + { + "bbox": [ + 107, + 253, + 200, + 263 + ], + "spans": [ + { + "bbox": [ + 107, + 253, + 200, + 263 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 284 + ], + "score": 1.0, + "content": "The author thanks Ludovic Berthier, Leon Bottou, Guy Gur-Ari, Kunihiko Kaneko, Ari Morcos, ´", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "Dheevatsa Mudigere, Yann Ollivier, Yuandong Tian, and Mark Tygert for discussions. 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0000000000000000000000000000000000000000..f6950d234bb74197bf0aad6e27940e2d4542850e --- /dev/null +++ b/parse/train/XJk19XzGq2J/XJk19XzGq2J.md @@ -0,0 +1,302 @@ +# THE INTRINSIC DIMENSION OF IMAGES AND ITS IMPACT ON LEARNING + +Phillip Pope1, Chen $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { 1 }$ , Ahmed Abdelkader2, Micah Goldblum1, Tom Goldstein1 1Department of Computer Science, University of Maryland, College Park 2Oden Institute for Computational Engineering and Sciences, University of Texas at Austin {pepope,chenzhu}@umd.edu, akader@utexas.edu, {goldblum,tomg}@umd.edu + +# ABSTRACT + +It is widely believed that natural image data exhibits low-dimensional structure despite the high dimensionality of conventional pixel representations. This idea underlies a common intuition for the remarkable success of deep learning in computer vision. In this work, we apply dimension estimation tools to popular datasets and investigate the role of low-dimensional structure in deep learning. We find that common natural image datasets indeed have very low intrinsic dimension relative to the high number of pixels in the images. Additionally, we find that low dimensional datasets are easier for neural networks to learn, and models solving these tasks generalize better from training to test data. Along the way, we develop a technique for validating our dimension estimation tools on synthetic data generated by GANs allowing us to actively manipulate the intrinsic dimension by controlling the image generation process. Code for our experiments may be found here. + +# 1 INTRODUCTION + +The idea that real-world data distributions can be described by very few variables underpins machine learning research from manifold learning to dimension reduction (Besold & Spokoiny, 2019; Fodor, 2002). The number of variables needed to describe a data distribution is known as its intrinsic dimension (ID). In applications, such as crystallography, computer graphics, and ecology, practitioners depend on data having low intrinsic dimension (Valle & Oganov, 2010; Desbrun et al., 2002; Laughlin, 2014). The utility of representations which are low-dimensional has motivated a variety of deep learning techniques including autoencoders and regularization methods (Hinton & Salakhutdinov, 2006; Vincent et al., 2010; Gonzalez & Balajewicz, 2018; Zhu et al., 2018). + +It is also known that dimensionality plays a strong role in learning function approximations and non-linear class boundaries. The exponential cost of learning in high dimensions is easily captured by the trivial case of sampling a function on a cube; in $d$ dimensions, sampling only the cube vertices would require $2 ^ { d }$ measurements. Similar behaviors emerge in learning theory. It is known that learning a manifold requires a number of samples that grows exponentially with the manifold’s intrinsic dimension (Narayanan & Mitter, 2010). Similarly, the number of samples needed to learn a well-conditioned decision boundary between two classes is an exponential function of the intrinsic dimension of the manifold on which the classes lie (Narayanan & Niyogi, 2009). Furthermore, these learning bounds have no dependence on the ambient dimension in which manifold-structured datasets live. + +In light of the exponentially large sample complexity of learning high-dimensional functions, the ability of neural networks to learn from image data is remarkable. Networks learn complex decision boundaries from small amounts of image data (often just a few hundred or thousand samples per class). At the same time, generative adversarial networks (GANs) are able to learn image “manifolds” from merely a few thousand samples. The seemingly low number of samples needed to learn these manifolds strongly suggests that image datasets have extremely low-dimensional structure. + +Despite the established role of low dimensional data in deep learning, little is known about the intrinsic dimension of popular datasets and the impact of dimensionality on the performance of neural networks. Computational methods for estimating intrinsic dimension enable these measurements. + +![](images/fcc707e5c48469d7362698f29421033c4b0c499f83ca0bca8273585e5b73ab72.jpg) +Figure 1: Estimates of the intrinsic dimension of commonly used datasets obtained using the MLE method with $k = 3 , 5 , 1 0 .$ , 20 nearest neighbors (left to right). The trends are consistent using different $k$ ’s. + +We adopt tools from the dimension estimation literature to shed light on dimensionality in settings of interest to the deep learning community. Our contributions can be summarized as follows: + +• We verify the reliability of intrinsic dimension estimation on high-dimensional data using generative adversarial networks (GANs), a setting in which we can a priori upper-bound the intrinsic dimension of generated data by the dimension of the latent noise vector. We measure the dimensionality of popular datasets such as MNIST, CIFAR-10, and ImageNet. In our experiments, we find that natural image datasets whose images contain thousands of pixels can, in fact, be described by orders of magnitude fewer variables. For example, we estimate that ImageNet, despite containing $2 2 4 \times 2 2 4 \times 3 = 1 5 0 5 2 8$ pixels per image, only has intrinsic dimension between 26 and 43; see Figure 1. • We train classifiers on data, synthetic and real, of various intrinsic dimension and find that this variable correlates closely with the number of samples needed for learning. On the other hand, we find that extrinsic dimension, the dimension of the ambient space in which data is embedded, has little impact on generalization. + +Together, these results put experimental weight behind the hypothesis that the unintuitively low dimensionality of natural images is being exploited by deep networks, and suggest that a characterization of this structure is an essential building block for a successful theory of deep learning. + +# 2 RELATED WORK + +While the hypothesis that natural images lie on or near a low-dimensional manifold is controversial, Goodfellow et al. (2016) argue that the low-dimensional manifold assumption is at least approximately correct for images, supported by two observations. First, natural images are locally connected, with each image surrounded by other highly similar images reachable through image transformations (e.g., contrast, brightness). Second, natural images seem to lie on a low-dimensional structure, as the probability distribution of images is highly concentrated; uniformly sampled pixels can hardly assemble a meaningful image. It is widely believed that the combination of natural scenes and sensor properties yields very sparse and concentrated image distributions, as has been supported by several empirical studies on image patches (Lee et al., 2003; Donoho & Grimes, 2005; Carlsson et al., 2008). This observation motivated work on efficient coding (Olshausen & Field, 1996) and served as a prior in computer vision (Peyre, 2009). Further, rigorous experiments have been con- ´ ducted clearly supporting the low-dimensional manifold hypothesis for many image datasets (Ruderman, 1994; Scholkopf et al., 1998; Roweis & Saul, 2000; Tenenbaum et al., 2000; Brand, 2003); ¨ see also (Fefferman et al., 2016) for principled algorithms on verifying the manifold hypothesis. + +The generalization literature seeks to understand why some models generalize better from training data to test data than others. One line of work suggests that the loss landscape geometry explains why neural networks generalize well (Huang et al., 2019). Other generalization work predicts that data with low dimension, along with other properties which do not include extrinsic dimension, characterize the generalization difficulty of classification problems (Narayanan & Niyogi, 2009). In the context of deep learning, Gong et al. (2019) found that neural network features are lowdimensional. Ansuini et al. (2019) further found that the intrinsic dimension of features decreases in late layers of neural networks and observed interesting trends in the dimension of features in early layers. In contrast to Gong et al. (2019) and Ansuini et al. (2019), who find that the intrinsic dimension of internal representations is inversely correlated with high performance, we study the dimensionality of data and its impact on performance, and we make a similar finding. Zhu et al. (2018) proposed a regularizer derived from the intrinsic dimension of images augmented with their corresponding feature vectors. Another line of work in deep learning has found that neural networks rely heavily on textures which are low-dimensional (Geirhos et al., 2018; Brendel & Bethge, 2019). Similarly, some have suggested that natural images can be represented as mixtures of textures which lie on a low-dimensional manifold (Vacher & Coen-Cagli, 2019; Vacher et al., 2020). + +# 3 INTRINSIC DIMENSION ESTIMATION + +Given a set of sample points $\mathcal { P } \subset \mathbb { R } ^ { N }$ , it is common to assume that $\mathcal { P }$ lies on or near a lowdimensional manifold $\bar { \mathcal { M } } \subseteq \mathbb { R } ^ { N }$ of intrinsic dimension $\dim ( { \mathcal { M } } ) = d \ll N$ . As a measure of the degrees of freedom in a dataset, as well as the information content, there is great interest in estimating the intrinsic dimension $d$ . In the remainder of this section, we briefly describe the dimension estimation method we use in this paper; for further information, see (Kim et al., 2019) and references therein. + +One of the main approaches to intrinsic dimension estimation is to examine a neighborhood around each point in the dataset, and compute the Euclidean distance to the $k ^ { t h }$ nearest neighbor. Assuming that density is constant within small neighborhoods, the Maximum Likelihood Estimation $( M L E )$ of Levina $\&$ Bickel (2005) uses a Poisson process to model the number of points found by random sampling within a given radius around each sample point. By relating the rate of this process to the surface area of the sphere, the likelihood equations yield an estimate of the ID at a given point $x$ as: + +$$ +\hat { m } _ { k } ( x ) = \left[ \frac { 1 } { k - 1 } \sum _ { j = 1 } ^ { k - 1 } \log \frac { T _ { k } ( x ) } { T _ { j } ( x ) } \right] ^ { - 1 } , +$$ + +where $T _ { j } ( x )$ is the Euclidean $( \ell _ { 2 } )$ distance from $x$ to its $j ^ { t h }$ nearest neighbor. Levina & Bickel to average the local estimates at each point to obtain a global estimate . MacKay & Ghahramani (2005) suggestion a correction based on avera $\begin{array} { r l } { \bar { m } _ { k } } & { { } = } \end{array}$ +$\textstyle { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } { \hat { m } } _ { k } ( x _ { i } )$ +inverses + +$$ +\bar { m } _ { k } = \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \hat { m } _ { k } ( x _ { i } ) ^ { - 1 } \right] ^ { - 1 } = \left[ \frac { 1 } { n ( k - 1 ) } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { k - 1 } \log \frac { T _ { k } ( x _ { i } ) } { T _ { j } ( x _ { i } ) } \right] ^ { - 1 } , +$$ + +where $n$ is the number of samples. We use Equation (2) as our MLE estimator throughout this paper. + +Since the geometry of natural images is complex and unknown, we face two challenges when verifying the accuracy of MLE on natural image datasets. First, we need to choose a proper value of $k$ . As shown by MacKay & Ghahramani (2005), the positive bias of the corrected estimator Equation (2) increases as $k$ increases, but the variance decreases. In order to navigate this bias-variance tradeoff, we try various values of $k$ in Section 4. Second, in addition to the aforementioned local uniformity assumption, MLE assumes that data arises as a sequence of i.i.d. random variables which can be written as a continuous and sufficiently smooth function of a random variable with smooth density, which may or may not be true for natural image datasets. While the truth of these assumptions is unknown on natural images, we verify the accuracy of our MLE estimates in a controlled setting in the following section. + +We briefly discuss other notable techniques for dimensionality estimation. GeoMLE (Gomtsyan et al., 2019) attempts to account for non-uniformity of density and nonlinearity of manifold using a polynomial regression of standard MLE based on distances to nearest neighbors in different sized neighborhoods. However, GeoMLE chooses to approximate averages of $\hat { m } _ { k } ( x _ { i } )$ , instead of averaging its reciprocal like Equation (2), resulting in a potentially wrong maximum likelihood estimator. As a result, we find its estimation deviates significantly from expected dimensionalities. TwoNN (Facco et al., 2017) is based on the ratio of the distances to the first and second nearest neighbors. Finally, the approach of (Granata & Carnevale, 2016) considers the distribution of geodesic distances over the data manifold, approximated by distances through kNN graphs, compared to the distribution of distances over hyperspheres of varying dimension. Unlike MLE, our preliminary experiments suggest that these techniques do not provide reasonable estimates for some natural and synthetic images which are key to this work; see Appendix A.5 for further discussion. + +# 4 VALIDATING DIMENSION ESTIMATION WITH SYNTHETIC DATA + +Dimensionality estimates are often applied on “simple” manifolds or toy datasets where the dimensionality is known, and so the accuracy of the methods can be validated. Image manifolds, by contrast, are highly complex, may contain many symmetries and modes, and are of unknown dimension. In principle, there is no reason why MLE-based dimensionality estimates cannot be applied to image datasets. However, because we lack knowledge of the exact dimensionality of image datasets, we cannot directly verify that MLE-based dimensionality estimates scale up to the complexity of image structures. + +There is an inherent uncertainty in estimating the ID of a given dataset. First, we cannot be sure if the dataset actually resembles a sampling of points on or near a manifold. Second, there are typically no guarantees that the sampling satisfies the conditions assumed by the ID estimators we are using. + +Towards a principled application of ID estimates in contexts of practical relevance to deep learning, we begin by validating that MLE methods can generate accurate dimensionality estimates for complex image data. We do this by generating synthetic image datasets using generative models for which the intrinsic dimensionality can be upper-bounded a priori. We believe such validations are essential to put recent findings in perspective (Gong et al., 2019; Ansuini et al., 2019). + +GAN Images We use the BigGAN variant with 128 latent entries and outputs of size $1 2 8 \times 1 2 8 \times 3$ trained on the ImageNet dataset (Deng et al., 2009). Using this GAN, we generate datasets with a varying number of images, where we fix most entries of the latent vectors to zero leaving only $\bar { d }$ free entries to be chosen at random. As we increase the number of free entries, we expect the intrinsic dimension to increase with $\bar { d }$ as an upper bound; see Section A.1 for further discussion. + +In particular, we create several synthetic datasets of varying intrinsic dimensionality using the ImageNet class, basen $\mathrm { j } \mathrm { i }$ , and check if the estimates match our expectation. As seen in Figure 2, we observe increasing diversity with increasing intrinsic dimension. In Figure 3, we show convergence of the MLE estimate on basenji data with dimension bounded above by $\bar { d } = 1 0$ . We observe that the estimates can be sensitive to the choice of $k$ as discussed in prior work; see Appendix A.2 for additional GAN classes. + +Scaling to large datasets. We develop a practical approach for estimating the ID of large datasets such as ImageNet. In this approach, we randomly select a fraction $\alpha$ of the dataset as anchors. Then, we evaluate the MLE estimate using only the anchor points, where nearest-neighbors are computed over the entire dataset. Note that, when anchors are chosen randomly, this acceleration has no impact on the expected value of the result. See Appendix A.3 for an evaluation of this approach. + +# 5 THE INTRINSIC DIMENSION OF POPULAR DATASETS + +In this section, we measure the intrinsic dimensions of a number of popular datasets including MNIST (Deng, 2012), SVHN (Netzer et al., 2011), CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009), ImageNet (Deng et al., 2009), MS-COCO (Lin et al., 2014), and CelebA (Liu et al., 2015). Using three different parameter settings for the MLE ID estimator, we find that the ID is indeed much smaller than the number of pixels; see Table 5. Notice that the rank order of datasets by dimension does not depend on the choice of $k$ . A comparison of state-of-the-art (SOTA) classification accuracy on each respective dataset1 with the dimension estimates suggests a negative correlation between the intrinsic dimension and test accuracy. In the next section, we take a closer look at this phenomenon through a series of dedicated experiments. + +![](images/4bf96c7b46ed3eb22d58dfee16c9ee0fa8a058f2ef00eeda468f7c786936eafd.jpg) +Figure 2: Visualization of basenji GAN samples of varying intrinsic dimension. + +![](images/eef4736776c29aebbfecbdb9cbdf9de49d6f13189d08e2ef3029e8033712ac31.jpg) +Figure 3: Validation of MLE estimate on synthetic basenji data with $\bar { d } = 1 0$ free entries. We observe the estimates to converge around the expected dimensionality of 10. Standard errors plotted with $N = 5$ replicates over random samples of the data. + +# 6 INTRINSIC DIMENSION AND GENERALIZATION + +Learning theory work has established that learning a manifold requires a number of samples that grows exponentially with the manifold’s intrinsic dimension (Narayanan & Mitter, 2010), but the required number of samples is independent of the extrinsic dimension. Specifically, the number of + +
DatasetMNISTSVHNCIFAR-100CelebACIFAR-10MS-COCOImageNet
MLE (k=3)79119132226
MLE (k=5)11141817213338
MLE (k=10)12182224253743
MLE (k=20)13192326263643
SOTA Accuracy99.8499.0193.51199.37188.55
+ +Table 1: The MLE estimates for practical image datasets, and the state-of-the-art test-set image classification accuracy (for classification problems only) for these datasets. + +samples needed to learn a well-conditioned decision boundary between two classes is exponential in the intrinsic dimension of the manifold on which the classes lie (Narayanan & Niyogi, 2009). + +We leverage dimension estimation tools to empirically verify these theoretical findings using a family of binary classification problems defined over both synthetic and real datasets of varying intrinsic dimension. In these experiments, we observe a connection between the intrinsic dimension of data and generalization. Specifically, we find that classification problems on data of lower intrinsic dimensionality are easier to solve. + +6.1 SYNTHETIC GAN DATA: SAMPLE COMPLEXITY DEPENDS ON INTRINSIC (NOT EXTRINSIC) DIMENSIONALITY + +The synthetic GAN data generation technique described in Section 4 provides a unique opportunity to test the relationship between generalization and the intrinsic/extrinsic dimensionality of images. By creating datasets with controlled intrinsic dimensionality, we may compare their sample complexity, that is the number of samples required to obtain a given level of test error. Specifically we test the following two hypotheses (1) data of lower intrinsic dimensionality has lower sample complexity than that of higher intrinsic dimensionality and (2) extrinsic dimensionality is irrelevant for sample complexity. + +To investigate hypothesis (1), we create four synthetic datasets of varying intrinsic dimensionality: 16, 32, 64, 128, fixed extrinsic dimensionality: $3 \times 3 2 \times 3 2$ , and two classes: basenji and beagle. For each dataset we fix a test set of size $N = 1 7 0 0$ . For all experiments, we use the ResNet-18 (width $= 6 4$ ) architecture (He et al., 2016). We then train models until they fit their entire training set with increasing amounts of training samples and measure the test error. We show these results in Figure 4. Observing the varying rates of growth, we see that data of higher intrinsic dimension requires more samples to achieve a given test error. + +For hypothesis (2), we carry out the same experiment with the roles of intrinsic and extrinsic dimension switched. We create four synthetic datasets of varying extrinsic dimensionality by resizing the images with nearest-neighbor interpolation. Specifically we create 6 datasets of square, 3-channel images of sizes 16, 32, 64, 128, 256, fixed intrinsic dimensionality of size 128, and all other experimental details the same. We show these results in Figure 5. Observing the lack of variable growth rates, we see that extrinsic dimension has little to no effect on sample complexity. + +To the best of our knowledge, this is the first experimental demonstration that intrinsic but not extrinsic dimensionality matters for the generalization of deep networks. + +# 6.2 REAL DATA: INTRINSIC DIMENSIONALITY MATTERS FOR GENERALIZATION + +Next, we examine the sample complexity of binary classification tasks from four common image datasets: MNIST, SVHN, CIFAR-10, and ImageNet. This case differs from the synthetic case in that we have no control over each dataset’s intrinsic dimension. Instead, we estimate it via the MLE method discussed in Section 3. To account for variable difficulty of classes, we randomly sample 5 class pairs from each dataset and run the previously described sample complexity experiment. Note that these subsets differ from those used in Table 5, where the estimates are taken from the entire dataset and across all classes. + +On these sampled subsets, we find the MLE estimates as shown in Table 2. Note that these estimates are consistent with expectation, e.g. MNIST is qualitatively simpler then SVHN or CIFAR-10. + +![](images/74fb9e6ad214cfae0b2079ae127d7f16bfbcb8e40925bbd46ffb410f6d03d5a6.jpg) +Figure 4: Sample complexity of synthetic datasets of varying intrinsic dimensionality. + +![](images/68d854a2a7dac5de7032cf261ec3d6876730e3c1b40ff24b700172449b97f0a5.jpg) +Figure 5: Sample complexity of synthetic datasets of varying extrinsic dimensionality. + +We conduct the same sample complexity experiment as the previous section on the datasets. Because these datasets are ordinarily of varying extrinsic dimensionality, we resize all to size $3 2 \times 3 2 \times 3$ (before applying MLE). We report results in Figure 6, where we overall observe trends ordered by intrinsic dimensionality estimate. These results are consistent with expectation of the relative hardness of each dataset. However, there are some notable differences from the synthetic case. Several unexpected cross-over points exist in the low-sample regime, and the gap between SVHN and CIFAR-10 is smaller than one may expect based on their estimated intrinsic dimension. + +From these observations we conclude that intrinsic dimensionality is indeed relevant to generalization on real data, but it is not the only feature of data that influences sample complexity. + +
MNISTSVHNCIFAR-10ImageNet
k=37.5 (0.2)8.5 (0.1)11.4 (0.2)15.4 (0.8)
k=49.8 (0.3)11.6 (0.1)15.9 (0.2)19.8 (0.9)
k=510.9 (0.4)13.2 (0.1)18.3 (0.3)21.6 (1.0)
+ +Table 2: Mean and standard error of estimated intrinsic dimensions for practical datasets under the same resolution $3 2 \times 3 2 \times 3$ using different $k$ ’s. These results are consistent with the test accuracies on these datasets under the same resolution. + +# 6.3 REAL DATA: ADDING NOISE CHANGES DIMENSIONALITY TO AFFECT GENERALIZATION + +In this section, we examine an alternative technique for changing the intrinsic dimension of a real dataset: adding noise to images. Here we leverage the fact that uniformly sampled noise in $[ 0 , 1 ] ^ { d }$ has dimension $\underline d$ . We thus add independent noise, drawn uniformly from a fixed randomly oriented $d _ { \underline { { \mathbf { \Pi } } } }$ -dimensional unit hypercube embedded in pixel space, to each sample in a dataset. This procedure ensures that the dataset has dimension at least $\underline d$ . Since the natural data we use has low dimension, and the hypercubes have high dimension, this procedure specifically increases dimensionality. We note that estimation error may occur when there is an insufficient number of samples to achieve a proper estimate. Since the variation in images in a dataset may still be dominated by non-noise directions, we expect to underestimate the new increased dimensions of these noised datasets. + +![](images/655a510373a3bd2e79ca7346c2749387a47b165b02494ada5c430f1eee984fae.jpg) +Figure 6: Sample complexity of real datasets. Standard errors are shown $N = 5$ class pairs. + +Starting with CIFAR-10 data, we add noise of varying dimensions, where we replace pixels at random in the image. We only add noise to an image once to keep the augmented dataset the same size as the original. We use the following noise dimensionalities: 256, 512, 1024, 2048, 2560. The estimated dimensions of the noised datasets are listed in Table 3. We see that intrinsic dimension increases with increasing noise dimensionality, but dimensionality does not saturate to the maximum true dimension, likely due to a poverty of samples. + +On these noisy CIFAR-10 datasets, we again carry out the sample complexity experiment of the previous sections. We show results in Figure 7. We observe sample complexity largely in the same order as intrinsic dimension. + +
d=256d=512d=1024d=1536d=2048d= 2560
k=319.730.957.177.8110.0136.1
k=425.239.172.8101.3142.1177.7
k=527.642.578.3110.2153.4196.6
+ +![](images/c541e44293bb8e715400d216feb8919c55c73fa830ce96423a4162e56e0c35ec.jpg) +Table 3: Estimated intrinsic dimension of the CIFAR-10 dataset after adding different dimensions $( d )$ of uniformly sampled noise using different $k$ ’s. The estimated dimension consistently increases with $d$ under different $k$ ’s. +Figure 7: Sample complexity of noisy datasets. Standard errors are shown $N = 5$ random subsets of the data. + +# .4 MANIPULATING THE INTRINSIC DIMENSIONALITY OF FONTS + +In this section, we describe a final technique for studying the effect of intrinsic dimensionality on sample complexity on the recently proposed FONTS dataset (Stutz et al., 2019). Beginning with a collection of characters and font types, termed a prototype set by the authors, FONTS datasets are constructed using a fixed set of data augmentations: scaling, translation, rotation, and sheering. In principle, these augmentations each increase the intrinsic dimension of the prototype set allowing us to synthetically alter the intrinsic dimension by varying the number of augmentations used. + +We construct 5 FONTS datasets in this way, FONTS- $\{ 0 , 1 , 2 , 3 , 4 \}$ , where the suffix denotes the number of transformations used in the data generation process. The MLE estimates on each of the datasets are given in Table 4. + +Consistent with expectation, we observe that MLE methods consistently resolve the increased dimesionality of transformed datasets. Carrying out the sample complexity experiment of the previous section, we report results in Figure 8. We observe again that, on the whole, sample complexity is ordered by intrinsic dimension. + +
FONTS-0FONTS-1FONTS-2FONTS-3FONTS-4
k=31.8 (0.1)3.8 (0.1)5.1 (0.2)5.8 (0.2)6.1 (0.3)
k=42.7 (0.1)5.2 (0.1)6.9 (0.2)7.8 (0.3)8.3 (0.4)
k=53.2 (0.1)5.9 (0.1)7.8 (0.3)8.8 (0.4)9.4 (0.4)
+ +Table 4: Mean and standard errror of MLE estimates on the FONTS dataset under different number of transforms using different $k$ ’s. Again, the ranks of the estimated IDs are consistent, and the estimated IDs increase with the number of transforms. + +![](images/f914034c6d90137ea05b5cf5b64e93a7f0c6e71a63b2592d064c5597155023b0.jpg) +Figure 8: Sample complexity of FONTS datasets. + +# 7 DISCUSSION + +In this work, we measure the intrinsic dimension of popular image datasets and show that the intrinsic dimension of data matters for deep learning. While there may be many factors, such as class separation and the number of classes, which determine generalization, we build the case that intrinsic dimension is one of these important factors. Along the way, we introduce a technique for using GANs to synthesize data while manipulating dimensionality. This technique is useful not only for validating dimension estimation methods but also for examining the learning behavior of neural networks under a dimension-controlled environment. In addition to synthetic data, we verify that dimension plays a large role in learning on natural data. Our results support the commonly held belief that low dimensional structure underpins the success of deep learning on high-resolution data. + +These findings raise a number of salient directions for future work. Methods for enhancing neural network learning on high-dimensional data could improve generalization on hard vision problems. To this end, a deeper understanding of the mechanisms by which dimensionality plays a role in learning may enable such methods. Additionally, our work indicates that different dimensionality estimation tools are better suited for different settings. Future work on computing tighter and more reliable estimates specific to image data would allow the community to more precisely study the relationship between the dimensionality of image datasets and learning. + +# ACKNOWLEDGEMENTS + +This work was supported by the DARPA GARD and DARPA QED programs. Further support was provided by the AFOSR MURI program, and the National Science Foundation’s DMS division. Computation resources were funded by the Sloan Foundation. + +# REFERENCES + +Alessio Ansuini, Alessandro Laio, Jakob H Macke, and Davide Zoccolan. Intrinsic dimension of data representations in deep neural networks. In Advances in Neural Information Processing Systems, pp. 6111–6122, 2019. +Franz Besold and Vladimir Spokoiny. Adaptive manifold clustering. arXiv preprint arXiv:1912.04869, 2019. +Matthew Brand. Charting a manifold. In Advances in neural information processing systems, pp. 985–992, 2003. +Wieland Brendel and Matthias Bethge. 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In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018. + +# A VALIDATION OF ID ESTIMATES + +In this section, we present additional discussion results and discussion relevant to the ID estimation and related validation experiments in Section 4. + +# A.1 GAN PROPERTIES + +We devise a method for validating ID measurements in a controlled setting using images generated by GANs. To justify this method, we first note that the image of $\mathbb { R } ^ { d }$ under a locally Lipschitz function can be a manifold with dimension at most $d$ . Then, consider that the BigGAN generator, a convolutional neural network with ReLU activations, is a function with this property (Brock et al., 2018). + +Specifically, BigGAN can be written as a composition of linear functions, translations, and ReLU activation functions. Individually, these operations do not increase dimension, and by a composition property, their composition cannot increase dimensionality either. The more general fact that the image of $\mathbb { R } ^ { d }$ under a locally Lipschitz function can be a manifold with dimension at most $d$ follows from Sard’s theorem (Sard, 1942). + +# A.2 CONVERGENCE FOR MORE GAN CLASSES + +We include additional results on the estimation of ID for synthetic GAN images from various ImageNet classes with $\bar { d } = 1 0$ free entries out of the 128-dimensional latent vector input to the GAN; see Figures 10 and 11 below. As observed earlier in Section 4, the MLE estimates are sensitive to the choice of $k$ , where we expect the ID to be close to 10 given the way we sample the latent vectors to use for the GAN. We note that for a number of classes, all choices of $k$ we considered seem to underestimate the ID. + +![](images/63be558951d943d505226e0278503d8d07ffea2e7281ef6f5dbab8c616c45ea1.jpg) +Figure 9: Validation of MLE estimates on synthetic daisy data with $\bar { d } = 1 0$ . + +# A.3 SUBSAMPLING FOR LARGE DATASETS + +In Figure 12 we validate the anchor approximation on basenji data of dimension 10 for varying anchor ratio $\alpha$ . Then, in Figure 13 we validate the anchor approximation on tree-frog data with $\bar { d } = 3 2$ for varying $k$ while fixing the anchor ratio at $\alpha = 0 . 0 0 1$ . + +# A.4 RELATIONSHIP BETWEEN $k$ IN THE MLE METHOD AND DATASET DIMENSION + +The choice of $k$ may affect the dimensionality estimates. To better understand this relationship, we conducted additional studies for the MLE method using synthetic GAN data generated as described in Section 4. Table 5 shows the estimation results for various $k$ and basen $\mathrm { j } \mathrm { i }$ data with $\bar { d } \in$ $\{ 2 , 4 , 8 , 1 6 , 3 2 , 6 4 , 1 2 8 \}$ . We use a fixed number of samples $\mathit { \Delta } n \ = \ 1 0 0 0 0 \rangle$ ). We observe that the estimated intrinsic dimension increases with $k$ , and large values can yield overestimates for $\mathrm { I D }$ . These results agree with the work of Levina & Bickel (2005) who suggest that low values of $k$ may produce an estimator with higher variance while higher values of $k$ produce an estimator with higher positive bias. Given that we have access to large amounts of synthetic data, we opt for lower values of $k$ in our work. We do not choose a particular value of $k$ , and we report all experiments with multiple values. + +![](images/71d04a891002126df16545d41e277f88bbb28494666cca53e8b86d98bf4b61e5.jpg) +Figure 10: Validation of MLE estimate on synthetic soap-bubbles data with $\bar { d } = 1 0$ . + +![](images/185a2f8d3955d06c26323c8fdb27a3e473b3404cad17e7f2acb9e1c7f2ce2f1b.jpg) +Figure 11: Validation of MLE estimate on synthetic coffee data with 10 free entries. Note that the estimates do not converge around the upper bound of $\bar { d } = 1 0$ , which suggests that data generated from this class is not of full dimension. + +![](images/ae2e5bdd4f0617662ea5ce28a0211af3afe1e01680a9c531a6c21ab684f0f09c.jpg) +Figure 12: Validation of anchor approximation on basenji with $\bar { d } = 1 0$ . + +![](images/409b518eaba102881cac07313ab1903575de99d7b649eebbb568047f0de99121.jpg) +Figure 13: Validation of anchor approximation on tree-frog with $\bar { d } = 3 2$ and $\alpha = 0 . 0 0 1$ +Table 5: Additional results on the relationship between $k$ and MLE dimensionality estimate for synthetic basenji images with varying $\bar { d }$ . + +
kd
248163264128
1.12.66.110.516.020.020.0
34-1.53.68.214.021.026.026.0
51.74.19.315.723.528.728.5
61.84.49.916.624.930.329.9
71.94.610.417.225.831.230.6
81.94.710.717.626.431.731.1
92.04.910.918.026.831.931.5
102.05.011.118.227.132.131.7
152.15.311.618.827.832.331.7
2.25.511.819.027.931.931.3
202.25.712.019.227.931.530.8
+ +# A.5 COMPARING MLE TO OTHER ESTIMATORS IN A CONTROLLED SETTING + +To validate MLE in comparison to other estimators, we evaluate three other dimensionality estimation methods: GeoMLE (Gomtsyan et al., 2019), TwoNN (Facco et al., 2017) and kNN graph distances (Granata & Carnevale, 2016). For GeoMLE, we sample a total of 20 bootstrap subsets $M = 2 0$ ) and use $k _ { 1 } = 2 0 , k _ { 2 } = 5 5$ as recommended by Gomtsyan et al. (2019). To extend the kNN graph distance method to large datasets, we randomly sample a subset of samples (fixed to 10,000 for datasets with more than 10,000 samples) and use shortest graph distances to their $k$ nearest neighbors to estimate the IDs. Other settings are as default in the implementations of Granata & Carnevale (2016). + +First, we validate each method on datasets of uniformly sampled from $d$ -dimensional hypercubes. We report these results in Figure 14. Each method works reasonable on low-dimensional cubes. We observed the Shortest Path method to give erratic estimates on cubes of higher dimension, and have omitted these. TwoNN has poor sample efficiency for higher dimensional cubes. Interestingly, GeoMLE estimates these high dimensional cubes well. + +Next, we report estimation results on basenji 10 for TwoNN, kNN graph distance, and GeoMLE methods in Figure 15. Comparing against the MLE results in Figure 3, we observe that each other method does not achieve an accurate estimate in this sample regime, thus motivating our focus on MLE. Notably, GeoMLE and TwoNN severely overestimate dimension, while the kNN graph distance method severely underestimates dimension. + +On MNIST, CIFAR-10, CIFAR-100, and SVHN, the results of these other estimation methods also deviate from expectation (Table 6). For example, TwoNN assigns a significantly higher dimension estimate to MNIST than to CIFAR-100, which contradicts both intuition and the results of other estimators. We set $k = 4$ and number of bins to 1000, and use default settings for all other parameters including $r _ { \mathrm { M A X } }$ . + +![](images/0c1cb62ac3d7868f164d4488598093dba23256647324c62368df5b7d307f77b0.jpg) +Figure 14: The TwoNN, Shortest-Path, and GeoMLE methods on $d$ -dimensional Hypercube data. Each method estimates low-dimensional cubes well, validating their implementation. + +Table 6: Additional ID estimators on popular datasets. + +
DatasetMNISTCIFAR-10CIFAR-100SVHN
MLE(k = 5)11211814
GeoMLE (k1= 20,k2 = 55)25969321
TwoNN151197
kNN Graph Distance7786
+ +![](images/5f2c77c808097691523633c1fe4e358c2efb6ea1af2b6ca13b61731afa91fb44.jpg) +Figure 15: The TwoNN, Shortest-Path, and GeoMLE methods on basenji 10 data. The estimates do not converge around the expected value of $\bar { d } = 1 0$ in this sample regime. \ No newline at end of file diff --git a/parse/train/XJk19XzGq2J/XJk19XzGq2J_content_list.json b/parse/train/XJk19XzGq2J/XJk19XzGq2J_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5598c181456f34a158721a17ec2ea8b0e3f0b6eb --- /dev/null +++ b/parse/train/XJk19XzGq2J/XJk19XzGq2J_content_list.json @@ -0,0 +1,1626 @@ +[ + { + "type": "text", + "text": "THE INTRINSIC DIMENSION OF IMAGES AND ITS IMPACT ON LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 651, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Phillip Pope1, Chen $\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { 1 }$ , Ahmed Abdelkader2, Micah Goldblum1, Tom Goldstein1 1Department of Computer Science, University of Maryland, College Park 2Oden Institute for Computational Engineering and Sciences, University of Texas at Austin {pepope,chenzhu}@umd.edu, akader@utexas.edu, {goldblum,tomg}@umd.edu ", + "bbox": [ + 184, + 167, + 781, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 263, + 544, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "It is widely believed that natural image data exhibits low-dimensional structure despite the high dimensionality of conventional pixel representations. This idea underlies a common intuition for the remarkable success of deep learning in computer vision. In this work, we apply dimension estimation tools to popular datasets and investigate the role of low-dimensional structure in deep learning. We find that common natural image datasets indeed have very low intrinsic dimension relative to the high number of pixels in the images. Additionally, we find that low dimensional datasets are easier for neural networks to learn, and models solving these tasks generalize better from training to test data. Along the way, we develop a technique for validating our dimension estimation tools on synthetic data generated by GANs allowing us to actively manipulate the intrinsic dimension by controlling the image generation process. Code for our experiments may be found here. ", + "bbox": [ + 233, + 292, + 764, + 472 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The idea that real-world data distributions can be described by very few variables underpins machine learning research from manifold learning to dimension reduction (Besold & Spokoiny, 2019; Fodor, 2002). The number of variables needed to describe a data distribution is known as its intrinsic dimension (ID). In applications, such as crystallography, computer graphics, and ecology, practitioners depend on data having low intrinsic dimension (Valle & Oganov, 2010; Desbrun et al., 2002; Laughlin, 2014). The utility of representations which are low-dimensional has motivated a variety of deep learning techniques including autoencoders and regularization methods (Hinton & Salakhutdinov, 2006; Vincent et al., 2010; Gonzalez & Balajewicz, 2018; Zhu et al., 2018). ", + "bbox": [ + 174, + 526, + 825, + 637 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "It is also known that dimensionality plays a strong role in learning function approximations and non-linear class boundaries. The exponential cost of learning in high dimensions is easily captured by the trivial case of sampling a function on a cube; in $d$ dimensions, sampling only the cube vertices would require $2 ^ { d }$ measurements. Similar behaviors emerge in learning theory. It is known that learning a manifold requires a number of samples that grows exponentially with the manifold’s intrinsic dimension (Narayanan & Mitter, 2010). Similarly, the number of samples needed to learn a well-conditioned decision boundary between two classes is an exponential function of the intrinsic dimension of the manifold on which the classes lie (Narayanan & Niyogi, 2009). Furthermore, these learning bounds have no dependence on the ambient dimension in which manifold-structured datasets live. ", + "bbox": [ + 174, + 645, + 825, + 784 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In light of the exponentially large sample complexity of learning high-dimensional functions, the ability of neural networks to learn from image data is remarkable. Networks learn complex decision boundaries from small amounts of image data (often just a few hundred or thousand samples per class). At the same time, generative adversarial networks (GANs) are able to learn image “manifolds” from merely a few thousand samples. The seemingly low number of samples needed to learn these manifolds strongly suggests that image datasets have extremely low-dimensional structure. ", + "bbox": [ + 174, + 791, + 823, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite the established role of low dimensional data in deep learning, little is known about the intrinsic dimension of popular datasets and the impact of dimensionality on the performance of neural networks. Computational methods for estimating intrinsic dimension enable these measurements. ", + "bbox": [ + 176, + 882, + 823, + 922 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/fcc707e5c48469d7362698f29421033c4b0c499f83ca0bca8273585e5b73ab72.jpg", + "image_caption": [ + "Figure 1: Estimates of the intrinsic dimension of commonly used datasets obtained using the MLE method with $k = 3 , 5 , 1 0 .$ , 20 nearest neighbors (left to right). The trends are consistent using different $k$ ’s. " + ], + "image_footnote": [], + "bbox": [ + 225, + 103, + 772, + 271 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We adopt tools from the dimension estimation literature to shed light on dimensionality in settings of interest to the deep learning community. Our contributions can be summarized as follows: ", + "bbox": [ + 173, + 340, + 823, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We verify the reliability of intrinsic dimension estimation on high-dimensional data using generative adversarial networks (GANs), a setting in which we can a priori upper-bound the intrinsic dimension of generated data by the dimension of the latent noise vector. We measure the dimensionality of popular datasets such as MNIST, CIFAR-10, and ImageNet. In our experiments, we find that natural image datasets whose images contain thousands of pixels can, in fact, be described by orders of magnitude fewer variables. For example, we estimate that ImageNet, despite containing $2 2 4 \\times 2 2 4 \\times 3 = 1 5 0 5 2 8$ pixels per image, only has intrinsic dimension between 26 and 43; see Figure 1. • We train classifiers on data, synthetic and real, of various intrinsic dimension and find that this variable correlates closely with the number of samples needed for learning. On the other hand, we find that extrinsic dimension, the dimension of the ambient space in which data is embedded, has little impact on generalization. ", + "bbox": [ + 215, + 381, + 825, + 559 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Together, these results put experimental weight behind the hypothesis that the unintuitively low dimensionality of natural images is being exploited by deep networks, and suggest that a characterization of this structure is an essential building block for a successful theory of deep learning. ", + "bbox": [ + 176, + 571, + 825, + 613 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 633, + 344, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While the hypothesis that natural images lie on or near a low-dimensional manifold is controversial, Goodfellow et al. (2016) argue that the low-dimensional manifold assumption is at least approximately correct for images, supported by two observations. First, natural images are locally connected, with each image surrounded by other highly similar images reachable through image transformations (e.g., contrast, brightness). Second, natural images seem to lie on a low-dimensional structure, as the probability distribution of images is highly concentrated; uniformly sampled pixels can hardly assemble a meaningful image. It is widely believed that the combination of natural scenes and sensor properties yields very sparse and concentrated image distributions, as has been supported by several empirical studies on image patches (Lee et al., 2003; Donoho & Grimes, 2005; Carlsson et al., 2008). This observation motivated work on efficient coding (Olshausen & Field, 1996) and served as a prior in computer vision (Peyre, 2009). Further, rigorous experiments have been con- ´ ducted clearly supporting the low-dimensional manifold hypothesis for many image datasets (Ruderman, 1994; Scholkopf et al., 1998; Roweis & Saul, 2000; Tenenbaum et al., 2000; Brand, 2003); ¨ see also (Fefferman et al., 2016) for principled algorithms on verifying the manifold hypothesis. ", + "bbox": [ + 174, + 666, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The generalization literature seeks to understand why some models generalize better from training data to test data than others. One line of work suggests that the loss landscape geometry explains why neural networks generalize well (Huang et al., 2019). Other generalization work predicts that data with low dimension, along with other properties which do not include extrinsic dimension, characterize the generalization difficulty of classification problems (Narayanan & Niyogi, 2009). In the context of deep learning, Gong et al. (2019) found that neural network features are lowdimensional. Ansuini et al. (2019) further found that the intrinsic dimension of features decreases in late layers of neural networks and observed interesting trends in the dimension of features in early layers. In contrast to Gong et al. (2019) and Ansuini et al. (2019), who find that the intrinsic dimension of internal representations is inversely correlated with high performance, we study the dimensionality of data and its impact on performance, and we make a similar finding. Zhu et al. (2018) proposed a regularizer derived from the intrinsic dimension of images augmented with their corresponding feature vectors. Another line of work in deep learning has found that neural networks rely heavily on textures which are low-dimensional (Geirhos et al., 2018; Brendel & Bethge, 2019). Similarly, some have suggested that natural images can be represented as mixtures of textures which lie on a low-dimensional manifold (Vacher & Coen-Cagli, 2019; Vacher et al., 2020). ", + "bbox": [ + 176, + 868, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 270 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 INTRINSIC DIMENSION ESTIMATION ", + "text_level": 1, + "bbox": [ + 174, + 291, + 508, + 309 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a set of sample points $\\mathcal { P } \\subset \\mathbb { R } ^ { N }$ , it is common to assume that $\\mathcal { P }$ lies on or near a lowdimensional manifold $\\bar { \\mathcal { M } } \\subseteq \\mathbb { R } ^ { N }$ of intrinsic dimension $\\dim ( { \\mathcal { M } } ) = d \\ll N$ . As a measure of the degrees of freedom in a dataset, as well as the information content, there is great interest in estimating the intrinsic dimension $d$ . In the remainder of this section, we briefly describe the dimension estimation method we use in this paper; for further information, see (Kim et al., 2019) and references therein. ", + "bbox": [ + 173, + 324, + 825, + 409 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "One of the main approaches to intrinsic dimension estimation is to examine a neighborhood around each point in the dataset, and compute the Euclidean distance to the $k ^ { t h }$ nearest neighbor. Assuming that density is constant within small neighborhoods, the Maximum Likelihood Estimation $( M L E )$ of Levina $\\&$ Bickel (2005) uses a Poisson process to model the number of points found by random sampling within a given radius around each sample point. By relating the rate of this process to the surface area of the sphere, the likelihood equations yield an estimate of the ID at a given point $x$ as: ", + "bbox": [ + 173, + 415, + 825, + 500 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/242af5134ae641e06d65866b5c92fcec6be0117aeae3be03f519a840a09d66fe.jpg", + "text": "$$\n\\hat { m } _ { k } ( x ) = \\left[ \\frac { 1 } { k - 1 } \\sum _ { j = 1 } ^ { k - 1 } \\log \\frac { T _ { k } ( x ) } { T _ { j } ( x ) } \\right] ^ { - 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 506, + 624, + 560 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $T _ { j } ( x )$ is the Euclidean $( \\ell _ { 2 } )$ distance from $x$ to its $j ^ { t h }$ nearest neighbor. Levina & Bickel to average the local estimates at each point to obtain a global estimate . MacKay & Ghahramani (2005) suggestion a correction based on avera $\\begin{array} { r l } { \\bar { m } _ { k } } & { { } = } \\end{array}$ \n$\\textstyle { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } { \\hat { m } } _ { k } ( x _ { i } )$ \ninverses ", + "bbox": [ + 174, + 568, + 825, + 625 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b19c30b6f73a472944280a299d1f0faedc2e9f5e7c741f11e02276ce87b4f475.jpg", + "text": "$$\n\\bar { m } _ { k } = \\left[ \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\hat { m } _ { k } ( x _ { i } ) ^ { - 1 } \\right] ^ { - 1 } = \\left[ \\frac { 1 } { n ( k - 1 ) } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { k - 1 } \\log \\frac { T _ { k } ( x _ { i } ) } { T _ { j } ( x _ { i } ) } \\right] ^ { - 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 271, + 643, + 723, + 696 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $n$ is the number of samples. We use Equation (2) as our MLE estimator throughout this paper. ", + "bbox": [ + 174, + 700, + 820, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Since the geometry of natural images is complex and unknown, we face two challenges when verifying the accuracy of MLE on natural image datasets. First, we need to choose a proper value of $k$ . As shown by MacKay & Ghahramani (2005), the positive bias of the corrected estimator Equation (2) increases as $k$ increases, but the variance decreases. In order to navigate this bias-variance tradeoff, we try various values of $k$ in Section 4. Second, in addition to the aforementioned local uniformity assumption, MLE assumes that data arises as a sequence of i.i.d. random variables which can be written as a continuous and sufficiently smooth function of a random variable with smooth density, which may or may not be true for natural image datasets. While the truth of these assumptions is unknown on natural images, we verify the accuracy of our MLE estimates in a controlled setting in the following section. ", + "bbox": [ + 174, + 720, + 825, + 861 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We briefly discuss other notable techniques for dimensionality estimation. GeoMLE (Gomtsyan et al., 2019) attempts to account for non-uniformity of density and nonlinearity of manifold using a polynomial regression of standard MLE based on distances to nearest neighbors in different sized neighborhoods. However, GeoMLE chooses to approximate averages of $\\hat { m } _ { k } ( x _ { i } )$ , instead of averaging its reciprocal like Equation (2), resulting in a potentially wrong maximum likelihood estimator. As a result, we find its estimation deviates significantly from expected dimensionalities. TwoNN (Facco et al., 2017) is based on the ratio of the distances to the first and second nearest neighbors. Finally, the approach of (Granata & Carnevale, 2016) considers the distribution of geodesic distances over the data manifold, approximated by distances through kNN graphs, compared to the distribution of distances over hyperspheres of varying dimension. Unlike MLE, our preliminary experiments suggest that these techniques do not provide reasonable estimates for some natural and synthetic images which are key to this work; see Appendix A.5 for further discussion. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 VALIDATING DIMENSION ESTIMATION WITH SYNTHETIC DATA", + "text_level": 1, + "bbox": [ + 176, + 234, + 728, + 251 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Dimensionality estimates are often applied on “simple” manifolds or toy datasets where the dimensionality is known, and so the accuracy of the methods can be validated. Image manifolds, by contrast, are highly complex, may contain many symmetries and modes, and are of unknown dimension. In principle, there is no reason why MLE-based dimensionality estimates cannot be applied to image datasets. However, because we lack knowledge of the exact dimensionality of image datasets, we cannot directly verify that MLE-based dimensionality estimates scale up to the complexity of image structures. ", + "bbox": [ + 174, + 266, + 825, + 364 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "There is an inherent uncertainty in estimating the ID of a given dataset. First, we cannot be sure if the dataset actually resembles a sampling of points on or near a manifold. Second, there are typically no guarantees that the sampling satisfies the conditions assumed by the ID estimators we are using. ", + "bbox": [ + 176, + 371, + 825, + 412 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Towards a principled application of ID estimates in contexts of practical relevance to deep learning, we begin by validating that MLE methods can generate accurate dimensionality estimates for complex image data. We do this by generating synthetic image datasets using generative models for which the intrinsic dimensionality can be upper-bounded a priori. We believe such validations are essential to put recent findings in perspective (Gong et al., 2019; Ansuini et al., 2019). ", + "bbox": [ + 174, + 420, + 823, + 489 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "GAN Images We use the BigGAN variant with 128 latent entries and outputs of size $1 2 8 \\times 1 2 8 \\times 3$ trained on the ImageNet dataset (Deng et al., 2009). Using this GAN, we generate datasets with a varying number of images, where we fix most entries of the latent vectors to zero leaving only $\\bar { d }$ free entries to be chosen at random. As we increase the number of free entries, we expect the intrinsic dimension to increase with $\\bar { d }$ as an upper bound; see Section A.1 for further discussion. ", + "bbox": [ + 174, + 506, + 825, + 574 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In particular, we create several synthetic datasets of varying intrinsic dimensionality using the ImageNet class, basen $\\mathrm { j } \\mathrm { i }$ , and check if the estimates match our expectation. As seen in Figure 2, we observe increasing diversity with increasing intrinsic dimension. In Figure 3, we show convergence of the MLE estimate on basenji data with dimension bounded above by $\\bar { d } = 1 0$ . We observe that the estimates can be sensitive to the choice of $k$ as discussed in prior work; see Appendix A.2 for additional GAN classes. ", + "bbox": [ + 174, + 582, + 825, + 665 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Scaling to large datasets. We develop a practical approach for estimating the ID of large datasets such as ImageNet. In this approach, we randomly select a fraction $\\alpha$ of the dataset as anchors. Then, we evaluate the MLE estimate using only the anchor points, where nearest-neighbors are computed over the entire dataset. Note that, when anchors are chosen randomly, this acceleration has no impact on the expected value of the result. See Appendix A.3 for an evaluation of this approach. ", + "bbox": [ + 174, + 681, + 825, + 751 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5 THE INTRINSIC DIMENSION OF POPULAR DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 771, + 640, + 787 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we measure the intrinsic dimensions of a number of popular datasets including MNIST (Deng, 2012), SVHN (Netzer et al., 2011), CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009), ImageNet (Deng et al., 2009), MS-COCO (Lin et al., 2014), and CelebA (Liu et al., 2015). Using three different parameter settings for the MLE ID estimator, we find that the ID is indeed much smaller than the number of pixels; see Table 5. Notice that the rank order of datasets by dimension does not depend on the choice of $k$ . A comparison of state-of-the-art (SOTA) classification accuracy on each respective dataset1 with the dimension estimates suggests a negative correlation between the intrinsic dimension and test accuracy. In the next section, we take a closer look at this phenomenon through a series of dedicated experiments. ", + "bbox": [ + 174, + 801, + 825, + 900 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4bf96c7b46ed3eb22d58dfee16c9ee0fa8a058f2ef00eeda468f7c786936eafd.jpg", + "image_caption": [ + "Figure 2: Visualization of basenji GAN samples of varying intrinsic dimension. " + ], + "image_footnote": [], + "bbox": [ + 246, + 102, + 753, + 500 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/eef4736776c29aebbfecbdb9cbdf9de49d6f13189d08e2ef3029e8033712ac31.jpg", + "image_caption": [ + "Figure 3: Validation of MLE estimate on synthetic basenji data with $\\bar { d } = 1 0$ free entries. We observe the estimates to converge around the expected dimensionality of 10. Standard errors plotted with $N = 5$ replicates over random samples of the data. " + ], + "image_footnote": [], + "bbox": [ + 225, + 555, + 772, + 707 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 792, + 823, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "6 INTRINSIC DIMENSION AND GENERALIZATION ", + "text_level": 1, + "bbox": [ + 173, + 847, + 596, + 863 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Learning theory work has established that learning a manifold requires a number of samples that grows exponentially with the manifold’s intrinsic dimension (Narayanan & Mitter, 2010), but the required number of samples is independent of the extrinsic dimension. Specifically, the number of ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/ecea9c74b1c03e67b316d1c4a64ca1a03bcbf670ac67f222ef51892661e9ef6b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DatasetMNISTSVHNCIFAR-100CelebACIFAR-10MS-COCOImageNet
MLE (k=3)79119132226
MLE (k=5)11141817213338
MLE (k=10)12182224253743
MLE (k=20)13192326263643
SOTA Accuracy99.8499.0193.51199.37188.55
", + "bbox": [ + 176, + 101, + 818, + 193 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1: The MLE estimates for practical image datasets, and the state-of-the-art test-set image classification accuracy (for classification problems only) for these datasets. ", + "bbox": [ + 174, + 204, + 823, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "samples needed to learn a well-conditioned decision boundary between two classes is exponential in the intrinsic dimension of the manifold on which the classes lie (Narayanan & Niyogi, 2009). ", + "bbox": [ + 173, + 256, + 823, + 285 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We leverage dimension estimation tools to empirically verify these theoretical findings using a family of binary classification problems defined over both synthetic and real datasets of varying intrinsic dimension. In these experiments, we observe a connection between the intrinsic dimension of data and generalization. Specifically, we find that classification problems on data of lower intrinsic dimensionality are easier to solve. ", + "bbox": [ + 174, + 291, + 825, + 361 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.1 SYNTHETIC GAN DATA: SAMPLE COMPLEXITY DEPENDS ON INTRINSIC (NOT EXTRINSIC) DIMENSIONALITY ", + "bbox": [ + 174, + 378, + 750, + 406 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The synthetic GAN data generation technique described in Section 4 provides a unique opportunity to test the relationship between generalization and the intrinsic/extrinsic dimensionality of images. By creating datasets with controlled intrinsic dimensionality, we may compare their sample complexity, that is the number of samples required to obtain a given level of test error. Specifically we test the following two hypotheses (1) data of lower intrinsic dimensionality has lower sample complexity than that of higher intrinsic dimensionality and (2) extrinsic dimensionality is irrelevant for sample complexity. ", + "bbox": [ + 173, + 419, + 825, + 517 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To investigate hypothesis (1), we create four synthetic datasets of varying intrinsic dimensionality: 16, 32, 64, 128, fixed extrinsic dimensionality: $3 \\times 3 2 \\times 3 2$ , and two classes: basenji and beagle. For each dataset we fix a test set of size $N = 1 7 0 0$ . For all experiments, we use the ResNet-18 (width $= 6 4$ ) architecture (He et al., 2016). We then train models until they fit their entire training set with increasing amounts of training samples and measure the test error. We show these results in Figure 4. Observing the varying rates of growth, we see that data of higher intrinsic dimension requires more samples to achieve a given test error. ", + "bbox": [ + 174, + 523, + 825, + 621 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For hypothesis (2), we carry out the same experiment with the roles of intrinsic and extrinsic dimension switched. We create four synthetic datasets of varying extrinsic dimensionality by resizing the images with nearest-neighbor interpolation. Specifically we create 6 datasets of square, 3-channel images of sizes 16, 32, 64, 128, 256, fixed intrinsic dimensionality of size 128, and all other experimental details the same. We show these results in Figure 5. Observing the lack of variable growth rates, we see that extrinsic dimension has little to no effect on sample complexity. ", + "bbox": [ + 174, + 628, + 823, + 712 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To the best of our knowledge, this is the first experimental demonstration that intrinsic but not extrinsic dimensionality matters for the generalization of deep networks. ", + "bbox": [ + 174, + 718, + 823, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.2 REAL DATA: INTRINSIC DIMENSIONALITY MATTERS FOR GENERALIZATION ", + "text_level": 1, + "bbox": [ + 178, + 765, + 735, + 780 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Next, we examine the sample complexity of binary classification tasks from four common image datasets: MNIST, SVHN, CIFAR-10, and ImageNet. This case differs from the synthetic case in that we have no control over each dataset’s intrinsic dimension. Instead, we estimate it via the MLE method discussed in Section 3. To account for variable difficulty of classes, we randomly sample 5 class pairs from each dataset and run the previously described sample complexity experiment. Note that these subsets differ from those used in Table 5, where the estimates are taken from the entire dataset and across all classes. ", + "bbox": [ + 174, + 790, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "On these sampled subsets, we find the MLE estimates as shown in Table 2. Note that these estimates are consistent with expectation, e.g. MNIST is qualitatively simpler then SVHN or CIFAR-10. ", + "bbox": [ + 176, + 895, + 821, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/74fb9e6ad214cfae0b2079ae127d7f16bfbcb8e40925bbd46ffb410f6d03d5a6.jpg", + "image_caption": [ + "Figure 4: Sample complexity of synthetic datasets of varying intrinsic dimensionality. " + ], + "image_footnote": [], + "bbox": [ + 225, + 103, + 771, + 255 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/68d854a2a7dac5de7032cf261ec3d6876730e3c1b40ff24b700172449b97f0a5.jpg", + "image_caption": [ + "Figure 5: Sample complexity of synthetic datasets of varying extrinsic dimensionality. " + ], + "image_footnote": [], + "bbox": [ + 225, + 303, + 771, + 452 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We conduct the same sample complexity experiment as the previous section on the datasets. Because these datasets are ordinarily of varying extrinsic dimensionality, we resize all to size $3 2 \\times 3 2 \\times 3$ (before applying MLE). We report results in Figure 6, where we overall observe trends ordered by intrinsic dimensionality estimate. These results are consistent with expectation of the relative hardness of each dataset. However, there are some notable differences from the synthetic case. Several unexpected cross-over points exist in the low-sample regime, and the gap between SVHN and CIFAR-10 is smaller than one may expect based on their estimated intrinsic dimension. ", + "bbox": [ + 173, + 510, + 825, + 608 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "From these observations we conclude that intrinsic dimensionality is indeed relevant to generalization on real data, but it is not the only feature of data that influences sample complexity. ", + "bbox": [ + 173, + 614, + 821, + 643 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/dc20ee9ad2309b53a2dfb84715aca439b2b80d0d6858d854043171f3629335af.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MNISTSVHNCIFAR-10ImageNet
k=37.5 (0.2)8.5 (0.1)11.4 (0.2)15.4 (0.8)
k=49.8 (0.3)11.6 (0.1)15.9 (0.2)19.8 (0.9)
k=510.9 (0.4)13.2 (0.1)18.3 (0.3)21.6 (1.0)
", + "bbox": [ + 312, + 657, + 684, + 724 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2: Mean and standard error of estimated intrinsic dimensions for practical datasets under the same resolution $3 2 \\times 3 2 \\times 3$ using different $k$ ’s. These results are consistent with the test accuracies on these datasets under the same resolution. ", + "bbox": [ + 176, + 736, + 821, + 773 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.3 REAL DATA: ADDING NOISE CHANGES DIMENSIONALITY TO AFFECT GENERALIZATION ", + "text_level": 1, + "bbox": [ + 171, + 800, + 815, + 814 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we examine an alternative technique for changing the intrinsic dimension of a real dataset: adding noise to images. Here we leverage the fact that uniformly sampled noise in $[ 0 , 1 ] ^ { d }$ has dimension $\\underline d$ . We thus add independent noise, drawn uniformly from a fixed randomly oriented $d _ { \\underline { { \\mathbf { \\Pi } } } }$ -dimensional unit hypercube embedded in pixel space, to each sample in a dataset. This procedure ensures that the dataset has dimension at least $\\underline d$ . Since the natural data we use has low dimension, and the hypercubes have high dimension, this procedure specifically increases dimensionality. We note that estimation error may occur when there is an insufficient number of samples to achieve a proper estimate. Since the variation in images in a dataset may still be dominated by non-noise directions, we expect to underestimate the new increased dimensions of these noised datasets. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/655a510373a3bd2e79ca7346c2749387a47b165b02494ada5c430f1eee984fae.jpg", + "image_caption": [ + "Figure 6: Sample complexity of real datasets. Standard errors are shown $N = 5$ class pairs. " + ], + "image_footnote": [], + "bbox": [ + 225, + 103, + 772, + 255 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 311, + 825, + 340 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Starting with CIFAR-10 data, we add noise of varying dimensions, where we replace pixels at random in the image. We only add noise to an image once to keep the augmented dataset the same size as the original. We use the following noise dimensionalities: 256, 512, 1024, 2048, 2560. The estimated dimensions of the noised datasets are listed in Table 3. We see that intrinsic dimension increases with increasing noise dimensionality, but dimensionality does not saturate to the maximum true dimension, likely due to a poverty of samples. ", + "bbox": [ + 174, + 347, + 825, + 431 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "On these noisy CIFAR-10 datasets, we again carry out the sample complexity experiment of the previous sections. We show results in Figure 7. We observe sample complexity largely in the same order as intrinsic dimension. ", + "bbox": [ + 176, + 438, + 823, + 479 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2111a51cd27b077902523f63bc2c604568fc825deaf1cc6c8efa3a35c6542e81.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
d=256d=512d=1024d=1536d=2048d= 2560
k=319.730.957.177.8110.0136.1
k=425.239.172.8101.3142.1177.7
k=527.642.578.3110.2153.4196.6
", + "bbox": [ + 243, + 491, + 754, + 558 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/c541e44293bb8e715400d216feb8919c55c73fa830ce96423a4162e56e0c35ec.jpg", + "image_caption": [ + "Table 3: Estimated intrinsic dimension of the CIFAR-10 dataset after adding different dimensions $( d )$ of uniformly sampled noise using different $k$ ’s. The estimated dimension consistently increases with $d$ under different $k$ ’s. ", + "Figure 7: Sample complexity of noisy datasets. Standard errors are shown $N = 5$ random subsets of the data. " + ], + "image_footnote": [], + "bbox": [ + 223, + 630, + 771, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": ".4 MANIPULATING THE INTRINSIC DIMENSIONALITY OF FONTS ", + "text_level": 1, + "bbox": [ + 184, + 842, + 640, + 856 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we describe a final technique for studying the effect of intrinsic dimensionality on sample complexity on the recently proposed FONTS dataset (Stutz et al., 2019). Beginning with a collection of characters and font types, termed a prototype set by the authors, FONTS datasets are constructed using a fixed set of data augmentations: scaling, translation, rotation, and sheering. In principle, these augmentations each increase the intrinsic dimension of the prototype set allowing us to synthetically alter the intrinsic dimension by varying the number of augmentations used. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We construct 5 FONTS datasets in this way, FONTS- $\\{ 0 , 1 , 2 , 3 , 4 \\}$ , where the suffix denotes the number of transformations used in the data generation process. The MLE estimates on each of the datasets are given in Table 4. ", + "bbox": [ + 176, + 138, + 823, + 180 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Consistent with expectation, we observe that MLE methods consistently resolve the increased dimesionality of transformed datasets. Carrying out the sample complexity experiment of the previous section, we report results in Figure 8. We observe again that, on the whole, sample complexity is ordered by intrinsic dimension. ", + "bbox": [ + 173, + 188, + 825, + 244 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/42dc6d1a7e3f0a34a215ccd04342179183fe2865a6dd137b54736c907e5f1301.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
FONTS-0FONTS-1FONTS-2FONTS-3FONTS-4
k=31.8 (0.1)3.8 (0.1)5.1 (0.2)5.8 (0.2)6.1 (0.3)
k=42.7 (0.1)5.2 (0.1)6.9 (0.2)7.8 (0.3)8.3 (0.4)
k=53.2 (0.1)5.9 (0.1)7.8 (0.3)8.8 (0.4)9.4 (0.4)
", + "bbox": [ + 272, + 267, + 725, + 334 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 4: Mean and standard errror of MLE estimates on the FONTS dataset under different number of transforms using different $k$ ’s. Again, the ranks of the estimated IDs are consistent, and the estimated IDs increase with the number of transforms. ", + "bbox": [ + 174, + 345, + 823, + 383 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/f914034c6d90137ea05b5cf5b64e93a7f0c6e71a63b2592d064c5597155023b0.jpg", + "image_caption": [ + "Figure 8: Sample complexity of FONTS datasets. " + ], + "image_footnote": [], + "bbox": [ + 225, + 424, + 771, + 574 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 655, + 310, + 671 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we measure the intrinsic dimension of popular image datasets and show that the intrinsic dimension of data matters for deep learning. While there may be many factors, such as class separation and the number of classes, which determine generalization, we build the case that intrinsic dimension is one of these important factors. Along the way, we introduce a technique for using GANs to synthesize data while manipulating dimensionality. This technique is useful not only for validating dimension estimation methods but also for examining the learning behavior of neural networks under a dimension-controlled environment. In addition to synthetic data, we verify that dimension plays a large role in learning on natural data. Our results support the commonly held belief that low dimensional structure underpins the success of deep learning on high-resolution data. ", + "bbox": [ + 173, + 694, + 825, + 819 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "These findings raise a number of salient directions for future work. Methods for enhancing neural network learning on high-dimensional data could improve generalization on hard vision problems. To this end, a deeper understanding of the mechanisms by which dimensionality plays a role in learning may enable such methods. Additionally, our work indicates that different dimensionality estimation tools are better suited for different settings. Future work on computing tighter and more reliable estimates specific to image data would allow the community to more precisely study the relationship between the dimensionality of image datasets and learning. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 367, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported by the DARPA GARD and DARPA QED programs. Further support was provided by the AFOSR MURI program, and the National Science Foundation’s DMS division. Computation resources were funded by the Sloan Foundation. ", + "bbox": [ + 174, + 133, + 825, + 176 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 196, + 285, + 212 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Alessio Ansuini, Alessandro Laio, Jakob H Macke, and Davide Zoccolan. Intrinsic dimension of data representations in deep neural networks. In Advances in Neural Information Processing Systems, pp. 6111–6122, 2019. \nFranz Besold and Vladimir Spokoiny. 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", + "bbox": [ + 176, + 381, + 825, + 424 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Wei Zhu, Qiang Qiu, Jiaji Huang, Robert Calderbank, Guillermo Sapiro, and Ingrid Daubechies. LDMNet: Low dimensional manifold regularized neural networks. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018. ", + "bbox": [ + 174, + 433, + 825, + 476 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A VALIDATION OF ID ESTIMATES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 467, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section, we present additional discussion results and discussion relevant to the ID estimation and related validation experiments in Section 4. ", + "bbox": [ + 173, + 133, + 823, + 161 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 GAN PROPERTIES", + "text_level": 1, + "bbox": [ + 176, + 179, + 343, + 194 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We devise a method for validating ID measurements in a controlled setting using images generated by GANs. To justify this method, we first note that the image of $\\mathbb { R } ^ { d }$ under a locally Lipschitz function can be a manifold with dimension at most $d$ . Then, consider that the BigGAN generator, a convolutional neural network with ReLU activations, is a function with this property (Brock et al., 2018). ", + "bbox": [ + 174, + 205, + 825, + 275 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Specifically, BigGAN can be written as a composition of linear functions, translations, and ReLU activation functions. Individually, these operations do not increase dimension, and by a composition property, their composition cannot increase dimensionality either. The more general fact that the image of $\\mathbb { R } ^ { d }$ under a locally Lipschitz function can be a manifold with dimension at most $d$ follows from Sard’s theorem (Sard, 1942). ", + "bbox": [ + 174, + 282, + 825, + 352 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 CONVERGENCE FOR MORE GAN CLASSES ", + "text_level": 1, + "bbox": [ + 174, + 369, + 516, + 383 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We include additional results on the estimation of ID for synthetic GAN images from various ImageNet classes with $\\bar { d } = 1 0$ free entries out of the 128-dimensional latent vector input to the GAN; see Figures 10 and 11 below. As observed earlier in Section 4, the MLE estimates are sensitive to the choice of $k$ , where we expect the ID to be close to 10 given the way we sample the latent vectors to use for the GAN. We note that for a number of classes, all choices of $k$ we considered seem to underestimate the ID. ", + "bbox": [ + 173, + 395, + 825, + 478 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/63be558951d943d505226e0278503d8d07ffea2e7281ef6f5dbab8c616c45ea1.jpg", + "image_caption": [ + "Figure 9: Validation of MLE estimates on synthetic daisy data with $\\bar { d } = 1 0$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 489, + 820, + 667 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.3 SUBSAMPLING FOR LARGE DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 728, + 482, + 742 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In Figure 12 we validate the anchor approximation on basenji data of dimension 10 for varying anchor ratio $\\alpha$ . Then, in Figure 13 we validate the anchor approximation on tree-frog data with $\\bar { d } = 3 2$ for varying $k$ while fixing the anchor ratio at $\\alpha = 0 . 0 0 1$ . ", + "bbox": [ + 173, + 755, + 826, + 796 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 RELATIONSHIP BETWEEN $k$ IN THE MLE METHOD AND DATASET DIMENSION ", + "text_level": 1, + "bbox": [ + 176, + 814, + 746, + 829 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The choice of $k$ may affect the dimensionality estimates. To better understand this relationship, we conducted additional studies for the MLE method using synthetic GAN data generated as described in Section 4. Table 5 shows the estimation results for various $k$ and basen $\\mathrm { j } \\mathrm { i }$ data with $\\bar { d } \\in$ $\\{ 2 , 4 , 8 , 1 6 , 3 2 , 6 4 , 1 2 8 \\}$ . We use a fixed number of samples $\\mathit { \\Delta } n \\ = \\ 1 0 0 0 0 \\rangle$ ). We observe that the estimated intrinsic dimension increases with $k$ , and large values can yield overestimates for $\\mathrm { I D }$ . These results agree with the work of Levina & Bickel (2005) who suggest that low values of $k$ may produce an estimator with higher variance while higher values of $k$ produce an estimator with higher positive bias. Given that we have access to large amounts of synthetic data, we opt for lower values of $k$ in our work. We do not choose a particular value of $k$ , and we report all experiments with multiple values. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/71d04a891002126df16545d41e277f88bbb28494666cca53e8b86d98bf4b61e5.jpg", + "image_caption": [ + "Figure 10: Validation of MLE estimate on synthetic soap-bubbles data with $\\bar { d } = 1 0$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 820, + 280 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/185a2f8d3955d06c26323c8fdb27a3e473b3404cad17e7f2acb9e1c7f2ce2f1b.jpg", + "image_caption": [ + "Figure 11: Validation of MLE estimate on synthetic coffee data with 10 free entries. Note that the estimates do not converge around the upper bound of $\\bar { d } = 1 0$ , which suggests that data generated from this class is not of full dimension. " + ], + "image_footnote": [], + "bbox": [ + 176, + 338, + 820, + 518 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 612, + 825, + 667 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/ae2e5bdd4f0617662ea5ce28a0211af3afe1e01680a9c531a6c21ab684f0f09c.jpg", + "image_caption": [ + "Figure 12: Validation of anchor approximation on basenji with $\\bar { d } = 1 0$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 708, + 821, + 890 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/409b518eaba102881cac07313ab1903575de99d7b649eebbb568047f0de99121.jpg", + "image_caption": [ + "Figure 13: Validation of anchor approximation on tree-frog with $\\bar { d } = 3 2$ and $\\alpha = 0 . 0 0 1$ ", + "Table 5: Additional results on the relationship between $k$ and MLE dimensionality estimate for synthetic basenji images with varying $\\bar { d }$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 820, + 280 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/808db043f4b64e63dbafc75228f9e42aa1835a2c991fa6af39ef4082c20e85ea.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
kd
248163264128
1.12.66.110.516.020.020.0
34-1.53.68.214.021.026.026.0
51.74.19.315.723.528.728.5
61.84.49.916.624.930.329.9
71.94.610.417.225.831.230.6
81.94.710.717.626.431.731.1
92.04.910.918.026.831.931.5
102.05.011.118.227.132.131.7
152.15.311.618.827.832.331.7
2.25.511.819.027.931.931.3
202.25.712.019.227.931.530.8
", + "bbox": [ + 318, + 325, + 679, + 527 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.5 COMPARING MLE TO OTHER ESTIMATORS IN A CONTROLLED SETTING ", + "text_level": 1, + "bbox": [ + 176, + 597, + 715, + 612 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To validate MLE in comparison to other estimators, we evaluate three other dimensionality estimation methods: GeoMLE (Gomtsyan et al., 2019), TwoNN (Facco et al., 2017) and kNN graph distances (Granata & Carnevale, 2016). For GeoMLE, we sample a total of 20 bootstrap subsets $M = 2 0$ ) and use $k _ { 1 } = 2 0 , k _ { 2 } = 5 5$ as recommended by Gomtsyan et al. (2019). To extend the kNN graph distance method to large datasets, we randomly sample a subset of samples (fixed to 10,000 for datasets with more than 10,000 samples) and use shortest graph distances to their $k$ nearest neighbors to estimate the IDs. Other settings are as default in the implementations of Granata & Carnevale (2016). ", + "bbox": [ + 174, + 623, + 825, + 734 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "First, we validate each method on datasets of uniformly sampled from $d$ -dimensional hypercubes. We report these results in Figure 14. Each method works reasonable on low-dimensional cubes. We observed the Shortest Path method to give erratic estimates on cubes of higher dimension, and have omitted these. TwoNN has poor sample efficiency for higher dimensional cubes. Interestingly, GeoMLE estimates these high dimensional cubes well. ", + "bbox": [ + 174, + 742, + 825, + 811 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Next, we report estimation results on basenji 10 for TwoNN, kNN graph distance, and GeoMLE methods in Figure 15. Comparing against the MLE results in Figure 3, we observe that each other method does not achieve an accurate estimate in this sample regime, thus motivating our focus on MLE. Notably, GeoMLE and TwoNN severely overestimate dimension, while the kNN graph distance method severely underestimates dimension. ", + "bbox": [ + 174, + 819, + 825, + 888 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "On MNIST, CIFAR-10, CIFAR-100, and SVHN, the results of these other estimation methods also deviate from expectation (Table 6). For example, TwoNN assigns a significantly higher dimension estimate to MNIST than to CIFAR-100, which contradicts both intuition and the results of other estimators. We set $k = 4$ and number of bins to 1000, and use default settings for all other parameters including $r _ { \\mathrm { M A X } }$ . ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/0c1cb62ac3d7868f164d4488598093dba23256647324c62368df5b7d307f77b0.jpg", + "image_caption": [ + "Figure 14: The TwoNN, Shortest-Path, and GeoMLE methods on $d$ -dimensional Hypercube data. Each method estimates low-dimensional cubes well, validating their implementation. " + ], + "image_footnote": [], + "bbox": [ + 173, + 160, + 823, + 659 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/2349d247687e0a97bd2f4ed741f1699298911d633950f59813c01f53fded8540.jpg", + "table_caption": [ + "Table 6: Additional ID estimators on popular datasets. " + ], + "table_footnote": [], + "table_body": "
DatasetMNISTCIFAR-10CIFAR-100SVHN
MLE(k = 5)11211814
GeoMLE (k1= 20,k2 = 55)25969321
TwoNN151197
kNN Graph Distance7786
", + "bbox": [ + 253, + 722, + 745, + 801 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/5f2c77c808097691523633c1fe4e358c2efb6ea1af2b6ca13b61731afa91fb44.jpg", + "image_caption": [ + "Figure 15: The TwoNN, Shortest-Path, and GeoMLE methods on basenji 10 data. The estimates do not converge around the expected value of $\\bar { d } = 1 0$ in this sample regime. " + ], + "image_footnote": [], + "bbox": [ + 173, + 244, + 823, + 739 + ], + "page_idx": 16 + } +] \ No newline at end of file diff --git a/parse/train/XJk19XzGq2J/XJk19XzGq2J_middle.json b/parse/train/XJk19XzGq2J/XJk19XzGq2J_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..1296de661284ef12a2454a3c5e79b2811ab17aff --- /dev/null +++ b/parse/train/XJk19XzGq2J/XJk19XzGq2J_middle.json @@ -0,0 +1,34349 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 399, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 399, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 399, + 96 + ], + "score": 1.0, + "content": "THE INTRINSIC DIMENSION OF IMAGES", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 98, + 340, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 98, + 340, + 117 + ], + "score": 1.0, + "content": "AND ITS IMPACT ON LEARNING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 133, + 478, + 180 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 470, + 147 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 198, + 147 + ], + "score": 1.0, + "content": "Phillip Pope1, Chen", + "type": "text" + }, + { + "bbox": [ + 199, + 134, + 222, + 146 + ], + "score": 0.81, + "content": "\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 134, + 470, + 147 + ], + "score": 1.0, + "content": ", Ahmed Abdelkader2, Micah Goldblum1, Tom Goldstein1", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 144, + 408, + 160 + ], + "spans": [ + { + "bbox": [ + 111, + 144, + 408, + 160 + ], + "score": 1.0, + "content": "1Department of Computer Science, University of Maryland, College Park", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 155, + 480, + 172 + ], + "spans": [ + { + "bbox": [ + 111, + 155, + 480, + 172 + ], + "score": 1.0, + "content": "2Oden Institute for Computational Engineering and Sciences, University of Texas at Austin", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 112, + 171, + 399, + 182 + ], + "spans": [ + { + "bbox": [ + 112, + 171, + 399, + 182 + ], + "score": 1.0, + "content": "{pepope,chenzhu}@umd.edu, akader@utexas.edu, {goldblum,tomg}@umd.edu", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 278, + 209, + 333, + 221 + ], + "lines": [ + { + "bbox": [ + 276, + 209, + 335, + 222 + ], + "spans": [ + { + "bbox": [ + 276, + 209, + 335, + 222 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 143, + 232, + 468, + 374 + ], + "lines": [ + { + "bbox": [ + 141, + 233, + 469, + 244 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 244 + ], + "score": 1.0, + "content": "It is widely believed that natural image data exhibits low-dimensional structure", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 255 + ], + "score": 1.0, + "content": "despite the high dimensionality of conventional pixel representations. 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Networks learn complex decision", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "boundaries from small amounts of image data (often just a few hundred or thousand samples per", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "class). At the same time, generative adversarial networks (GANs) are able to learn image “mani-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "folds” from merely a few thousand samples. The seemingly low number of samples needed to learn", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 494, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 494, + 694 + ], + "score": 1.0, + "content": "these manifolds strongly suggests that image datasets have extremely low-dimensional structure.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Despite the established role of low dimensional data in deep learning, little is known about the in-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "trinsic dimension of popular datasets and the impact of dimensionality on the performance of neural", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "networks. 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Additionally, we find that low", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 321 + ], + "score": 1.0, + "content": "dimensional datasets are easier for neural networks to learn, and models solving", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 469, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 469, + 332 + ], + "score": 1.0, + "content": "these tasks generalize better from training to test data. Along the way, we de-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "velop a technique for validating our dimension estimation tools on synthetic data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "score": 1.0, + "content": "generated by GANs allowing us to actively manipulate the intrinsic dimension by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 366 + ], + "score": 1.0, + "content": "controlling the image generation process. Code for our experiments may be found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 165, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 165, + 376 + ], + "score": 1.0, + "content": "here.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 233, + 470, + 376 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 393, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 208, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 208, + 409 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "The idea that real-world data distributions can be described by very few variables underpins ma-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "chine learning research from manifold learning to dimension reduction (Besold & Spokoiny, 2019;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Fodor, 2002). The number of variables needed to describe a data distribution is known as its in-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "trinsic dimension (ID). In applications, such as crystallography, computer graphics, and ecology,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "practitioners depend on data having low intrinsic dimension (Valle & Oganov, 2010; Desbrun et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "2002; Laughlin, 2014). The utility of representations which are low-dimensional has motivated a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "variety of deep learning techniques including autoencoders and regularization methods (Hinton &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 473, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 473, + 507 + ], + "score": 1.0, + "content": "Salakhutdinov, 2006; Vincent et al., 2010; Gonzalez & Balajewicz, 2018; Zhu et al., 2018).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 417, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "It is also known that dimensionality plays a strong role in learning function approximations and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "non-linear class boundaries. The exponential cost of learning in high dimensions is easily captured", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 343, + 546 + ], + "score": 1.0, + "content": "by the trivial case of sampling a function on a cube; in", + "type": "text" + }, + { + "bbox": [ + 344, + 534, + 351, + 543 + ], + "score": 0.74, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "dimensions, sampling only the cube", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 198, + 557 + ], + "score": 1.0, + "content": "vertices would require", + "type": "text" + }, + { + "bbox": [ + 199, + 543, + 210, + 555 + ], + "score": 0.85, + "content": "2 ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "measurements. Similar behaviors emerge in learning theory. It is known", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "that learning a manifold requires a number of samples that grows exponentially with the manifold’s", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "score": 1.0, + "content": "intrinsic dimension (Narayanan & Mitter, 2010). Similarly, the number of samples needed to learn a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "well-conditioned decision boundary between two classes is an exponential function of the intrinsic", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "dimension of the manifold on which the classes lie (Narayanan & Niyogi, 2009). Furthermore,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "these learning bounds have no dependence on the ambient dimension in which manifold-structured", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 160, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 160, + 622 + ], + "score": 1.0, + "content": "datasets live.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 510, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "In light of the exponentially large sample complexity of learning high-dimensional functions, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "ability of neural networks to learn from image data is remarkable. Networks learn complex decision", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "boundaries from small amounts of image data (often just a few hundred or thousand samples per", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "class). At the same time, generative adversarial networks (GANs) are able to learn image “mani-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "folds” from merely a few thousand samples. The seemingly low number of samples needed to learn", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 494, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 494, + 694 + ], + "score": 1.0, + "content": "these manifolds strongly suggests that image datasets have extremely low-dimensional structure.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 626, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Despite the established role of low dimensional data in deep learning, little is known about the in-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "trinsic dimension of popular datasets and the impact of dimensionality on the performance of neural", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "networks. Computational methods for estimating intrinsic dimension enable these measurements.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 699, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 138, + 82, + 473, + 215 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 82, + 473, + 215 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 82, + 473, + 215 + ], + "spans": [ + { + "bbox": [ + 138, + 82, + 473, + 215 + ], + "score": 0.972, + "type": "image", + "image_path": "fcc707e5c48469d7362698f29421033c4b0c499f83ca0bca8273585e5b73ab72.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 138, + 82, + 473, + 126.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 138, + 126.33333333333334, + 473, + 170.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 170.66666666666669, + 473, + 215.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 228, + 504, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "Figure 1: Estimates of the intrinsic dimension of commonly used datasets obtained using the MLE method", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 239, + 462, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 124, + 250 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 125, + 239, + 171, + 249 + ], + "score": 0.6, + "content": "k = 3 , 5 , 1 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 239, + 447, + 250 + ], + "score": 1.0, + "content": ", 20 nearest neighbors (left to right). The trends are consistent using different", + "type": "text" + }, + { + "bbox": [ + 447, + 239, + 453, + 248 + ], + "score": 0.6, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 239, + 462, + 250 + ], + "score": 1.0, + "content": "’s.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 504, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "We adopt tools from the dimension estimation literature to shed light on dimensionality in settings", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 280, + 479, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 479, + 293 + ], + "score": 1.0, + "content": "of interest to the deep learning community. Our contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 132, + 302, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 131, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 131, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "• We verify the reliability of intrinsic dimension estimation on high-dimensional data using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "generative adversarial networks (GANs), a setting in which we can a priori upper-bound", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 325, + 480, + 336 + ], + "spans": [ + { + "bbox": [ + 142, + 325, + 480, + 336 + ], + "score": 1.0, + "content": "the intrinsic dimension of generated data by the dimension of the latent noise vector.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 140, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 140, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "We measure the dimensionality of popular datasets such as MNIST, CIFAR-10, and Im-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "ageNet. In our experiments, we find that natural image datasets whose images contain", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 142, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "thousands of pixels can, in fact, be described by orders of magnitude fewer variables. For", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 142, + 373, + 370, + 385 + ], + "score": 1.0, + "content": "example, we estimate that ImageNet, despite containing", + "type": "text" + }, + { + "bbox": [ + 370, + 372, + 477, + 383 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4 \\times 3 = 1 5 0 5 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "pixels", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 384, + 436, + 395 + ], + "spans": [ + { + "bbox": [ + 141, + 384, + 436, + 395 + ], + "score": 1.0, + "content": "per image, only has intrinsic dimension between 26 and 43; see Figure 1.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 135, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 135, + 403, + 138, + 407 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 138, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "We train classifiers on data, synthetic and real, of various intrinsic dimension and find that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "this variable correlates closely with the number of samples needed for learning. On the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 142, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "other hand, we find that extrinsic dimension, the dimension of the ambient space in which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 432, + 355, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 432, + 355, + 444 + ], + "score": 1.0, + "content": "data is embedded, has little impact on generalization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 108, + 453, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "Together, these results put experimental weight behind the hypothesis that the unintuitively low", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "dimensionality of natural images is being exploited by deep networks, and suggest that a character-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 474, + 481, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 481, + 489 + ], + "score": 1.0, + "content": "ization of this structure is an essential building block for a successful theory of deep learning.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 502, + 211, + 515 + ], + "lines": [ + { + "bbox": [ + 104, + 501, + 213, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 213, + 518 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "While the hypothesis that natural images lie on or near a low-dimensional manifold is controversial,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Goodfellow et al. (2016) argue that the low-dimensional manifold assumption is at least approx-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "imately correct for images, supported by two observations. First, natural images are locally con-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "nected, with each image surrounded by other highly similar images reachable through image trans-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "formations (e.g., contrast, brightness). Second, natural images seem to lie on a low-dimensional", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "structure, as the probability distribution of images is highly concentrated; uniformly sampled pixels", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "can hardly assemble a meaningful image. It is widely believed that the combination of natural scenes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "and sensor properties yields very sparse and concentrated image distributions, as has been supported", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "by several empirical studies on image patches (Lee et al., 2003; Donoho & Grimes, 2005; Carlsson", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "et al., 2008). This observation motivated work on efficient coding (Olshausen & Field, 1996) and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "served as a prior in computer vision (Peyre, 2009). Further, rigorous experiments have been con- ´", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "ducted clearly supporting the low-dimensional manifold hypothesis for many image datasets (Rud-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "erman, 1994; Scholkopf et al., 1998; Roweis & Saul, 2000; Tenenbaum et al., 2000; Brand, 2003); ¨", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 491, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 491, + 684 + ], + "score": 1.0, + "content": "see also (Fefferman et al., 2016) for principled algorithms on verifying the manifold hypothesis.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 108, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "The generalization literature seeks to understand why some models generalize better from training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "data to test data than others. One line of work suggests that the loss landscape geometry explains", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "why neural networks generalize well (Huang et al., 2019). Other generalization work predicts that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "data with low dimension, along with other properties which do not include extrinsic dimension,", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 138, + 82, + 473, + 215 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 82, + 473, + 215 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 82, + 473, + 215 + ], + "spans": [ + { + "bbox": [ + 138, + 82, + 473, + 215 + ], + "score": 0.972, + "type": "image", + "image_path": "fcc707e5c48469d7362698f29421033c4b0c499f83ca0bca8273585e5b73ab72.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 138, + 82, + 473, + 126.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 138, + 126.33333333333334, + 473, + 170.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 138, + 170.66666666666669, + 473, + 215.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 228, + 504, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "Figure 1: Estimates of the intrinsic dimension of commonly used datasets obtained using the MLE method", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 239, + 462, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 124, + 250 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 125, + 239, + 171, + 249 + ], + "score": 0.6, + "content": "k = 3 , 5 , 1 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 239, + 447, + 250 + ], + "score": 1.0, + "content": ", 20 nearest neighbors (left to right). The trends are consistent using different", + "type": "text" + }, + { + "bbox": [ + 447, + 239, + 453, + 248 + ], + "score": 0.6, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 239, + 462, + 250 + ], + "score": 1.0, + "content": "’s.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 504, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "We adopt tools from the dimension estimation literature to shed light on dimensionality in settings", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 280, + 479, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 479, + 293 + ], + "score": 1.0, + "content": "of interest to the deep learning community. Our contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 269, + 505, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 302, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 131, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 131, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "• We verify the reliability of intrinsic dimension estimation on high-dimensional data using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "generative adversarial networks (GANs), a setting in which we can a priori upper-bound", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 325, + 480, + 336 + ], + "spans": [ + { + "bbox": [ + 142, + 325, + 480, + 336 + ], + "score": 1.0, + "content": "the intrinsic dimension of generated data by the dimension of the latent noise vector.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 140, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 140, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "We measure the dimensionality of popular datasets such as MNIST, CIFAR-10, and Im-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "ageNet. In our experiments, we find that natural image datasets whose images contain", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 142, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "thousands of pixels can, in fact, be described by orders of magnitude fewer variables. For", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 142, + 373, + 370, + 385 + ], + "score": 1.0, + "content": "example, we estimate that ImageNet, despite containing", + "type": "text" + }, + { + "bbox": [ + 370, + 372, + 477, + 383 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4 \\times 3 = 1 5 0 5 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "pixels", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 384, + 436, + 395 + ], + "spans": [ + { + "bbox": [ + 141, + 384, + 436, + 395 + ], + "score": 1.0, + "content": "per image, only has intrinsic dimension between 26 and 43; see Figure 1.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 135, + 399, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 135, + 403, + 138, + 407 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 138, + 399, + 506, + 411 + ], + "score": 1.0, + "content": "We train classifiers on data, synthetic and real, of various intrinsic dimension and find that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "this variable correlates closely with the number of samples needed for learning. On the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 142, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "other hand, we find that extrinsic dimension, the dimension of the ambient space in which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 432, + 355, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 432, + 355, + 444 + ], + "score": 1.0, + "content": "data is embedded, has little impact on generalization.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 131, + 301, + 506, + 444 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 453, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "Together, these results put experimental weight behind the hypothesis that the unintuitively low", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "dimensionality of natural images is being exploited by deep networks, and suggest that a character-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 474, + 481, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 481, + 489 + ], + "score": 1.0, + "content": "ization of this structure is an essential building block for a successful theory of deep learning.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 452, + 505, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 502, + 211, + 515 + ], + "lines": [ + { + "bbox": [ + 104, + 501, + 213, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 213, + 518 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "While the hypothesis that natural images lie on or near a low-dimensional manifold is controversial,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Goodfellow et al. (2016) argue that the low-dimensional manifold assumption is at least approx-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "imately correct for images, supported by two observations. First, natural images are locally con-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "nected, with each image surrounded by other highly similar images reachable through image trans-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "formations (e.g., contrast, brightness). Second, natural images seem to lie on a low-dimensional", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "structure, as the probability distribution of images is highly concentrated; uniformly sampled pixels", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "can hardly assemble a meaningful image. It is widely believed that the combination of natural scenes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "and sensor properties yields very sparse and concentrated image distributions, as has been supported", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "by several empirical studies on image patches (Lee et al., 2003; Donoho & Grimes, 2005; Carlsson", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "et al., 2008). This observation motivated work on efficient coding (Olshausen & Field, 1996) and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "served as a prior in computer vision (Peyre, 2009). Further, rigorous experiments have been con- ´", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "ducted clearly supporting the low-dimensional manifold hypothesis for many image datasets (Rud-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "erman, 1994; Scholkopf et al., 1998; Roweis & Saul, 2000; Tenenbaum et al., 2000; Brand, 2003); ¨", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 491, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 491, + 684 + ], + "score": 1.0, + "content": "see also (Fefferman et al., 2016) for principled algorithms on verifying the manifold hypothesis.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 527, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "The generalization literature seeks to understand why some models generalize better from training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "data to test data than others. One line of work suggests that the loss landscape geometry explains", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "why neural networks generalize well (Huang et al., 2019). Other generalization work predicts that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "data with low dimension, along with other properties which do not include extrinsic dimension,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "characterize the generalization difficulty of classification problems (Narayanan & Niyogi, 2009).", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "In the context of deep learning, Gong et al. (2019) found that neural network features are low-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "dimensional. Ansuini et al. (2019) further found that the intrinsic dimension of features decreases", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "in late layers of neural networks and observed interesting trends in the dimension of features in", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "early layers. In contrast to Gong et al. (2019) and Ansuini et al. (2019), who find that the intrinsic", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "dimension of internal representations is inversely correlated with high performance, we study the", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "dimensionality of data and its impact on performance, and we make a similar finding. Zhu et al.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "(2018) proposed a regularizer derived from the intrinsic dimension of images augmented with their", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "corresponding feature vectors. Another line of work in deep learning has found that neural networks", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "score": 1.0, + "content": "rely heavily on textures which are low-dimensional (Geirhos et al., 2018; Brendel & Bethge, 2019).", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "Similarly, some have suggested that natural images can be represented as mixtures of textures which", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 448, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 448, + 215 + ], + "score": 1.0, + "content": "lie on a low-dimensional manifold (Vacher & Coen-Cagli, 2019; Vacher et al., 2020).", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "characterize the generalization difficulty of classification problems (Narayanan & Niyogi, 2009).", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "In the context of deep learning, Gong et al. (2019) found that neural network features are low-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "dimensional. Ansuini et al. (2019) further found that the intrinsic dimension of features decreases", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "in late layers of neural networks and observed interesting trends in the dimension of features in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "early layers. In contrast to Gong et al. (2019) and Ansuini et al. (2019), who find that the intrinsic", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "dimension of internal representations is inversely correlated with high performance, we study the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "dimensionality of data and its impact on performance, and we make a similar finding. Zhu et al.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "(2018) proposed a regularizer derived from the intrinsic dimension of images augmented with their", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "corresponding feature vectors. Another line of work in deep learning has found that neural networks", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "score": 1.0, + "content": "rely heavily on textures which are low-dimensional (Geirhos et al., 2018; Brendel & Bethge, 2019).", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "Similarly, some have suggested that natural images can be represented as mixtures of textures which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 448, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 448, + 215 + ], + "score": 1.0, + "content": "lie on a low-dimensional manifold (Vacher & Coen-Cagli, 2019; Vacher et al., 2020).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 231, + 311, + 245 + ], + "lines": [ + { + "bbox": [ + 104, + 231, + 313, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 313, + 247 + ], + "score": 1.0, + "content": "3 INTRINSIC DIMENSION ESTIMATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 230, + 270 + ], + "score": 1.0, + "content": "Given a set of sample points", + "type": "text" + }, + { + "bbox": [ + 230, + 257, + 272, + 268 + ], + "score": 0.91, + "content": "\\mathcal { P } \\subset \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 256, + 399, + 270 + ], + "score": 1.0, + "content": ", it is common to assume that", + "type": "text" + }, + { + "bbox": [ + 400, + 258, + 409, + 268 + ], + "score": 0.81, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "lies on or near a low-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 197, + 281 + ], + "score": 1.0, + "content": "dimensional manifold", + "type": "text" + }, + { + "bbox": [ + 197, + 268, + 239, + 280 + ], + "score": 0.92, + "content": "\\bar { \\mathcal { M } } \\subseteq \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 267, + 333, + 281 + ], + "score": 1.0, + "content": "of intrinsic dimension", + "type": "text" + }, + { + "bbox": [ + 333, + 268, + 416, + 281 + ], + "score": 0.93, + "content": "\\dim ( { \\mathcal { M } } ) = d \\ll N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 267, + 505, + 281 + ], + "score": 1.0, + "content": ". As a measure of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "degrees of freedom in a dataset, as well as the information content, there is great interest in estimat-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 218, + 303 + ], + "score": 1.0, + "content": "ing the intrinsic dimension", + "type": "text" + }, + { + "bbox": [ + 218, + 291, + 225, + 300 + ], + "score": 0.73, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 291, + 505, + 303 + ], + "score": 1.0, + "content": ". In the remainder of this section, we briefly describe the dimension", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "estimation method we use in this paper; for further information, see (Kim et al., 2019) and references", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 140, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 140, + 325 + ], + "score": 1.0, + "content": "therein.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "score": 1.0, + "content": "One of the main approaches to intrinsic dimension estimation is to examine a neighborhood around", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 375, + 354 + ], + "score": 1.0, + "content": "each point in the dataset, and compute the Euclidean distance to the", + "type": "text" + }, + { + "bbox": [ + 375, + 340, + 390, + 351 + ], + "score": 0.89, + "content": "k ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 338, + 506, + 354 + ], + "score": 1.0, + "content": "nearest neighbor. Assuming", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 476, + 364 + ], + "score": 1.0, + "content": "that density is constant within small neighborhoods, the Maximum Likelihood Estimation", + "type": "text" + }, + { + "bbox": [ + 476, + 352, + 504, + 362 + ], + "score": 0.26, + "content": "( M L E )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 147, + 375 + ], + "score": 1.0, + "content": "of Levina", + "type": "text" + }, + { + "bbox": [ + 147, + 363, + 157, + 372 + ], + "score": 0.3, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "Bickel (2005) uses a Poisson process to model the number of points found by random", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "sampling within a given radius around each sample point. 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We use Equation (2) as our MLE estimator throughout this paper.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Since the geometry of natural images is complex and unknown, we face two challenges when verify-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 481, + 595 + ], + "score": 1.0, + "content": "ing the accuracy of MLE on natural image datasets. First, we need to choose a proper value of", + "type": "text" + }, + { + "bbox": [ + 481, + 585, + 487, + 593 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 583, + 505, + 595 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "shown by MacKay & Ghahramani (2005), the positive bias of the corrected estimator Equation (2)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 156, + 617 + ], + "score": 1.0, + "content": "increases as", + "type": "text" + }, + { + "bbox": [ + 156, + 605, + 163, + 615 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "increases, but the variance decreases. In order to navigate this bias-variance tradeoff,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 205, + 628 + ], + "score": 1.0, + "content": "we try various values of", + "type": "text" + }, + { + "bbox": [ + 205, + 617, + 212, + 626 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "in Section 4. Second, in addition to the aforementioned local uniformity", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "assumption, MLE assumes that data arises as a sequence of i.i.d. random variables which can be", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "written as a continuous and sufficiently smooth function of a random variable with smooth density,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "which may or may not be true for natural image datasets. While the truth of these assumptions is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "unknown on natural images, we verify the accuracy of our MLE estimates in a controlled setting in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 670, + 195, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 195, + 684 + ], + "score": 1.0, + "content": "the following section.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "We briefly discuss other notable techniques for dimensionality estimation. 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However, GeoMLE chooses to approximate averages of", + "type": "text" + }, + { + "bbox": [ + 426, + 721, + 458, + 732 + ], + "score": 0.92, + "content": "\\hat { m } _ { k } ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 721, + 506, + 734 + ], + "score": 1.0, + "content": ", instead of", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 214 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 105, + 82, + 506, + 215 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 231, + 311, + 245 + ], + "lines": [ + { + "bbox": [ + 104, + 231, + 313, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 313, + 247 + ], + "score": 1.0, + "content": "3 INTRINSIC DIMENSION ESTIMATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 230, + 270 + ], + "score": 1.0, + "content": "Given a set of sample points", + "type": "text" + }, + { + "bbox": [ + 230, + 257, + 272, + 268 + ], + "score": 0.91, + "content": "\\mathcal { P } \\subset \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 256, + 399, + 270 + ], + "score": 1.0, + "content": ", it is common to assume that", + "type": "text" + }, + { + "bbox": [ + 400, + 258, + 409, + 268 + ], + "score": 0.81, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "lies on or near a low-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 197, + 281 + ], + "score": 1.0, + "content": "dimensional manifold", + "type": "text" + }, + { + "bbox": [ + 197, + 268, + 239, + 280 + ], + "score": 0.92, + "content": "\\bar { \\mathcal { M } } \\subseteq \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 267, + 333, + 281 + ], + "score": 1.0, + "content": "of intrinsic dimension", + "type": "text" + }, + { + "bbox": [ + 333, + 268, + 416, + 281 + ], + "score": 0.93, + "content": "\\dim ( { \\mathcal { M } } ) = d \\ll N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 267, + 505, + 281 + ], + "score": 1.0, + "content": ". As a measure of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "degrees of freedom in a dataset, as well as the information content, there is great interest in estimat-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 218, + 303 + ], + "score": 1.0, + "content": "ing the intrinsic dimension", + "type": "text" + }, + { + "bbox": [ + 218, + 291, + 225, + 300 + ], + "score": 0.73, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 291, + 505, + 303 + ], + "score": 1.0, + "content": ". In the remainder of this section, we briefly describe the dimension", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "estimation method we use in this paper; for further information, see (Kim et al., 2019) and references", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 140, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 140, + 325 + ], + "score": 1.0, + "content": "therein.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 256, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "score": 1.0, + "content": "One of the main approaches to intrinsic dimension estimation is to examine a neighborhood around", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 338, + 375, + 354 + ], + "score": 1.0, + "content": "each point in the dataset, and compute the Euclidean distance to the", + "type": "text" + }, + { + "bbox": [ + 375, + 340, + 390, + 351 + ], + "score": 0.89, + "content": "k ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 338, + 506, + 354 + ], + "score": 1.0, + "content": "nearest neighbor. Assuming", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 476, + 364 + ], + "score": 1.0, + "content": "that density is constant within small neighborhoods, the Maximum Likelihood Estimation", + "type": "text" + }, + { + "bbox": [ + 476, + 352, + 504, + 362 + ], + "score": 0.26, + "content": "( M L E )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 147, + 375 + ], + "score": 1.0, + "content": "of Levina", + "type": "text" + }, + { + "bbox": [ + 147, + 363, + 157, + 372 + ], + "score": 0.3, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "Bickel (2005) uses a Poisson process to model the number of points found by random", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "sampling within a given radius around each sample point. 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Levina & Bickel", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 173, + 461, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 173, + 461, + 475, + 490 + ], + "score": 1.0, + "content": "to average the local estimates at each point to obtain a global estimate . 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We use Equation (2) as our MLE estimator throughout this paper.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 552, + 505, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Since the geometry of natural images is complex and unknown, we face two challenges when verify-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 481, + 595 + ], + "score": 1.0, + "content": "ing the accuracy of MLE on natural image datasets. First, we need to choose a proper value of", + "type": "text" + }, + { + "bbox": [ + 481, + 585, + 487, + 593 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 583, + 505, + 595 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "shown by MacKay & Ghahramani (2005), the positive bias of the corrected estimator Equation (2)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 156, + 617 + ], + "score": 1.0, + "content": "increases as", + "type": "text" + }, + { + "bbox": [ + 156, + 605, + 163, + 615 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "increases, but the variance decreases. 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Second, in addition to the aforementioned local uniformity", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "assumption, MLE assumes that data arises as a sequence of i.i.d. random variables which can be", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "written as a continuous and sufficiently smooth function of a random variable with smooth density,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "which may or may not be true for natural image datasets. While the truth of these assumptions is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "unknown on natural images, we verify the accuracy of our MLE estimates in a controlled setting in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 670, + 195, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 195, + 684 + ], + "score": 1.0, + "content": "the following section.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 572, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "We briefly discuss other notable techniques for dimensionality estimation. 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However, GeoMLE chooses to approximate averages of", + "type": "text" + }, + { + "bbox": [ + 426, + 721, + 458, + 732 + ], + "score": 0.92, + "content": "\\hat { m } _ { k } ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 721, + 506, + 734 + ], + "score": 1.0, + "content": ", instead of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "averaging its reciprocal like Equation (2), resulting in a potentially wrong maximum likelihood es-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "timator. As a result, we find its estimation deviates significantly from expected dimensionalities.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 116 + ], + "score": 1.0, + "content": "TwoNN (Facco et al., 2017) is based on the ratio of the distances to the first and second near-", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "est neighbors. Finally, the approach of (Granata & Carnevale, 2016) considers the distribution of", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "geodesic distances over the data manifold, approximated by distances through kNN graphs, com-", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "pared to the distribution of distances over hyperspheres of varying dimension. Unlike MLE, our", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "preliminary experiments suggest that these techniques do not provide reasonable estimates for some", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 498, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 498, + 172 + ], + "score": 1.0, + "content": "natural and synthetic images which are key to this work; see Appendix A.5 for further discussion.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "averaging its reciprocal like Equation (2), resulting in a potentially wrong maximum likelihood es-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "timator. As a result, we find its estimation deviates significantly from expected dimensionalities.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 116 + ], + "score": 1.0, + "content": "TwoNN (Facco et al., 2017) is based on the ratio of the distances to the first and second near-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "est neighbors. Finally, the approach of (Granata & Carnevale, 2016) considers the distribution of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "geodesic distances over the data manifold, approximated by distances through kNN graphs, com-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "pared to the distribution of distances over hyperspheres of varying dimension. Unlike MLE, our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "preliminary experiments suggest that these techniques do not provide reasonable estimates for some", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 498, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 498, + 172 + ], + "score": 1.0, + "content": "natural and synthetic images which are key to this work; see Appendix A.5 for further discussion.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 186, + 446, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 448, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 448, + 200 + ], + "score": 1.0, + "content": "4 VALIDATING DIMENSION ESTIMATION WITH SYNTHETIC DATA", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "Dimensionality estimates are often applied on “simple” manifolds or toy datasets where the dimen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "sionality is known, and so the accuracy of the methods can be validated. Image manifolds, by con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "trast, are highly complex, may contain many symmetries and modes, and are of unknown dimension.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "In principle, there is no reason why MLE-based dimensionality estimates cannot be applied to image", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "datasets. However, because we lack knowledge of the exact dimensionality of image datasets, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "cannot directly verify that MLE-based dimensionality estimates scale up to the complexity of image", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 279, + 150, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 150, + 289 + ], + "score": 1.0, + "content": "structures.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 294, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 506, + 306 + ], + "score": 1.0, + "content": "There is an inherent uncertainty in estimating the ID of a given dataset. First, we cannot be sure if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "score": 1.0, + "content": "the dataset actually resembles a sampling of points on or near a manifold. Second, there are typically", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 504, + 330 + ], + "score": 1.0, + "content": "no guarantees that the sampling satisfies the conditions assumed by the ID estimators we are using.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "Towards a principled application of ID estimates in contexts of practical relevance to deep learn-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "ing, we begin by validating that MLE methods can generate accurate dimensionality estimates for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "complex image data. We do this by generating synthetic image datasets using generative models for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "which the intrinsic dimensionality can be upper-bounded a priori. We believe such validations are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 451, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 451, + 390 + ], + "score": 1.0, + "content": "essential to put recent findings in perspective (Gong et al., 2019; Ansuini et al., 2019).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 449, + 413 + ], + "score": 1.0, + "content": "GAN Images We use the BigGAN variant with 128 latent entries and outputs of size", + "type": "text" + }, + { + "bbox": [ + 450, + 401, + 504, + 411 + ], + "score": 0.91, + "content": "1 2 8 \\times 1 2 8 \\times 3", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "trained on the ImageNet dataset (Deng et al., 2009). Using this GAN, we generate datasets with a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 479, + 435 + ], + "score": 1.0, + "content": "varying number of images, where we fix most entries of the latent vectors to zero leaving only", + "type": "text" + }, + { + "bbox": [ + 480, + 422, + 487, + 432 + ], + "score": 0.82, + "content": "\\bar { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "free", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 434, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 445 + ], + "score": 1.0, + "content": "entries to be chosen at random. As we increase the number of free entries, we expect the intrinsic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 444, + 456, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 216, + 456 + ], + "score": 1.0, + "content": "dimension to increase with", + "type": "text" + }, + { + "bbox": [ + 216, + 444, + 223, + 454 + ], + "score": 0.83, + "content": "\\bar { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 444, + 456, + 456 + ], + "score": 1.0, + "content": "as an upper bound; see Section A.1 for further discussion.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "In particular, we create several synthetic datasets of varying intrinsic dimensionality using the Ima-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 189, + 486 + ], + "score": 1.0, + "content": "geNet class, basen", + "type": "text" + }, + { + "bbox": [ + 189, + 473, + 200, + 484 + ], + "score": 0.33, + "content": "\\mathrm { j } \\mathrm { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 472, + 506, + 486 + ], + "score": 1.0, + "content": ", and check if the estimates match our expectation. As seen in Figure 2, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "observe increasing diversity with increasing intrinsic dimension. In Figure 3, we show convergence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 405, + 506 + ], + "score": 1.0, + "content": "of the MLE estimate on basenji data with dimension bounded above by", + "type": "text" + }, + { + "bbox": [ + 405, + 493, + 435, + 505 + ], + "score": 0.9, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 495, + 505, + 506 + ], + "score": 1.0, + "content": ". We observe that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 294, + 517 + ], + "score": 1.0, + "content": "the estimates can be sensitive to the choice of", + "type": "text" + }, + { + "bbox": [ + 295, + 506, + 302, + 515 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "as discussed in prior work; see Appendix A.2 for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 206, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 206, + 528 + ], + "score": 1.0, + "content": "additional GAN classes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Scaling to large datasets. We develop a practical approach for estimating the ID of large datasets", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 369, + 563 + ], + "score": 1.0, + "content": "such as ImageNet. In this approach, we randomly select a fraction", + "type": "text" + }, + { + "bbox": [ + 370, + 553, + 377, + 561 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "of the dataset as anchors. Then,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "we evaluate the MLE estimate using only the anchor points, where nearest-neighbors are computed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "over the entire dataset. Note that, when anchors are chosen randomly, this acceleration has no impact", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 584, + 462, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 462, + 596 + ], + "score": 1.0, + "content": "on the expected value of the result. See Appendix A.3 for an evaluation of this approach.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 611, + 392, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 394, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 394, + 626 + ], + "score": 1.0, + "content": "5 THE INTRINSIC DIMENSION OF POPULAR DATASETS", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "score": 1.0, + "content": "In this section, we measure the intrinsic dimensions of a number of popular datasets including", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "MNIST (Deng, 2012), SVHN (Netzer et al., 2011), CIFAR-10 and CIFAR-100 (Krizhevsky et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 658, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 670 + ], + "score": 1.0, + "content": "2009), ImageNet (Deng et al., 2009), MS-COCO (Lin et al., 2014), and CelebA (Liu et al., 2015).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "Using three different parameter settings for the MLE ID estimator, we find that the ID is indeed", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "much smaller than the number of pixels; see Table 5. Notice that the rank order of datasets by di-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 275, + 703 + ], + "score": 1.0, + "content": "mension does not depend on the choice of", + "type": "text" + }, + { + "bbox": [ + 276, + 691, + 282, + 701 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 691, + 505, + 703 + ], + "score": 1.0, + "content": ". A comparison of state-of-the-art (SOTA) classification", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "score": 1.0, + "content": "accuracy on each respective dataset1 with the dimension estimates suggests a negative correlation", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 722, + 452, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 454, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 454, + 734 + ], + "score": 1.0, + "content": "1Values from https://paperswithcode.com/task/image-classification.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 172 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 186, + 446, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 448, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 448, + 200 + ], + "score": 1.0, + "content": "4 VALIDATING DIMENSION ESTIMATION WITH SYNTHETIC DATA", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "Dimensionality estimates are often applied on “simple” manifolds or toy datasets where the dimen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "sionality is known, and so the accuracy of the methods can be validated. Image manifolds, by con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "trast, are highly complex, may contain many symmetries and modes, and are of unknown dimension.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "In principle, there is no reason why MLE-based dimensionality estimates cannot be applied to image", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "datasets. However, because we lack knowledge of the exact dimensionality of image datasets, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "cannot directly verify that MLE-based dimensionality estimates scale up to the complexity of image", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 279, + 150, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 150, + 289 + ], + "score": 1.0, + "content": "structures.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 212, + 505, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 294, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 506, + 306 + ], + "score": 1.0, + "content": "There is an inherent uncertainty in estimating the ID of a given dataset. First, we cannot be sure if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "score": 1.0, + "content": "the dataset actually resembles a sampling of points on or near a manifold. Second, there are typically", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 504, + 330 + ], + "score": 1.0, + "content": "no guarantees that the sampling satisfies the conditions assumed by the ID estimators we are using.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 294, + 506, + 330 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "Towards a principled application of ID estimates in contexts of practical relevance to deep learn-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "ing, we begin by validating that MLE methods can generate accurate dimensionality estimates for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "complex image data. We do this by generating synthetic image datasets using generative models for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "which the intrinsic dimensionality can be upper-bounded a priori. We believe such validations are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 451, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 451, + 390 + ], + "score": 1.0, + "content": "essential to put recent findings in perspective (Gong et al., 2019; Ansuini et al., 2019).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 333, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 449, + 413 + ], + "score": 1.0, + "content": "GAN Images We use the BigGAN variant with 128 latent entries and outputs of size", + "type": "text" + }, + { + "bbox": [ + 450, + 401, + 504, + 411 + ], + "score": 0.91, + "content": "1 2 8 \\times 1 2 8 \\times 3", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "trained on the ImageNet dataset (Deng et al., 2009). Using this GAN, we generate datasets with a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 479, + 435 + ], + "score": 1.0, + "content": "varying number of images, where we fix most entries of the latent vectors to zero leaving only", + "type": "text" + }, + { + "bbox": [ + 480, + 422, + 487, + 432 + ], + "score": 0.82, + "content": "\\bar { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "free", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 434, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 445 + ], + "score": 1.0, + "content": "entries to be chosen at random. As we increase the number of free entries, we expect the intrinsic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 444, + 456, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 216, + 456 + ], + "score": 1.0, + "content": "dimension to increase with", + "type": "text" + }, + { + "bbox": [ + 216, + 444, + 223, + 454 + ], + "score": 0.83, + "content": "\\bar { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 444, + 456, + 456 + ], + "score": 1.0, + "content": "as an upper bound; see Section A.1 for further discussion.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 401, + 505, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "In particular, we create several synthetic datasets of varying intrinsic dimensionality using the Ima-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 189, + 486 + ], + "score": 1.0, + "content": "geNet class, basen", + "type": "text" + }, + { + "bbox": [ + 189, + 473, + 200, + 484 + ], + "score": 0.33, + "content": "\\mathrm { j } \\mathrm { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 472, + 506, + 486 + ], + "score": 1.0, + "content": ", and check if the estimates match our expectation. As seen in Figure 2, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "observe increasing diversity with increasing intrinsic dimension. In Figure 3, we show convergence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 405, + 506 + ], + "score": 1.0, + "content": "of the MLE estimate on basenji data with dimension bounded above by", + "type": "text" + }, + { + "bbox": [ + 405, + 493, + 435, + 505 + ], + "score": 0.9, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 495, + 505, + 506 + ], + "score": 1.0, + "content": ". We observe that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 294, + 517 + ], + "score": 1.0, + "content": "the estimates can be sensitive to the choice of", + "type": "text" + }, + { + "bbox": [ + 295, + 506, + 302, + 515 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "as discussed in prior work; see Appendix A.2 for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 206, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 206, + 528 + ], + "score": 1.0, + "content": "additional GAN classes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 461, + 506, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Scaling to large datasets. We develop a practical approach for estimating the ID of large datasets", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 369, + 563 + ], + "score": 1.0, + "content": "such as ImageNet. In this approach, we randomly select a fraction", + "type": "text" + }, + { + "bbox": [ + 370, + 553, + 377, + 561 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "of the dataset as anchors. Then,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "we evaluate the MLE estimate using only the anchor points, where nearest-neighbors are computed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "over the entire dataset. Note that, when anchors are chosen randomly, this acceleration has no impact", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 584, + 462, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 462, + 596 + ], + "score": 1.0, + "content": "on the expected value of the result. See Appendix A.3 for an evaluation of this approach.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 539, + 506, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 611, + 392, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 394, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 394, + 626 + ], + "score": 1.0, + "content": "5 THE INTRINSIC DIMENSION OF POPULAR DATASETS", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "score": 1.0, + "content": "In this section, we measure the intrinsic dimensions of a number of popular datasets including", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "MNIST (Deng, 2012), SVHN (Netzer et al., 2011), CIFAR-10 and CIFAR-100 (Krizhevsky et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 658, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 670 + ], + "score": 1.0, + "content": "2009), ImageNet (Deng et al., 2009), MS-COCO (Lin et al., 2014), and CelebA (Liu et al., 2015).", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "Using three different parameter settings for the MLE ID estimator, we find that the ID is indeed", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "much smaller than the number of pixels; see Table 5. Notice that the rank order of datasets by di-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 691, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 275, + 703 + ], + "score": 1.0, + "content": "mension does not depend on the choice of", + "type": "text" + }, + { + "bbox": [ + 276, + 691, + 282, + 701 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 691, + 505, + 703 + ], + "score": 1.0, + "content": ". A comparison of state-of-the-art (SOTA) classification", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "score": 1.0, + "content": "accuracy on each respective dataset1 with the dimension estimates suggests a negative correlation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "between the intrinsic dimension and test accuracy. In the next section, we take a closer look at this", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 638, + 331, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 331, + 651 + ], + "score": 1.0, + "content": "phenomenon through a series of dedicated experiments.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 634, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 151, + 81, + 461, + 396 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 151, + 81, + 461, + 396 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 151, + 81, + 461, + 396 + ], + "spans": [ + { + "bbox": [ + 151, + 81, + 461, + 396 + ], + "score": 0.976, + "type": "image", + "image_path": "4bf96c7b46ed3eb22d58dfee16c9ee0fa8a058f2ef00eeda468f7c786936eafd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 151, + 81, + 461, + 186.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 151, + 186.0, + 461, + 291.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 151, + 291.0, + 461, + 396.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 151, + 412, + 457, + 424 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 150, + 411, + 458, + 425 + ], + "spans": [ + { + "bbox": [ + 150, + 411, + 458, + 425 + ], + "score": 1.0, + "content": "Figure 2: Visualization of basenji GAN samples of varying intrinsic dimension.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "image", + "bbox": [ + 138, + 440, + 473, + 560 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 440, + 473, + 560 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 138, + 440, + 473, + 560 + ], + "spans": [ + { + "bbox": [ + 138, + 440, + 473, + 560 + ], + "score": 0.971, + "type": "image", + "image_path": "eef4736776c29aebbfecbdb9cbdf9de49d6f13189d08e2ef3029e8033712ac31.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 138, + 440, + 473, + 480.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 138, + 480.0, + 473, + 520.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 138, + 520.0, + 473, + 560.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 572, + 505, + 604 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 373, + 584 + ], + "score": 1.0, + "content": "Figure 3: Validation of MLE estimate on synthetic basenji data with", + "type": "text" + }, + { + "bbox": [ + 373, + 572, + 401, + 582 + ], + "score": 0.91, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "free entries. We observe the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 441, + 595 + ], + "score": 1.0, + "content": "estimates to converge around the expected dimensionality of 10. Standard errors plotted with", + "type": "text" + }, + { + "bbox": [ + 441, + 583, + 468, + 592 + ], + "score": 0.9, + "content": "N = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "replicates", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 593, + 227, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 227, + 604 + ], + "score": 1.0, + "content": "over random samples of the data.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 105, + 628, + 504, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "between the intrinsic dimension and test accuracy. In the next section, we take a closer look at this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 638, + 331, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 331, + 651 + ], + "score": 1.0, + "content": "phenomenon through a series of dedicated experiments.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 106, + 671, + 365, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 366, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 366, + 686 + ], + "score": 1.0, + "content": "6 INTRINSIC DIMENSION AND GENERALIZATION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Learning theory work has established that learning a manifold requires a number of samples that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "grows exponentially with the manifold’s intrinsic dimension (Narayanan & Mitter, 2010), but the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "required number of samples is independent of the extrinsic dimension. 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DatasetMNISTSVHNCIFAR-100CelebACIFAR-10MS-COCOImageNet
MLE (k=3)79119132226
MLE (k=5)11141817213338
MLE (k=10)12182224253743
MLE (k=20)13192326263643
SOTA Accuracy99.8499.0193.51199.37188.55
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In these experiments, we observe a connection between the intrinsic dimension of data", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "and generalization. Specifically, we find that classification problems on data of lower intrinsic di-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 276, + 237, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 237, + 288 + ], + "score": 1.0, + "content": "mensionality are easier to solve.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 459, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 460, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 460, + 313 + ], + "score": 1.0, + "content": "6.1 SYNTHETIC GAN DATA: SAMPLE COMPLEXITY DEPENDS ON INTRINSIC (NOT", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 130, + 312, + 259, + 324 + ], + "spans": [ + { + "bbox": [ + 130, + 312, + 259, + 324 + ], + "score": 1.0, + "content": "EXTRINSIC) DIMENSIONALITY", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 346 + ], + "score": 1.0, + "content": "The synthetic GAN data generation technique described in Section 4 provides a unique opportunity", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "score": 1.0, + "content": "to test the relationship between generalization and the intrinsic/extrinsic dimensionality of images.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "By creating datasets with controlled intrinsic dimensionality, we may compare their sample com-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "plexity, that is the number of samples required to obtain a given level of test error. Specifically we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "test the following two hypotheses (1) data of lower intrinsic dimensionality has lower sample com-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "plexity than that of higher intrinsic dimensionality and (2) extrinsic dimensionality is irrelevant for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 398, + 186, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 186, + 411 + ], + "score": 1.0, + "content": "sample complexity.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "To investigate hypothesis (1), we create four synthetic datasets of varying intrinsic dimensionality:", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 291, + 439 + ], + "score": 1.0, + "content": "16, 32, 64, 128, fixed extrinsic dimensionality:", + "type": "text" + }, + { + "bbox": [ + 291, + 426, + 334, + 437 + ], + "score": 0.9, + "content": "3 \\times 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 426, + 505, + 439 + ], + "score": 1.0, + "content": ", and two classes: basenji and beagle.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 270, + 450 + ], + "score": 1.0, + "content": "For each dataset we fix a test set of size", + "type": "text" + }, + { + "bbox": [ + 270, + 438, + 315, + 447 + ], + "score": 0.9, + "content": "N = 1 7 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 436, + 505, + 450 + ], + "score": 1.0, + "content": ". For all experiments, we use the ResNet-18", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 134, + 461 + ], + "score": 1.0, + "content": "(width", + "type": "text" + }, + { + "bbox": [ + 135, + 448, + 159, + 459 + ], + "score": 0.79, + "content": "= 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 447, + 506, + 461 + ], + "score": 1.0, + "content": ") architecture (He et al., 2016). We then train models until they fit their entire training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "set with increasing amounts of training samples and measure the test error. We show these results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "in Figure 4. Observing the varying rates of growth, we see that data of higher intrinsic dimension", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 313, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 313, + 495 + ], + "score": 1.0, + "content": "requires more samples to achieve a given test error.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 504, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "For hypothesis (2), we carry out the same experiment with the roles of intrinsic and extrinsic dimen-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "sion switched. We create four synthetic datasets of varying extrinsic dimensionality by resizing the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "images with nearest-neighbor interpolation. Specifically we create 6 datasets of square, 3-channel", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "images of sizes 16, 32, 64, 128, 256, fixed intrinsic dimensionality of size 128, and all other experi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "mental details the same. We show these results in Figure 5. Observing the lack of variable growth", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 552, + 434, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 434, + 565 + ], + "score": 1.0, + "content": "rates, we see that extrinsic dimension has little to no effect on sample complexity.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "To the best of our knowledge, this is the first experimental demonstration that intrinsic but not", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 397, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 397, + 593 + ], + "score": 1.0, + "content": "extrinsic dimensionality matters for the generalization of deep networks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 109, + 606, + 450, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 452, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 452, + 618 + ], + "score": 1.0, + "content": "6.2 REAL DATA: INTRINSIC DIMENSIONALITY MATTERS FOR GENERALIZATION", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Next, we examine the sample complexity of binary classification tasks from four common image", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "datasets: MNIST, SVHN, CIFAR-10, and ImageNet. This case differs from the synthetic case in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "that we have no control over each dataset’s intrinsic dimension. Instead, we estimate it via the MLE", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "method discussed in Section 3. To account for variable difficulty of classes, we randomly sample 5", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "class pairs from each dataset and run the previously described sample complexity experiment. Note", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 682, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 693 + ], + "score": 1.0, + "content": "that these subsets differ from those used in Table 5, where the estimates are taken from the entire", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 693, + 227, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 227, + 704 + ], + "score": 1.0, + "content": "dataset and across all classes.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "On these sampled subsets, we find the MLE estimates as shown in Table 2. Note that these estimates", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 486, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 486, + 732 + ], + "score": 1.0, + "content": "are consistent with expectation, e.g. MNIST is qualitatively simpler then SVHN or CIFAR-10.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 80, + 501, + 153 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 80, + 501, + 153 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 501, + 153 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 501, + 153 + ], + "score": 0.979, + "html": "
DatasetMNISTSVHNCIFAR-100CelebACIFAR-10MS-COCOImageNet
MLE (k=3)79119132226
MLE (k=5)11141817213338
MLE (k=10)12182224253743
MLE (k=20)13192326263643
SOTA Accuracy99.8499.0193.51199.37188.55
", + "type": "table", + "image_path": "ecea9c74b1c03e67b316d1c4a64ca1a03bcbf670ac67f222ef51892661e9ef6b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 501, + 104.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 104.33333333333333, + 501, + 128.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 128.66666666666666, + 501, + 153.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 504, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "Table 1: The MLE estimates for practical image datasets, and the state-of-the-art test-set image classification", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 172, + 329, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 329, + 183 + ], + "score": 1.0, + "content": "accuracy (for classification problems only) for these datasets.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 161, + 505, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 504, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "samples needed to learn a well-conditioned decision boundary between two classes is exponential", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 214, + 490, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 490, + 227 + ], + "score": 1.0, + "content": "in the intrinsic dimension of the manifold on which the classes lie (Narayanan & Niyogi, 2009).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 203, + 505, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 231, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "We leverage dimension estimation tools to empirically verify these theoretical findings using a fam-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "score": 1.0, + "content": "ily of binary classification problems defined over both synthetic and real datasets of varying intrinsic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "dimension. In these experiments, we observe a connection between the intrinsic dimension of data", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "and generalization. Specifically, we find that classification problems on data of lower intrinsic di-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 276, + 237, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 237, + 288 + ], + "score": 1.0, + "content": "mensionality are easier to solve.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 231, + 505, + 288 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 459, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 460, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 460, + 313 + ], + "score": 1.0, + "content": "6.1 SYNTHETIC GAN DATA: SAMPLE COMPLEXITY DEPENDS ON INTRINSIC (NOT", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 130, + 312, + 259, + 324 + ], + "spans": [ + { + "bbox": [ + 130, + 312, + 259, + 324 + ], + "score": 1.0, + "content": "EXTRINSIC) DIMENSIONALITY", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 300, + 460, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 346 + ], + "score": 1.0, + "content": "The synthetic GAN data generation technique described in Section 4 provides a unique opportunity", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "score": 1.0, + "content": "to test the relationship between generalization and the intrinsic/extrinsic dimensionality of images.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "By creating datasets with controlled intrinsic dimensionality, we may compare their sample com-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "plexity, that is the number of samples required to obtain a given level of test error. Specifically we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "test the following two hypotheses (1) data of lower intrinsic dimensionality has lower sample com-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 401 + ], + "score": 1.0, + "content": "plexity than that of higher intrinsic dimensionality and (2) extrinsic dimensionality is irrelevant for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 398, + 186, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 186, + 411 + ], + "score": 1.0, + "content": "sample complexity.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 331, + 506, + 411 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "To investigate hypothesis (1), we create four synthetic datasets of varying intrinsic dimensionality:", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 291, + 439 + ], + "score": 1.0, + "content": "16, 32, 64, 128, fixed extrinsic dimensionality:", + "type": "text" + }, + { + "bbox": [ + 291, + 426, + 334, + 437 + ], + "score": 0.9, + "content": "3 \\times 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 426, + 505, + 439 + ], + "score": 1.0, + "content": ", and two classes: basenji and beagle.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 270, + 450 + ], + "score": 1.0, + "content": "For each dataset we fix a test set of size", + "type": "text" + }, + { + "bbox": [ + 270, + 438, + 315, + 447 + ], + "score": 0.9, + "content": "N = 1 7 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 436, + 505, + 450 + ], + "score": 1.0, + "content": ". For all experiments, we use the ResNet-18", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 134, + 461 + ], + "score": 1.0, + "content": "(width", + "type": "text" + }, + { + "bbox": [ + 135, + 448, + 159, + 459 + ], + "score": 0.79, + "content": "= 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 447, + 506, + 461 + ], + "score": 1.0, + "content": ") architecture (He et al., 2016). We then train models until they fit their entire training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "set with increasing amounts of training samples and measure the test error. We show these results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "in Figure 4. Observing the varying rates of growth, we see that data of higher intrinsic dimension", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 313, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 313, + 495 + ], + "score": 1.0, + "content": "requires more samples to achieve a given test error.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 414, + 506, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 504, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "For hypothesis (2), we carry out the same experiment with the roles of intrinsic and extrinsic dimen-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "sion switched. We create four synthetic datasets of varying extrinsic dimensionality by resizing the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "images with nearest-neighbor interpolation. Specifically we create 6 datasets of square, 3-channel", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "images of sizes 16, 32, 64, 128, 256, fixed intrinsic dimensionality of size 128, and all other experi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "mental details the same. We show these results in Figure 5. Observing the lack of variable growth", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 552, + 434, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 434, + 565 + ], + "score": 1.0, + "content": "rates, we see that extrinsic dimension has little to no effect on sample complexity.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 497, + 506, + 565 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "To the best of our knowledge, this is the first experimental demonstration that intrinsic but not", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 397, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 397, + 593 + ], + "score": 1.0, + "content": "extrinsic dimensionality matters for the generalization of deep networks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 569, + 506, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 606, + 450, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 452, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 452, + 618 + ], + "score": 1.0, + "content": "6.2 REAL DATA: INTRINSIC DIMENSIONALITY MATTERS FOR GENERALIZATION", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Next, we examine the sample complexity of binary classification tasks from four common image", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "datasets: MNIST, SVHN, CIFAR-10, and ImageNet. This case differs from the synthetic case in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "that we have no control over each dataset’s intrinsic dimension. Instead, we estimate it via the MLE", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "method discussed in Section 3. To account for variable difficulty of classes, we randomly sample 5", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "class pairs from each dataset and run the previously described sample complexity experiment. Note", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 682, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 693 + ], + "score": 1.0, + "content": "that these subsets differ from those used in Table 5, where the estimates are taken from the entire", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 693, + 227, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 227, + 704 + ], + "score": 1.0, + "content": "dataset and across all classes.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 626, + 506, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "On these sampled subsets, we find the MLE estimates as shown in Table 2. 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Because", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 415, + 504, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 451, + 428 + ], + "score": 1.0, + "content": "these datasets are ordinarily of varying extrinsic dimensionality, we resize all to size", + "type": "text" + }, + { + "bbox": [ + 451, + 415, + 504, + 426 + ], + "score": 0.9, + "content": "3 2 \\times 3 2 \\times 3", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "(before applying MLE). We report results in Figure 6, where we overall observe trends ordered", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "by intrinsic dimensionality estimate. 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These results are consistent with the test accuracies on these datasets", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 603, + 203, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 203, + 614 + ], + "score": 1.0, + "content": "under the same resolution.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 105, + 634, + 499, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 500, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 500, + 645 + ], + "score": 1.0, + "content": "6.3 REAL DATA: ADDING NOISE CHANGES DIMENSIONALITY TO AFFECT GENERALIZATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this section, we examine an alternative technique for changing the intrinsic dimension of a real", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 478, + 678 + ], + "score": 1.0, + "content": "dataset: adding noise to images. 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MNISTSVHNCIFAR-10ImageNet
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These results are consistent with the test accuracies on these datasets", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 603, + 203, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 203, + 614 + ], + "score": 1.0, + "content": "under the same resolution.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 582, + 505, + 614 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 634, + 499, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 500, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 500, + 645 + ], + "score": 1.0, + "content": "6.3 REAL DATA: ADDING NOISE CHANGES DIMENSIONALITY TO AFFECT GENERALIZATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this section, we examine an alternative technique for changing the intrinsic dimension of a real", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 665, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 478, + 678 + ], + "score": 1.0, + "content": "dataset: adding noise to images. Here we leverage the fact that uniformly sampled noise in", + "type": "text" + }, + { + "bbox": [ + 478, + 665, + 504, + 678 + ], + "score": 0.9, + "content": "[ 0 , 1 ] ^ { d }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 166, + 689 + ], + "score": 1.0, + "content": "has dimension", + "type": "text" + }, + { + "bbox": [ + 167, + 677, + 173, + 688 + ], + "score": 0.75, + "content": "\\underline d", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 677, + 505, + 689 + ], + "score": 1.0, + "content": ". We thus add independent noise, drawn uniformly from a fixed randomly oriented", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 113, + 699 + ], + "score": 0.83, + "content": "d _ { \\underline { { \\mathbf { \\Pi } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "-dimensional unit hypercube embedded in pixel space, to each sample in a dataset. This procedure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 293, + 711 + ], + "score": 1.0, + "content": "ensures that the dataset has dimension at least", + "type": "text" + }, + { + "bbox": [ + 294, + 700, + 300, + 710 + ], + "score": 0.79, + "content": "\\underline d", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". Since the natural data we use has low dimension,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "and the hypercubes have high dimension, this procedure specifically increases dimensionality. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "note that estimation error may occur when there is an insufficient number of samples to achieve", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "a proper estimate. 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We only add noise to an image once to keep the augmented dataset the same", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "size as the original. We use the following noise dimensionalities: 256, 512, 1024, 2048, 2560. The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "estimated dimensions of the noised datasets are listed in Table 3. We see that intrinsic dimension in-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "score": 1.0, + "content": "creases with increasing noise dimensionality, but dimensionality does not saturate to the maximum", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 331, + 311, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 311, + 343 + ], + "score": 1.0, + "content": "true dimension, likely due to a poverty of samples.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 347, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "On these noisy CIFAR-10 datasets, we again carry out the sample complexity experiment of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "previous sections. 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Standard errors are shown", + "type": "text" + }, + { + "bbox": [ + 379, + 633, + 405, + 642 + ], + "score": 0.89, + "content": "N = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "random subsets of the data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 113, + 667, + 392, + 678 + ], + "lines": [ + { + "bbox": [ + 111, + 667, + 394, + 679 + ], + "spans": [ + { + "bbox": [ + 111, + 667, + 394, + 679 + ], + "score": 1.0, + "content": ".4 MANIPULATING THE INTRINSIC DIMENSIONALITY OF FONTS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "In this section, we describe a final technique for studying the effect of intrinsic dimensionality on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "sample complexity on the recently proposed FONTS dataset (Stutz et al., 2019). Beginning with a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "collection of characters and font types, termed a prototype set by the authors, FONTS datasets are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "constructed using a fixed set of data augmentations: scaling, translation, rotation, and sheering. 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We show results in Figure 7. We observe sample complexity largely in the same", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 370, + 222, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 222, + 381 + ], + "score": 1.0, + "content": "order as intrinsic dimension.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 347, + 505, + 381 + ] + }, + { + "type": "table", + "bbox": [ + 149, + 389, + 462, + 442 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 149, + 389, + 462, + 442 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 389, + 462, + 442 + ], + "spans": [ + { + "bbox": [ + 149, + 389, + 462, + 442 + ], + "score": 0.969, + "html": "
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The estimated dimension consistently increases with", + "type": "text" + }, + { + "bbox": [ + 474, + 462, + 480, + 470 + ], + "score": 0.71, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 462, + 505, + 472 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 470, + 156, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 139, + 483 + ], + "score": 1.0, + "content": "different", + "type": "text" + }, + { + "bbox": [ + 140, + 472, + 146, + 480 + ], + "score": 0.6, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 470, + 156, + 483 + ], + "score": 1.0, + "content": "’s.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "image_body", + "bbox": [ + 137, + 499, + 472, + 620 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 137, + 499, + 472, + 620 + ], + "spans": [ + { + "bbox": [ + 137, + 499, + 472, + 620 + ], + "score": 0.971, + "type": "image", + "image_path": "c541e44293bb8e715400d216feb8919c55c73fa830ce96423a4162e56e0c35ec.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 137, + 499, + 472, + 539.3333333333334 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 137, + 539.3333333333334, + 472, + 579.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 137, + 579.6666666666667, + 472, + 620.0000000000001 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 633, + 504, + 644 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 378, + 645 + ], + "score": 1.0, + "content": "Figure 7: Sample complexity of noisy datasets. Standard errors are shown", + "type": "text" + }, + { + "bbox": [ + 379, + 633, + 405, + 642 + ], + "score": 0.89, + "content": "N = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "random subsets of the data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 113, + 667, + 392, + 678 + ], + "lines": [ + { + "bbox": [ + 111, + 667, + 394, + 679 + ], + "spans": [ + { + "bbox": [ + 111, + 667, + 394, + 679 + ], + "score": 1.0, + "content": ".4 MANIPULATING THE INTRINSIC DIMENSIONALITY OF FONTS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "In this section, we describe a final technique for studying the effect of intrinsic dimensionality on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "sample complexity on the recently proposed FONTS dataset (Stutz et al., 2019). Beginning with a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "collection of characters and font types, termed a prototype set by the authors, FONTS datasets are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "constructed using a fixed set of data augmentations: scaling, translation, rotation, and sheering. In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "principle, these augmentations each increase the intrinsic dimension of the prototype set allowing us", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 472, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 472, + 107 + ], + "score": 1.0, + "content": "to synthetically alter the intrinsic dimension by varying the number of augmentations used.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "principle, these augmentations each increase the intrinsic dimension of the prototype set allowing us", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 472, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 472, + 107 + ], + "score": 1.0, + "content": "to synthetically alter the intrinsic dimension by varying the number of augmentations used.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 108, + 110, + 504, + 143 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 328, + 123 + ], + "score": 1.0, + "content": "We construct 5 FONTS datasets in this way, FONTS-", + "type": "text" + }, + { + "bbox": [ + 329, + 110, + 381, + 123 + ], + "score": 0.9, + "content": "\\{ 0 , 1 , 2 , 3 , 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 110, + 505, + 123 + ], + "score": 1.0, + "content": ", where the suffix denotes the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "number of transformations used in the data generation process. The MLE estimates on each of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 133, + 224, + 144 + ], + "spans": [ + { + "bbox": [ + 107, + 133, + 224, + 144 + ], + "score": 1.0, + "content": "datasets are given in Table 4.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "Consistent with expectation, we observe that MLE methods consistently resolve the increased dime-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "sionality of transformed datasets. 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While there may be many factors, such as class", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "separation and the number of classes, which determine generalization, we build the case that in-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "trinsic dimension is one of these important factors. Along the way, we introduce a technique for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "using GANs to synthesize data while manipulating dimensionality. This technique is useful not only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "for validating dimension estimation methods but also for examining the learning behavior of neural", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "networks under a dimension-controlled environment. In addition to synthetic data, we verify that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "dimension plays a large role in learning on natural data. Our results support the commonly held", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "belief that low dimensional structure underpins the success of deep learning on high-resolution data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "These findings raise a number of salient directions for future work. 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FONTS-0FONTS-1FONTS-2FONTS-3FONTS-4
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k=42.7 (0.1)5.2 (0.1)6.9 (0.2)7.8 (0.3)8.3 (0.4)
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Future work on computing tighter and more", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "reliable estimates specific to image data would allow the community to more precisely study the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 394, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 394, + 734 + ], + "score": 1.0, + "content": "relationship between the dimensionality of image datasets and learning.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 655, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 225, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "spans": [ + { + "bbox": [ + 107, + 80, + 226, + 96 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "This work was supported by the DARPA GARD and DARPA QED programs. Further support was", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "provided by the AFOSR MURI program, and the National Science Foundation’s DMS division.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 356, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 356, + 141 + ], + "score": 1.0, + "content": "Computation resources were funded by the Sloan Foundation.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 156, + 175, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 176, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 176, + 171 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 161, + 507, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 187 + ], + "score": 1.0, + "content": "Alessio Ansuini, Alessandro Laio, Jakob H Macke, and Davide Zoccolan. 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In The IEEE Conference on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 116, + 364, + 368, + 379 + ], + "spans": [ + { + "bbox": [ + 116, + 364, + 368, + 379 + ], + "score": 1.0, + "content": "Computer Vision and Pattern Recognition (CVPR), June 2018.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 343, + 505, + 379 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 286, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 289, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 289, + 96 + ], + "score": 1.0, + "content": "A VALIDATION OF ID ESTIMATES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "In this section, we present additional discussion results and discussion relevant to the ID estimation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 299, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 299, + 129 + ], + "score": 1.0, + "content": "and related validation experiments in Section 4.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 210, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 212, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 212, + 155 + ], + "score": 1.0, + "content": "A.1 GAN PROPERTIES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "We devise a method for validating ID measurements in a controlled setting using images generated", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 384, + 186 + ], + "score": 1.0, + "content": "by GANs. To justify this method, we first note that the image of", + "type": "text" + }, + { + "bbox": [ + 384, + 173, + 397, + 184 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "under a locally Lipschitz", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 507, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 312, + 199 + ], + "score": 1.0, + "content": "function can be a manifold with dimension at most", + "type": "text" + }, + { + "bbox": [ + 312, + 186, + 319, + 195 + ], + "score": 0.6, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 183, + 507, + 199 + ], + "score": 1.0, + "content": ". Then, consider that the BigGAN generator, a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "convolutional neural network with ReLU activations, is a function with this property (Brock et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 205, + 136, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 136, + 221 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 224, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "Specifically, BigGAN can be written as a composition of linear functions, translations, and ReLU", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "activation functions. Individually, these operations do not increase dimension, and by a composition", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "property, their composition cannot increase dimensionality either. The more general fact that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 144, + 269 + ], + "score": 1.0, + "content": "image of", + "type": "text" + }, + { + "bbox": [ + 144, + 256, + 157, + 267 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 256, + 465, + 269 + ], + "score": 1.0, + "content": "under a locally Lipschitz function can be a manifold with dimension at most", + "type": "text" + }, + { + "bbox": [ + 465, + 257, + 472, + 267 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 256, + 506, + 269 + ], + "score": 1.0, + "content": "follows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 244, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 244, + 279 + ], + "score": 1.0, + "content": "from Sard’s theorem (Sard, 1942).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 293, + 316, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 317, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 317, + 306 + ], + "score": 1.0, + "content": "A.2 CONVERGENCE FOR MORE GAN CLASSES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "We include additional results on the estimation of ID for synthetic GAN images from various Ima-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 184, + 338 + ], + "score": 1.0, + "content": "geNet classes with", + "type": "text" + }, + { + "bbox": [ + 185, + 324, + 216, + 335 + ], + "score": 0.91, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "free entries out of the 128-dimensional latent vector input to the GAN;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "see Figures 10 and 11 below. As observed earlier in Section 4, the MLE estimates are sensitive to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 160, + 358 + ], + "score": 1.0, + "content": "the choice of", + "type": "text" + }, + { + "bbox": [ + 160, + 347, + 167, + 357 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 346, + 505, + 358 + ], + "score": 1.0, + "content": ", where we expect the ID to be close to 10 given the way we sample the latent vectors", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 402, + 370 + ], + "score": 1.0, + "content": "to use for the GAN. We note that for a number of classes, all choices of", + "type": "text" + }, + { + "bbox": [ + 402, + 358, + 409, + 367 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "we considered seem to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 369, + 194, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 194, + 380 + ], + "score": 1.0, + "content": "underestimate the ID.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "image", + "bbox": [ + 108, + 388, + 502, + 529 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 388, + 502, + 529 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 388, + 502, + 529 + ], + "spans": [ + { + "bbox": [ + 108, + 388, + 502, + 529 + ], + "score": 0.971, + "type": "image", + "image_path": "63be558951d943d505226e0278503d8d07ffea2e7281ef6f5dbab8c616c45ea1.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 108, + 388, + 502, + 435.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 108, + 435.0, + 502, + 482.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 108, + 482.0, + 502, + 529.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 161, + 544, + 450, + 556 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 160, + 543, + 450, + 556 + ], + "spans": [ + { + "bbox": [ + 160, + 543, + 419, + 556 + ], + "score": 1.0, + "content": "Figure 9: Validation of MLE estimates on synthetic daisy data with", + "type": "text" + }, + { + "bbox": [ + 420, + 543, + 447, + 554 + ], + "score": 0.89, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 543, + 450, + 556 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "index": 23.0 + }, + { + "type": "title", + "bbox": [ + 107, + 577, + 295, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 297, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 297, + 590 + ], + "score": 1.0, + "content": "A.3 SUBSAMPLING FOR LARGE DATASETS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 506, + 631 + ], + "lines": [ + { + "bbox": [ + 104, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "In Figure 12 we validate the anchor approximation on basenji data of dimension 10 for varying", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 156, + 621 + ], + "score": 1.0, + "content": "anchor ratio", + "type": "text" + }, + { + "bbox": [ + 157, + 610, + 164, + 619 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 609, + 505, + 621 + ], + "score": 1.0, + "content": ". Then, in Figure 13 we validate the anchor approximation on tree-frog data with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 618, + 365, + 632 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 136, + 630 + ], + "score": 0.91, + "content": "\\bar { d } = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 618, + 184, + 632 + ], + "score": 1.0, + "content": "for varying", + "type": "text" + }, + { + "bbox": [ + 184, + 621, + 191, + 630 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 618, + 317, + 632 + ], + "score": 1.0, + "content": "while fixing the anchor ratio at", + "type": "text" + }, + { + "bbox": [ + 317, + 621, + 361, + 630 + ], + "score": 0.89, + "content": "\\alpha = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 618, + 365, + 632 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 645, + 457, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 459, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 242, + 657 + ], + "score": 1.0, + "content": "A.4 RELATIONSHIP BETWEEN", + "type": "text" + }, + { + "bbox": [ + 243, + 646, + 249, + 655 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 645, + 459, + 657 + ], + "score": 1.0, + "content": "IN THE MLE METHOD AND DATASET DIMENSION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 164, + 678 + ], + "score": 1.0, + "content": "The choice of", + "type": "text" + }, + { + "bbox": [ + 164, + 666, + 171, + 676 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "may affect the dimensionality estimates. To better understand this relationship, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "conducted additional studies for the MLE method using synthetic GAN data generated as described", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 369, + 700 + ], + "score": 1.0, + "content": "in Section 4. Table 5 shows the estimation results for various", + "type": "text" + }, + { + "bbox": [ + 370, + 688, + 377, + 698 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 687, + 428, + 700 + ], + "score": 1.0, + "content": "and basen", + "type": "text" + }, + { + "bbox": [ + 428, + 689, + 441, + 699 + ], + "score": 0.3, + "content": "\\mathrm { j } \\mathrm { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 687, + 485, + 700 + ], + "score": 1.0, + "content": "data with", + "type": "text" + }, + { + "bbox": [ + 486, + 687, + 505, + 699 + ], + "score": 0.87, + "content": "\\bar { d } \\in", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 204, + 711 + ], + "score": 0.87, + "content": "\\{ 2 , 4 , 8 , 1 6 , 3 2 , 6 4 , 1 2 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 699, + 360, + 712 + ], + "score": 1.0, + "content": ". We use a fixed number of samples", + "type": "text" + }, + { + "bbox": [ + 360, + 699, + 412, + 709 + ], + "score": 0.86, + "content": "\\mathit { \\Delta } n \\ = \\ 1 0 0 0 0 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "). We observe that the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 291, + 721 + ], + "score": 1.0, + "content": "estimated intrinsic dimension increases with", + "type": "text" + }, + { + "bbox": [ + 292, + 710, + 299, + 720 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 709, + 489, + 721 + ], + "score": 1.0, + "content": ", and large values can yield overestimates for", + "type": "text" + }, + { + "bbox": [ + 489, + 710, + 501, + 720 + ], + "score": 0.27, + "content": "\\mathrm { I D }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 709, + 505, + 721 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 477, + 733 + ], + "score": 1.0, + "content": "These results agree with the work of Levina & Bickel (2005) who suggest that low values of", + "type": "text" + }, + { + "bbox": [ + 478, + 721, + 484, + 730 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "may", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 286, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 289, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 289, + 96 + ], + "score": 1.0, + "content": "A VALIDATION OF ID ESTIMATES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "In this section, we present additional discussion results and discussion relevant to the ID estimation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 299, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 299, + 129 + ], + "score": 1.0, + "content": "and related validation experiments in Section 4.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 106, + 505, + 129 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 210, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 212, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 212, + 155 + ], + "score": 1.0, + "content": "A.1 GAN PROPERTIES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "We devise a method for validating ID measurements in a controlled setting using images generated", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 384, + 186 + ], + "score": 1.0, + "content": "by GANs. To justify this method, we first note that the image of", + "type": "text" + }, + { + "bbox": [ + 384, + 173, + 397, + 184 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "under a locally Lipschitz", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 507, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 312, + 199 + ], + "score": 1.0, + "content": "function can be a manifold with dimension at most", + "type": "text" + }, + { + "bbox": [ + 312, + 186, + 319, + 195 + ], + "score": 0.6, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 183, + 507, + 199 + ], + "score": 1.0, + "content": ". Then, consider that the BigGAN generator, a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "convolutional neural network with ReLU activations, is a function with this property (Brock et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 205, + 136, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 136, + 221 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 162, + 507, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 224, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "Specifically, BigGAN can be written as a composition of linear functions, translations, and ReLU", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "activation functions. Individually, these operations do not increase dimension, and by a composition", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "property, their composition cannot increase dimensionality either. The more general fact that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 144, + 269 + ], + "score": 1.0, + "content": "image of", + "type": "text" + }, + { + "bbox": [ + 144, + 256, + 157, + 267 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 256, + 465, + 269 + ], + "score": 1.0, + "content": "under a locally Lipschitz function can be a manifold with dimension at most", + "type": "text" + }, + { + "bbox": [ + 465, + 257, + 472, + 267 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 256, + 506, + 269 + ], + "score": 1.0, + "content": "follows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 244, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 244, + 279 + ], + "score": 1.0, + "content": "from Sard’s theorem (Sard, 1942).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 223, + 506, + 279 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 293, + 316, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 317, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 317, + 306 + ], + "score": 1.0, + "content": "A.2 CONVERGENCE FOR MORE GAN CLASSES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "We include additional results on the estimation of ID for synthetic GAN images from various Ima-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 184, + 338 + ], + "score": 1.0, + "content": "geNet classes with", + "type": "text" + }, + { + "bbox": [ + 185, + 324, + 216, + 335 + ], + "score": 0.91, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "free entries out of the 128-dimensional latent vector input to the GAN;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "see Figures 10 and 11 below. As observed earlier in Section 4, the MLE estimates are sensitive to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 160, + 358 + ], + "score": 1.0, + "content": "the choice of", + "type": "text" + }, + { + "bbox": [ + 160, + 347, + 167, + 357 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 346, + 505, + 358 + ], + "score": 1.0, + "content": ", where we expect the ID to be close to 10 given the way we sample the latent vectors", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 402, + 370 + ], + "score": 1.0, + "content": "to use for the GAN. We note that for a number of classes, all choices of", + "type": "text" + }, + { + "bbox": [ + 402, + 358, + 409, + 367 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "we considered seem to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 369, + 194, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 194, + 380 + ], + "score": 1.0, + "content": "underestimate the ID.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 312, + 506, + 380 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 388, + 502, + 529 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 388, + 502, + 529 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 388, + 502, + 529 + ], + "spans": [ + { + "bbox": [ + 108, + 388, + 502, + 529 + ], + "score": 0.971, + "type": "image", + "image_path": "63be558951d943d505226e0278503d8d07ffea2e7281ef6f5dbab8c616c45ea1.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 108, + 388, + 502, + 435.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 108, + 435.0, + 502, + 482.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 108, + 482.0, + 502, + 529.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 161, + 544, + 450, + 556 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 160, + 543, + 450, + 556 + ], + "spans": [ + { + "bbox": [ + 160, + 543, + 419, + 556 + ], + "score": 1.0, + "content": "Figure 9: Validation of MLE estimates on synthetic daisy data with", + "type": "text" + }, + { + "bbox": [ + 420, + 543, + 447, + 554 + ], + "score": 0.89, + "content": "\\bar { d } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 543, + 450, + 556 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + } + ], + "index": 23.0 + }, + { + "type": "title", + "bbox": [ + 107, + 577, + 295, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 297, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 297, + 590 + ], + "score": 1.0, + "content": "A.3 SUBSAMPLING FOR LARGE DATASETS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 506, + 631 + ], + "lines": [ + { + "bbox": [ + 104, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 104, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "In Figure 12 we validate the anchor approximation on basenji data of dimension 10 for varying", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 156, + 621 + ], + "score": 1.0, + "content": "anchor ratio", + "type": "text" + }, + { + "bbox": [ + 157, + 610, + 164, + 619 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 609, + 505, + 621 + ], + "score": 1.0, + "content": ". 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To better understand this relationship, we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "conducted additional studies for the MLE method using synthetic GAN data generated as described", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 369, + 700 + ], + "score": 1.0, + "content": "in Section 4. Table 5 shows the estimation results for various", + "type": "text" + }, + { + "bbox": [ + 370, + 688, + 377, + 698 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 687, + 428, + 700 + ], + "score": 1.0, + "content": "and basen", + "type": "text" + }, + { + "bbox": [ + 428, + 689, + 441, + 699 + ], + "score": 0.3, + "content": "\\mathrm { j } \\mathrm { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 687, + 485, + 700 + ], + "score": 1.0, + "content": "data with", + "type": "text" + }, + { + "bbox": [ + 486, + 687, + 505, + 699 + ], + "score": 0.87, + "content": "\\bar { d } \\in", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 204, + 711 + ], + "score": 0.87, + "content": "\\{ 2 , 4 , 8 , 1 6 , 3 2 , 6 4 , 1 2 8 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 699, + 360, + 712 + ], + "score": 1.0, + "content": ". We use a fixed number of samples", + "type": "text" + }, + { + "bbox": [ + 360, + 699, + 412, + 709 + ], + "score": 0.86, + "content": "\\mathit { \\Delta } n \\ = \\ 1 0 0 0 0 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "). 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Each method works reasonable on low-dimensional cubes.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "We observed the Shortest Path method to give erratic estimates on cubes of higher dimension, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "have omitted these. TwoNN has poor sample efficiency for higher dimensional cubes. 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Each method works reasonable on low-dimensional cubes.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "We observed the Shortest Path method to give erratic estimates on cubes of higher dimension, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 634 + ], + "score": 1.0, + "content": "have omitted these. TwoNN has poor sample efficiency for higher dimensional cubes. 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Comparing against the MLE results in Figure 3, we observe that each other", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "method does not achieve an accurate estimate in this sample regime, thus motivating our focus", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "on MLE. 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For example, TwoNN assigns a significantly higher dimension", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "estimate to MNIST than to CIFAR-100, which contradicts both intuition and the results of other", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 180, + 106 + ], + "score": 1.0, + "content": "estimators. We set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 181, + 94, + 206, + 104 + ], + "score": 0.89, + "content": "k = 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 207, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and number of bins to 1000, and use default settings for all other parameters", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 174, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 146, + 118 + ], + "score": 1.0, + "content": "including", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 146, + 106, + 169, + 116 + ], + "score": 0.48, + "content": "r _ { \\mathrm { M A X } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 169, + 103, + 174, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "estimate to MNIST than to CIFAR-100, which contradicts both intuition and the results of other", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 180, + 106 + ], + "score": 1.0, + "content": "estimators. We set", + "type": "text" + }, + { + "bbox": [ + 181, + 94, + 206, + 104 + ], + "score": 0.89, + "content": "k = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and number of bins to 1000, and use default settings for all other parameters", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 174, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 146, + 118 + ], + "score": 1.0, + "content": "including", + "type": "text" + }, + { + "bbox": [ + 146, + 106, + 169, + 116 + ], + "score": 0.48, + "content": "r _ { \\mathrm { M A X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 103, + 174, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 106, + 127, + 504, + 522 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 127, + 504, + 522 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 127, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 504, + 522 + ], + "score": 0.941, + "type": "image", + "image_path": "0c1cb62ac3d7868f164d4488598093dba23256647324c62368df5b7d307f77b0.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 106, + 127, + 504, + 258.66666666666663 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 106, + 258.66666666666663, + 504, + 390.33333333333326 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 106, + 390.33333333333326, + 504, + 521.9999999999999 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 533, + 504, + 557 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 376, + 547 + ], + "score": 1.0, + "content": "Figure 14: The TwoNN, Shortest-Path, and GeoMLE methods on", + "type": "text" + }, + { + "bbox": [ + 377, + 534, + 383, + 544 + ], + "score": 0.75, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "-dimensional Hypercube data.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 544, + 447, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 447, + 558 + ], + "score": 1.0, + "content": "Each method estimates low-dimensional cubes well, validating their implementation.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + } + ], + "index": 5.25 + }, + { + "type": "table", + "bbox": [ + 155, + 572, + 456, + 635 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 155, + 572, + 456, + 635 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 572, + 456, + 635 + ], + "spans": [ + { + "bbox": [ + 155, + 572, + 456, + 635 + ], + "score": 0.969, + "html": "
DatasetMNISTCIFAR-10CIFAR-100SVHN
MLE(k = 5)11211814
GeoMLE (k1= 20,k2 = 55)25969321
TwoNN151197
kNN Graph Distance7786
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DatasetMNISTCIFAR-10CIFAR-100SVHN
MLE(k = 5)11211814
GeoMLE (k1= 20,k2 = 55)25969321
TwoNN151197
kNN Graph Distance7786
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DatasetMNISTSVHNCIFAR-100CelebACIFAR-10MS-COCOImageNet
MLE (k=3)79119132226
MLE (k=5)11141817213338
MLE (k=10)12182224253743
MLE (k=20)13192326263643
SOTA Accuracy99.8499.0193.51199.37188.55
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DatasetMNISTCIFAR-10CIFAR-100SVHN
MLE(k = 5)11211814
GeoMLE (k1= 20,k2 = 55)25969321
TwoNN151197
kNN Graph Distance7786
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Bajor, Thomas A. Lasko +Department of Biomedical Informatics +Vanderbilt University School of Medicine +Nashville, TN 37203, USA +{jacek.m.bajor,tom.lasko}@vanderbilt.edu + +# ABSTRACT + +It is a surprising fact that electronic medical records are failing at one of their primary purposes, that of tracking the set of medications that the patient is actively taking. Studies estimate that up to $50 \%$ of such lists omit active drugs, and that up to $2 5 \%$ of all active medications do not appear on the appropriate patient list. Manual efforts to maintain these lists involve a great deal of tedious human labor, which could be reduced by computational tools to suggest likely missing or incorrect medications on a patient’s list. We report here an application of recurrent neural networks to predict the likely therapeutic classes of medications that a patient is taking, given a sequence of the last 100 billing codes in their record. Our best model was a GRU that achieved high prediction accuracy (micro-averaged AUC 0.93, Label Ranking Loss 0.076), limited by hardware constraints on model size. Additionally, examining individual cases revealed that many of the predictions marked incorrect were likely to be examples of either omitted medications or omitted billing codes, supporting our assertion of a substantial number of errors and omissions in the data, and the likelihood of models such as these to help correct them. + +# 1 INTRODUCTION + +The idea of exploiting the large amounts of data captured in electronic medical records for both clinical care and secondary research holds great promise, but its potential is weakened by errors and omissions in those records (Safran et al., 2007; de Lusignan & van Weel, 2006). Among many other problems, accurately capturing the list of medications currently taken by a given patient is extremely challenging (Velo & Minuz, 2009). In one study, over $50 \%$ of electronic medication lists contained omissions (Caglar et al., 2011), and in another, $2 5 \%$ of all medications taken by patients were not recorded (Kaboli et al., 2004). Even medication lists provided by the patients themselves contain multiple errors and omissions (Green et al., 2010) . + +Many efforts have been made to ensure the correctness of medication lists, most of them involving improved communication between patients and providers (Keogh et al., 2016), but these efforts have not yet been successful, and incorrect or incomplete medication documentation continues to be a source of error in computational medical research. In this work we attempt to identify likely errors and omissions in the record, predicting the set of active medications from the sequence of most recent disease-based billing codes in the record. Predictions from such a model could be used either in manual medication reconciliation (a common process undertaken to correct the medication record) or to provide a prior to other models, such as an NLP model attempting to extract medication use from the narrative clinical text. + +Given the sequential nature of clinical data, we suspected that recurrent neural networks would be a good architecture for making these predictions. In this work we investigate this potential, comparing the performance of recurrent networks to that of similarly-configured feed forward networks. + +The input for each case is a sequence of ICD-9 billing codes (Section 2.1), for which the model produces a single, multi-label prediction of the therapeutic classes (Section 3.1) of medications taken by the patient during the period of time covered by the billing code sequence. + +This work is designed to test how well the complete set of medications a patient is actively taking at a given moment can be predicted by the sequence of diagnostic billing codes leading up to that moment, in the context of non-trivial label noise. It also explores whether sequence-oriented recursive neural nets can do a better job of that prediction than standard feed-forward networks. + +# 2 BACKGROUND + +# 2.1 MEDICAL BILLING CODES + +Each time a patient has billable contact with the healthcare system, one or more date-stamped billing codes are attached to the patient record, indicating the medical conditions that are associated (or suspected to be associated) with the reason for the visit. While these codes are notoriously unreliable because they are only used for billing and not actual clinical practice (O’Malley et al., 2005), they are nevertheless useful in a research context (Bastarache & Denny, 2011; Denny et al., 2010), especially if they are used probabilistically (Lasko, 2014). In our institution, codes from the International Classification of Diseases, Ninth Revision (ICD-9) have historically been used, although we have recently transitioned to the tenth revision (ICD-10). For this project, we used ICD-9 codes. + +The ICD-9 hierarchy consists of 21 chapters roughly corresponding to a single organ system or pathologic class (Appendix B). Leaf-level codes in that tree represent single diseases or disease subtypes. For this project, we used a subset of the two thousand most common leaf-level codes as our input data. + +# 2.2 RECURRENT NEURAL NETWORKS AND VARIATIONS + +Most of the ICLR community are very familiar with recurrent neural networks and their variations, but we include a conceptual description of them here for readers coming from other fields. More thorough descriptions are available elsewhere (Graves, 2012; Olah, 2015). + +A recurrent neural network is a variation in which the output of one node on input $x _ { t }$ loops around to become an input to another node on input $x _ { t + 1 }$ , allowing information to be preserved as it iterates over an input data sequence (Figure 1). They were introduced in the 1980s (Rumelhart et al., 1986), but achieved explosive popularity only recently, after the development of methods to more reliably capture long-term dependencies, which significantly improved their performance on sequence-tosequence mapping (Hochreiter & Schmidhuber, 1997; Sutskever et al., 2014). + +The basic RNN unit has a simple internal structure (Figure 2a). Output from the previous iteration $h _ { t - 1 }$ and the next input in a sequence $x _ { t }$ are both fed to the network on the next iteration. The Long Short-Term Memory configuration (LSTM) introduces new, more complex internal structure (Figure 2b) consisting of four neural network layers and a cell state $\left( c _ { t } \right)$ , which is carried from one iteration to another. The additional layers form forget, input and output gates, which allow for the information to be forgotten (reset) or passed on to varying degrees. + +The LSTM model and its variations are commonly used in applications where sequence and temporal data are involved, such as in image captioning (Vinyals et al., 2014), language translation (Sutskever et al., 2014), and speech recognition (Graves et al., 2013). In many cases LSTM models define the state of the art, such as with a recent conversational speech recognizer that (slightly) outperforms professional transcriptionists (Xiong et al., 2016). + +A recent variation on the LSTM architecture is the Gated Recurrent Unit (GRU) (Cho et al., 2014), which introduces a single update gate in place of input and forget gates (Figure 2c). GRUs perform as well as or better than LSTMs in many cases (Chung et al., 2014; Jozefowicz et al., 2015), and have the additional advantage of a simpler structure. + +In this work we try both an LSTM and a GRU on our learning problem. + +# 2.3 RELATED WORK + +Little research in the computational medical domain has used recurrent neural networks. The earliest example we are aware of is the use of an LSTM model that produced reasonable accuracy (micro-AUC 0.86) in a 128-dimensional multi-label prediction of diagnoses from regularly sampled, continuously-monitored, real-valued physiologic variables in an Intensive Care Unit setting. This was an interesting initial application, but it turned out to be only 0.001 better than the baseline classifier, which was a multi-layer perceptron with expert-designed features (Lipton et al., 2016). Given the dataset size (10,401 patient records) the lack of improvement may have been due to insufficient data to power accurate feature learning in the recurrent network. + +![](images/4635a9ad051e7fb99b59eb58e4560782a699b4018a7fd120b96c900025575f43.jpg) +Figure 1: Simplified representation of a recurrent neural network (left) and an unrolled recurrent neural network (right). $x _ { i }$ is a single element in an input sequence $x$ , $h _ { i }$ is an output after a single pass through the recurrent unit. Adapted from Olah (2015). + +![](images/1d24c8bfe88b1af3a64d8f47db85ee31caded177bb64afbe606f8dd1a0af679c.jpg) +Figure 2: Architectures of (a) Simple RNN, (b) LSTM, and (c) GRU units. $x _ { t }$ : a single element in an input sequence being considered in the current iteration, $h _ { t - 1 } , h _ { t }$ : the output from the previous and current iterations, $c _ { t - 1 } , c _ { t }$ : the cell states of the previous and current iterations. Adapted from Olah (2015). + +Very recent work, contemporary with ours, used a GRU model with a semantic embedding in 32,787 patient records to predict the development of heart failure 3 - 6 months in the future, from medication orders and billing codes in an 18-month window. The model achieved respectable accuracy (0.88 AUC), and demonstrated a meaningful 0.05 AUC improvement over a deep feedforward network (Choi et al., 2016b). + +Other recent work from the same group used a GRU model in a multi-label context to predict the medications, billing codes, and time of the next patient visit from a sequence of that same information for previous visits, using 263,706 patient records. It achieved a recall $@ 3 0$ of 72.4 for the task, an improvement of 20 over a single-hidden-layer MLP with 2000 units (Choi et al., 2016a). This is an example of using one of the strengths of a recurrent network - predicting the next element in a sequence. It contrasts with our work that exploits a different strength of recurrent networks - predicting a sequence or class that is semantically distinct from but parallel to the elements of the input sequence. + +The closest work to ours from a medical domain perspective is a series of collaborative filter models (including co-occurrence counting, $\mathbf { k }$ -nearest neighbors, and logistic regression) that predict missing medications using a leave-one-drug-out evaluation design, with predictions based on the rest of the medications, ICD-9 billing codes, and demographic data. The models were trained and tested on data from 419 patients in three different clinics, with accuracy varying by clinic, as expected, but not appreciably by model. Most models ranked the missing drug in the top 10 results between 40 and $50 \%$ of the time, and ranked the therapeutic class of the drug in the top 10 results between 50 and $65 \%$ of the time. + +Many aspects of our work can be found in these prior efforts, but none addresses our particular problem in the same way. Our work is unique in its learning problem of identifying all drugs a patient is likely to be taking, based only on the billing codes in the record. Like most others cited, we use recurrent neural networks in a multi-label predictive context, but in contrast to them we compare to the most similar non-recurrent model we can construct, in order to evaluate the contribution of the temporal sequence information to the solution. Finally, we use one to four orders of magnitude more data (3.3 million instances, see Section 3.1) than these prior efforts, which we hope will give us a more realistic assessment of the various deep architectures we use on our problem. + +# 3 EXPERIMENTS + +# 3.1 DATA + +Our source database was the deidentified mirror of Vanderbilt’s Electronic Medical Record, which contains billing codes, medication histories, laboratory test results, narrative text and medical imaging data for over 2 million patients, reaching back nearly 30 years (Roden et al., 2008). We obtained IRB approval to use this data in this research. + +For this experiment we filtered all records in our database to include only the top 1,000 most common medications and the top $m = 2 0 0 0$ most common billing codes, which cover $9 9 . 5 \%$ of all medication occurrences and $8 5 . 1 \%$ of all billing code occurrences. We then included all records from the filtered data that had at least one medication occurrence and at least ten billing code occurrences. This resulted in 610,076 complete patient records, which we divided 80/5/15 into training, validation, and final test sets. + +A data instance $d = \{ E , T , y \}$ consisted of a sequence $E = \{ e _ { 1 } , \ldots , e _ { n } \}$ , of one-hot billing code vectors $e _ { i } \in \{ 0 , 1 \} ^ { m }$ and their associated times $T = \{ t _ { 1 } , \dots , t _ { n } \} , t _ { i } \in \mathbb { R }$ as input, and a multi-label vector $y \in \{ 0 , 1 \} ^ { k }$ of medication classes as the output target. The most recent $n = 1 0 0$ billing codes to a selected reference time point in a given patient record were collected into the input sequence $E$ , and their occurrence times into $T$ , zero padding if necessary. All medications that occurred during the time span of $T$ were then collected into the output vector $y$ . Practice patterns change over time, so simply taking the most recent 100 codes for each patient could produce a biased result. To avoid this, we chose random reference points, stratified by medication. In other words, the reference points were randomly chosen from the occurrences of each medication in the entire dataset, up to 10,000 points per medication. This resulted in 3.3 million data instances, an average of 5.4 instances per patient record. Each patient’s data was included in at most one of the training, validation, or test sets. + +Because there are often many approximately equivalent medication choices for a given therapeutic purpose, we converted medication names to their therapeutic class (beta blocker, immunosuppressant, corticosteroid, etc.) as a synonym reduction step. This step also aggregated generic with brand names, as well as different formulations of the same active ingredient. For this task we used the Anatomical Chemical Classification System $( \mathrm { A T C } ) ^ { 1 }$ , which is a multi-level ontology of medications, organized by both anatomic and therapeutic class. The top level is a broad categorization of medications (Appendix B), the bottom (fifth) level is individual medications, and we used the third level, which contains 287 therapeutic classes of the approximately appropriate abstraction level for our purpose. We used a publicly available mapping2 to translate between our medication names and ATC codes, with manual mapping for the minority of medications that had no mapping entry. Our set of medications used $k = 1 8 2$ third-level ATC codes, rendering our output label a 182-elementlong multi-label vector, in which an element is set $y _ { i } = 1$ if a medication in that class appeared in the set of medications identified for that instance, $y _ { i } = 0$ otherwise. Some medications mapped to more than one class, and we set $y _ { i } = 1$ for all of them. + +Our medication data was collected from structured order entry records and extracted using NLP (Xu et al., 2010) from mentions in the narrative text of a patient record that included the medication name, dose, route and frequency. As discussed above, we assumed (and our results demonstrate) that the medication data is incomplete, and our hope was that a model learned from a sufficiently large dataset will be robust to the missing data. + +This configuration represents the input billing codes in a sequence, but the output medications as a multi-label vector. This is because ICD-9 codes are represented sequentially in our source data, but medications are not. They are represented as a list that changes over time in the record. The usual goal of clinicians is to verify the list of medications at each visit, and if omissions or additions are indicated by the patient, to change the list to reflect that. But in the time-constrained reality of clinical practice, this reconciliation happens sporadically, and many clinicians are hesitant to change an entry on the medication list for which they were not the original prescriber, so the timing of the changes in the documentation do not reflect the timing of changes in reality. Therefore we are reduced to predicting a single multi-label vector, representing the medications that the patient probably took during the span of time represented by the input codes. (We actually did attempt some full sequence-to-sequence mappings, with various orderings of the medication sequences, but we did not achieve any promising results in that direction.) + +# 3.2 CLASSIFIERS + +Our main technical goal was to test the performance of recurrent neural networks on this sequencecentric prediction problem. To evaluate the specific gains provided by the recurrent architectures, we compare performance against a fully connected feed-forward network configured as similarly as possible to the recurrent networks, and (as baselines) a random forest and a constant-prevalence model. We discuss the specific configurations of these classifiers in this section. + +# 3.2.1 RECURRENT NEURAL NETWORKS + +We tested both LSTMs and GRUs in this experiment. We configured both architectures to first compute a semantic embedding $x _ { i } \in \mathbb { R } ^ { b }$ of each input $e _ { i }$ vector, before appending the times $t _ { i }$ (Figure 3) and feeding the result to three layers of recurrent units. The final output from the last pass of recurrent unit is as a multi-label prediction for each candidate medication. + +The optimal hyperparameters for the model were selected in the randomized parameter optimization (Bergstra & Bengio, 2012), with the embedding dimension $b = 3 2$ , number of layers, and number of nodes optimized by a few trials of human-guided search. Other optimized parameters included the fraction of dropout (between layers, input gates and recurrent connections), and L1 and L2 regularization coefficients (final values are presented in Appendix A). + +Both models were implemented using Keras (Chollet, 2015) and trained for 300 iterations using cross-entropy under the Adadelta optimizer (Zeiler, 2012). + +![](images/a01cf382c11d59c2f3c23cad91f68c47ef22ec05a1b7184eb1a19ec121f8c698.jpg) +Figure 3: Recurrent (left) and feed-forward (right) neural network architectures. Arrows indicate the flow of information. Input for both models is sequence of billing code observations $e$ and sequence of corresponding timestamps $t$ . A code observation $e _ { i }$ passes through an embedding layer, producing an embedding vector $x _ { i }$ , which is then appended with time $t$ . The processed matrix then passes through either recurrent layers or feed-forward layers. The output in both cases is a single vector $y$ of label probabilities. + +# 3.2.2 FULLY CONNECTED NEURAL NETWORK + +The fully connected network used as similar an architecture as possible to the recurrent networks, in an attempt to isolate the gain achieved from the recurrence property. Specifically, we used the same architecture for embedding and timestamp appending (Figure 3). + +Hyperparameters were optimized using random search over the number of layers, number of nodes, dropout, activation function between layers, L1 and L2 regularization coefficients (Appendix A). (Surprisingly, the optimizer chose tanh over ReLU as the optimal activation function.) + +The models were also implemented using Keras, and were trained using cross-entropy for 500 iterations under the Adadelta optimizer. + +# 3.2.3 RANDOM FOREST + +Because the random forest model is not easily structured to operate on sequences, we represented the input data as either binary occurrence vectors $v \in \{ 0 , 1 \} ^ { m }$ , or bag-of-codes vectors $w \in \mathbb { N } ^ { m }$ (counts of each code value in the sequence) rather than as sequences of codes with associated times. No embedding was used, because random forest code was not able to cope with the large size of the data in the (dense) embedded space. + +Even in the (sparse) original space, the full dataset was too large for the random forest code, so we implemented it as an ensemble of ten independent forests, each trained on one tenth of the training data, and their average score used for test predictions. + +Models were implemented using scikit-learn (Pedregosa et al., 2011) with parameters optimized under random search (Appendix A). + +While other models could reasonably serve as a baseline for this work, we chose a random forest because they tend to perform well on widely varying datasets (Fernandez-Delgado et al., 2014), they ´ are efficient to train and test, and they don’t require a huge effort to optimize (in order to produce a fair comparison). + +# 3.3 CONSTANT-PREVALENCE MODEL + +This minimum baseline model simply predicts the prevalence of each label for all instances. For example, if there were three possible medications, with prevalences of 0.3, 0.9, and 0.2, then the prediction of this model would be a constant [0.3, 0.9, 0.2] for each instance. We include this model in order to mitigate the fact that while all of our evaluation measures are suitable for comparing models on the same data, some are not well suited for external comparison because they depend, for example, on the prevalence of positive labels (Section 3.4). By including this model we can at least establish a true minimum baseline for reference. + +# 3.4 EVALUATION + +Our main evaluation focused on the models, although we also performed a separate evaluation of the embedding. + +# 3.4.1 MODELS + +There are several possibilities for evaluation in a multi-label classification context (Sechidis et al., 2011; Zhang & Zhou, 2014). We chose micro-averaged area under the ROC curve (AUC) and label ranking loss as the primary methods of evaluation, because they treat each instance with equal weight, regardless of the nature of the positive labels for that instance. In other words, we wanted primary measures that did not give a scoring advantage to instances with either very many or very few positive labels, or that included very rare or very prevalent labels. Additionally, both of these measures appeal to us as intuitive extensions of the usual binary AUC, when seen from the perspective of a single instance. However, because these two measures don’t reflect all aspects of multi-label prediction performance, we also include macro-averaged AUC, label ranking average precision and coverage error measures. + +Micro-averaged AUC considers each of the multiple label predictions in each instance as either true or false, and then computes the binary AUC as if they all belonged to the same 2-class problem (Zhang & Zhou, 2014). In other words, micro-averaged AUC $A _ { \mu }$ is: + +$$ +A _ { \mu } = \frac { \big | ( x , x ^ { \prime } , l , l ^ { \prime } ) : f ( x , l ) \geq f ( x ^ { \prime } , l ^ { \prime } ) , ( x , l ) , \in S , ( x ^ { \prime } , l ^ { \prime } ) \in \bar { S } \big | } { \big | S \big | \big | \bar { S } \big | } , +$$ + +where $S = \{ ( x , l ) : l \in Y \}$ is the set of (instance, label) pairs with a positive label, and $Y = \{ y _ { i } :$ $y _ { i } = 1 , i = 1 \ldots k \}$ is the set of positive labels for input $x$ . + +Label ranking loss $L _ { R }$ gives the average fraction of all possible (positive, negative) label pairs for each instance in which the negative label has a higher score than the positive label (Tsoumakas et al., 2010): + +$$ +L _ { R } = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \frac { 1 } { | Y ^ { ( j ) } | | \overline { { Y ^ { ( j ) } } } | } | \{ ( l , l ^ { \prime } ) : r ^ { ( j ) } ( l ) > r ^ { ( j ) } ( l ^ { \prime } ) , ( l , l ^ { \prime } ) \in Y ^ { ( j ) } \times \overline { { Y ^ { ( j ) } } } \} | +$$ + +where the superscript $( j )$ refers to the $j$ th test instance (of $N$ total instances) and $r ( l )$ is the predicted rank of a label $l$ . + +Macro-averaged AUC can be thought of as averaging the AUC performance of several one-vs-all classifiers, one model for each label. It treats each model equally, regardless of the prevalence of positive labels for that model. This gives a score of 0.5 to the constant-prevalence model, at the cost of weighting instances differently in order to achieve that. This is in contrast to micro-averaged AUC, which can be thought of as averaging across instances rather than labels. It weighs each instance equally, at the cost of a 0.5 score no longer being the random-guessing baseline. + +Label ranking average precision gives the mean fraction of correct positive labels among all positive labels with lower scores for each label. The coverage error function calculates the mean number of labels on the ranked list that are needed to cover all the positive labels of the sample. Both of these depend on the prevalence of positive labels in a test instance. + +# 3.4.2 EMBEDDING + +We evaluated the embedding based on how strongly related in a clinical semantic sense the nearest neighbor to each code is (in the embedding space). A licensed physician manually annotated the list of all 2000 codes with its match category $m \in$ {strongly related, loosely related, unrelated}, and we computed the empirical marginal probability $P ( m )$ of each category, the empirical conditional probability $P ( m | d )$ of the match category given the nearest neighbor (Manhattan) distance $d$ and the empirical marginal probability $P ( \bar { d } )$ . For comparison, we computed $P ( m )$ under 100 random code pairings. + +# 4 RESULTS AND DISCUSSION + +The GRU model had the top performance by all measures, although the LSTM was a close second (Table 1), a performance pattern consistent with previous reports (Chung et al., 2014). The deep neural net performance was about 0.01 worse in both measures, suggesting that the recurrent models were able to use the sequence information, but only to a small advantage over the most similar nontemporal architecture. However, we note that both RNNs’ performance peaked at the top end of our tractable range for model size, while the feed-forward network peaked using a model about one third that size (Appendix A). Experimenting with the architecture, we found that increasing the number of nodes or layers for the feed-forward network increased training time but not performance. This suggests that the RNN performance was limited by the hardware available, and increasing the size of the model may further increase performance, and that the feed-forward network was limited by something else. + +Both random forest models were weaker than the deep neural net, as might be expected from the need to resort to binary and bag-of-codes representations of the input data. + +Table 1: Results of multi-label classification for each model. Baseline is the constant-prevalence model. Perfect is the best possible performance for our data under the given measure. + +
ModelMicro-AUCLabel Ranking LossMacro-AUCLabel Ranking Avg. PrecisionCoverage Error
0.861
GRU LSTM0.927 0.9260.076 0.0770.8590.603 0.60062.6 63.0
NN0.9160.0860.8350.57067.3
0.9030.1020.8040.52373.7
RF (binary) RF (counts)0.8940.1110.7870.49777.3
Baseline0.8280.1720.5000.35597.2
Perfect1.00.01.01.015.0
+ +A natural question is what performance is good enough for clinical use. While there is little clinical experience with multi-label classifiers, we would generally expect clinicians using a binary classifier in an advisory role to find an $\mathrm { A U C } \gtrsim 0 . 9$ to be useful, and $\mathrm { A U C } \gtrsim 0 . 9 5$ to be very useful. An AUC difference of 0.01, and perhaps 0.005 are potentially noticeable in clinical use. + +This $0 . 9 / 0 . 0 1$ rule of thumb may loosely translate to our AUC variants, but it can directly translate to Label Ranking Loss $L _ { R }$ (2). If we think of a single output prediction $\hat { y } \in [ 0 , 1 ] ^ { k }$ as a set of predictions for $k$ binary labels, then $1 - \mathrm { A U C }$ for that set of predictions is equivalent to $L _ { R }$ for the original instance $\hat { y }$ . Therefore, values of $L _ { R } \lesssim 0 . 1$ may be clinically useful, and $L _ { R } \lesssim 0 . 0 5$ may be very useful. + +Subjectively examining performance on 20 randomly selected cases, we find very good detailed predictions, but also evidence of both missing medications and missing billing codes. An example of a good set of detailed predictions is from a complex patient suffering from multiple myeloma (a type of cancer) with various complications. This patient was taking 26 medications, 24 of which had moderate to high probability predictions (Figure 4). (We have found by eyeball that a prediction cutoff of 0.2 gives a reasonable balance between sensitivity and specificity for our model.) In the other direction, only two of the high-prediction classes were not actually being taken, but those classes, along with several of the other moderately-predicted classes, are commonly used for cancer and are clinically reasonable for the case. (Details of this and the two cases below are in Appendix C). + +A good example of missing medications is a case in which the record has multiple billing codes for both osteoporosis (which is very commonly treated with medication) and postablative hypothyroidism (a deliberately induced condition that is always treated with medication), but no medications of the appropriate classes were in the record. The GRU model predicted both of these classes, which the patient was almost surely taking. + +A good example of either missing billing codes or discontinued medications that remain documented as active is a case in which the record has at least five years of data consisting only of codes for Parkinson’s disease, but which lists medications for high cholesterol, hypertension, and other heart disease. The GRU model predicted a reasonable set of medications for Parkinson’s disease and its complications, but did not predict the other medications that are not suggested by the record. + +Given how easy it was to find cases with apparently missing codes and medications, we conclude that there is indeed a substantial amount of label noise in our data, and we therefore interpret our models’ performance as lower bounds on the actual performance. We are encouraged that this kind of a model may actually be useful for identifying missing medications in the record, but of course a more thorough validation, and possibly a more accurate model, would be necessary before using in a clinical scenario. A definitive experiment would use off-line research, including reconciling information from various electronic and human sources to establish the ground truth of which medications were being taken on a particular day, but such efforts are labor intensive and expensive, and can only be conducted on a very small scale. + +An interesting byproduct of these models is the semantic embedding of ICD-9 codes used in the recurrent networks (Figure 5). Transforming input to a semantic embedding is a common preprocessing step to improve performance, but clearly the semantic understanding it provides to an algorithm can be useful beyond the immediate learning problem (Mikolov et al., 2013). Investigating the embedding learned in this experiment shows some generalizable potential, but it also reveals the need for further refinement before it can be truly useful. Specifically, while it’s easy to find tight groups of ICD-9 codes that are strongly clinically related in our embedding, we also find groups for which we cannot see a meaningful clinical relationship. + +![](images/312c72aba26595ef96d357d8ff0979e219e91e21d0a62461d8b7896a84f472e1.jpg) +Figure 4: Medication predictions for a complicated patient. Each vertical bar represents the prediction for a single medication class, with the height of the bar representing the confidence of the prediction. Black labels with arrows indicate ATC therapeutic classes for medications the patient was actually taking. Colors and letters below the axis indicate organ system groups. More detail in Appendix C. + +For example, we see two groups of codes relating to kidney failure and diabetes mellitus, two classes of very prevalent disease (Figure 5, insets). In other iterations with different parameter settings, the kidney failure codes were even embedded in a sequence reflecting the natural progression of the disease, with the code for dialysis (an intensive treatment for end-stage kidney failure) embedded at the appropriate place. Interestingly, these were not the parameter settings that optimized overall prediction performance. In other settings, such as our performance-optimal setting, the sequence is close to the natural progression of the disease, but not quite identical. Nevertheless, this is an exciting result that suggests great potential. + +Further evaluation of the embedding found that $49 \%$ of codes were strongly related semantically to their nearest neighbor, $10 \%$ were loosely related, and $41 \%$ unrelated. This fraction of strongly related nearest neighbors was lower than we had hoped, but much higher than expected by chance (Figure 6), and it definitely improved classification performance. Furthermore, it was obvious by inspection that in general, codes closer in the embedding were more semantically related than distant codes, but interestingly, the distance to the nearest such neighbor showed the opposite relationship nearest neighbors that were very close were less likely to be semantically related than nearest neighbors that were far, and this trend is roughly linear across the full range of $d$ (Figure 6). So the sparser the points are in the embedded space, the more semantically related they are to their nearest neighbor, but the causal direction of that effect and the technical reason for it are beyond the scope of this initial work. + +For this prediction problem, we settled on predicting the medications that occurred in the record during the same time span as the billing codes used. Originally, we intended to predict only the medications listed on the day of the reference point, but that turned out to greatly exacerbate the missing medication problem. After trying medications that fell on the reference day only, the week prior to the reference day, and the six months prior, our best performance both subjectively and objectively was achieved using the full time range of the input data. + +While the performance of the recurrent networks was quite good, we believe it could be improved by including additional input data, such as laboratory test results, demographics, and perhaps vital signs. We also suspect that if we can devise a way to convert our medication data into reliablyordered sequences, we can more fully exploit the strengths of recurrent networks for medication prediction. We look forward to trying these and other variations in future work. + +![](images/7b5765589f26e16732fff295acfd6634b214b494aa7f74ec4d2e028652ea64b4.jpg) +Figure 5: A t-SNE representation of our final embedding. The insets highlight two groups of codes (diabetes mellitus and kidney failure) that are strongly related clinically, and a third group that is not. Codes are colored by whether their nearest neighbor in the embedding space (which may be different from the nearest neighbor in this t-SNE space) is strongly related (blue), loosely related (orange), or unrelated (gray) from a clinical perspective. + +![](images/c3e0f01d9cde5a243cb56f7a206c204b22dc956b1d2d5ed9220429d5162ce1f8.jpg) +Figure 6: Semantic relatedness of nearest neighbors vs. the distance between them. Solid lines are the conditional probabilities $P ( m | d )$ for the three values of $m$ , dashed line is the marginal probability $P ( d )$ of nearest neighbor distances $d$ . Surprisingly, nearest neighbors that are farther away (but still the nearest neighbor) are more strongly related than nearest neighbors that are closer in the embedding space. Shaded regions, colored to correspond to the three values of $m$ , are the $9 5 \%$ CI for empirically estimated $P ( m )$ under random pairings, and represent the expected null result. + +# ACKNOWLEDGMENTS + +This work was funded by grants from the Edward Mallinckrodt, Jr. Foundation and the National Institutes of Health R21LM011664 and R01EB020666. Clinical data was provided by the Vanderbilt Synthetic Derivative, which is supported by institutional funding and by the Vanderbilt CTSA grant ULTR000445. + +# REFERENCES + +Lisa Bastarache and Joshua C. Denny. The use of ICD-9 codes in genetic association studies. In AMIA Annu Symp Proc, volume 2011, pp. 1738, 2011. + +James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. The Journal of Machine Learning Research, 13(1):281–305, 2012. + +Selin Caglar, Philip L Henneman, Fidela S Blank, Howard A Smithline, and Elizabeth A Henneman. Emergency department medication lists are not accurate. 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The use of routinely collected computer data for research in primary care: opportunities and challenges. Family practice, 23:253–263, Apr 2006. + +Joshua C. Denny, Marylyn D. Ritchie, Melissa A. Basford, Jill M. Pulley, Lisa Bastarache, Kristin Brown-Gentry, Deede Wang, Dan R. Masys, Dan M. Roden, and Dana C. Crawford. Phewas: demonstrating the feasibility of a phenome-wide scan to discover gene–disease associations. Bioinformatics, 26(9):1205–1210, 2010. + +Manuel Fernandez-Delgado, Eva Cernadas, Sen ´ en Barro, and Dinani Amorim. Do we need hun- ´ dreds of classifiers to solve real world classification problems? Journal of Machine Learning Research, 15:3133–3181, 2014. + +Alex Graves. Supervised Sequence Labelling with Recurrent Neural Networks. Springer, 2012. + +Alex Graves, Abdel rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. arXiv preprint, 1303.5778, 2013. + +Christopher F Green, Kirti Burgul, and David J Armstrong. 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Efficient inference of Gaussian process modulated renewal processes with application to medical event data. In Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence (UAI), July 2014. +Zachary C Lipton, David C Kale, Charles Elkan, and Randall Wetzell. Learning to diagnose with LSTM recurrent neural networks. In Proceedings of the International Conference on Learning Representaitons (ICLR 2016), 2016. +Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 26, pp. 3111–3119. Curran Associates, Inc., 2013. +Christopher Olah. Understanding lstm networks, 2015. URL https://colah.github.io/ posts/2015-08-Understanding-LSTMs/. Accessed: 2016-10-19. +Kimberly J. O’Malley, Karon F. Cook, Matt D. Price, Kimberly Raiford Wildes, John F. Hurdle, and Carol M. Ashton. Measuring diagnoses: ICD code accuracy. Health Serv Res, 40(5 Pt 2): 1620–1639, Oct 2005. +F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011. +D. M. Roden, J. M. Pulley, M. A. Basford, G. R. Bernard, E. W. Clayton, J. R. Balser, and D. R. Masys. Development of a large-scale de-identified dna biobank to enable personalized medicine. Clin Pharmacol Ther, 84(3):362–369, Sep 2008. +D. G. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland (eds.), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 1: Foundations, pp. 318 – 362. MIT Press, 1986. +Charles Safran, Meryl Bloomrosen, W Edward Hammond, Steven Labkoff, Suzanne Markel-Fox, Paul C. Tang, Don E. Detmer, and Expert Panel. Toward a national framework for the secondary use of health data: an american medical informatics association white paper. J Am Med Inform Assoc, 14(1):1–9, 2007. +Konstantinos Sechidis, Grigorios Tsoumakas, and Ioannis Vlahavas. On the stratification of multilabel data. In Proceedings of the 2011 European Conference on Machine Learning and Knowledge Discovery in Databases - Volume Part III, ECML PKDD’11, pp. 145–158, Berlin, Heidelberg, 2011. Springer-Verlag. +Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 3104–3112. Curran Associates, Inc., 2014. +Grigorios Tsoumakas, Ioannis Katakis, and Ioannis Vlahavas. Mining multi-label data. In Oded Maimon and Lior Rokach (eds.), Data Mining and Knowledge Discovery Handbook, pp. 667– 685. Springer US, Boston, MA, 2010. +G. P. Velo and P. Minuz. Medication errors: prescribing faults and prescription errors. Br J Clin Pharmacol, 67(6):624–628, Jun 2009. +Oriol Vinyals, Alexander Toshev, Samy Bengio, and Dumitru Erhan. Show and tell: A neural image caption generator. CoRR, abs/1411.4555, 2014. +W. Xiong, J. Droppo, X. Huang, F. Seide, M. Seltzer, A. Stolcke, D. Yu, and G. Zweig. Achieving human parity in conversational speech recognition, 2016. +Hua Xu, Shane P Stenner, Son Doan, Kevin B Johnson, Lemuel R Waitman, and Joshua C Denny. Medex: a medication information extraction system for clinical narratives. J Am Med Inform Assoc, 17(1):19–24, 2010. + +Matthew D. Zeiler. ADADELTA: an adaptive learning rate method. CoRR, abs/1212.5701, 2012. + +M. L. Zhang and Z. H. Zhou. A review on multi-label learning algorithms. IEEE Transactions on Knowledge and Data Engineering, 26(8):1819–1837, Aug 2014. + +# APPENDIX A. + +This appendix lists the optimized parameters for the different models. Except where noted, parameters were optimized under random search. + +Recurrent Neural Network Models: (parameters marked with an asterisk were optimized with human-guided search.) + +Feed Forward Neural Network Model: + +
ParameterModel
GRULSTM
Dropout for input gates0.10.25
Dropout for recurrent connections0.750.75
L1 applied to the input weights matrices00
L1 applied to the recurrent weights matrices00
L2 applied to the input weights matrices0.00010.0001
L2 applied to the recurrent weights matrices0.00010.001
L2 applied to the output layer's weights matrices0.00010.001
Dropout before the output layer0.50.5
*Number of recurrent layers33
400
*Number of nodes in recurrent units400
+ +
ParameterValue
Dropout before the output layer0.1
Dropout between feed-forward layers Numberof feed-forward layers0.1
Activation function between feed-forward layers3
tanh
Number of nodes in feed-forward layers128
+ +Random Forest Model (binary input): + +# Parameter + +Value + +Number of estimators 800 +Ratio of features to consider when looking for the best split 0.4666 +Minimum number of samples required to split an internal node 87 +Minimum number of samples required to be at a leaf node 3 +The function to measure the quality of a split entropy + +# APPENDIX B. + +This appendix lists the top level classes for International Statistical Classification of Diseases and Related Health Problems, Ninth Revision (ICD-9) and Anatomical Chemical Classification System (ATC). + +ICD-9 chapters. + +# Code range + +# Description + +001-139 Infectious and parasitic diseases 140-239 Neoplasms 240-279 Endocrine, nutritional and metabolic diseases, and immunity disorders 280-289 Diseases of the blood and blood-forming organs 290-319 Mental disorders 320-359 Diseases of the nervous system 360-389 Diseases of the sense organs 390-459 Diseases of the circulatory system 460-519 Diseases of the respiratory system 520-579 Diseases of the digestive system 580-629 Diseases of the genitourinary system 630-679 Complications of pregnancy, childbirth, and the puerperium 680-709 Diseases of the skin and subcutaneous tissue 710-739 Diseases of the musculoskeletal system and connective tissue 740-759 Congenital anomalies 760-779 Certain conditions originating in the perinatal period 780-799 Symptoms, signs, and ill-defined conditions 800-999 Injury and poisoning V01-V91 Supplementary - factors influencing health status and contact with health services E000-E999 Supplementary - external causes of injury and poisoning + +Top level groups ATC codes and their corresponding colors used in Figure 4 and Appendix C. + +# Code + +# Contents + +A Alimentary tract and metabolism +B Blood and blood forming organs +C Cardiovascular system +D Dermatologicals +G Genito-urinary system and sex hormones +H Systemic hormonal preparations, excluding sex hormones and insulins +J Antiinfectives for systemic use +L Antineoplastic and immunomodulating agents +M Musculo-skeletal system +N Nervous system +P Antiparasitic products, insecticides and repellents +R Respiratory system +S Sensory organs +V Various + +# APPENDIX C. + +This appendix presents results from three illustrative cases from the dozen cases randomly selected for individual evaluation. + +CASE 1. + +
ICD-9 codeCode descriptionTime estimate (ago)
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
273.1Monoclonal paraproteinemia4.8 months ago
285.9Anemia, unspecified4.8 months ago
276.50Volume depletion, unspecified4.8 months ago
733.00Osteoporosis,unspecified4.8 months ago
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
203.00Multiple myeloma,without mention of having achieved remission2.9 months ago
203.01Multiple myeloma, in remission2.9 months ago
273.1Monoclonal paraproteinemia2.9 months ago
273.1Monoclonal paraproteinemia1.6 months ago
279.3Unspecified immunity deficiency1.6 months ago
203.00Multiple myeloma, without mention of having achieved remission1.6 months ago
781.2Abnormality of gait3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.7 weeks ago
401.9Unspecified essential hypertension3.7 weeks ago
V12.54Personal history of transient ischemic attack (TIA),and cerebral infarction without residual deficits3.7 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.7 weeks ago
786.09Otherrespiratoryabnormalities3.7 weeks ago
273.1Monoclonal paraproteinemia3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.6 weeks ago
V58.69Long-term (current) use of other medications3.6 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.4 weeks ago
203.00Multiple myeloma, without mention of having achieved remission4 days ago
V42.82Peripheral stem cells replaced by transplant4 days ago
203.01Multiple myeloma, in remission3 days ago
38.97Central venous catheter placement with guidance3 days ago
V42.82Peripheral stem cells replaced by transplant3 daysago
V58.81Fitting and adjustment of vascular catheter3 days ago
203.00Multiple myeloma, without mention of having achieved remission3 daysago
V42.82Peripheral stem cells replaced by transplant2 days ago
203.01Multiple myeloma, in remission2 days ago
203.00Multiple myeloma, without mention of having achieved remission1 day ago
V42.82Peripheral stem cells replaced by transplant1 day ago
203.00Multiple myeloma, without mention of having achieved remissionnow
V42.82Peripheral stem cells replaced by transplantnow
+ +![](images/601a66fe0b0ec2214e2f5d221c00f14d95ed27ad2e8346ef5834b595295e14b4.jpg) + +Medication predictions for a complicated patient. Each vertical bar represents the prediction for a single medication class, with the height of the bar representing the confidence of the prediction. Black labels above arrows indicate ATC therapeutic classes for medications the patient was actually taking. Colors and letters below the axis indicate high-level therapeutic class groups. + +Predicted vs. actual medication classes for the patient in Case 1. The four-character sequence in the first and fourth columns is the ATC code for the medication therapeutic class, and an asterisk in the first column indicates that the predicted medication is in the actual medication list. Probabilities listed are the model predictions for the listed therapeutic class. In the predicted medications column, all predictions with probability at least 0.2 are listed. + +
Top predictionsProb.True labelsProb.
Corticosteroids97.01%
S03B* S01C*Antiinflammatory agents and antiinfectives in combi-97.01% 95.54%S03B S01CCorticosteroids Antiinflammatory agents and antiinfectives in combi-95.54%
nationnation
S02B*Corticosteroids95.54%S02BCorticosteroids95.54%
L01AAlkylating agents94.00%D07XCorticosteroids,other combinations93.37%
D07X*Corticosteroids,other combinations93.37%H02ACorticosteroids for systemic use, plain91.06%
H02A*Corticosteroids for systemic use, plain91.06%D07ACorticosteroids, plain90.83%
D07A*Corticosteroids,plain90.83%S01BAntiinflammatory agents90.79%
S01B*Antiinflammatory agents90.79%D10AAnti-acne preparations for topical use88.56%
D10A*Anti-acne preparations for topical use88.56%C05AAgents for treatment of hemorrhoids and anal fissures for topical use88.52%
C05A*Agents for treatment of hemorrhoids and anal fissures for topical use88.52%R01ADecongestants and other nasal preparations for topi-87.02%
A04AAntiemetics and antinauseants87.95%J05Acal use Direct acting antivirals86.83%
R01A*Decongestants and other nasal preparations for topi- cal use87.02%A01AStomatological preparations86.11%
J05A*Direct acting antivirals86.8%N02AOpioids84.86%
A01A*Stomatological preparations86.11%B05CIrrigating solutions82.56%
N02A*Opioids84.86%A12COther mineral supplements79.50%
B05C*Irrigating solutions82.56%B05XI.V. solution additives74.84%
A12C*Other mineral supplements79.50%L04AImmunosuppressants68.76%
B05X*Iv. solution additives74.84%N02BOther analgesics and antipyretics57.24%
L04A*Immunosuppressants68.76%S01AAntiinfectives54.59%
N05AAntipsychotics58.64%J01DOther beta-lactam antibacterials43.40%
N02B*Other analgesics and antipyretics57.24%C03CHigh-ceiling diuretics39.88%
S01A*Antiinfectives54.59%J01MQuinolone antibacterials29.78%
L03AImmunostimulants45.96%C07ABeta blocking agents27.08%
A02BDrugs for peptic ulcer and gastro-oesophageal reflux44.56%
J01D*disease Other beta-lactam antibacterials43.40%N03AAntiepileptics
C03C*High-ceiling diuretics39.88%J01XOther antibacterials20.00%
B01AAntithrombotic agents37.80%M03BMuscle relaxants,centrally acting agents5.88%
V03AAll other therapeutic products34.18%5.09%
R06AAntihistamines for systemic use31.78%
A06ADrugs for constipation31.57%
J01M*Quinolone antibacterials29.78%
N05BAnxiolytics29.42%
D04AAntipruritics,incl. antihistamines,anesthetics, etc.27.62%
C07A*Beta blocking agents27.08%
L01XOther antineoplastic agents24.72%
R05CExpectorants,excl. combinations with cough sup-20.43%
N03A*pressants
+ +CASE 2. + +
ICD-9 codeCode descriptionTime estimate (ago)
735.4Other hammer toe (acquired)2.4 years ago
729.5Pain in limb2.4 years ago
244.1Other postablative hypothyroidism1.5 years ago
285.9Anemia, unspecified1.5 years ago
244.1Other postablative hypothyroidism1.2 years ago
244.1Other postablative hypothyroidism11.5 months ago
733.00Osteoporosis,unspecified11.5 months ago
733.01Senile osteoporosis7.7 months ago
268.9Unspecified vitamin D deficiency7.7 months ago
729.5Pain in limb7.7 months ago
174.9Malignant neoplasm of breast (female),unspecified7.7 months ago
722.52Degeneration of lumbar or lumbosacral intervertebral disc7.7 months ago
279.3Unspecified immunity deficiency7.7 months ago
733.01Senile osteoporosis6.4 months ago
733.01Senile osteoporosis6.2 months ago
244.1Other postablative hypothyroidism6.0 months ago
401.1Benign essential hypertension6.0 months ago
V58.69Long-term (current) use of other medications1.9 weeks ago
733.01Senile osteoporosisnow
244.1Other postablative hypothyroidismnow
V58.69Long-term (current) use of other medicationsnow
+ +Predicted vs. actual medication classes for Case 2. Table structure as in case 1. + +
Top predictionsProb.True labelsProb.
M05BDrugs affecting bone structure and mineralization88.18%A11CVitamin a and d, incl. combinations of the two39.42%
H03AThyroid preparations84.82%N06AAntidepressants20.88%
H05AParathyroid hormones and analogues66.33%C10ALipid modifying agents,plain17.05%
A11C*39.42%N03AAntiepileptics15.61%
N02BVitamin a and d, incl. combinations of the two Other analgesics and antipyretics37.58%C09C
A01AStomatological preparations23.05%L02BAngiotensin ii antagonists,plain10.38% 4.22%
A12A21.59%Hormone antagonists and related agents
N06A*Calcium Antidepressants20.88%
C07A
Beta blocking agents20.81%
+ +![](images/ced88c51ba51d971d4a1ae6b9036511aa51abe45adb9d0f96e717197c40ff15d.jpg) + +Medication predictions for a simpler patient. Note that the high-prediction medications are clinically reasonable given the billing codes in the sequence. Figure representation as in case 1. + +CASE 3. +Predicted vs. actual medication classes for Case 3. Table structure as in case 1. + +
ICD-9 codeCodedescriptionTime estimate (ago)
332.0Paralysis agitans5.0 years ago
332.0Paralysis agitans4.7 years ago
332.0Paralysis agitans4.5 years ago
332.0Paralysis agitans4.0 years ago
332.0Paralysis agitans3.5 years ago
332.0Paralysis agitans3.0 years ago
332.0Paralysis agitans2.7 years ago
332.0Paralysis agitans2.4 years ago
332.0Paralysis agitans2.0 years ago
332.0Paralysis agitans1.7 years ago
332.0Paralysis agitans
332.0Paralysis agitans1.0 years ago
332.09.9 months ago
Paralysis agitans4.1 months ago
332.0Paralysis agitansnow
+ +![](images/647f6f150e88634b289fe4474f49576c0f5151752fa240698a1ce54917ac379d.jpg) + +Medication predictions for a patient with only one ICD-9 code, repeated many times over five years. The medications listed under true labels are not indicated for paralysis agitans (Parkinson’s disease), but the patient was surely taking them for reasons not documented in the ICD-9 sequence. The model predicted mostly reasonable medications for a patient with Parkinson’s disease, especially Dopaminergic agents, which is the primary treatment for the disease. Figure representation as in case 1, above. \ No newline at end of file diff --git a/parse/train/rJEgeXFex/rJEgeXFex_content_list.json b/parse/train/rJEgeXFex/rJEgeXFex_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..de2cf6399e434b168aca48a2d9ac7a4f34b5b9e6 --- /dev/null +++ b/parse/train/rJEgeXFex/rJEgeXFex_content_list.json @@ -0,0 +1,1740 @@ +[ + { + "type": "text", + "text": "PREDICTING MEDICATIONS FROM DIAGNOSTIC CODES WITH RECURRENT NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jacek M. Bajor, Thomas A. Lasko \nDepartment of Biomedical Informatics \nVanderbilt University School of Medicine \nNashville, TN 37203, USA \n{jacek.m.bajor,tom.lasko}@vanderbilt.edu ", + "bbox": [ + 184, + 170, + 575, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "It is a surprising fact that electronic medical records are failing at one of their primary purposes, that of tracking the set of medications that the patient is actively taking. Studies estimate that up to $50 \\%$ of such lists omit active drugs, and that up to $2 5 \\%$ of all active medications do not appear on the appropriate patient list. Manual efforts to maintain these lists involve a great deal of tedious human labor, which could be reduced by computational tools to suggest likely missing or incorrect medications on a patient’s list. We report here an application of recurrent neural networks to predict the likely therapeutic classes of medications that a patient is taking, given a sequence of the last 100 billing codes in their record. Our best model was a GRU that achieved high prediction accuracy (micro-averaged AUC 0.93, Label Ranking Loss 0.076), limited by hardware constraints on model size. Additionally, examining individual cases revealed that many of the predictions marked incorrect were likely to be examples of either omitted medications or omitted billing codes, supporting our assertion of a substantial number of errors and omissions in the data, and the likelihood of models such as these to help correct them. ", + "bbox": [ + 233, + 306, + 764, + 526 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 551, + 336, + 568 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The idea of exploiting the large amounts of data captured in electronic medical records for both clinical care and secondary research holds great promise, but its potential is weakened by errors and omissions in those records (Safran et al., 2007; de Lusignan & van Weel, 2006). Among many other problems, accurately capturing the list of medications currently taken by a given patient is extremely challenging (Velo & Minuz, 2009). In one study, over $50 \\%$ of electronic medication lists contained omissions (Caglar et al., 2011), and in another, $2 5 \\%$ of all medications taken by patients were not recorded (Kaboli et al., 2004). Even medication lists provided by the patients themselves contain multiple errors and omissions (Green et al., 2010) . ", + "bbox": [ + 174, + 582, + 823, + 693 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many efforts have been made to ensure the correctness of medication lists, most of them involving improved communication between patients and providers (Keogh et al., 2016), but these efforts have not yet been successful, and incorrect or incomplete medication documentation continues to be a source of error in computational medical research. In this work we attempt to identify likely errors and omissions in the record, predicting the set of active medications from the sequence of most recent disease-based billing codes in the record. Predictions from such a model could be used either in manual medication reconciliation (a common process undertaken to correct the medication record) or to provide a prior to other models, such as an NLP model attempting to extract medication use from the narrative clinical text. ", + "bbox": [ + 174, + 700, + 825, + 825 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Given the sequential nature of clinical data, we suspected that recurrent neural networks would be a good architecture for making these predictions. In this work we investigate this potential, comparing the performance of recurrent networks to that of similarly-configured feed forward networks. ", + "bbox": [ + 176, + 833, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The input for each case is a sequence of ICD-9 billing codes (Section 2.1), for which the model produces a single, multi-label prediction of the therapeutic classes (Section 3.1) of medications taken by the patient during the period of time covered by the billing code sequence. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This work is designed to test how well the complete set of medications a patient is actively taking at a given moment can be predicted by the sequence of diagnostic billing codes leading up to that moment, in the context of non-trivial label noise. It also explores whether sequence-oriented recursive neural nets can do a better job of that prediction than standard feed-forward networks. ", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 183, + 326, + 199 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 MEDICAL BILLING CODES ", + "text_level": 1, + "bbox": [ + 176, + 215, + 400, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Each time a patient has billable contact with the healthcare system, one or more date-stamped billing codes are attached to the patient record, indicating the medical conditions that are associated (or suspected to be associated) with the reason for the visit. While these codes are notoriously unreliable because they are only used for billing and not actual clinical practice (O’Malley et al., 2005), they are nevertheless useful in a research context (Bastarache & Denny, 2011; Denny et al., 2010), especially if they are used probabilistically (Lasko, 2014). In our institution, codes from the International Classification of Diseases, Ninth Revision (ICD-9) have historically been used, although we have recently transitioned to the tenth revision (ICD-10). For this project, we used ICD-9 codes. ", + "bbox": [ + 174, + 242, + 825, + 354 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The ICD-9 hierarchy consists of 21 chapters roughly corresponding to a single organ system or pathologic class (Appendix B). Leaf-level codes in that tree represent single diseases or disease subtypes. For this project, we used a subset of the two thousand most common leaf-level codes as our input data. ", + "bbox": [ + 176, + 361, + 825, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 RECURRENT NEURAL NETWORKS AND VARIATIONS ", + "text_level": 1, + "bbox": [ + 174, + 436, + 576, + 452 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Most of the ICLR community are very familiar with recurrent neural networks and their variations, but we include a conceptual description of them here for readers coming from other fields. More thorough descriptions are available elsewhere (Graves, 2012; Olah, 2015). ", + "bbox": [ + 178, + 464, + 825, + 506 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A recurrent neural network is a variation in which the output of one node on input $x _ { t }$ loops around to become an input to another node on input $x _ { t + 1 }$ , allowing information to be preserved as it iterates over an input data sequence (Figure 1). They were introduced in the 1980s (Rumelhart et al., 1986), but achieved explosive popularity only recently, after the development of methods to more reliably capture long-term dependencies, which significantly improved their performance on sequence-tosequence mapping (Hochreiter & Schmidhuber, 1997; Sutskever et al., 2014). ", + "bbox": [ + 174, + 512, + 825, + 597 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The basic RNN unit has a simple internal structure (Figure 2a). Output from the previous iteration $h _ { t - 1 }$ and the next input in a sequence $x _ { t }$ are both fed to the network on the next iteration. The Long Short-Term Memory configuration (LSTM) introduces new, more complex internal structure (Figure 2b) consisting of four neural network layers and a cell state $\\left( c _ { t } \\right)$ , which is carried from one iteration to another. The additional layers form forget, input and output gates, which allow for the information to be forgotten (reset) or passed on to varying degrees. ", + "bbox": [ + 174, + 603, + 823, + 688 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The LSTM model and its variations are commonly used in applications where sequence and temporal data are involved, such as in image captioning (Vinyals et al., 2014), language translation (Sutskever et al., 2014), and speech recognition (Graves et al., 2013). In many cases LSTM models define the state of the art, such as with a recent conversational speech recognizer that (slightly) outperforms professional transcriptionists (Xiong et al., 2016). ", + "bbox": [ + 174, + 694, + 823, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A recent variation on the LSTM architecture is the Gated Recurrent Unit (GRU) (Cho et al., 2014), which introduces a single update gate in place of input and forget gates (Figure 2c). GRUs perform as well as or better than LSTMs in many cases (Chung et al., 2014; Jozefowicz et al., 2015), and have the additional advantage of a simpler structure. ", + "bbox": [ + 174, + 770, + 825, + 827 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work we try both an LSTM and a GRU on our learning problem. ", + "bbox": [ + 174, + 833, + 642, + 848 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.3 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 868, + 330, + 882 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Little research in the computational medical domain has used recurrent neural networks. The earliest example we are aware of is the use of an LSTM model that produced reasonable accuracy (micro-AUC 0.86) in a 128-dimensional multi-label prediction of diagnoses from regularly sampled, continuously-monitored, real-valued physiologic variables in an Intensive Care Unit setting. This was an interesting initial application, but it turned out to be only 0.001 better than the baseline classifier, which was a multi-layer perceptron with expert-designed features (Lipton et al., 2016). Given the dataset size (10,401 patient records) the lack of improvement may have been due to insufficient data to power accurate feature learning in the recurrent network. ", + "bbox": [ + 174, + 895, + 823, + 922 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/4635a9ad051e7fb99b59eb58e4560782a699b4018a7fd120b96c900025575f43.jpg", + "image_caption": [ + "Figure 1: Simplified representation of a recurrent neural network (left) and an unrolled recurrent neural network (right). $x _ { i }$ is a single element in an input sequence $x$ , $h _ { i }$ is an output after a single pass through the recurrent unit. Adapted from Olah (2015). " + ], + "image_footnote": [], + "bbox": [ + 334, + 98, + 663, + 179 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/1d24c8bfe88b1af3a64d8f47db85ee31caded177bb64afbe606f8dd1a0af679c.jpg", + "image_caption": [ + "Figure 2: Architectures of (a) Simple RNN, (b) LSTM, and (c) GRU units. $x _ { t }$ : a single element in an input sequence being considered in the current iteration, $h _ { t - 1 } , h _ { t }$ : the output from the previous and current iterations, $c _ { t - 1 } , c _ { t }$ : the cell states of the previous and current iterations. Adapted from Olah (2015). " + ], + "image_footnote": [], + "bbox": [ + 171, + 247, + 826, + 367 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 463, + 825, + 547 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Very recent work, contemporary with ours, used a GRU model with a semantic embedding in 32,787 patient records to predict the development of heart failure 3 - 6 months in the future, from medication orders and billing codes in an 18-month window. The model achieved respectable accuracy (0.88 AUC), and demonstrated a meaningful 0.05 AUC improvement over a deep feedforward network (Choi et al., 2016b). ", + "bbox": [ + 174, + 554, + 825, + 625 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Other recent work from the same group used a GRU model in a multi-label context to predict the medications, billing codes, and time of the next patient visit from a sequence of that same information for previous visits, using 263,706 patient records. It achieved a recall $@ 3 0$ of 72.4 for the task, an improvement of 20 over a single-hidden-layer MLP with 2000 units (Choi et al., 2016a). This is an example of using one of the strengths of a recurrent network - predicting the next element in a sequence. It contrasts with our work that exploits a different strength of recurrent networks - predicting a sequence or class that is semantically distinct from but parallel to the elements of the input sequence. ", + "bbox": [ + 174, + 631, + 825, + 742 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The closest work to ours from a medical domain perspective is a series of collaborative filter models (including co-occurrence counting, $\\mathbf { k }$ -nearest neighbors, and logistic regression) that predict missing medications using a leave-one-drug-out evaluation design, with predictions based on the rest of the medications, ICD-9 billing codes, and demographic data. The models were trained and tested on data from 419 patients in three different clinics, with accuracy varying by clinic, as expected, but not appreciably by model. Most models ranked the missing drug in the top 10 results between 40 and $50 \\%$ of the time, and ranked the therapeutic class of the drug in the top 10 results between 50 and $65 \\%$ of the time. ", + "bbox": [ + 174, + 750, + 825, + 861 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Many aspects of our work can be found in these prior efforts, but none addresses our particular problem in the same way. Our work is unique in its learning problem of identifying all drugs a patient is likely to be taking, based only on the billing codes in the record. Like most others cited, we use recurrent neural networks in a multi-label predictive context, but in contrast to them we compare to the most similar non-recurrent model we can construct, in order to evaluate the contribution of the temporal sequence information to the solution. Finally, we use one to four orders of magnitude more data (3.3 million instances, see Section 3.1) than these prior efforts, which we hope will give us a more realistic assessment of the various deep architectures we use on our problem. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 180, + 326, + 196 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 DATA ", + "text_level": 1, + "bbox": [ + 176, + 212, + 253, + 227 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our source database was the deidentified mirror of Vanderbilt’s Electronic Medical Record, which contains billing codes, medication histories, laboratory test results, narrative text and medical imaging data for over 2 million patients, reaching back nearly 30 years (Roden et al., 2008). We obtained IRB approval to use this data in this research. ", + "bbox": [ + 174, + 238, + 825, + 294 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For this experiment we filtered all records in our database to include only the top 1,000 most common medications and the top $m = 2 0 0 0$ most common billing codes, which cover $9 9 . 5 \\%$ of all medication occurrences and $8 5 . 1 \\%$ of all billing code occurrences. We then included all records from the filtered data that had at least one medication occurrence and at least ten billing code occurrences. This resulted in 610,076 complete patient records, which we divided 80/5/15 into training, validation, and final test sets. ", + "bbox": [ + 174, + 301, + 825, + 385 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A data instance $d = \\{ E , T , y \\}$ consisted of a sequence $E = \\{ e _ { 1 } , \\ldots , e _ { n } \\}$ , of one-hot billing code vectors $e _ { i } \\in \\{ 0 , 1 \\} ^ { m }$ and their associated times $T = \\{ t _ { 1 } , \\dots , t _ { n } \\} , t _ { i } \\in \\mathbb { R }$ as input, and a multi-label vector $y \\in \\{ 0 , 1 \\} ^ { k }$ of medication classes as the output target. The most recent $n = 1 0 0$ billing codes to a selected reference time point in a given patient record were collected into the input sequence $E$ , and their occurrence times into $T$ , zero padding if necessary. All medications that occurred during the time span of $T$ were then collected into the output vector $y$ . Practice patterns change over time, so simply taking the most recent 100 codes for each patient could produce a biased result. To avoid this, we chose random reference points, stratified by medication. In other words, the reference points were randomly chosen from the occurrences of each medication in the entire dataset, up to 10,000 points per medication. This resulted in 3.3 million data instances, an average of 5.4 instances per patient record. Each patient’s data was included in at most one of the training, validation, or test sets. ", + "bbox": [ + 173, + 391, + 825, + 558 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Because there are often many approximately equivalent medication choices for a given therapeutic purpose, we converted medication names to their therapeutic class (beta blocker, immunosuppressant, corticosteroid, etc.) as a synonym reduction step. This step also aggregated generic with brand names, as well as different formulations of the same active ingredient. For this task we used the Anatomical Chemical Classification System $( \\mathrm { A T C } ) ^ { 1 }$ , which is a multi-level ontology of medications, organized by both anatomic and therapeutic class. The top level is a broad categorization of medications (Appendix B), the bottom (fifth) level is individual medications, and we used the third level, which contains 287 therapeutic classes of the approximately appropriate abstraction level for our purpose. We used a publicly available mapping2 to translate between our medication names and ATC codes, with manual mapping for the minority of medications that had no mapping entry. Our set of medications used $k = 1 8 2$ third-level ATC codes, rendering our output label a 182-elementlong multi-label vector, in which an element is set $y _ { i } = 1$ if a medication in that class appeared in the set of medications identified for that instance, $y _ { i } = 0$ otherwise. Some medications mapped to more than one class, and we set $y _ { i } = 1$ for all of them. ", + "bbox": [ + 173, + 565, + 825, + 760 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our medication data was collected from structured order entry records and extracted using NLP (Xu et al., 2010) from mentions in the narrative text of a patient record that included the medication name, dose, route and frequency. As discussed above, we assumed (and our results demonstrate) that the medication data is incomplete, and our hope was that a model learned from a sufficiently large dataset will be robust to the missing data. ", + "bbox": [ + 174, + 766, + 825, + 837 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This configuration represents the input billing codes in a sequence, but the output medications as a multi-label vector. This is because ICD-9 codes are represented sequentially in our source data, but medications are not. They are represented as a list that changes over time in the record. The usual goal of clinicians is to verify the list of medications at each visit, and if omissions or additions are indicated by the patient, to change the list to reflect that. But in the time-constrained reality of clinical practice, this reconciliation happens sporadically, and many clinicians are hesitant to change an entry on the medication list for which they were not the original prescriber, so the timing of the changes in the documentation do not reflect the timing of changes in reality. Therefore we are reduced to predicting a single multi-label vector, representing the medications that the patient probably took during the span of time represented by the input codes. (We actually did attempt some full sequence-to-sequence mappings, with various orderings of the medication sequences, but we did not achieve any promising results in that direction.) ", + "bbox": [ + 176, + 843, + 825, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 174, + 250, + 305, + 263 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our main technical goal was to test the performance of recurrent neural networks on this sequencecentric prediction problem. To evaluate the specific gains provided by the recurrent architectures, we compare performance against a fully connected feed-forward network configured as similarly as possible to the recurrent networks, and (as baselines) a random forest and a constant-prevalence model. We discuss the specific configurations of these classifiers in this section. ", + "bbox": [ + 174, + 277, + 825, + 347 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2.1 RECURRENT NEURAL NETWORKS", + "text_level": 1, + "bbox": [ + 176, + 367, + 464, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We tested both LSTMs and GRUs in this experiment. We configured both architectures to first compute a semantic embedding $x _ { i } \\in \\mathbb { R } ^ { b }$ of each input $e _ { i }$ vector, before appending the times $t _ { i }$ (Figure 3) and feeding the result to three layers of recurrent units. The final output from the last pass of recurrent unit is as a multi-label prediction for each candidate medication. ", + "bbox": [ + 174, + 392, + 825, + 448 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The optimal hyperparameters for the model were selected in the randomized parameter optimization (Bergstra & Bengio, 2012), with the embedding dimension $b = 3 2$ , number of layers, and number of nodes optimized by a few trials of human-guided search. Other optimized parameters included the fraction of dropout (between layers, input gates and recurrent connections), and L1 and L2 regularization coefficients (final values are presented in Appendix A). ", + "bbox": [ + 174, + 455, + 825, + 525 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Both models were implemented using Keras (Chollet, 2015) and trained for 300 iterations using cross-entropy under the Adadelta optimizer (Zeiler, 2012). ", + "bbox": [ + 174, + 531, + 823, + 560 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/a01cf382c11d59c2f3c23cad91f68c47ef22ec05a1b7184eb1a19ec121f8c698.jpg", + "image_caption": [ + "Figure 3: Recurrent (left) and feed-forward (right) neural network architectures. Arrows indicate the flow of information. Input for both models is sequence of billing code observations $e$ and sequence of corresponding timestamps $t$ . A code observation $e _ { i }$ passes through an embedding layer, producing an embedding vector $x _ { i }$ , which is then appended with time $t$ . The processed matrix then passes through either recurrent layers or feed-forward layers. The output in both cases is a single vector $y$ of label probabilities. " + ], + "image_footnote": [], + "bbox": [ + 174, + 575, + 833, + 815 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2.2 FULLY CONNECTED NEURAL NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 508, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The fully connected network used as similar an architecture as possible to the recurrent networks, in an attempt to isolate the gain achieved from the recurrence property. Specifically, we used the same architecture for embedding and timestamp appending (Figure 3). ", + "bbox": [ + 176, + 128, + 823, + 171 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Hyperparameters were optimized using random search over the number of layers, number of nodes, dropout, activation function between layers, L1 and L2 regularization coefficients (Appendix A). (Surprisingly, the optimizer chose tanh over ReLU as the optimal activation function.) ", + "bbox": [ + 176, + 178, + 823, + 220 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The models were also implemented using Keras, and were trained using cross-entropy for 500 iterations under the Adadelta optimizer. ", + "bbox": [ + 173, + 227, + 821, + 256 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2.3 RANDOM FOREST ", + "text_level": 1, + "bbox": [ + 176, + 275, + 352, + 289 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Because the random forest model is not easily structured to operate on sequences, we represented the input data as either binary occurrence vectors $v \\in \\{ 0 , 1 \\} ^ { m }$ , or bag-of-codes vectors $w \\in \\mathbb { N } ^ { m }$ (counts of each code value in the sequence) rather than as sequences of codes with associated times. No embedding was used, because random forest code was not able to cope with the large size of the data in the (dense) embedded space. ", + "bbox": [ + 174, + 300, + 825, + 369 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Even in the (sparse) original space, the full dataset was too large for the random forest code, so we implemented it as an ensemble of ten independent forests, each trained on one tenth of the training data, and their average score used for test predictions. ", + "bbox": [ + 176, + 377, + 821, + 419 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Models were implemented using scikit-learn (Pedregosa et al., 2011) with parameters optimized under random search (Appendix A). ", + "bbox": [ + 173, + 426, + 823, + 454 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "While other models could reasonably serve as a baseline for this work, we chose a random forest because they tend to perform well on widely varying datasets (Fernandez-Delgado et al., 2014), they ´ are efficient to train and test, and they don’t require a huge effort to optimize (in order to produce a fair comparison). ", + "bbox": [ + 176, + 462, + 825, + 517 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 CONSTANT-PREVALENCE MODEL ", + "text_level": 1, + "bbox": [ + 178, + 539, + 446, + 553 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This minimum baseline model simply predicts the prevalence of each label for all instances. For example, if there were three possible medications, with prevalences of 0.3, 0.9, and 0.2, then the prediction of this model would be a constant [0.3, 0.9, 0.2] for each instance. We include this model in order to mitigate the fact that while all of our evaluation measures are suitable for comparing models on the same data, some are not well suited for external comparison because they depend, for example, on the prevalence of positive labels (Section 3.4). By including this model we can at least establish a true minimum baseline for reference. ", + "bbox": [ + 174, + 565, + 825, + 664 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 EVALUATION ", + "text_level": 1, + "bbox": [ + 174, + 684, + 305, + 698 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our main evaluation focused on the models, although we also performed a separate evaluation of the embedding. ", + "bbox": [ + 176, + 712, + 823, + 739 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4.1 MODELS ", + "text_level": 1, + "bbox": [ + 174, + 758, + 289, + 773 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "There are several possibilities for evaluation in a multi-label classification context (Sechidis et al., 2011; Zhang & Zhou, 2014). We chose micro-averaged area under the ROC curve (AUC) and label ranking loss as the primary methods of evaluation, because they treat each instance with equal weight, regardless of the nature of the positive labels for that instance. In other words, we wanted primary measures that did not give a scoring advantage to instances with either very many or very few positive labels, or that included very rare or very prevalent labels. Additionally, both of these measures appeal to us as intuitive extensions of the usual binary AUC, when seen from the perspective of a single instance. However, because these two measures don’t reflect all aspects of multi-label prediction performance, we also include macro-averaged AUC, label ranking average precision and coverage error measures. ", + "bbox": [ + 174, + 785, + 825, + 922 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Micro-averaged AUC considers each of the multiple label predictions in each instance as either true or false, and then computes the binary AUC as if they all belonged to the same 2-class problem (Zhang & Zhou, 2014). In other words, micro-averaged AUC $A _ { \\mu }$ is: ", + "bbox": [ + 173, + 103, + 825, + 147 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/7de24e8fa7ae0a697593adfafbd009e5dfced8b11835ba4fea1f746afa1c3654.jpg", + "text": "$$\nA _ { \\mu } = \\frac { \\big | ( x , x ^ { \\prime } , l , l ^ { \\prime } ) : f ( x , l ) \\geq f ( x ^ { \\prime } , l ^ { \\prime } ) , ( x , l ) , \\in S , ( x ^ { \\prime } , l ^ { \\prime } ) \\in \\bar { S } \\big | } { \\big | S \\big | \\big | \\bar { S } \\big | } ,\n$$", + "text_format": "latex", + "bbox": [ + 284, + 164, + 712, + 204 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $S = \\{ ( x , l ) : l \\in Y \\}$ is the set of (instance, label) pairs with a positive label, and $Y = \\{ y _ { i } :$ $y _ { i } = 1 , i = 1 \\ldots k \\}$ is the set of positive labels for input $x$ . ", + "bbox": [ + 173, + 207, + 820, + 236 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Label ranking loss $L _ { R }$ gives the average fraction of all possible (positive, negative) label pairs for each instance in which the negative label has a higher score than the positive label (Tsoumakas et al., 2010): ", + "bbox": [ + 173, + 242, + 825, + 285 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/afbc19f2aa2b42f0bf78e4521ee4202ce1fe769acabe40d66828e8db817de5b2.jpg", + "text": "$$\nL _ { R } = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\frac { 1 } { | Y ^ { ( j ) } | | \\overline { { Y ^ { ( j ) } } } | } | \\{ ( l , l ^ { \\prime } ) : r ^ { ( j ) } ( l ) > r ^ { ( j ) } ( l ^ { \\prime } ) , ( l , l ^ { \\prime } ) \\in Y ^ { ( j ) } \\times \\overline { { Y ^ { ( j ) } } } \\} |\n$$", + "text_format": "latex", + "bbox": [ + 238, + 303, + 758, + 348 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where the superscript $( j )$ refers to the $j$ th test instance (of $N$ total instances) and $r ( l )$ is the predicted rank of a label $l$ . ", + "bbox": [ + 176, + 359, + 825, + 388 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Macro-averaged AUC can be thought of as averaging the AUC performance of several one-vs-all classifiers, one model for each label. It treats each model equally, regardless of the prevalence of positive labels for that model. This gives a score of 0.5 to the constant-prevalence model, at the cost of weighting instances differently in order to achieve that. This is in contrast to micro-averaged AUC, which can be thought of as averaging across instances rather than labels. It weighs each instance equally, at the cost of a 0.5 score no longer being the random-guessing baseline. ", + "bbox": [ + 173, + 395, + 825, + 479 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Label ranking average precision gives the mean fraction of correct positive labels among all positive labels with lower scores for each label. The coverage error function calculates the mean number of labels on the ranked list that are needed to cover all the positive labels of the sample. Both of these depend on the prevalence of positive labels in a test instance. ", + "bbox": [ + 174, + 486, + 825, + 541 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.4.2 EMBEDDING", + "text_level": 1, + "bbox": [ + 174, + 559, + 316, + 573 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluated the embedding based on how strongly related in a clinical semantic sense the nearest neighbor to each code is (in the embedding space). A licensed physician manually annotated the list of all 2000 codes with its match category $m \\in$ {strongly related, loosely related, unrelated}, and we computed the empirical marginal probability $P ( m )$ of each category, the empirical conditional probability $P ( m | d )$ of the match category given the nearest neighbor (Manhattan) distance $d$ and the empirical marginal probability $P ( \\bar { d } )$ . For comparison, we computed $P ( m )$ under 100 random code pairings. ", + "bbox": [ + 174, + 584, + 825, + 681 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 RESULTS AND DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 703, + 431, + 719 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The GRU model had the top performance by all measures, although the LSTM was a close second (Table 1), a performance pattern consistent with previous reports (Chung et al., 2014). The deep neural net performance was about 0.01 worse in both measures, suggesting that the recurrent models were able to use the sequence information, but only to a small advantage over the most similar nontemporal architecture. However, we note that both RNNs’ performance peaked at the top end of our tractable range for model size, while the feed-forward network peaked using a model about one third that size (Appendix A). Experimenting with the architecture, we found that increasing the number of nodes or layers for the feed-forward network increased training time but not performance. This suggests that the RNN performance was limited by the hardware available, and increasing the size of the model may further increase performance, and that the feed-forward network was limited by something else. ", + "bbox": [ + 174, + 734, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Both random forest models were weaker than the deep neural net, as might be expected from the need to resort to binary and bag-of-codes representations of the input data. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f4d0c69c2675d7da7a32eff980cc32128dcebf9bfe9ca17a6347f3bc58aec382.jpg", + "table_caption": [ + "Table 1: Results of multi-label classification for each model. Baseline is the constant-prevalence model. Perfect is the best possible performance for our data under the given measure. " + ], + "table_footnote": [], + "table_body": "
ModelMicro-AUCLabel Ranking LossMacro-AUCLabel Ranking Avg. PrecisionCoverage Error
0.861
GRU LSTM0.927 0.9260.076 0.0770.8590.603 0.60062.6 63.0
NN0.9160.0860.8350.57067.3
0.9030.1020.8040.52373.7
RF (binary) RF (counts)0.8940.1110.7870.49777.3
Baseline0.8280.1720.5000.35597.2
Perfect1.00.01.01.015.0
", + "bbox": [ + 174, + 145, + 825, + 290 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A natural question is what performance is good enough for clinical use. While there is little clinical experience with multi-label classifiers, we would generally expect clinicians using a binary classifier in an advisory role to find an $\\mathrm { A U C } \\gtrsim 0 . 9$ to be useful, and $\\mathrm { A U C } \\gtrsim 0 . 9 5$ to be very useful. An AUC difference of 0.01, and perhaps 0.005 are potentially noticeable in clinical use. ", + "bbox": [ + 174, + 324, + 825, + 381 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This $0 . 9 / 0 . 0 1$ rule of thumb may loosely translate to our AUC variants, but it can directly translate to Label Ranking Loss $L _ { R }$ (2). If we think of a single output prediction $\\hat { y } \\in [ 0 , 1 ] ^ { k }$ as a set of predictions for $k$ binary labels, then $1 - \\mathrm { A U C }$ for that set of predictions is equivalent to $L _ { R }$ for the original instance $\\hat { y }$ . Therefore, values of $L _ { R } \\lesssim 0 . 1$ may be clinically useful, and $L _ { R } \\lesssim 0 . 0 5$ may be very useful. ", + "bbox": [ + 174, + 387, + 825, + 457 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Subjectively examining performance on 20 randomly selected cases, we find very good detailed predictions, but also evidence of both missing medications and missing billing codes. An example of a good set of detailed predictions is from a complex patient suffering from multiple myeloma (a type of cancer) with various complications. This patient was taking 26 medications, 24 of which had moderate to high probability predictions (Figure 4). (We have found by eyeball that a prediction cutoff of 0.2 gives a reasonable balance between sensitivity and specificity for our model.) In the other direction, only two of the high-prediction classes were not actually being taken, but those classes, along with several of the other moderately-predicted classes, are commonly used for cancer and are clinically reasonable for the case. (Details of this and the two cases below are in Appendix C). ", + "bbox": [ + 173, + 463, + 825, + 602 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A good example of missing medications is a case in which the record has multiple billing codes for both osteoporosis (which is very commonly treated with medication) and postablative hypothyroidism (a deliberately induced condition that is always treated with medication), but no medications of the appropriate classes were in the record. The GRU model predicted both of these classes, which the patient was almost surely taking. ", + "bbox": [ + 174, + 609, + 823, + 680 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A good example of either missing billing codes or discontinued medications that remain documented as active is a case in which the record has at least five years of data consisting only of codes for Parkinson’s disease, but which lists medications for high cholesterol, hypertension, and other heart disease. The GRU model predicted a reasonable set of medications for Parkinson’s disease and its complications, but did not predict the other medications that are not suggested by the record. ", + "bbox": [ + 174, + 686, + 825, + 756 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Given how easy it was to find cases with apparently missing codes and medications, we conclude that there is indeed a substantial amount of label noise in our data, and we therefore interpret our models’ performance as lower bounds on the actual performance. We are encouraged that this kind of a model may actually be useful for identifying missing medications in the record, but of course a more thorough validation, and possibly a more accurate model, would be necessary before using in a clinical scenario. A definitive experiment would use off-line research, including reconciling information from various electronic and human sources to establish the ground truth of which medications were being taken on a particular day, but such efforts are labor intensive and expensive, and can only be conducted on a very small scale. ", + "bbox": [ + 174, + 762, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "An interesting byproduct of these models is the semantic embedding of ICD-9 codes used in the recurrent networks (Figure 5). Transforming input to a semantic embedding is a common preprocessing step to improve performance, but clearly the semantic understanding it provides to an algorithm can be useful beyond the immediate learning problem (Mikolov et al., 2013). Investigating the embedding learned in this experiment shows some generalizable potential, but it also reveals the need for further refinement before it can be truly useful. Specifically, while it’s easy to find tight groups of ICD-9 codes that are strongly clinically related in our embedding, we also find groups for which we cannot see a meaningful clinical relationship. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/312c72aba26595ef96d357d8ff0979e219e91e21d0a62461d8b7896a84f472e1.jpg", + "image_caption": [ + "Figure 4: Medication predictions for a complicated patient. Each vertical bar represents the prediction for a single medication class, with the height of the bar representing the confidence of the prediction. Black labels with arrows indicate ATC therapeutic classes for medications the patient was actually taking. Colors and letters below the axis indicate organ system groups. More detail in Appendix C. " + ], + "image_footnote": [], + "bbox": [ + 174, + 104, + 818, + 320 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 436, + 825, + 520 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For example, we see two groups of codes relating to kidney failure and diabetes mellitus, two classes of very prevalent disease (Figure 5, insets). In other iterations with different parameter settings, the kidney failure codes were even embedded in a sequence reflecting the natural progression of the disease, with the code for dialysis (an intensive treatment for end-stage kidney failure) embedded at the appropriate place. Interestingly, these were not the parameter settings that optimized overall prediction performance. In other settings, such as our performance-optimal setting, the sequence is close to the natural progression of the disease, but not quite identical. Nevertheless, this is an exciting result that suggests great potential. ", + "bbox": [ + 174, + 526, + 825, + 638 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Further evaluation of the embedding found that $49 \\%$ of codes were strongly related semantically to their nearest neighbor, $10 \\%$ were loosely related, and $41 \\%$ unrelated. This fraction of strongly related nearest neighbors was lower than we had hoped, but much higher than expected by chance (Figure 6), and it definitely improved classification performance. Furthermore, it was obvious by inspection that in general, codes closer in the embedding were more semantically related than distant codes, but interestingly, the distance to the nearest such neighbor showed the opposite relationship nearest neighbors that were very close were less likely to be semantically related than nearest neighbors that were far, and this trend is roughly linear across the full range of $d$ (Figure 6). So the sparser the points are in the embedded space, the more semantically related they are to their nearest neighbor, but the causal direction of that effect and the technical reason for it are beyond the scope of this initial work. ", + "bbox": [ + 173, + 645, + 825, + 797 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For this prediction problem, we settled on predicting the medications that occurred in the record during the same time span as the billing codes used. Originally, we intended to predict only the medications listed on the day of the reference point, but that turned out to greatly exacerbate the missing medication problem. After trying medications that fell on the reference day only, the week prior to the reference day, and the six months prior, our best performance both subjectively and objectively was achieved using the full time range of the input data. ", + "bbox": [ + 174, + 805, + 825, + 888 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "While the performance of the recurrent networks was quite good, we believe it could be improved by including additional input data, such as laboratory test results, demographics, and perhaps vital signs. We also suspect that if we can devise a way to convert our medication data into reliablyordered sequences, we can more fully exploit the strengths of recurrent networks for medication prediction. We look forward to trying these and other variations in future work. ", + "bbox": [ + 174, + 896, + 821, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/7b5765589f26e16732fff295acfd6634b214b494aa7f74ec4d2e028652ea64b4.jpg", + "image_caption": [ + "Figure 5: A t-SNE representation of our final embedding. The insets highlight two groups of codes (diabetes mellitus and kidney failure) that are strongly related clinically, and a third group that is not. Codes are colored by whether their nearest neighbor in the embedding space (which may be different from the nearest neighbor in this t-SNE space) is strongly related (blue), loosely related (orange), or unrelated (gray) from a clinical perspective. " + ], + "image_footnote": [], + "bbox": [ + 178, + 102, + 825, + 383 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/c3e0f01d9cde5a243cb56f7a206c204b22dc956b1d2d5ed9220429d5162ce1f8.jpg", + "image_caption": [ + "Figure 6: Semantic relatedness of nearest neighbors vs. the distance between them. Solid lines are the conditional probabilities $P ( m | d )$ for the three values of $m$ , dashed line is the marginal probability $P ( d )$ of nearest neighbor distances $d$ . Surprisingly, nearest neighbors that are farther away (but still the nearest neighbor) are more strongly related than nearest neighbors that are closer in the embedding space. Shaded regions, colored to correspond to the three values of $m$ , are the $9 5 \\%$ CI for empirically estimated $P ( m )$ under random pairings, and represent the expected null result. " + ], + "image_footnote": [], + "bbox": [ + 264, + 481, + 732, + 648 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 773, + 825, + 816 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 838, + 356, + 852 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was funded by grants from the Edward Mallinckrodt, Jr. Foundation and the National Institutes of Health R21LM011664 and R01EB020666. Clinical data was provided by the Vanderbilt Synthetic Derivative, which is supported by institutional funding and by the Vanderbilt CTSA grant ULTR000445. ", + "bbox": [ + 176, + 867, + 825, + 922 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Lisa Bastarache and Joshua C. Denny. The use of ICD-9 codes in genetic association studies. 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Sequence to sequence learning with neural networks. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 3104–3112. Curran Associates, Inc., 2014. \nGrigorios Tsoumakas, Ioannis Katakis, and Ioannis Vlahavas. Mining multi-label data. In Oded Maimon and Lior Rokach (eds.), Data Mining and Knowledge Discovery Handbook, pp. 667– 685. Springer US, Boston, MA, 2010. \nG. P. Velo and P. Minuz. Medication errors: prescribing faults and prescription errors. Br J Clin Pharmacol, 67(6):624–628, Jun 2009. \nOriol Vinyals, Alexander Toshev, Samy Bengio, and Dumitru Erhan. Show and tell: A neural image caption generator. CoRR, abs/1411.4555, 2014. \nW. Xiong, J. Droppo, X. Huang, F. Seide, M. Seltzer, A. Stolcke, D. Yu, and G. Zweig. Achieving human parity in conversational speech recognition, 2016. \nHua Xu, Shane P Stenner, Son Doan, Kevin B Johnson, Lemuel R Waitman, and Joshua C Denny. Medex: a medication information extraction system for clinical narratives. J Am Med Inform Assoc, 17(1):19–24, 2010. ", + "bbox": [ + 171, + 93, + 826, + 928 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Matthew D. Zeiler. ADADELTA: an adaptive learning rate method. CoRR, abs/1212.5701, 2012. ", + "bbox": [ + 169, + 103, + 810, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "M. L. Zhang and Z. H. Zhou. A review on multi-label learning algorithms. IEEE Transactions on Knowledge and Data Engineering, 26(8):1819–1837, Aug 2014. ", + "bbox": [ + 174, + 127, + 821, + 156 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX A. ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This appendix lists the optimized parameters for the different models. Except where noted, parameters were optimized under random search. ", + "bbox": [ + 176, + 133, + 821, + 161 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Recurrent Neural Network Models: (parameters marked with an asterisk were optimized with human-guided search.) ", + "bbox": [ + 173, + 174, + 823, + 203 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/9adf5af71fcf883dea74d8ee5583bd23c8d851fa45e0e908260b63d335c0f282.jpg", + "table_caption": [ + "Feed Forward Neural Network Model: " + ], + "table_footnote": [], + "table_body": "
ParameterModel
GRULSTM
Dropout for input gates0.10.25
Dropout for recurrent connections0.750.75
L1 applied to the input weights matrices00
L1 applied to the recurrent weights matrices00
L2 applied to the input weights matrices0.00010.0001
L2 applied to the recurrent weights matrices0.00010.001
L2 applied to the output layer's weights matrices0.00010.001
Dropout before the output layer0.50.5
*Number of recurrent layers33
400
*Number of nodes in recurrent units400
", + "bbox": [ + 266, + 213, + 735, + 411 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/d978db678b75b57f4381f75ea4dc39162ba044d4e34b42bc5032fd4b33e40d9d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ParameterValue
Dropout before the output layer0.1
Dropout between feed-forward layers Numberof feed-forward layers0.1
Activation function between feed-forward layers3
tanh
Number of nodes in feed-forward layers128
", + "bbox": [ + 300, + 463, + 696, + 564 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Random Forest Model (binary input): ", + "bbox": [ + 375, + 590, + 622, + 606 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Parameter ", + "text_level": 1, + "bbox": [ + 426, + 617, + 501, + 631 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Value ", + "bbox": [ + 692, + 617, + 735, + 631 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Number of estimators 800 \nRatio of features to consider when looking for the best split 0.4666 \nMinimum number of samples required to split an internal node 87 \nMinimum number of samples required to be at a leaf node 3 \nThe function to measure the quality of a split entropy ", + "bbox": [ + 256, + 645, + 743, + 717 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "APPENDIX B. ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This appendix lists the top level classes for International Statistical Classification of Diseases and Related Health Problems, Ninth Revision (ICD-9) and Anatomical Chemical Classification System (ATC). ", + "bbox": [ + 174, + 133, + 825, + 176 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "ICD-9 chapters. ", + "bbox": [ + 446, + 188, + 550, + 203 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Code range ", + "text_level": 1, + "bbox": [ + 181, + 215, + 266, + 229 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Description ", + "text_level": 1, + "bbox": [ + 506, + 215, + 589, + 229 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "001-139 Infectious and parasitic diseases 140-239 Neoplasms 240-279 Endocrine, nutritional and metabolic diseases, and immunity disorders 280-289 Diseases of the blood and blood-forming organs 290-319 Mental disorders 320-359 Diseases of the nervous system 360-389 Diseases of the sense organs 390-459 Diseases of the circulatory system 460-519 Diseases of the respiratory system 520-579 Diseases of the digestive system 580-629 Diseases of the genitourinary system 630-679 Complications of pregnancy, childbirth, and the puerperium 680-709 Diseases of the skin and subcutaneous tissue 710-739 Diseases of the musculoskeletal system and connective tissue 740-759 Congenital anomalies 760-779 Certain conditions originating in the perinatal period 780-799 Symptoms, signs, and ill-defined conditions 800-999 Injury and poisoning V01-V91 Supplementary - factors influencing health status and contact with health services E000-E999 Supplementary - external causes of injury and poisoning ", + "bbox": [ + 186, + 241, + 816, + 522 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Top level groups ATC codes and their corresponding colors used in Figure 4 and Appendix C. ", + "bbox": [ + 187, + 547, + 805, + 564 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Code ", + "text_level": 1, + "bbox": [ + 183, + 575, + 222, + 588 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Contents ", + "text_level": 1, + "bbox": [ + 464, + 575, + 529, + 589 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A Alimentary tract and metabolism \nB Blood and blood forming organs \nC Cardiovascular system \nD Dermatologicals \nG Genito-urinary system and sex hormones \nH Systemic hormonal preparations, excluding sex hormones and insulins \nJ Antiinfectives for systemic use \nL Antineoplastic and immunomodulating agents \nM Musculo-skeletal system \nN Nervous system \nP Antiparasitic products, insecticides and repellents \nR Respiratory system \nS Sensory organs \nV Various ", + "bbox": [ + 196, + 603, + 823, + 799 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "APPENDIX C. ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This appendix presents results from three illustrative cases from the dozen cases randomly selected for individual evaluation. ", + "bbox": [ + 176, + 133, + 823, + 161 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/792951becd11f4bc6d5ea7520d846bdf34c7ed8ad1c790667c2f6c104d2299a8.jpg", + "table_caption": [ + "CASE 1. " + ], + "table_footnote": [], + "table_body": "
ICD-9 codeCode descriptionTime estimate (ago)
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
273.1Monoclonal paraproteinemia4.8 months ago
285.9Anemia, unspecified4.8 months ago
276.50Volume depletion, unspecified4.8 months ago
733.00Osteoporosis,unspecified4.8 months ago
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
203.00Multiple myeloma,without mention of having achieved remission2.9 months ago
203.01Multiple myeloma, in remission2.9 months ago
273.1Monoclonal paraproteinemia2.9 months ago
273.1Monoclonal paraproteinemia1.6 months ago
279.3Unspecified immunity deficiency1.6 months ago
203.00Multiple myeloma, without mention of having achieved remission1.6 months ago
781.2Abnormality of gait3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.7 weeks ago
401.9Unspecified essential hypertension3.7 weeks ago
V12.54Personal history of transient ischemic attack (TIA),and cerebral infarction without residual deficits3.7 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.7 weeks ago
786.09Otherrespiratoryabnormalities3.7 weeks ago
273.1Monoclonal paraproteinemia3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.6 weeks ago
V58.69Long-term (current) use of other medications3.6 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.4 weeks ago
203.00Multiple myeloma, without mention of having achieved remission4 days ago
V42.82Peripheral stem cells replaced by transplant4 days ago
203.01Multiple myeloma, in remission3 days ago
38.97Central venous catheter placement with guidance3 days ago
V42.82Peripheral stem cells replaced by transplant3 daysago
V58.81Fitting and adjustment of vascular catheter3 days ago
203.00Multiple myeloma, without mention of having achieved remission3 daysago
V42.82Peripheral stem cells replaced by transplant2 days ago
203.01Multiple myeloma, in remission2 days ago
203.00Multiple myeloma, without mention of having achieved remission1 day ago
V42.82Peripheral stem cells replaced by transplant1 day ago
203.00Multiple myeloma, without mention of having achieved remissionnow
V42.82Peripheral stem cells replaced by transplantnow
", + "bbox": [ + 209, + 207, + 782, + 536 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/601a66fe0b0ec2214e2f5d221c00f14d95ed27ad2e8346ef5834b595295e14b4.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 174, + 590, + 818, + 806 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Medication predictions for a complicated patient. Each vertical bar represents the prediction for a single medication class, with the height of the bar representing the confidence of the prediction. Black labels above arrows indicate ATC therapeutic classes for medications the patient was actually taking. Colors and letters below the axis indicate high-level therapeutic class groups. ", + "bbox": [ + 174, + 818, + 825, + 875 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Predicted vs. actual medication classes for the patient in Case 1. The four-character sequence in the first and fourth columns is the ATC code for the medication therapeutic class, and an asterisk in the first column indicates that the predicted medication is in the actual medication list. Probabilities listed are the model predictions for the listed therapeutic class. In the predicted medications column, all predictions with probability at least 0.2 are listed. ", + "bbox": [ + 173, + 263, + 825, + 334 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/66be88aba6d21e0354ed9549d1616088fbf0ebda91f054da89323f91ce73ba31.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Top predictionsProb.True labelsProb.
Corticosteroids97.01%
S03B* S01C*Antiinflammatory agents and antiinfectives in combi-97.01% 95.54%S03B S01CCorticosteroids Antiinflammatory agents and antiinfectives in combi-95.54%
nationnation
S02B*Corticosteroids95.54%S02BCorticosteroids95.54%
L01AAlkylating agents94.00%D07XCorticosteroids,other combinations93.37%
D07X*Corticosteroids,other combinations93.37%H02ACorticosteroids for systemic use, plain91.06%
H02A*Corticosteroids for systemic use, plain91.06%D07ACorticosteroids, plain90.83%
D07A*Corticosteroids,plain90.83%S01BAntiinflammatory agents90.79%
S01B*Antiinflammatory agents90.79%D10AAnti-acne preparations for topical use88.56%
D10A*Anti-acne preparations for topical use88.56%C05AAgents for treatment of hemorrhoids and anal fissures for topical use88.52%
C05A*Agents for treatment of hemorrhoids and anal fissures for topical use88.52%R01ADecongestants and other nasal preparations for topi-87.02%
A04AAntiemetics and antinauseants87.95%J05Acal use Direct acting antivirals86.83%
R01A*Decongestants and other nasal preparations for topi- cal use87.02%A01AStomatological preparations86.11%
J05A*Direct acting antivirals86.8%N02AOpioids84.86%
A01A*Stomatological preparations86.11%B05CIrrigating solutions82.56%
N02A*Opioids84.86%A12COther mineral supplements79.50%
B05C*Irrigating solutions82.56%B05XI.V. solution additives74.84%
A12C*Other mineral supplements79.50%L04AImmunosuppressants68.76%
B05X*Iv. solution additives74.84%N02BOther analgesics and antipyretics57.24%
L04A*Immunosuppressants68.76%S01AAntiinfectives54.59%
N05AAntipsychotics58.64%J01DOther beta-lactam antibacterials43.40%
N02B*Other analgesics and antipyretics57.24%C03CHigh-ceiling diuretics39.88%
S01A*Antiinfectives54.59%J01MQuinolone antibacterials29.78%
L03AImmunostimulants45.96%C07ABeta blocking agents27.08%
A02BDrugs for peptic ulcer and gastro-oesophageal reflux44.56%
J01D*disease Other beta-lactam antibacterials43.40%N03AAntiepileptics
C03C*High-ceiling diuretics39.88%J01XOther antibacterials20.00%
B01AAntithrombotic agents37.80%M03BMuscle relaxants,centrally acting agents5.88%
V03AAll other therapeutic products34.18%5.09%
R06AAntihistamines for systemic use31.78%
A06ADrugs for constipation31.57%
J01M*Quinolone antibacterials29.78%
N05BAnxiolytics29.42%
D04AAntipruritics,incl. antihistamines,anesthetics, etc.27.62%
C07A*Beta blocking agents27.08%
L01XOther antineoplastic agents24.72%
R05CExpectorants,excl. combinations with cough sup-20.43%
N03A*pressants
", + "bbox": [ + 176, + 349, + 823, + 752 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/c7f36a14b7084031fbf068723cafe1b007cb10bb15113c53489b532efaf242d7.jpg", + "table_caption": [ + "CASE 2. " + ], + "table_footnote": [], + "table_body": "
ICD-9 codeCode descriptionTime estimate (ago)
735.4Other hammer toe (acquired)2.4 years ago
729.5Pain in limb2.4 years ago
244.1Other postablative hypothyroidism1.5 years ago
285.9Anemia, unspecified1.5 years ago
244.1Other postablative hypothyroidism1.2 years ago
244.1Other postablative hypothyroidism11.5 months ago
733.00Osteoporosis,unspecified11.5 months ago
733.01Senile osteoporosis7.7 months ago
268.9Unspecified vitamin D deficiency7.7 months ago
729.5Pain in limb7.7 months ago
174.9Malignant neoplasm of breast (female),unspecified7.7 months ago
722.52Degeneration of lumbar or lumbosacral intervertebral disc7.7 months ago
279.3Unspecified immunity deficiency7.7 months ago
733.01Senile osteoporosis6.4 months ago
733.01Senile osteoporosis6.2 months ago
244.1Other postablative hypothyroidism6.0 months ago
401.1Benign essential hypertension6.0 months ago
V58.69Long-term (current) use of other medications1.9 weeks ago
733.01Senile osteoporosisnow
244.1Other postablative hypothyroidismnow
V58.69Long-term (current) use of other medicationsnow
", + "bbox": [ + 289, + 132, + 702, + 338 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/6287f63c5c8aeeed296b53e77356fc5cd1f158e5b75c60bcf4f81d5fc25919b1.jpg", + "table_caption": [ + "Predicted vs. actual medication classes for Case 2. Table structure as in case 1. " + ], + "table_footnote": [], + "table_body": "
Top predictionsProb.True labelsProb.
M05BDrugs affecting bone structure and mineralization88.18%A11CVitamin a and d, incl. combinations of the two39.42%
H03AThyroid preparations84.82%N06AAntidepressants20.88%
H05AParathyroid hormones and analogues66.33%C10ALipid modifying agents,plain17.05%
A11C*39.42%N03AAntiepileptics15.61%
N02BVitamin a and d, incl. combinations of the two Other analgesics and antipyretics37.58%C09C
A01AStomatological preparations23.05%L02BAngiotensin ii antagonists,plain10.38% 4.22%
A12A21.59%Hormone antagonists and related agents
N06A*Calcium Antidepressants20.88%
C07A
Beta blocking agents20.81%
", + "bbox": [ + 176, + 421, + 823, + 522 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/ced88c51ba51d971d4a1ae6b9036511aa51abe45adb9d0f96e717197c40ff15d.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 173, + 551, + 823, + 752 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Medication predictions for a simpler patient. Note that the high-prediction medications are clinically reasonable given the billing codes in the sequence. Figure representation as in case 1. ", + "bbox": [ + 171, + 763, + 823, + 794 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/bf23979fd5de43a049055f6a88c60f604dbbd2ae9cdadc9cddd8495dd18e8eb3.jpg", + "table_caption": [ + "CASE 3. ", + "Predicted vs. actual medication classes for Case 3. Table structure as in case 1. " + ], + "table_footnote": [], + "table_body": "
ICD-9 codeCodedescriptionTime estimate (ago)
332.0Paralysis agitans5.0 years ago
332.0Paralysis agitans4.7 years ago
332.0Paralysis agitans4.5 years ago
332.0Paralysis agitans4.0 years ago
332.0Paralysis agitans3.5 years ago
332.0Paralysis agitans3.0 years ago
332.0Paralysis agitans2.7 years ago
332.0Paralysis agitans2.4 years ago
332.0Paralysis agitans2.0 years ago
332.0Paralysis agitans1.7 years ago
332.0Paralysis agitans
332.0Paralysis agitans1.0 years ago
332.09.9 months ago
Paralysis agitans4.1 months ago
332.0Paralysis agitansnow
", + "bbox": [ + 367, + 132, + 627, + 277 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/647f6f150e88634b289fe4474f49576c0f5151752fa240698a1ce54917ac379d.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 171, + 356, + 825, + 664 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Medication predictions for a patient with only one ICD-9 code, repeated many times over five years. The medications listed under true labels are not indicated for paralysis agitans (Parkinson’s disease), but the patient was surely taking them for reasons not documented in the ICD-9 sequence. The model predicted mostly reasonable medications for a patient with Parkinson’s disease, especially Dopaminergic agents, which is the primary treatment for the disease. Figure representation as in case 1, above. 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Additionally, examining individual cases revealed that many of the predic-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 375, + 469, + 386 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 469, + 386 + ], + "score": 1.0, + "content": "tions marked incorrect were likely to be examples of either omitted medications", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 385, + 469, + 397 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 469, + 397 + ], + "score": 1.0, + "content": "or omitted billing codes, supporting our assertion of a substantial number of er-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 396, + 469, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 396, + 469, + 408 + ], + "score": 1.0, + "content": "rors and omissions in the data, and the likelihood of models such as these to help", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 408, + 197, + 418 + ], + "spans": [ + { + "bbox": [ + 142, + 408, + 197, + 418 + ], + "score": 1.0, + "content": "correct them.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 15.5, + "bbox_fs": [ + 141, + 243, + 470, + 418 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 206, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 208, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 208, + 452 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 504, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "The idea of exploiting the large amounts of data captured in electronic medical records for both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "clinical care and secondary research holds great promise, but its potential is weakened by errors and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "omissions in those records (Safran et al., 2007; de Lusignan & van Weel, 2006). Among many other", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "problems, accurately capturing the list of medications currently taken by a given patient is extremely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 324, + 518 + ], + "score": 1.0, + "content": "challenging (Velo & Minuz, 2009). In one study, over", + "type": "text" + }, + { + "bbox": [ + 325, + 506, + 344, + 516 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "of electronic medication lists contained", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 299, + 528 + ], + "score": 1.0, + "content": "omissions (Caglar et al., 2011), and in another,", + "type": "text" + }, + { + "bbox": [ + 299, + 516, + 319, + 527 + ], + "score": 0.88, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "of all medications taken by patients were not", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "recorded (Kaboli et al., 2004). Even medication lists provided by the patients themselves contain", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 538, + 313, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 313, + 550 + ], + "score": 1.0, + "content": "multiple errors and omissions (Green et al., 2010) .", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 461, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Many efforts have been made to ensure the correctness of medication lists, most of them involving", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "improved communication between patients and providers (Keogh et al., 2016), but these efforts", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "have not yet been successful, and incorrect or incomplete medication documentation continues to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "be a source of error in computational medical research. In this work we attempt to identify likely", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "errors and omissions in the record, predicting the set of active medications from the sequence of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "most recent disease-based billing codes in the record. Predictions from such a model could be used", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "either in manual medication reconciliation (a common process undertaken to correct the medication", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "record) or to provide a prior to other models, such as an NLP model attempting to extract medication", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 248, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 248, + 655 + ], + "score": 1.0, + "content": "use from the narrative clinical text.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 554, + 505, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "Given the sequential nature of clinical data, we suspected that recurrent neural networks would be a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "good architecture for making these predictions. In this work we investigate this potential, comparing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 479, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 479, + 694 + ], + "score": 1.0, + "content": "the performance of recurrent networks to that of similarly-configured feed forward networks.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 660, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "The input for each case is a sequence of ICD-9 billing codes (Section 2.1), for which the model", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "produces a single, multi-label prediction of the therapeutic classes (Section 3.1) of medications", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 442, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 442, + 734 + ], + "score": 1.0, + "content": "taken by the patient during the period of time covered by the billing code sequence.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 698, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "This work is designed to test how well the complete set of medications a patient is actively taking at", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "a given moment can be predicted by the sequence of diagnostic billing codes leading up to that mo-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "ment, in the context of non-trivial label noise. It also explores whether sequence-oriented recursive", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 451, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 451, + 127 + ], + "score": 1.0, + "content": "neural nets can do a better job of that prediction than standard feed-forward networks.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 145, + 200, + 158 + ], + "lines": [ + { + "bbox": [ + 104, + 143, + 202, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 143, + 202, + 161 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 171, + 245, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 246, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 246, + 185 + ], + "score": 1.0, + "content": "2.1 MEDICAL BILLING CODES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "Each time a patient has billable contact with the healthcare system, one or more date-stamped billing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "codes are attached to the patient record, indicating the medical conditions that are associated (or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "suspected to be associated) with the reason for the visit. While these codes are notoriously unreliable", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "because they are only used for billing and not actual clinical practice (O’Malley et al., 2005), they are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 235, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 250 + ], + "score": 1.0, + "content": "nevertheless useful in a research context (Bastarache & Denny, 2011; Denny et al., 2010), especially", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "if they are used probabilistically (Lasko, 2014). In our institution, codes from the International", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "Classification of Diseases, Ninth Revision (ICD-9) have historically been used, although we have", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 270, + 471, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 471, + 282 + ], + "score": 1.0, + "content": "recently transitioned to the tenth revision (ICD-10). For this project, we used ICD-9 codes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 108, + 286, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 300 + ], + "score": 1.0, + "content": "The ICD-9 hierarchy consists of 21 chapters roughly corresponding to a single organ system or", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "pathologic class (Appendix B). Leaf-level codes in that tree represent single diseases or disease", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "subtypes. For this project, we used a subset of the two thousand most common leaf-level codes as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 320, + 167, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 167, + 332 + ], + "score": 1.0, + "content": "our input data.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 353, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 354, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 354, + 360 + ], + "score": 1.0, + "content": "2.2 RECURRENT NEURAL NETWORKS AND VARIATIONS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 109, + 368, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "Most of the ICLR community are very familiar with recurrent neural networks and their variations,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 107, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "but we include a conceptual description of them here for readers coming from other fields. More", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 390, + 404, + 402 + ], + "spans": [ + { + "bbox": [ + 107, + 390, + 404, + 402 + ], + "score": 1.0, + "content": "thorough descriptions are available elsewhere (Graves, 2012; Olah, 2015).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 438, + 419 + ], + "score": 1.0, + "content": "A recurrent neural network is a variation in which the output of one node on input", + "type": "text" + }, + { + "bbox": [ + 439, + 408, + 449, + 418 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "loops around", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 282, + 430 + ], + "score": 1.0, + "content": "to become an input to another node on input", + "type": "text" + }, + { + "bbox": [ + 283, + 420, + 303, + 430 + ], + "score": 0.89, + "content": "x _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 418, + 505, + 430 + ], + "score": 1.0, + "content": ", allowing information to be preserved as it iterates", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "over an input data sequence (Figure 1). They were introduced in the 1980s (Rumelhart et al., 1986),", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "but achieved explosive popularity only recently, after the development of methods to more reliably", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "capture long-term dependencies, which significantly improved their performance on sequence-to-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 419, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 419, + 474 + ], + "score": 1.0, + "content": "sequence mapping (Hochreiter & Schmidhuber, 1997; Sutskever et al., 2014).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "The basic RNN unit has a simple internal structure (Figure 2a). Output from the previous iteration", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 127, + 501 + ], + "score": 0.9, + "content": "h _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 490, + 266, + 502 + ], + "score": 1.0, + "content": "and the next input in a sequence", + "type": "text" + }, + { + "bbox": [ + 266, + 492, + 277, + 501 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "are both fed to the network on the next iteration. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Long Short-Term Memory configuration (LSTM) introduces new, more complex internal structure", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 380, + 524 + ], + "score": 1.0, + "content": "(Figure 2b) consisting of four neural network layers and a cell state", + "type": "text" + }, + { + "bbox": [ + 380, + 512, + 395, + 523 + ], + "score": 0.8, + "content": "\\left( c _ { t } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ", which is carried from one", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "iteration to another. The additional layers form forget, input and output gates, which allow for the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 375, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 375, + 546 + ], + "score": 1.0, + "content": "information to be forgotten (reset) or passed on to varying degrees.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "The LSTM model and its variations are commonly used in applications where sequence and temporal", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "data are involved, such as in image captioning (Vinyals et al., 2014), language translation (Sutskever", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "et al., 2014), and speech recognition (Graves et al., 2013). In many cases LSTM models define the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "state of the art, such as with a recent conversational speech recognizer that (slightly) outperforms", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 594, + 306, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 306, + 607 + ], + "score": 1.0, + "content": "professional transcriptionists (Xiong et al., 2016).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "A recent variation on the LSTM architecture is the Gated Recurrent Unit (GRU) (Cho et al., 2014),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "which introduces a single update gate in place of input and forget gates (Figure 2c). GRUs perform", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "as well as or better than LSTMs in many cases (Chung et al., 2014; Jozefowicz et al., 2015), and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 317, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 317, + 657 + ], + "score": 1.0, + "content": "have the additional advantage of a simpler structure.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 393, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 394, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 394, + 675 + ], + "score": 1.0, + "content": "In this work we try both an LSTM and a GRU on our learning problem.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 108, + 688, + 202, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 205, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 205, + 701 + ], + "score": 1.0, + "content": "2.3 RELATED WORK", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Little research in the computational medical domain has used recurrent neural networks. The ear-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "liest example we are aware of is the use of an LSTM model that produced reasonable accuracy", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "This work is designed to test how well the complete set of medications a patient is actively taking at", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "a given moment can be predicted by the sequence of diagnostic billing codes leading up to that mo-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "ment, in the context of non-trivial label noise. It also explores whether sequence-oriented recursive", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 451, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 451, + 127 + ], + "score": 1.0, + "content": "neural nets can do a better job of that prediction than standard feed-forward networks.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 127 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 145, + 200, + 158 + ], + "lines": [ + { + "bbox": [ + 104, + 143, + 202, + 161 + ], + "spans": [ + { + "bbox": [ + 104, + 143, + 202, + 161 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 171, + 245, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 246, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 246, + 185 + ], + "score": 1.0, + "content": "2.1 MEDICAL BILLING CODES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 192, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "Each time a patient has billable contact with the healthcare system, one or more date-stamped billing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "codes are attached to the patient record, indicating the medical conditions that are associated (or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "suspected to be associated) with the reason for the visit. While these codes are notoriously unreliable", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "because they are only used for billing and not actual clinical practice (O’Malley et al., 2005), they are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 235, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 250 + ], + "score": 1.0, + "content": "nevertheless useful in a research context (Bastarache & Denny, 2011; Denny et al., 2010), especially", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "if they are used probabilistically (Lasko, 2014). In our institution, codes from the International", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "Classification of Diseases, Ninth Revision (ICD-9) have historically been used, although we have", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 270, + 471, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 471, + 282 + ], + "score": 1.0, + "content": "recently transitioned to the tenth revision (ICD-10). 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For this project, we used a subset of the two thousand most common leaf-level codes as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 320, + 167, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 167, + 332 + ], + "score": 1.0, + "content": "our input data.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 285, + 506, + 332 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 353, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 354, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 354, + 360 + ], + "score": 1.0, + "content": "2.2 RECURRENT NEURAL NETWORKS AND VARIATIONS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 109, + 368, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "Most of the ICLR community are very familiar with recurrent neural networks and their variations,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 107, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "but we include a conceptual description of them here for readers coming from other fields. More", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 390, + 404, + 402 + ], + "spans": [ + { + "bbox": [ + 107, + 390, + 404, + 402 + ], + "score": 1.0, + "content": "thorough descriptions are available elsewhere (Graves, 2012; Olah, 2015).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 367, + 505, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 438, + 419 + ], + "score": 1.0, + "content": "A recurrent neural network is a variation in which the output of one node on input", + "type": "text" + }, + { + "bbox": [ + 439, + 408, + 449, + 418 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "loops around", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 282, + 430 + ], + "score": 1.0, + "content": "to become an input to another node on input", + "type": "text" + }, + { + "bbox": [ + 283, + 420, + 303, + 430 + ], + "score": 0.89, + "content": "x _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 418, + 505, + 430 + ], + "score": 1.0, + "content": ", allowing information to be preserved as it iterates", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "over an input data sequence (Figure 1). They were introduced in the 1980s (Rumelhart et al., 1986),", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "but achieved explosive popularity only recently, after the development of methods to more reliably", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "capture long-term dependencies, which significantly improved their performance on sequence-to-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 419, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 419, + 474 + ], + "score": 1.0, + "content": "sequence mapping (Hochreiter & Schmidhuber, 1997; Sutskever et al., 2014).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 407, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "The basic RNN unit has a simple internal structure (Figure 2a). Output from the previous iteration", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 127, + 501 + ], + "score": 0.9, + "content": "h _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 490, + 266, + 502 + ], + "score": 1.0, + "content": "and the next input in a sequence", + "type": "text" + }, + { + "bbox": [ + 266, + 492, + 277, + 501 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "are both fed to the network on the next iteration. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Long Short-Term Memory configuration (LSTM) introduces new, more complex internal structure", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 380, + 524 + ], + "score": 1.0, + "content": "(Figure 2b) consisting of four neural network layers and a cell state", + "type": "text" + }, + { + "bbox": [ + 380, + 512, + 395, + 523 + ], + "score": 0.8, + "content": "\\left( c _ { t } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ", which is carried from one", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "iteration to another. 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We obtained", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 222, + 289, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 289, + 234 + ], + "score": 1.0, + "content": "IRB approval to use this data in this research.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "For this experiment we filtered all records in our database to include only the top 1,000 most common", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 201, + 262 + ], + "score": 1.0, + "content": "medications and the top", + "type": "text" + }, + { + "bbox": [ + 201, + 250, + 245, + 260 + ], + "score": 0.86, + "content": "m = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 249, + 408, + 262 + ], + "score": 1.0, + "content": "most common billing codes, which cover", + "type": "text" + }, + { + "bbox": [ + 408, + 250, + 435, + 261 + ], + "score": 0.87, + "content": "9 9 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "of all medication", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 172, + 273 + ], + "score": 1.0, + "content": "occurrences and", + "type": "text" + }, + { + "bbox": [ + 173, + 261, + 199, + 271 + ], + "score": 0.87, + "content": "8 5 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "of all billing code occurrences. We then included all records from the filtered", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 272, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 504, + 284 + ], + "score": 1.0, + "content": "data that had at least one medication occurrence and at least ten billing code occurrences. This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 281, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 506, + 296 + ], + "score": 1.0, + "content": "resulted in 610,076 complete patient records, which we divided 80/5/15 into training, validation,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 294, + 180, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 180, + 306 + ], + "score": 1.0, + "content": "and final test sets.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 172, + 324 + ], + "score": 1.0, + "content": "A data instance", + "type": "text" + }, + { + "bbox": [ + 172, + 311, + 231, + 322 + ], + "score": 0.93, + "content": "d = \\{ E , T , y \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 310, + 330, + 324 + ], + "score": 1.0, + "content": "consisted of a sequence", + "type": "text" + }, + { + "bbox": [ + 331, + 311, + 406, + 322 + ], + "score": 0.91, + "content": "E = \\{ e _ { 1 } , \\ldots , e _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 310, + 506, + 324 + ], + "score": 1.0, + "content": ", of one-hot billing code", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 319, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 319, + 137, + 335 + ], + "score": 1.0, + "content": "vectors", + "type": "text" + }, + { + "bbox": [ + 138, + 322, + 190, + 333 + ], + "score": 0.9, + "content": "e _ { i } \\in \\{ 0 , 1 \\} ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 319, + 297, + 335 + ], + "score": 1.0, + "content": "and their associated times", + "type": "text" + }, + { + "bbox": [ + 297, + 322, + 397, + 334 + ], + "score": 0.86, + "content": "T = \\{ t _ { 1 } , \\dots , t _ { n } \\} , t _ { i } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 319, + 506, + 335 + ], + "score": 1.0, + "content": "as input, and a multi-label", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 331, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 133, + 345 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 134, + 333, + 181, + 344 + ], + "score": 0.92, + "content": "y \\in \\{ 0 , 1 \\} ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 331, + 415, + 345 + ], + "score": 1.0, + "content": "of medication classes as the output target. 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All medications that occurred during", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 173, + 378 + ], + "score": 1.0, + "content": "the time span of", + "type": "text" + }, + { + "bbox": [ + 173, + 366, + 182, + 375 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 365, + 351, + 378 + ], + "score": 1.0, + "content": "were then collected into the output vector", + "type": "text" + }, + { + "bbox": [ + 352, + 367, + 358, + 377 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 365, + 505, + 378 + ], + "score": 1.0, + "content": ". Practice patterns change over time,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "so simply taking the most recent 100 codes for each patient could produce a biased result. To avoid", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "this, we chose random reference points, stratified by medication. 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The top level is a broad categorization of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "medications (Appendix B), the bottom (fifth) level is individual medications, and we used the third", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "level, which contains 287 therapeutic classes of the approximately appropriate abstraction level for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "our purpose. We used a publicly available mapping2 to translate between our medication names and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "ATC codes, with manual mapping for the minority of medications that had no mapping entry. Our", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 203, + 569 + ], + "score": 1.0, + "content": "set of medications used", + "type": "text" + }, + { + "bbox": [ + 204, + 558, + 240, + 568 + ], + "score": 0.9, + "content": "k = 1 8 2", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "third-level ATC codes, rendering our output label a 182-element-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 311, + 581 + ], + "score": 1.0, + "content": "long multi-label vector, in which an element is set", + "type": "text" + }, + { + "bbox": [ + 311, + 569, + 340, + 580 + ], + "score": 0.91, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "if a medication in that class appeared in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 308, + 593 + ], + "score": 1.0, + "content": "the set of medications identified for that instance,", + "type": "text" + }, + { + "bbox": [ + 308, + 581, + 337, + 591 + ], + "score": 0.9, + "content": "y _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "otherwise. Some medications mapped to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 591, + 326, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 235, + 603 + ], + "score": 1.0, + "content": "more than one class, and we set", + "type": "text" + }, + { + "bbox": [ + 235, + 591, + 263, + 602 + ], + "score": 0.91, + "content": "y _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 591, + 326, + 603 + ], + "score": 1.0, + "content": "for all of them.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "Our medication data was collected from structured order entry records and extracted using NLP (Xu", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "et al., 2010) from mentions in the narrative text of a patient record that included the medication", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "name, dose, route and frequency. 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This is because ICD-9 codes are represented sequentially in our source data,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "but medications are not. They are represented as a list that changes over time in the record. 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We obtained", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 222, + 289, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 289, + 234 + ], + "score": 1.0, + "content": "IRB approval to use this data in this research.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 189, + 506, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 239, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "For this experiment we filtered all records in our database to include only the top 1,000 most common", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 201, + 262 + ], + "score": 1.0, + "content": "medications and the top", + "type": "text" + }, + { + "bbox": [ + 201, + 250, + 245, + 260 + ], + "score": 0.86, + "content": "m = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 249, + 408, + 262 + ], + "score": 1.0, + "content": "most common billing codes, which cover", + "type": "text" + }, + { + "bbox": [ + 408, + 250, + 435, + 261 + ], + "score": 0.87, + "content": "9 9 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "of all medication", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 172, + 273 + ], + "score": 1.0, + "content": "occurrences and", + "type": "text" + }, + { + "bbox": [ + 173, + 261, + 199, + 271 + ], + "score": 0.87, + "content": "8 5 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "of all billing code occurrences. 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Practice patterns change over time,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "so simply taking the most recent 100 codes for each patient could produce a biased result. To avoid", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "this, we chose random reference points, stratified by medication. 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376, + 505, + 390 + ], + "score": 1.0, + "content": "because they tend to perform well on widely varying datasets (Fernandez-Delgado et al., 2014), they ´", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "are efficient to train and test, and they don’t require a huge effort to optimize (in order to produce a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 399, + 177, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 177, + 411 + ], + "score": 1.0, + "content": "fair comparison).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 109, + 427, + 273, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 275, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 275, + 439 + ], + "score": 1.0, + "content": "3.3 CONSTANT-PREVALENCE MODEL", + "type": "text" + } + ], + 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For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "example, if there were three possible medications, with prevalences of 0.3, 0.9, and 0.2, then the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "prediction of this model would be a constant [0.3, 0.9, 0.2] for each instance. We include this model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "in order to mitigate the fact that while all of our evaluation measures are suitable for comparing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "models on the same data, some are not well suited for external comparison because they depend, for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "example, on the prevalence of positive labels (Section 3.4). 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For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "example, if there were three possible medications, with prevalences of 0.3, 0.9, and 0.2, then the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "prediction of this model would be a constant [0.3, 0.9, 0.2] for each instance. We include this model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "in order to mitigate the fact that while all of our evaluation measures are suitable for comparing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "models on the same data, some are not well suited for external comparison because they depend, for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "example, on the prevalence of positive labels (Section 3.4). By including this model we can at least", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 514, + 302, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 302, + 527 + ], + "score": 1.0, + "content": "establish a true minimum baseline for reference.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 448, + 506, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 542, + 187, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 189, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 189, + 555 + ], + "score": 1.0, + "content": "3.4 EVALUATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 564, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "Our main evaluation focused on the models, although we also performed a separate evaluation of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 572, + 171, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 171, + 589 + ], + "score": 1.0, + "content": "the embedding.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 563, + 505, + 589 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 601, + 177, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 179, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 179, + 614 + ], + "score": 1.0, + "content": "3.4.1 MODELS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "There are several possibilities for evaluation in a multi-label classification context (Sechidis et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "2011; Zhang & Zhou, 2014). We chose micro-averaged area under the ROC curve (AUC) and la-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "bel ranking loss as the primary methods of evaluation, because they treat each instance with equal", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "weight, regardless of the nature of the positive labels for that instance. In other words, we wanted", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "primary measures that did not give a scoring advantage to instances with either very many or very", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "few positive labels, or that included very rare or very prevalent labels. 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Both of these", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 352, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 352, + 431 + ], + "score": 1.0, + "content": "depend on the prevalence of positive labels in a test instance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 194, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 195, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 195, + 456 + ], + "score": 1.0, + "content": "3.4.2 EMBEDDING", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "We evaluated the embedding based on how strongly related in a clinical semantic sense", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "the nearest neighbor to each code is (in the embedding space). A licensed physi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 504, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 473, + 498 + ], + "score": 1.0, + "content": "cian manually annotated the list of all 2000 codes with its match category", + "type": "text" + }, + { + "bbox": [ + 474, + 486, + 504, + 496 + ], + "score": 0.82, + "content": "m \\in", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "{strongly related, loosely related, unrelated}, and we computed the empirical", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 192, + 519 + ], + "score": 1.0, + "content": "marginal probability", + "type": "text" + }, + { + "bbox": [ + 192, + 507, + 217, + 519 + ], + "score": 0.92, + "content": "P ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 506, + 443, + 519 + ], + "score": 1.0, + "content": "of each category, the empirical conditional probability", + "type": "text" + }, + { + "bbox": [ + 443, + 507, + 477, + 519 + ], + "score": 0.95, + "content": "P ( m | d )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 364, + 531 + ], + "score": 1.0, + "content": "match category given the nearest neighbor (Manhattan) distance", + "type": "text" + }, + { + "bbox": [ + 364, + 518, + 371, + 528 + ], + "score": 0.73, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "and the empirical marginal prob-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 527, + 445, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 134, + 542 + ], + "score": 1.0, + "content": "ability", + "type": "text" + }, + { + "bbox": [ + 135, + 528, + 156, + 541 + ], + "score": 0.91, + "content": "P ( \\bar { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 527, + 284, + 542 + ], + "score": 1.0, + "content": ". For comparison, we computed", + "type": "text" + }, + { + "bbox": [ + 285, + 529, + 309, + 540 + ], + "score": 0.92, + "content": "P ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 527, + 445, + 542 + ], + "score": 1.0, + "content": "under 100 random code pairings.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 557, + 264, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 266, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 266, + 573 + ], + "score": 1.0, + "content": "4 RESULTS AND DISCUSSION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "The GRU model had the top performance by all measures, although the LSTM was a close second", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "(Table 1), a performance pattern consistent with previous reports (Chung et al., 2014). The deep", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "neural net performance was about 0.01 worse in both measures, suggesting that the recurrent models", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "were able to use the sequence information, but only to a small advantage over the most similar non-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "temporal architecture. 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For comparison, we computed", + "type": "text" + }, + { + "bbox": [ + 285, + 529, + 309, + 540 + ], + "score": 0.92, + "content": "P ( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 527, + 445, + 542 + ], + "score": 1.0, + "content": "under 100 random code pairings.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 462, + 505, + 542 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 557, + 264, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 266, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 266, + 573 + ], + "score": 1.0, + "content": "4 RESULTS AND DISCUSSION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "The GRU model had the top performance by all measures, although the LSTM was a close second", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "(Table 1), a performance pattern consistent with previous reports (Chung et al., 2014). The deep", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "neural net performance was about 0.01 worse in both measures, suggesting that the recurrent models", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "were able to use the sequence information, but only to a small advantage over the most similar non-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "temporal architecture. However, we note that both RNNs’ performance peaked at the top end of our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "tractable range for model size, while the feed-forward network peaked using a model about one third", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "that size (Appendix A). Experimenting with the architecture, we found that increasing the number", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "of nodes or layers for the feed-forward network increased training time but not performance. This", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "suggests that the RNN performance was limited by the hardware available, and increasing the size", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "of the model may further increase performance, and that the feed-forward network was limited by", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 171, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 171, + 705 + ], + "score": 1.0, + "content": "something else.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 582, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "Both random forest models were weaker than the deep neural net, as might be expected from the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 405, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 405, + 733 + ], + "score": 1.0, + "content": "need to resort to binary and bag-of-codes representations of the input data.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 115, + 505, + 230 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 1: Results of multi-label classification for each model. Baseline is the constant-prevalence", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 449, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 449, + 104 + ], + "score": 1.0, + "content": "model. Perfect is the best possible performance for our data under the given measure.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 115, + 505, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 115, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 115, + 505, + 230 + ], + "score": 0.982, + "html": "
ModelMicro-AUCLabel Ranking LossMacro-AUCLabel Ranking Avg. PrecisionCoverage Error
0.861
GRU LSTM0.927 0.9260.076 0.0770.8590.603 0.60062.6 63.0
NN0.9160.0860.8350.57067.3
0.9030.1020.8040.52373.7
RF (binary) RF (counts)0.8940.1110.7870.49777.3
Baseline0.8280.1720.5000.35597.2
Perfect1.00.01.01.015.0
", + "type": "table", + "image_path": "f4d0c69c2675d7da7a32eff980cc32128dcebf9bfe9ca17a6347f3bc58aec382.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 107, + 115, + 505, + 153.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 107, + 153.33333333333334, + 505, + 191.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 107, + 191.66666666666669, + 505, + 230.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "A natural question is what performance is good enough for clinical use. While there is little clinical", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "experience with multi-label classifiers, we would generally expect clinicians using a binary classifier", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 224, + 292 + ], + "score": 1.0, + "content": "in an advisory role to find an", + "type": "text" + }, + { + "bbox": [ + 225, + 279, + 272, + 291 + ], + "score": 0.84, + "content": "\\mathrm { A U C } \\gtrsim 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 279, + 341, + 292 + ], + "score": 1.0, + "content": "to be useful, and", + "type": "text" + }, + { + "bbox": [ + 342, + 279, + 394, + 291 + ], + "score": 0.84, + "content": "\\mathrm { A U C } \\gtrsim 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "to be very useful. An AUC", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 290, + 421, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 421, + 302 + ], + "score": 1.0, + "content": "difference of 0.01, and perhaps 0.005 are potentially noticeable in clinical use.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 127, + 318 + ], + "score": 1.0, + "content": "This", + "type": "text" + }, + { + "bbox": [ + 127, + 307, + 161, + 318 + ], + "score": 0.44, + "content": "0 . 9 / 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "rule of thumb may loosely translate to our AUC variants, but it can directly translate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 203, + 331 + ], + "score": 1.0, + "content": "to Label Ranking Loss", + "type": "text" + }, + { + "bbox": [ + 204, + 318, + 218, + 329 + ], + "score": 0.88, + "content": "L _ { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 317, + 410, + 331 + ], + "score": 1.0, + "content": "(2). If we think of a single output prediction", + "type": "text" + }, + { + "bbox": [ + 410, + 317, + 457, + 330 + ], + "score": 0.94, + "content": "\\hat { y } \\in [ 0 , 1 ] ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "as a set of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 167, + 341 + ], + "score": 1.0, + "content": "predictions for", + "type": "text" + }, + { + "bbox": [ + 168, + 330, + 174, + 339 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 329, + 252, + 341 + ], + "score": 1.0, + "content": "binary labels, then", + "type": "text" + }, + { + "bbox": [ + 252, + 329, + 293, + 340 + ], + "score": 0.51, + "content": "1 - \\mathrm { A U C }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 329, + 460, + 341 + ], + "score": 1.0, + "content": "for that set of predictions is equivalent to", + "type": "text" + }, + { + "bbox": [ + 460, + 330, + 475, + 340 + ], + "score": 0.88, + "content": "L _ { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 174, + 353 + ], + "score": 1.0, + "content": "original instance", + "type": "text" + }, + { + "bbox": [ + 175, + 340, + 181, + 352 + ], + "score": 0.8, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 339, + 268, + 353 + ], + "score": 1.0, + "content": ". Therefore, values of", + "type": "text" + }, + { + "bbox": [ + 268, + 340, + 309, + 352 + ], + "score": 0.91, + "content": "L _ { R } \\lesssim 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 339, + 426, + 353 + ], + "score": 1.0, + "content": "may be clinically useful, and", + "type": "text" + }, + { + "bbox": [ + 427, + 340, + 473, + 352 + ], + "score": 0.9, + "content": "L _ { R } \\lesssim 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "may be", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 351, + 156, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 156, + 365 + ], + "score": 1.0, + "content": "very useful.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "Subjectively examining performance on 20 randomly selected cases, we find very good detailed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "predictions, but also evidence of both missing medications and missing billing codes. An example", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "of a good set of detailed predictions is from a complex patient suffering from multiple myeloma (a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "type of cancer) with various complications. This patient was taking 26 medications, 24 of which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "had moderate to high probability predictions (Figure 4). (We have found by eyeball that a prediction", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "cutoff of 0.2 gives a reasonable balance between sensitivity and specificity for our model.) In the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "other direction, only two of the high-prediction classes were not actually being taken, but those", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "classes, along with several of the other moderately-predicted classes, are commonly used for cancer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "and are clinically reasonable for the case. (Details of this and the two cases below are in Appendix", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 466, + 123, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 123, + 480 + ], + "score": 1.0, + "content": "C).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "A good example of missing medications is a case in which the record has multiple billing codes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 508 + ], + "score": 1.0, + "content": "for both osteoporosis (which is very commonly treated with medication) and postablative hypothy-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "roidism (a deliberately induced condition that is always treated with medication), but no medications", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "of the appropriate classes were in the record. The GRU model predicted both of these classes, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 526, + 254, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 254, + 541 + ], + "score": 1.0, + "content": "the patient was almost surely taking.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "A good example of either missing billing codes or discontinued medications that remain documented", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "as active is a case in which the record has at least five years of data consisting only of codes for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Parkinson’s disease, but which lists medications for high cholesterol, hypertension, and other heart", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "disease. The GRU model predicted a reasonable set of medications for Parkinson’s disease and its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 477, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 477, + 601 + ], + "score": 1.0, + "content": "complications, but did not predict the other medications that are not suggested by the record.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "Given how easy it was to find cases with apparently missing codes and medications, we conclude", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "that there is indeed a substantial amount of label noise in our data, and we therefore interpret our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "models’ performance as lower bounds on the actual performance. We are encouraged that this kind", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "of a model may actually be useful for identifying missing medications in the record, but of course", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "a more thorough validation, and possibly a more accurate model, would be necessary before using", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "in a clinical scenario. A definitive experiment would use off-line research, including reconciling", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "information from various electronic and human sources to establish the ground truth of which med-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "ications were being taken on a particular day, but such efforts are labor intensive and expensive, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 693, + 286, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 286, + 705 + ], + "score": 1.0, + "content": "can only be conducted on a very small scale.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "An interesting byproduct of these models is the semantic embedding of ICD-9 codes used in the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "recurrent networks (Figure 5). Transforming input to a semantic embedding is a common pre-", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 115, + 505, + 230 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 1: Results of multi-label classification for each model. Baseline is the constant-prevalence", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 449, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 449, + 104 + ], + "score": 1.0, + "content": "model. Perfect is the best possible performance for our data under the given measure.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 115, + 505, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 115, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 115, + 505, + 230 + ], + "score": 0.982, + "html": "
ModelMicro-AUCLabel Ranking LossMacro-AUCLabel Ranking Avg. PrecisionCoverage Error
0.861
GRU LSTM0.927 0.9260.076 0.0770.8590.603 0.60062.6 63.0
NN0.9160.0860.8350.57067.3
0.9030.1020.8040.52373.7
RF (binary) RF (counts)0.8940.1110.7870.49777.3
Baseline0.8280.1720.5000.35597.2
Perfect1.00.01.01.015.0
", + "type": "table", + "image_path": "f4d0c69c2675d7da7a32eff980cc32128dcebf9bfe9ca17a6347f3bc58aec382.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 107, + 115, + 505, + 153.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 107, + 153.33333333333334, + 505, + 191.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 107, + 191.66666666666669, + 505, + 230.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "A natural question is what performance is good enough for clinical use. While there is little clinical", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "experience with multi-label classifiers, we would generally expect clinicians using a binary classifier", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 279, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 224, + 292 + ], + "score": 1.0, + "content": "in an advisory role to find an", + "type": "text" + }, + { + "bbox": [ + 225, + 279, + 272, + 291 + ], + "score": 0.84, + "content": "\\mathrm { A U C } \\gtrsim 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 279, + 341, + 292 + ], + "score": 1.0, + "content": "to be useful, and", + "type": "text" + }, + { + "bbox": [ + 342, + 279, + 394, + 291 + ], + "score": 0.84, + "content": "\\mathrm { A U C } \\gtrsim 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 279, + 506, + 292 + ], + "score": 1.0, + "content": "to be very useful. An AUC", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 290, + 421, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 421, + 302 + ], + "score": 1.0, + "content": "difference of 0.01, and perhaps 0.005 are potentially noticeable in clinical use.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 257, + 506, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 127, + 318 + ], + "score": 1.0, + "content": "This", + "type": "text" + }, + { + "bbox": [ + 127, + 307, + 161, + 318 + ], + "score": 0.44, + "content": "0 . 9 / 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "rule of thumb may loosely translate to our AUC variants, but it can directly translate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 203, + 331 + ], + "score": 1.0, + "content": "to Label Ranking Loss", + "type": "text" + }, + { + "bbox": [ + 204, + 318, + 218, + 329 + ], + "score": 0.88, + "content": "L _ { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 317, + 410, + 331 + ], + "score": 1.0, + "content": "(2). If we think of a single output prediction", + "type": "text" + }, + { + "bbox": [ + 410, + 317, + 457, + 330 + ], + "score": 0.94, + "content": "\\hat { y } \\in [ 0 , 1 ] ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "as a set of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 167, + 341 + ], + "score": 1.0, + "content": "predictions for", + "type": "text" + }, + { + "bbox": [ + 168, + 330, + 174, + 339 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 329, + 252, + 341 + ], + "score": 1.0, + "content": "binary labels, then", + "type": "text" + }, + { + "bbox": [ + 252, + 329, + 293, + 340 + ], + "score": 0.51, + "content": "1 - \\mathrm { A U C }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 329, + 460, + 341 + ], + "score": 1.0, + "content": "for that set of predictions is equivalent to", + "type": "text" + }, + { + "bbox": [ + 460, + 330, + 475, + 340 + ], + "score": 0.88, + "content": "L _ { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 174, + 353 + ], + "score": 1.0, + "content": "original instance", + "type": "text" + }, + { + "bbox": [ + 175, + 340, + 181, + 352 + ], + "score": 0.8, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 339, + 268, + 353 + ], + "score": 1.0, + "content": ". Therefore, values of", + "type": "text" + }, + { + "bbox": [ + 268, + 340, + 309, + 352 + ], + "score": 0.91, + "content": "L _ { R } \\lesssim 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 339, + 426, + 353 + ], + "score": 1.0, + "content": "may be clinically useful, and", + "type": "text" + }, + { + "bbox": [ + 427, + 340, + 473, + 352 + ], + "score": 0.9, + "content": "L _ { R } \\lesssim 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "may be", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 351, + 156, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 156, + 365 + ], + "score": 1.0, + "content": "very useful.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 307, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 367, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "Subjectively examining performance on 20 randomly selected cases, we find very good detailed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "predictions, but also evidence of both missing medications and missing billing codes. An example", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "of a good set of detailed predictions is from a complex patient suffering from multiple myeloma (a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "type of cancer) with various complications. This patient was taking 26 medications, 24 of which", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "had moderate to high probability predictions (Figure 4). (We have found by eyeball that a prediction", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "cutoff of 0.2 gives a reasonable balance between sensitivity and specificity for our model.) In the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "other direction, only two of the high-prediction classes were not actually being taken, but those", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "classes, along with several of the other moderately-predicted classes, are commonly used for cancer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "and are clinically reasonable for the case. (Details of this and the two cases below are in Appendix", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 466, + 123, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 123, + 480 + ], + "score": 1.0, + "content": "C).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 368, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "A good example of missing medications is a case in which the record has multiple billing codes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 508 + ], + "score": 1.0, + "content": "for both osteoporosis (which is very commonly treated with medication) and postablative hypothy-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "roidism (a deliberately induced condition that is always treated with medication), but no medications", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "of the appropriate classes were in the record. The GRU model predicted both of these classes, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 526, + 254, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 254, + 541 + ], + "score": 1.0, + "content": "the patient was almost surely taking.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 483, + 506, + 541 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "A good example of either missing billing codes or discontinued medications that remain documented", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "as active is a case in which the record has at least five years of data consisting only of codes for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Parkinson’s disease, but which lists medications for high cholesterol, hypertension, and other heart", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "disease. The GRU model predicted a reasonable set of medications for Parkinson’s disease and its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 477, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 477, + 601 + ], + "score": 1.0, + "content": "complications, but did not predict the other medications that are not suggested by the record.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 544, + 505, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "Given how easy it was to find cases with apparently missing codes and medications, we conclude", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "that there is indeed a substantial amount of label noise in our data, and we therefore interpret our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "models’ performance as lower bounds on the actual performance. We are encouraged that this kind", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "of a model may actually be useful for identifying missing medications in the record, but of course", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "a more thorough validation, and possibly a more accurate model, would be necessary before using", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "in a clinical scenario. A definitive experiment would use off-line research, including reconciling", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "information from various electronic and human sources to establish the ground truth of which med-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "ications were being taken on a particular day, but such efforts are labor intensive and expensive, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 693, + 286, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 286, + 705 + ], + "score": 1.0, + "content": "can only be conducted on a very small scale.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 604, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "An interesting byproduct of these models is the semantic embedding of ICD-9 codes used in the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "recurrent networks (Figure 5). Transforming input to a semantic embedding is a common pre-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "processing step to improve performance, but clearly the semantic understanding it provides to an", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 356, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 371 + ], + "score": 1.0, + "content": "algorithm can be useful beyond the immediate learning problem (Mikolov et al., 2013). Investigating", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "the embedding learned in this experiment shows some generalizable potential, but it also reveals the", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "need for further refinement before it can be truly useful. Specifically, while it’s easy to find tight", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "groups of ICD-9 codes that are strongly clinically related in our embedding, we also find groups for", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 400, + 330, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 330, + 414 + ], + "score": 1.0, + "content": "which we cannot see a meaningful clinical relationship.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 83, + 501, + 254 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 83, + 501, + 254 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 83, + 501, + 254 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 501, + 254 + ], + "score": 0.963, + "type": "image", + "image_path": "312c72aba26595ef96d357d8ff0979e219e91e21d0a62461d8b7896a84f472e1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 83, + 501, + 140.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 140.0, + 501, + 197.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 197.0, + 501, + 254.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 263, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "Figure 4: Medication predictions for a complicated patient. Each vertical bar represents the pre-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "diction for a single medication class, with the height of the bar representing the confidence of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "prediction. Black labels with arrows indicate ATC therapeutic classes for medications the patient", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "was actually taking. Colors and letters below the axis indicate organ system groups. More detail in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 307, + 160, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 160, + 319 + ], + "score": 1.0, + "content": "Appendix C.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "processing step to improve performance, but clearly the semantic understanding it provides to an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 356, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 371 + ], + "score": 1.0, + "content": "algorithm can be useful beyond the immediate learning problem (Mikolov et al., 2013). Investigating", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "the embedding learned in this experiment shows some generalizable potential, but it also reveals the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "need for further refinement before it can be truly useful. Specifically, while it’s easy to find tight", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "groups of ICD-9 codes that are strongly clinically related in our embedding, we also find groups for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 400, + 330, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 330, + 414 + ], + "score": 1.0, + "content": "which we cannot see a meaningful clinical relationship.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "For example, we see two groups of codes relating to kidney failure and diabetes mellitus, two classes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "of very prevalent disease (Figure 5, insets). In other iterations with different parameter settings, the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "kidney failure codes were even embedded in a sequence reflecting the natural progression of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "disease, with the code for dialysis (an intensive treatment for end-stage kidney failure) embedded", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "at the appropriate place. Interestingly, these were not the parameter settings that optimized overall", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "prediction performance. In other settings, such as our performance-optimal setting, the sequence", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "is close to the natural progression of the disease, but not quite identical. Nevertheless, this is an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 495, + 281, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 281, + 507 + ], + "score": 1.0, + "content": "exciting result that suggests great potential.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 303, + 523 + ], + "score": 1.0, + "content": "Further evaluation of the embedding found that", + "type": "text" + }, + { + "bbox": [ + 303, + 511, + 323, + 522 + ], + "score": 0.86, + "content": "49 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 511, + 504, + 523 + ], + "score": 1.0, + "content": "of codes were strongly related semantically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 211, + 536 + ], + "score": 1.0, + "content": "to their nearest neighbor,", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 231, + 533 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 521, + 337, + 536 + ], + "score": 1.0, + "content": "were loosely related, and", + "type": "text" + }, + { + "bbox": [ + 337, + 523, + 357, + 533 + ], + "score": 0.87, + "content": "41 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "unrelated. This fraction of strongly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "related nearest neighbors was lower than we had hoped, but much higher than expected by chance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 544, + 504, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 504, + 557 + ], + "score": 1.0, + "content": "(Figure 6), and it definitely improved classification performance. Furthermore, it was obvious by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "inspection that in general, codes closer in the embedding were more semantically related than distant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "codes, but interestingly, the distance to the nearest such neighbor showed the opposite relationship", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 117, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 117, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "nearest neighbors that were very close were less likely to be semantically related than nearest", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 423, + 601 + ], + "score": 1.0, + "content": "neighbors that were far, and this trend is roughly linear across the full range of", + "type": "text" + }, + { + "bbox": [ + 424, + 588, + 430, + 598 + ], + "score": 0.63, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "(Figure 6). So the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "sparser the points are in the embedded space, the more semantically related they are to their nearest", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "neighbor, but the causal direction of that effect and the technical reason for it are beyond the scope", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 621, + 185, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 185, + 632 + ], + "score": 1.0, + "content": "of this initial work.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "For this prediction problem, we settled on predicting the medications that occurred in the record", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "during the same time span as the billing codes used. Originally, we intended to predict only the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "medications listed on the day of the reference point, but that turned out to greatly exacerbate the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "missing medication problem. After trying medications that fell on the reference day only, the week", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "prior to the reference day, and the six months prior, our best performance both subjectively and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 693, + 378, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 378, + 705 + ], + "score": 1.0, + "content": "objectively was achieved using the full time range of the input data.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 710, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "While the performance of the recurrent networks was quite good, we believe it could be improved", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "by including additional input data, such as laboratory test results, demographics, and perhaps vital", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "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": [ + 107, + 83, + 501, + 254 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 83, + 501, + 254 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 83, + 501, + 254 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 501, + 254 + ], + "score": 0.963, + "type": "image", + "image_path": "312c72aba26595ef96d357d8ff0979e219e91e21d0a62461d8b7896a84f472e1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 83, + 501, + 140.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 140.0, + 501, + 197.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 197.0, + 501, + 254.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 263, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "Figure 4: Medication predictions for a complicated patient. Each vertical bar represents the pre-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "diction for a single medication class, with the height of the bar representing the confidence of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "prediction. Black labels with arrows indicate ATC therapeutic classes for medications the patient", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "was actually taking. Colors and letters below the axis indicate organ system groups. More detail in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 307, + 160, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 160, + 319 + ], + "score": 1.0, + "content": "Appendix C.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 412 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 105, + 345, + 506, + 414 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "For example, we see two groups of codes relating to kidney failure and diabetes mellitus, two classes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "of very prevalent disease (Figure 5, insets). In other iterations with different parameter settings, the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "kidney failure codes were even embedded in a sequence reflecting the natural progression of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "disease, with the code for dialysis (an intensive treatment for end-stage kidney failure) embedded", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "at the appropriate place. Interestingly, these were not the parameter settings that optimized overall", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "prediction performance. In other settings, such as our performance-optimal setting, the sequence", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "is close to the natural progression of the disease, but not quite identical. Nevertheless, this is an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 495, + 281, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 281, + 507 + ], + "score": 1.0, + "content": "exciting result that suggests great potential.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 417, + 505, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 303, + 523 + ], + "score": 1.0, + "content": "Further evaluation of the embedding found that", + "type": "text" + }, + { + "bbox": [ + 303, + 511, + 323, + 522 + ], + "score": 0.86, + "content": "49 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 511, + 504, + 523 + ], + "score": 1.0, + "content": "of codes were strongly related semantically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 211, + 536 + ], + "score": 1.0, + "content": "to their nearest neighbor,", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 231, + 533 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 521, + 337, + 536 + ], + "score": 1.0, + "content": "were loosely related, and", + "type": "text" + }, + { + "bbox": [ + 337, + 523, + 357, + 533 + ], + "score": 0.87, + "content": "41 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "unrelated. This fraction of strongly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "related nearest neighbors was lower than we had hoped, but much higher than expected by chance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 544, + 504, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 504, + 557 + ], + "score": 1.0, + "content": "(Figure 6), and it definitely improved classification performance. Furthermore, it was obvious by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "inspection that in general, codes closer in the embedding were more semantically related than distant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "codes, but interestingly, the distance to the nearest such neighbor showed the opposite relationship", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 117, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 117, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "nearest neighbors that were very close were less likely to be semantically related than nearest", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 423, + 601 + ], + "score": 1.0, + "content": "neighbors that were far, and this trend is roughly linear across the full range of", + "type": "text" + }, + { + "bbox": [ + 424, + 588, + 430, + 598 + ], + "score": 0.63, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "(Figure 6). So the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "sparser the points are in the embedded space, the more semantically related they are to their nearest", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "neighbor, but the causal direction of that effect and the technical reason for it are beyond the scope", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 621, + 185, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 185, + 632 + ], + "score": 1.0, + "content": "of this initial work.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 511, + 506, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "For this prediction problem, we settled on predicting the medications that occurred in the record", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "during the same time span as the billing codes used. 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We also suspect that if we can devise a way to convert our medication data into reliably-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "ordered sequences, we can more fully exploit the strengths of recurrent networks for medication", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 635, + 425, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 425, + 647 + ], + "score": 1.0, + "content": "prediction. 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Clinical data was provided by the Vanderbilt", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Synthetic Derivative, which is supported by institutional funding and by the Vanderbilt CTSA grant", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 721, + 166, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 166, + 731 + ], + "score": 1.0, + "content": "ULTR000445.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 81, + 505, + 304 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 81, + 505, + 304 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 81, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 109, + 81, + 505, + 304 + ], + "score": 0.955, + "type": "image", + "image_path": "7b5765589f26e16732fff295acfd6634b214b494aa7f74ec4d2e028652ea64b4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 81, + 505, + 155.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 155.33333333333331, + 505, + 229.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 229.66666666666663, + 505, + 303.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 313, + 506, + 369 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "Figure 5: A t-SNE representation of our final embedding. 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ParameterModel
GRULSTM
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Dropout for recurrent connections0.750.75
L1 applied to the input weights matrices00
L1 applied to the recurrent weights matrices00
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L2 applied to the recurrent weights matrices0.00010.001
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Dropout before the output layer0.50.5
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400
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ParameterValue
Dropout before the output layer0.1
Dropout between feed-forward layers Numberof feed-forward layers0.1
Activation function between feed-forward layers3
tanh
Number of nodes in feed-forward layers128
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ParameterModel
GRULSTM
Dropout for input gates0.10.25
Dropout for recurrent connections0.750.75
L1 applied to the input weights matrices00
L1 applied to the recurrent weights matrices00
L2 applied to the input weights matrices0.00010.0001
L2 applied to the recurrent weights matrices0.00010.001
L2 applied to the output layer's weights matrices0.00010.001
Dropout before the output layer0.50.5
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400
*Number of nodes in recurrent units400
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ParameterValue
Dropout before the output layer0.1
Dropout between feed-forward layers Numberof feed-forward layers0.1
Activation function between feed-forward layers3
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ICD-9 codeCode descriptionTime estimate (ago)
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
273.1Monoclonal paraproteinemia4.8 months ago
285.9Anemia, unspecified4.8 months ago
276.50Volume depletion, unspecified4.8 months ago
733.00Osteoporosis,unspecified4.8 months ago
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
203.00Multiple myeloma,without mention of having achieved remission2.9 months ago
203.01Multiple myeloma, in remission2.9 months ago
273.1Monoclonal paraproteinemia2.9 months ago
273.1Monoclonal paraproteinemia1.6 months ago
279.3Unspecified immunity deficiency1.6 months ago
203.00Multiple myeloma, without mention of having achieved remission1.6 months ago
781.2Abnormality of gait3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.7 weeks ago
401.9Unspecified essential hypertension3.7 weeks ago
V12.54Personal history of transient ischemic attack (TIA),and cerebral infarction without residual deficits3.7 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.7 weeks ago
786.09Otherrespiratoryabnormalities3.7 weeks ago
273.1Monoclonal paraproteinemia3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.6 weeks ago
V58.69Long-term (current) use of other medications3.6 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.4 weeks ago
203.00Multiple myeloma, without mention of having achieved remission4 days ago
V42.82Peripheral stem cells replaced by transplant4 days ago
203.01Multiple myeloma, in remission3 days ago
38.97Central venous catheter placement with guidance3 days ago
V42.82Peripheral stem cells replaced by transplant3 daysago
V58.81Fitting and adjustment of vascular catheter3 days ago
203.00Multiple myeloma, without mention of having achieved remission3 daysago
V42.82Peripheral stem cells replaced by transplant2 days ago
203.01Multiple myeloma, in remission2 days ago
203.00Multiple myeloma, without mention of having achieved remission1 day ago
V42.82Peripheral stem cells replaced by transplant1 day ago
203.00Multiple myeloma, without mention of having achieved remissionnow
V42.82Peripheral stem cells replaced by transplantnow
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ICD-9 codeCode descriptionTime estimate (ago)
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
273.1Monoclonal paraproteinemia4.8 months ago
285.9Anemia, unspecified4.8 months ago
276.50Volume depletion, unspecified4.8 months ago
733.00Osteoporosis,unspecified4.8 months ago
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
203.00Multiple myeloma,without mention of having achieved remission2.9 months ago
203.01Multiple myeloma, in remission2.9 months ago
273.1Monoclonal paraproteinemia2.9 months ago
273.1Monoclonal paraproteinemia1.6 months ago
279.3Unspecified immunity deficiency1.6 months ago
203.00Multiple myeloma, without mention of having achieved remission1.6 months ago
781.2Abnormality of gait3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.7 weeks ago
401.9Unspecified essential hypertension3.7 weeks ago
V12.54Personal history of transient ischemic attack (TIA),and cerebral infarction without residual deficits3.7 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.7 weeks ago
786.09Otherrespiratoryabnormalities3.7 weeks ago
273.1Monoclonal paraproteinemia3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.6 weeks ago
V58.69Long-term (current) use of other medications3.6 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.4 weeks ago
203.00Multiple myeloma, without mention of having achieved remission4 days ago
V42.82Peripheral stem cells replaced by transplant4 days ago
203.01Multiple myeloma, in remission3 days ago
38.97Central venous catheter placement with guidance3 days ago
V42.82Peripheral stem cells replaced by transplant3 daysago
V58.81Fitting and adjustment of vascular catheter3 days ago
203.00Multiple myeloma, without mention of having achieved remission3 daysago
V42.82Peripheral stem cells replaced by transplant2 days ago
203.01Multiple myeloma, in remission2 days ago
203.00Multiple myeloma, without mention of having achieved remission1 day ago
V42.82Peripheral stem cells replaced by transplant1 day ago
203.00Multiple myeloma, without mention of having achieved remissionnow
V42.82Peripheral stem cells replaced by transplantnow
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Top predictionsProb.True labelsProb.
Corticosteroids97.01%
S03B* S01C*Antiinflammatory agents and antiinfectives in combi-97.01% 95.54%S03B S01CCorticosteroids Antiinflammatory agents and antiinfectives in combi-95.54%
nationnation
S02B*Corticosteroids95.54%S02BCorticosteroids95.54%
L01AAlkylating agents94.00%D07XCorticosteroids,other combinations93.37%
D07X*Corticosteroids,other combinations93.37%H02ACorticosteroids for systemic use, plain91.06%
H02A*Corticosteroids for systemic use, plain91.06%D07ACorticosteroids, plain90.83%
D07A*Corticosteroids,plain90.83%S01BAntiinflammatory agents90.79%
S01B*Antiinflammatory agents90.79%D10AAnti-acne preparations for topical use88.56%
D10A*Anti-acne preparations for topical use88.56%C05AAgents for treatment of hemorrhoids and anal fissures for topical use88.52%
C05A*Agents for treatment of hemorrhoids and anal fissures for topical use88.52%R01ADecongestants and other nasal preparations for topi-87.02%
A04AAntiemetics and antinauseants87.95%J05Acal use Direct acting antivirals86.83%
R01A*Decongestants and other nasal preparations for topi- cal use87.02%A01AStomatological preparations86.11%
J05A*Direct acting antivirals86.8%N02AOpioids84.86%
A01A*Stomatological preparations86.11%B05CIrrigating solutions82.56%
N02A*Opioids84.86%A12COther mineral supplements79.50%
B05C*Irrigating solutions82.56%B05XI.V. solution additives74.84%
A12C*Other mineral supplements79.50%L04AImmunosuppressants68.76%
B05X*Iv. solution additives74.84%N02BOther analgesics and antipyretics57.24%
L04A*Immunosuppressants68.76%S01AAntiinfectives54.59%
N05AAntipsychotics58.64%J01DOther beta-lactam antibacterials43.40%
N02B*Other analgesics and antipyretics57.24%C03CHigh-ceiling diuretics39.88%
S01A*Antiinfectives54.59%J01MQuinolone antibacterials29.78%
L03AImmunostimulants45.96%C07ABeta blocking agents27.08%
A02BDrugs for peptic ulcer and gastro-oesophageal reflux44.56%
J01D*disease Other beta-lactam antibacterials43.40%N03AAntiepileptics
C03C*High-ceiling diuretics39.88%J01XOther antibacterials20.00%
B01AAntithrombotic agents37.80%M03BMuscle relaxants,centrally acting agents5.88%
V03AAll other therapeutic products34.18%5.09%
R06AAntihistamines for systemic use31.78%
A06ADrugs for constipation31.57%
J01M*Quinolone antibacterials29.78%
N05BAnxiolytics29.42%
D04AAntipruritics,incl. antihistamines,anesthetics, etc.27.62%
C07A*Beta blocking agents27.08%
L01XOther antineoplastic agents24.72%
R05CExpectorants,excl. combinations with cough sup-20.43%
N03A*pressants
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Top predictionsProb.True labelsProb.
Corticosteroids97.01%
S03B* S01C*Antiinflammatory agents and antiinfectives in combi-97.01% 95.54%S03B S01CCorticosteroids Antiinflammatory agents and antiinfectives in combi-95.54%
nationnation
S02B*Corticosteroids95.54%S02BCorticosteroids95.54%
L01AAlkylating agents94.00%D07XCorticosteroids,other combinations93.37%
D07X*Corticosteroids,other combinations93.37%H02ACorticosteroids for systemic use, plain91.06%
H02A*Corticosteroids for systemic use, plain91.06%D07ACorticosteroids, plain90.83%
D07A*Corticosteroids,plain90.83%S01BAntiinflammatory agents90.79%
S01B*Antiinflammatory agents90.79%D10AAnti-acne preparations for topical use88.56%
D10A*Anti-acne preparations for topical use88.56%C05AAgents for treatment of hemorrhoids and anal fissures for topical use88.52%
C05A*Agents for treatment of hemorrhoids and anal fissures for topical use88.52%R01ADecongestants and other nasal preparations for topi-87.02%
A04AAntiemetics and antinauseants87.95%J05Acal use Direct acting antivirals86.83%
R01A*Decongestants and other nasal preparations for topi- cal use87.02%A01AStomatological preparations86.11%
J05A*Direct acting antivirals86.8%N02AOpioids84.86%
A01A*Stomatological preparations86.11%B05CIrrigating solutions82.56%
N02A*Opioids84.86%A12COther mineral supplements79.50%
B05C*Irrigating solutions82.56%B05XI.V. solution additives74.84%
A12C*Other mineral supplements79.50%L04AImmunosuppressants68.76%
B05X*Iv. solution additives74.84%N02BOther analgesics and antipyretics57.24%
L04A*Immunosuppressants68.76%S01AAntiinfectives54.59%
N05AAntipsychotics58.64%J01DOther beta-lactam antibacterials43.40%
N02B*Other analgesics and antipyretics57.24%C03CHigh-ceiling diuretics39.88%
S01A*Antiinfectives54.59%J01MQuinolone antibacterials29.78%
L03AImmunostimulants45.96%C07ABeta blocking agents27.08%
A02BDrugs for peptic ulcer and gastro-oesophageal reflux44.56%
J01D*disease Other beta-lactam antibacterials43.40%N03AAntiepileptics
C03C*High-ceiling diuretics39.88%J01XOther antibacterials20.00%
B01AAntithrombotic agents37.80%M03BMuscle relaxants,centrally acting agents5.88%
V03AAll other therapeutic products34.18%5.09%
R06AAntihistamines for systemic use31.78%
A06ADrugs for constipation31.57%
J01M*Quinolone antibacterials29.78%
N05BAnxiolytics29.42%
D04AAntipruritics,incl. antihistamines,anesthetics, etc.27.62%
C07A*Beta blocking agents27.08%
L01XOther antineoplastic agents24.72%
R05CExpectorants,excl. combinations with cough sup-20.43%
N03A*pressants
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ICD-9 codeCode descriptionTime estimate (ago)
735.4Other hammer toe (acquired)2.4 years ago
729.5Pain in limb2.4 years ago
244.1Other postablative hypothyroidism1.5 years ago
285.9Anemia, unspecified1.5 years ago
244.1Other postablative hypothyroidism1.2 years ago
244.1Other postablative hypothyroidism11.5 months ago
733.00Osteoporosis,unspecified11.5 months ago
733.01Senile osteoporosis7.7 months ago
268.9Unspecified vitamin D deficiency7.7 months ago
729.5Pain in limb7.7 months ago
174.9Malignant neoplasm of breast (female),unspecified7.7 months ago
722.52Degeneration of lumbar or lumbosacral intervertebral disc7.7 months ago
279.3Unspecified immunity deficiency7.7 months ago
733.01Senile osteoporosis6.4 months ago
733.01Senile osteoporosis6.2 months ago
244.1Other postablative hypothyroidism6.0 months ago
401.1Benign essential hypertension6.0 months ago
V58.69Long-term (current) use of other medications1.9 weeks ago
733.01Senile osteoporosisnow
244.1Other postablative hypothyroidismnow
V58.69Long-term (current) use of other medicationsnow
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Top predictionsProb.True labelsProb.
M05BDrugs affecting bone structure and mineralization88.18%A11CVitamin a and d, incl. combinations of the two39.42%
H03AThyroid preparations84.82%N06AAntidepressants20.88%
H05AParathyroid hormones and analogues66.33%C10ALipid modifying agents,plain17.05%
A11C*39.42%N03AAntiepileptics15.61%
N02BVitamin a and d, incl. combinations of the two Other analgesics and antipyretics37.58%C09C
A01AStomatological preparations23.05%L02BAngiotensin ii antagonists,plain10.38% 4.22%
A12A21.59%Hormone antagonists and related agents
N06A*Calcium Antidepressants20.88%
C07A
Beta blocking agents20.81%
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ICD-9 codeCode descriptionTime estimate (ago)
735.4Other hammer toe (acquired)2.4 years ago
729.5Pain in limb2.4 years ago
244.1Other postablative hypothyroidism1.5 years ago
285.9Anemia, unspecified1.5 years ago
244.1Other postablative hypothyroidism1.2 years ago
244.1Other postablative hypothyroidism11.5 months ago
733.00Osteoporosis,unspecified11.5 months ago
733.01Senile osteoporosis7.7 months ago
268.9Unspecified vitamin D deficiency7.7 months ago
729.5Pain in limb7.7 months ago
174.9Malignant neoplasm of breast (female),unspecified7.7 months ago
722.52Degeneration of lumbar or lumbosacral intervertebral disc7.7 months ago
279.3Unspecified immunity deficiency7.7 months ago
733.01Senile osteoporosis6.4 months ago
733.01Senile osteoporosis6.2 months ago
244.1Other postablative hypothyroidism6.0 months ago
401.1Benign essential hypertension6.0 months ago
V58.69Long-term (current) use of other medications1.9 weeks ago
733.01Senile osteoporosisnow
244.1Other postablative hypothyroidismnow
V58.69Long-term (current) use of other medicationsnow
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Top predictionsProb.True labelsProb.
M05BDrugs affecting bone structure and mineralization88.18%A11CVitamin a and d, incl. combinations of the two39.42%
H03AThyroid preparations84.82%N06AAntidepressants20.88%
H05AParathyroid hormones and analogues66.33%C10ALipid modifying agents,plain17.05%
A11C*39.42%N03AAntiepileptics15.61%
N02BVitamin a and d, incl. combinations of the two Other analgesics and antipyretics37.58%C09C
A01AStomatological preparations23.05%L02BAngiotensin ii antagonists,plain10.38% 4.22%
A12A21.59%Hormone antagonists and related agents
N06A*Calcium Antidepressants20.88%
C07A
Beta blocking agents20.81%
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ICD-9 codeCodedescriptionTime estimate (ago)
332.0Paralysis agitans5.0 years ago
332.0Paralysis agitans4.7 years ago
332.0Paralysis agitans4.5 years ago
332.0Paralysis agitans4.0 years ago
332.0Paralysis agitans3.5 years ago
332.0Paralysis agitans3.0 years ago
332.0Paralysis agitans2.7 years ago
332.0Paralysis agitans2.4 years ago
332.0Paralysis agitans2.0 years ago
332.0Paralysis agitans1.7 years ago
332.0Paralysis agitans
332.0Paralysis agitans1.0 years ago
332.09.9 months ago
Paralysis agitans4.1 months ago
332.0Paralysis agitansnow
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Table structure as in case 1.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "image", + "bbox": [ + 105, + 282, + 505, + 526 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 282, + 505, + 526 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 526 + ], + "score": 0.46, + "type": "image", + "image_path": "647f6f150e88634b289fe4474f49576c0f5151752fa240698a1ce54917ac379d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 105, + 282, + 505, + 363.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 105, + 363.3333333333333, + 505, + 444.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 105, + 444.66666666666663, + 505, + 526.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "Medication predictions for a patient with only one ICD-9 code, repeated many times over five years.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "The medications listed under true labels are not indicated for paralysis agitans (Parkinson’s disease),", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "but the patient was surely taking them for reasons not documented in the ICD-9 sequence. The", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "model predicted mostly reasonable medications for a patient with Parkinson’s disease, especially", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "score": 1.0, + "content": "Dopaminergic agents, which is the primary treatment for the disease. Figure representation as in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 591, + 165, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 165, + 603 + ], + "score": 1.0, + "content": "case 1, above.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 225, + 105, + 384, + 220 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 83, + 142, + 93 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 145, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 145, + 95 + ], + "score": 1.0, + "content": "CASE 3.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 225, + 105, + 384, + 220 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 225, + 105, + 384, + 220 + ], + "spans": [ + { + "bbox": [ + 225, + 105, + 384, + 220 + ], + "score": 0.954, + "html": "
ICD-9 codeCodedescriptionTime estimate (ago)
332.0Paralysis agitans5.0 years ago
332.0Paralysis agitans4.7 years ago
332.0Paralysis agitans4.5 years ago
332.0Paralysis agitans4.0 years ago
332.0Paralysis agitans3.5 years ago
332.0Paralysis agitans3.0 years ago
332.0Paralysis agitans2.7 years ago
332.0Paralysis agitans2.4 years ago
332.0Paralysis agitans2.0 years ago
332.0Paralysis agitans1.7 years ago
332.0Paralysis agitans
332.0Paralysis agitans1.0 years ago
332.09.9 months ago
Paralysis agitans4.1 months ago
332.0Paralysis agitansnow
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Table structure as in case 1.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "image", + "bbox": [ + 105, + 282, + 505, + 526 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 282, + 505, + 526 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 526 + ], + "score": 0.46, + "type": "image", + "image_path": "647f6f150e88634b289fe4474f49576c0f5151752fa240698a1ce54917ac379d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 105, + 282, + 505, + 363.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 105, + 363.3333333333333, + 505, + 444.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 105, + 444.66666666666663, + 505, + 526.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "Medication predictions for a patient with only one ICD-9 code, repeated many times over five years.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "The medications listed under true labels are not indicated for paralysis agitans (Parkinson’s disease),", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "but the patient was surely taking them for reasons not documented in the ICD-9 sequence. 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ModelMicro-AUCLabel Ranking LossMacro-AUCLabel Ranking Avg. PrecisionCoverage Error
0.861
GRU LSTM0.927 0.9260.076 0.0770.8590.603 0.60062.6 63.0
NN0.9160.0860.8350.57067.3
0.9030.1020.8040.52373.7
RF (binary) RF (counts)0.8940.1110.7870.49777.3
Baseline0.8280.1720.5000.35597.2
Perfect1.00.01.01.015.0
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ParameterModel
GRULSTM
Dropout for input gates0.10.25
Dropout for recurrent connections0.750.75
L1 applied to the input weights matrices00
L1 applied to the recurrent weights matrices00
L2 applied to the input weights matrices0.00010.0001
L2 applied to the recurrent weights matrices0.00010.001
L2 applied to the output layer's weights matrices0.00010.001
Dropout before the output layer0.50.5
*Number of recurrent layers33
400
*Number of nodes in recurrent units400
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ParameterValue
Dropout before the output layer0.1
Dropout between feed-forward layers Numberof feed-forward layers0.1
Activation function between feed-forward layers3
tanh
Number of nodes in feed-forward layers128
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ICD-9 codeCode descriptionTime estimate (ago)
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
273.1Monoclonal paraproteinemia4.8 months ago
285.9Anemia, unspecified4.8 months ago
276.50Volume depletion, unspecified4.8 months ago
733.00Osteoporosis,unspecified4.8 months ago
203.00Multiple myeloma, without mention of having achieved remission4.8 months ago
203.00Multiple myeloma,without mention of having achieved remission2.9 months ago
203.01Multiple myeloma, in remission2.9 months ago
273.1Monoclonal paraproteinemia2.9 months ago
273.1Monoclonal paraproteinemia1.6 months ago
279.3Unspecified immunity deficiency1.6 months ago
203.00Multiple myeloma, without mention of having achieved remission1.6 months ago
781.2Abnormality of gait3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.7 weeks ago
401.9Unspecified essential hypertension3.7 weeks ago
V12.54Personal history of transient ischemic attack (TIA),and cerebral infarction without residual deficits3.7 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.7 weeks ago
786.09Otherrespiratoryabnormalities3.7 weeks ago
273.1Monoclonal paraproteinemia3.7 weeks ago
203.00Multiple myeloma, without mention of having achieved remission3.6 weeks ago
V58.69Long-term (current) use of other medications3.6 weeks ago
794.31Nonspecific abnormal electrocardiogram [ECG] [EKG]3.4 weeks ago
203.00Multiple myeloma, without mention of having achieved remission4 days ago
V42.82Peripheral stem cells replaced by transplant4 days ago
203.01Multiple myeloma, in remission3 days ago
38.97Central venous catheter placement with guidance3 days ago
V42.82Peripheral stem cells replaced by transplant3 daysago
V58.81Fitting and adjustment of vascular catheter3 days ago
203.00Multiple myeloma, without mention of having achieved remission3 daysago
V42.82Peripheral stem cells replaced by transplant2 days ago
203.01Multiple myeloma, in remission2 days ago
203.00Multiple myeloma, without mention of having achieved remission1 day ago
V42.82Peripheral stem cells replaced by transplant1 day ago
203.00Multiple myeloma, without mention of having achieved remissionnow
V42.82Peripheral stem cells replaced by transplantnow
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Top predictionsProb.True labelsProb.
Corticosteroids97.01%
S03B* S01C*Antiinflammatory agents and antiinfectives in combi-97.01% 95.54%S03B S01CCorticosteroids Antiinflammatory agents and antiinfectives in combi-95.54%
nationnation
S02B*Corticosteroids95.54%S02BCorticosteroids95.54%
L01AAlkylating agents94.00%D07XCorticosteroids,other combinations93.37%
D07X*Corticosteroids,other combinations93.37%H02ACorticosteroids for systemic use, plain91.06%
H02A*Corticosteroids for systemic use, plain91.06%D07ACorticosteroids, plain90.83%
D07A*Corticosteroids,plain90.83%S01BAntiinflammatory agents90.79%
S01B*Antiinflammatory agents90.79%D10AAnti-acne preparations for topical use88.56%
D10A*Anti-acne preparations for topical use88.56%C05AAgents for treatment of hemorrhoids and anal fissures for topical use88.52%
C05A*Agents for treatment of hemorrhoids and anal fissures for topical use88.52%R01ADecongestants and other nasal preparations for topi-87.02%
A04AAntiemetics and antinauseants87.95%J05Acal use Direct acting antivirals86.83%
R01A*Decongestants and other nasal preparations for topi- cal use87.02%A01AStomatological preparations86.11%
J05A*Direct acting antivirals86.8%N02AOpioids84.86%
A01A*Stomatological preparations86.11%B05CIrrigating solutions82.56%
N02A*Opioids84.86%A12COther mineral supplements79.50%
B05C*Irrigating solutions82.56%B05XI.V. solution additives74.84%
A12C*Other mineral supplements79.50%L04AImmunosuppressants68.76%
B05X*Iv. solution additives74.84%N02BOther analgesics and antipyretics57.24%
L04A*Immunosuppressants68.76%S01AAntiinfectives54.59%
N05AAntipsychotics58.64%J01DOther beta-lactam antibacterials43.40%
N02B*Other analgesics and antipyretics57.24%C03CHigh-ceiling diuretics39.88%
S01A*Antiinfectives54.59%J01MQuinolone antibacterials29.78%
L03AImmunostimulants45.96%C07ABeta blocking agents27.08%
A02BDrugs for peptic ulcer and gastro-oesophageal reflux44.56%
J01D*disease Other beta-lactam antibacterials43.40%N03AAntiepileptics
C03C*High-ceiling diuretics39.88%J01XOther antibacterials20.00%
B01AAntithrombotic agents37.80%M03BMuscle relaxants,centrally acting agents5.88%
V03AAll other therapeutic products34.18%5.09%
R06AAntihistamines for systemic use31.78%
A06ADrugs for constipation31.57%
J01M*Quinolone antibacterials29.78%
N05BAnxiolytics29.42%
D04AAntipruritics,incl. antihistamines,anesthetics, etc.27.62%
C07A*Beta blocking agents27.08%
L01XOther antineoplastic agents24.72%
R05CExpectorants,excl. combinations with cough sup-20.43%
N03A*pressants
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ICD-9 codeCode descriptionTime estimate (ago)
735.4Other hammer toe (acquired)2.4 years ago
729.5Pain in limb2.4 years ago
244.1Other postablative hypothyroidism1.5 years ago
285.9Anemia, unspecified1.5 years ago
244.1Other postablative hypothyroidism1.2 years ago
244.1Other postablative hypothyroidism11.5 months ago
733.00Osteoporosis,unspecified11.5 months ago
733.01Senile osteoporosis7.7 months ago
268.9Unspecified vitamin D deficiency7.7 months ago
729.5Pain in limb7.7 months ago
174.9Malignant neoplasm of breast (female),unspecified7.7 months ago
722.52Degeneration of lumbar or lumbosacral intervertebral disc7.7 months ago
279.3Unspecified immunity deficiency7.7 months ago
733.01Senile osteoporosis6.4 months ago
733.01Senile osteoporosis6.2 months ago
244.1Other postablative hypothyroidism6.0 months ago
401.1Benign essential hypertension6.0 months ago
V58.69Long-term (current) use of other medications1.9 weeks ago
733.01Senile osteoporosisnow
244.1Other postablative hypothyroidismnow
V58.69Long-term (current) use of other medicationsnow
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Top predictionsProb.True labelsProb.
M05BDrugs affecting bone structure and mineralization88.18%A11CVitamin a and d, incl. combinations of the two39.42%
H03AThyroid preparations84.82%N06AAntidepressants20.88%
H05AParathyroid hormones and analogues66.33%C10ALipid modifying agents,plain17.05%
A11C*39.42%N03AAntiepileptics15.61%
N02BVitamin a and d, incl. combinations of the two Other analgesics and antipyretics37.58%C09C
A01AStomatological preparations23.05%L02BAngiotensin ii antagonists,plain10.38% 4.22%
A12A21.59%Hormone antagonists and related agents
N06A*Calcium Antidepressants20.88%
C07A
Beta blocking agents20.81%
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ICD-9 codeCodedescriptionTime estimate (ago)
332.0Paralysis agitans5.0 years ago
332.0Paralysis agitans4.7 years ago
332.0Paralysis agitans4.5 years ago
332.0Paralysis agitans4.0 years ago
332.0Paralysis agitans3.5 years ago
332.0Paralysis agitans3.0 years ago
332.0Paralysis agitans2.7 years ago
332.0Paralysis agitans2.4 years ago
332.0Paralysis agitans2.0 years ago
332.0Paralysis agitans1.7 years ago
332.0Paralysis agitans
332.0Paralysis agitans1.0 years ago
332.09.9 months ago
Paralysis agitans4.1 months ago
332.0Paralysis agitansnow
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0000000000000000000000000000000000000000..e121715eda3e06cedbd38c9759786dee7848b0a6 --- /dev/null +++ b/parse/train/wWtk6GxJB2x/wWtk6GxJB2x.md @@ -0,0 +1,316 @@ +# PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations + +Moshe Eliasof Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva, Israel eliasof@post.bgu.ac.il + +Eldad Haber Department of Earth, Ocean and Atmospheric Sciences University of British Columbia Vancouver, Canada ehaber@eoas.ubc.ca + +# Eran Treister + +Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva, Israel erant@cs.bgu.ac.il + +# Abstract + +Graph neural networks are increasingly becoming the go-to approach in various fields such as computer vision, computational biology and chemistry, where data are naturally explained by graphs. However, unlike traditional convolutional neural networks, deep graph networks do not necessarily yield better performance than shallow graph networks. This behavior usually stems from the over-smoothing phenomenon. In this work, we propose a family of architectures to control this behavior by design. Our networks are motivated by numerical methods for solving Partial Differential Equations (PDEs) on manifolds, and as such, their behavior can be explained by similar analysis. Moreover, as we demonstrate using an extensive set of experiments, our PDE-motivated networks can generalize and be effective for various types of problems from different fields. Our architectures obtain better or on par with the current state-of-the-art results for problems that are typically approached using different architectures. + +# 1 Introduction + +In recent years, Graph Convolutional Networks (GCNs) [1, 2, 3] have drawn the attention of researchers and practitioners in a variety of domains and applications, ranging from computer vision and graphics [4, 5, 6, 7, 8] to computational biology [9, 10, 11], recommendation systems [12] and social network analysis [13, 14]. However, GCNs still suffer from two main problems. First, they are usually shallow as opposed to the concept of deep convolutional neural networks (CNNs) [15, 16] due to the over-smoothing phenomenon [17, 18, 19], where the node feature vectors become almost identical, such that they are indistinguishable, which yields non-optimal performance. Furthermore, GCNs are typically customized to a specific domain and application. That is, as we demonstrate in Sec. 4.1, a successful point-cloud classification network [5] can perform poorly on a citation graph node-classification problem [19], and vice-versa. Furthermore, because many GCNs lack theoretical guarantees, it is difficult to reason about their success on one problem and lack on others. These observations motivate us to develop a profound understanding of graph networks and their dynamics. + +To this end, we suggest a novel, universal approach to the design of GCN architectures. Our inspiration stems from the similarities and equivalence between Partial Differential Equations (PDEs) and deep networks explored in [20, 21, 22]. Furthermore, as GCNs can be thought of as a generalization of + +CNNs, and a standard convolution can be represented as a combination of differential operators on a structured grid [22], we adopt this interpretation to explore versions of GCNs as PDEs on graphs or manifolds. We therefore call our network architectures PDE-GCN, and demonstrate that our approach is general with respect to the given task. That is, our architectures behave similarly for different problems, and their performance is on par or better than other domain-specific GCNs. Furthermore, our family of architectures are backed by theoretical guarantees that allow us to explain the behavior of the GCNs in some of the results that we present. To be more specific, our contribution is as follows: + +• We introduce and implement general graph convolution operators, based on graph gradient and divergence. This abstraction of the spatial operation on the graph leads to a more general and flexible approach for architecture design. +• We propose treating a variety of graph related problems as discretized PDEs, and formulate the dynamics that match different problems such as node-classification and dense shape-correspondence. This is in direct effort to propose a family of networks that can solve multiple problems, instead of GCNs which are tailored for a specific application. +• Our method allows constructing a deep GCN without over-smoothing, with theoretical guarantees. +• We validate our method by conducting numerous experiments on multiple datasets and applications, achieving significantly better or similar accuracy compared to state-of-the-art models. + +# 2 Related work + +Graph Convolutional Networks: GCNs are typically divided into spectral [1, 3, 2] and spatial [23, 24, 25, 4] categories. Most of those can be implemented using the Message-Passing Neural Network paradigm [25], where each node aggregates features (messages) from its neighbors, according to some scheme. The works [3, 2] use polynomials of the graph Laplacian to parameterize the convolution operator. DGCNN [5] constructs a $\mathbf { k }$ -nearest-neighbors graph from point-clouds and dynamically updates it. MoNet [4] learns a Gaussian mixture model to weight the edges of meshes for shape analysis tasks. Works like GraphSAGE [26] and GAT [27] propose methods for inductive and transductive learning on non-geometrical graphs. + +Several of the methods above suffer from over-smoothing [17, 19], leading to undesired node features similarity for deep networks. To overcome this problem, some approaches rely on imposing regularization and augmentation. For example, PairNorm [17] introduces a novel normalization layer, and DropEdge [28] randomly removes edges to decrease the degree of nodes. Other methods prevent over-smoothing by dedicated construction. For instance, JKNet [29] combines all intermediate representations at each stage of the network. APPNP [30] replaces learnt convolutional layers with a pre-defined kernel based on PageRank, yielding a shallow network which preserves locality, making it robust to over-smoothing. GCNII [19] proposes to add an identity residual where the initial features of the networks are added to the features of the $l$ -th layer, scaled by some coefficient. + +Another approach is to construct a network that inherently does not over-smooth, as we suggest in this work. Our network is based on discretized PDEs, hence we are able to motivate our choices by well studied theory and numerical experiments [31]. On a similar note, the recent DiffGCN [8] also makes use of discretized operators to approximate the graph gradient and Laplacian. However, DiffGCN is specifically tailored for geometric tasks since it projects its components on the $x , y , z$ axes, and is applied using a ResNet [16] (diffusion) structure only. The recent GRAND [32] applies attention mechanism with diffusive dynamics, using several integration schemes. Here we propose a network for both geometric and non-geometric tasks, and also utilize both the diffusion or hyperbolic layer dynamics, including a mixture of the two. + +PDEs and CNNs: In a recent series of works, the connection between PDEs and CNNs was studied [20, 21, 22, 33, 34, 35, 36, 37]. It was shown that it is possible to treat a deep neural network as a dynamical system driven by some PDE, where each convolution layer is considered a time step of a discretized PDE. The connection between PDEs and CNNs was also used to reduce the computational burden [38]. Besides the interpretation of CNNs as a PDE solver, it was shown that it is also possible to learn a symbolic representation of PDEs in a task-and data driven approach in [39, 40]. In the context of GCNs and PDEs, it was recently shown [41, 42] that GCNs can be utilized to enhance the solution of PDEs for problems like fluid flow dynamics. However, in this work, we harness PDE concepts to design and construct GCNs for a variety of applications. + +![](images/84fba16b4d384652779751fd38ae51dc9abe79770ca77ec1b8facd8fdbdfb1ea.jpg) +Figure 1: Feature evolvement on an input mesh (left). Propagation in time is from left to right. Hyperbolic and diffusion equation dynamics are on the top and bottom row, respectively. While a diffusive graph network (similar to most common GCNs that rely on ResNets) smooths the information on the manifold, the hyperbolic network yields a non-uniform field. + +# 3 Methods + +# 3.1 Partial Differential Equations on manifolds + +We show now that GCNs can be viewed as discretizations of PDEs on manifolds, similarly to CNNs that are viewed as discretized PDEs on regular grids [22]. Consider a general manifold $\mathcal { M }$ where a vector function $f$ resides (also dubbed the features function), along with its continuous differential operators such as the gradient $\nabla$ , divergence $\nabla$ · and the Laplacian $\Delta$ that reside on the manifold $\mathcal { M }$ + +Given these differential operators, one can model different processes on $\mathcal { M }$ . In particular, we consider two PDEs – the non-linear diffusion and the non-linear hyperbolic equations + +$$ +\begin{array} { r l } & { f _ { t } = \nabla \cdot K ^ { * } \sigma ( K \nabla f ) , \quad f ( t = 0 ) = f ^ { 0 } , \quad t \in [ 0 , T ] , } \\ & { f _ { t t } = \nabla \cdot K ^ { * } \sigma ( K \nabla f ) , \quad f ( t = 0 ) = f ^ { 0 } , \quad f _ { t } ( t = 0 ) = 0 , \quad t \in [ 0 , T ] , } \end{array} +$$ + +respectively, equipped with appropriate boundary conditions. Here $K$ is a coefficient matrix that can change in time and represents the propagation over the manifold $\mathcal { M }$ , $K ^ { * }$ is its conjugate transpose and $\sigma ( \cdot )$ is a non-linear activation function. Eq. (1)-(2) define a non-linear operator that takes initial feature vectors $f ^ { 0 }$ at time 0 and propagates them to time $T$ , yielding $f ^ { T }$ where they can be used for different tasks. We now provide two theorems that characterize the behavior of Eq. (1)-(2), based on ideas from [22]1. + +Theorem 1. If the activation function $\sigma ( \cdot )$ is monotonically non-decreasing and sign-preserving, then the forward propagation through the diffusive PDE in (1) for $t \in [ 0 , \infty )$ yields a non-increasing feature norm, that is, + +$$ +\frac { \partial } { \partial t } \lvert | f \rvert | ^ { 2 } \leq 0 . +$$ + +Theorem 2. Assume that the activation function $\sigma ( \cdot )$ is monotonically non-decreasing, signpreserving and satisfies $| \sigma ( x ) | \leq | x |$ , and define energy + +$$ +\mathcal { E } _ { n e t } = \| f _ { t } \| ^ { 2 } + \left( K \nabla f , \sigma ( K \nabla f ) \right) , +$$ + +then the forward propagation through the hyperbolic PDE in (2) satisfies $\mathcal { E } _ { n e t } \leq c _ { K }$ , where $c _ { K }$ is $a$ constant that depends on $K$ but independent of time. + +The outcome of those theorems is that the dynamics described in Eq. (1) is smoothing, while the one in Eq. (2) is bounded by a conserving mapping. An illustration of this behavior is presented in Fig. 1. + +In the physical world, diffusion and hyperbolic equations are used for different applications. Similarly, many computational models for image segmentation [43], denoising [44] and deblurring are based on anisotropic diffusion which are similar to the model in Eq. (1). On the other hand, applications that require conservation such as volume/distance preservation as in the dense shape correspondence task [45] and protein folding [11], are typically better treated using a hyperbolic equation as in Eq. (2). Those insights motivate us to construct two types of layers according to Eq. (1)-(2) using discretized differential operators on graphs. + +# 3.2 Discretized differential operators on graphs + +The models (1) and (2) reside in a continuous manifold $\mathcal { M }$ , on which a continuous function vector $f$ is defined. A graph can be thought of as a discretization of that manifold to a finite space. Assume we are given an undirected graph $\mathcal { G } = ( \nu , \mathcal { E } )$ where $\nu \in \mathcal { M }$ is the set of $n$ vertices of the graph and $\mathcal { E }$ is the set of $m$ edges of the graph. Let us denote by $\mathbf { f } _ { i } \in \mathbb { R } ^ { c }$ the value of the discrete version of $f$ , on the $i$ -th node of $\mathcal { G }$ . $c$ is the number of channels, which is the width of the neural network. We define $\mathbf { G }$ , the discrete gradient operator on the graph, also known as the incidence matrix, as follows: + +$$ +( \mathbf { G f } ) _ { i j } = \mathbf { W } _ { i j } ( \mathbf { f } _ { i } - \mathbf { f } _ { j } ) , +$$ + +where nodes $i$ and $j$ are connected via the $( i , j )$ -th edge, $\mathbf { W } _ { i j }$ is an edge weight matrix which can be learnt, and $\mathbf { f } _ { i }$ and $\mathbf { f } _ { j }$ are the features on the $i$ -th and $j$ -th nodes, respectively. The gradient operator can be thought of as a weighted directional derivative of the function $f$ in the direction defined by the nodes $i$ and $j$ . Furthermore, if we choose the scaled identity matrix $\mathbf { W } _ { i j } = d _ { i j } ^ { - 1 } \mathbf { I }$ , where $d _ { i j }$ is the distance between the two nodes, then the discrete gradient is a second order approximation to the true gradient of the function $f$ on the edges of the graph. In this work, we use $\mathbf { W } _ { i j } = \gamma _ { i j } \mathbf { I }$ , where the scale $\gamma _ { i j }$ is the geometric mean of the degree of the nodes $i , j$ . Note, that the gradient operator is a mapping from the vertex space to the edge space. + +Given the gradient matrix, it is possible to define the divergence matrix [46], which is an approach that is used extensively in mimetic discretizations of PDEs. To this end, we define the inner product between an edge feature vector $\mathbf { q }$ and the gradient of a node feature vector f as + +$$ +\begin{array} { r } { ( \mathbf { q } , \mathbf { G } \mathbf { f } ) = \mathbf { q } ^ { \top } \mathbf { G } \mathbf { f } = \mathbf { f } ^ { \top } \mathbf { G } ^ { \top } \mathbf { q } . } \end{array} +$$ + +The divergence is naturally defined as the operator that maps edge operator $\mathbf { q }$ to the node space, that is $\nabla \cdot \approx - \mathbf { G } ^ { \top }$ . As usual, the graph Laplacian operator can be obtained by taking the divergence of the gradient. In graph theory it is defined as a positive matrix that is, $\Delta \widetilde { \approx } \mathbf { G } ^ { \intercal } \mathbf { G } ^ { \intercal }$ . + +We also define the weighted line integral over an edge. Similarly to Eq. (3)-(4), we define + +$$ +( \mathbf { A } \mathbf { f } ) _ { i j } = { \frac { 1 } { 2 } } \mathbf { W } _ { i j } ( \mathbf { f } _ { i } + \mathbf { f } _ { j } ) , \quad ( \mathbf { q } , \mathbf { A } \mathbf { f } ) = \mathbf { q } ^ { \top } \mathbf { A } \mathbf { f } = \mathbf { f } ^ { \top } \mathbf { A } ^ { \top } \mathbf { q } . +$$ + +The operator $\mathbf { A }$ approximates the mass operator on the edges. The right equation in Eq. (5) suggests that an appropriate averaging operator for edge features is the transpose of the nodal edge average. + +The advantage of defining such operators is that we are able to design networks with architectures that mimic the continuous operators (1) and (2) on the discrete level, as we show in the next section. + +# 3.3 PDE-GCN: Graph Convolutional Networks by Partial Differential Equations + +In order to use the computational models in Eq. (1)-(2), we form their discrete versions: + +$$ +\begin{array} { r l } & { { \bf f } ^ { ( l + 1 ) } = { \bf f } ^ { ( l ) } - h { \bf G } ^ { \top } { \bf K } _ { l } ^ { \top } \boldsymbol { \sigma } ( { \bf K } _ { l } { \bf G } { \bf f } ^ { ( l ) } ) , } \\ & { { \bf f } ^ { ( l + 1 ) } = 2 { \bf f } ^ { ( l ) } - { \bf f } ^ { ( l - 1 ) } - h ^ { 2 } { \bf G } ^ { \top } { \bf K } _ { l } ^ { \top } \boldsymbol { \sigma } ( { \bf K } _ { l } { \bf G } { \bf f } ^ { ( l ) } ) . } \end{array} +$$ + +Here, in Eq. (6) we use the forward Euler to discretize Eq. (1), and in Eq. (7) we discretize the second order time derivative in Eq. (2), using the leapfrog method. In both cases, $\mathbf { f } ^ { ( l ) }$ are the node features and $\mathbf { K } _ { l }$ is a $1 \times 1$ trainable convolution of the $l$ -th layer. The similarity between (6) and ResNet is well documented in the context of CNNs [22]. The hyper-parameter $h$ is the step-size, and it is chosen such that the stability of the discretization is kept. We use $\sigma = \operatorname { t a n h }$ for the activation function as it yields slightly better results in our experiments, although other functions such as ReLU can also be used, as reported in Sec. 4.7. Each of Eq. (6)-(7) defines a PDE-GCN block. We denote the former (diffusive equation) by PDE- $\mathrm { G C N _ { D } }$ and the latter (hyperbolic equation) by PDE- $\mathrm { G C N _ { H } }$ . + +To complete the description of our network, a few more details are required, as follows: + +The (opening) embedding layer. The input vertex features $\mathbf { u } _ { \nu }$ are fed through an embedding $( 1 \times 1$ convolution) layer ${ \bf K } _ { o }$ to obtain the initial features $\mathbf { f } _ { 0 }$ of our PDE-GCN network: $\mathbf { f } _ { 0 } = \mathbf { K } _ { o } \mathbf { u } _ { \mathcal { V } }$ . In cases where input edge attributes (features) $\mathbf { u } _ { \mathcal { E } }$ are available (as in the experiment in Sec. 4.6), we transform them to the vertex space by taking their divergence and average, and concatenate them to the input of the embedding layer ${ \bf K } _ { o }$ as follows: $\mathbf { f } _ { 0 } = \bar { \mathbf { K } _ { o } } ( \mathbf { u } _ { \mathcal { V } } \oplus \mathbf { A } ^ { \top } \mathbf { u } _ { \mathcal { E } } \oplus \mathbf { G } ^ { \top } \mathbf { u } _ { \mathcal { E } } )$ . + +The (closing) embedding layer. Given the final vertex features $\mathbf { f } ^ { ( L ) }$ of our PDE-GCN with $L$ layers, we obtain the output of the network by performing: ${ \bf u } _ { o u t } = { \bf K } _ { c } { \bf f } ^ { ( L ) }$ . Here $\mathbf { K } _ { c }$ is a $1 \times 1$ convolution layer mapping the hidden feature space to the output shape. + +Initialization. The $1 \times 1$ convolutions $\mathbf { K } _ { l }$ in Eq. (6)-(7) are initialized as identity. This way, the network begins from a diffusion/hyperbolic equation, which serves as a prior and guides the network to initially behave like classical methods [43, 45] and to further improve during training. + +The choice of dynamics. For some applications, anisotropic diffusion is appropriate, while for others conservation is more important. However, in some applications this may not be clear. To this end, it is possible to combine Eq. (1)-(2) to obtain the continuous process + +$$ +\alpha f _ { t t } + ( 1 - \alpha ) f _ { t } = \nabla \cdot K ^ { * } \sigma ( K \nabla f ) , \quad f ( t = 0 ) = f ^ { 0 } , \quad f _ { t } ( t = 0 ) = 0 \quad t \in [ 0 , T ] , +$$ + +where $\alpha = s i g m o i d ( \beta )$ , meaning $0 \leq \alpha \leq 1$ , and $\beta$ is a single trainable parameter. The discretization of this PDE leads to the following network dynamics: + +$$ +\alpha ( { \bf f } ^ { ( l + 1 ) } - 2 { \bf f } ^ { ( l ) } + { \bf f } ^ { ( l - 1 ) } ) + h ( 1 - \alpha ) ( { \bf f } ^ { ( l + 1 ) } - { \bf f } ^ { ( l ) } ) = - h ^ { 2 } { \bf G } ^ { \top } { \bf K } _ { l } ^ { \top } \sigma ( { \bf K } _ { l } { \bf G } { \bf f } ^ { ( l ) } ) , +$$ + +where $\mathbf { f } ^ { ( l + 1 ) }$ is updated by the known $\mathbf { f } ^ { ( l ) }$ and $\mathbf { f } ^ { ( l - 1 ) }$ . We denote a layer that is governed by Eq. (9) by PDE- $\mathrm { \bf G C N _ { M } }$ (where $\mathbf { M }$ stands for mixture). Note, that it is also possible to learn a combination coefficient $\alpha _ { i }$ per layer, although we did not read a benefit from such scheme. + +As we show in our numerical experiments, learning $\alpha$ yields results that are consistent with our understanding of the problem, that is, graph node-classification gravitates towards no second order derivatives while applications that require conservation gravitate towards the hyperbolic equation. + +# 4 Experiments + +In this section we demonstrate our approach on various problems from different fields and applications ranging from 3D shape-classification [47] to protein-protein interaction [26] and node-classification [48]. The experiments also vary in their output type. Node classification is similar to segmentation problems that are typically solved by anisotropic diffusion while the dense shape correspondence problem is conservative and therefore can be thought of as a problem that does not require smoothing. + +In all experiments, we use the suitable PDE-GCN (D, H or M) block as described in Sec. 3, with various depths (number of layers) and widths (number of channels), as well as the appropriate final convolution steps, depending on the task at hand. A detailed description of the architectures used in our experiments is given in Appendix B. We use the Adam [49] optimizer in all experiments, and perform grid search over the hyper-parameters of our network. The selected hyper-parameters are reported in Appendix C. Our objective function in all experiments is the cross-entropy loss, besides inductive learning on PPI where we use the binary cross-entropy loss. Our code is implemented using PyTorch [50], trained on an Nvidia Titan RTX GPU. + +We show that for all the considered tasks and datasets, our method is either remarkably better or on par with state-of-the-art models. + +Table 1: Generalization of GCNs to different domains and applications. $( L )$ denotes $L$ layers. + +
MethodDatasetAccuracy (%)
DGCNN (4) [5]ModelNet-1092.8
DGCNN (2)/ (4)Cora34.9 /25.2
DGCNN + Diffusion (2) /(4)Cora71.0 / 66.1
GCNII (4) [19]ModelNet-1065.4
Cora82.6
PDE-GCND(4) (Ours)ModelNet-1092.2
Cora83.6
+ +![](images/046ce85cdd3e4f12bc605c2fa6f9a0bc85b25d19266eee7bfd91a71978336f2a.jpg) +Figure 2: Learnt mixture of the hyperbolic equation $\alpha$ , to the dynamics of the network. The diffusion equation contribution is $1 - \alpha$ . + +# 4.1 Generalization of GCNs to different applications + +To gain deeper understanding about the effectiveness and generalization of various GCN methods to different tasks, we start by picking two datasets - ModelNet-10 [47] for 3D shape-classification, and Cora [48] for semi-supervised node classification. Those datasets are not only different in terms of application (global versus local classification), but also stem from different domains. While in ModelNet-10 the data has geometrical meaning, the data in Cora has no obvious geometrical interpretation. Therefore, we suggest that success in both applications should be obtained from a generalizable GCN. We compare our PDE- $\mathrm { G C N _ { D } }$ with two recent and popular networks - DGCNN [5] and GCNII [19]. For ModelNet-10 shape-classification, we randomly sample 1,024 points from each shape to form a point cloud, and connect its points using k-nearest-neighbors (k-nn) algorithm with $k = 1 0$ to obtain a graph and follow the training scheme of [5]. On Cora, we follow the same procedure as in [19]. We evaluate all models with 4 layers, as well as 2 layers for DGCNN on Cora. + +Our results, reported in Tab. 1 suggest that while each of the considered methods obtains high accuracy on the dataset it originally was tested on (Cora for GCNII and ModelNet-10 for DGCNN), obtaining a similar measure of success on a different dataset was not possible when using the very same networks. Additionally, while our attempts to add a diffusion-equation dynamics to DGCNN (i.e., updating features as in Eq. (6)) showed an increase in performance – a large gap to state-of-theart model still exists. On top of that, we also see that DGCNN suffers from over-smoothing, as its accuracy significantly decreases when adding more layers. Last but not least, we observe that our PDE- $\mathrm { \cdot G C N _ { D } }$ obtains high accuracy on both datasets, similar or better than state-of-the-art models. + +# 4.2 Learning PDE network dynamics + +In this experiment, we delve on the ability to learn the appropriate PDE that better models a given problem. To this end, we use the mixture model from Eq. (9) so that the resulting PDE is a combination of the diffusion and hyperbolic dynamics. We use a 8 layer mixed PDE-GCN, starting with $\alpha = 0 . 5$ , such that it is balanced between a PDE- $\mathrm { \cdot G C N _ { D } }$ and PDE- $\mathrm { \cdot G C N _ { H } }$ . By learning the parameter $\alpha$ in (9), we allow to choose a mixed PDE between a purely conservative network and a diffusive one. We consider two problems: semi-supervised node classification on Cora, and dense shape correspondence on FAUST [51]. + +Our results, reported in Fig. 2 suggest that just as in classical works [43, 52], problems like nodeclassification obtain better performance with an anisotropic diffusion like in Eq. (6), and for problems involving dense-correspondences like in [45, 11] that tend to conserve the energy of the underlying problem, a hyperbolic equation type of PDE as in Eq. (7) is more appropriate. + +Table 2: Statistics of datasets used in our semi-and fully supervised node-classification experiments. + +
DatasetCoraCiteSeerPubMedChameleonCornellTexasWisconsinPPI
Classes7635555121
Nodes2,7083,32719,7172,27718318325156,944
Edges5,4294,73244,33836,101295309499818,716
Features1,4333,7035002,3251,7031,7031,70350
+ +# 4.3 Semi-supervised node classification + +In this set of experiments we use three datasets – Cora, Citeseer and PubMed [48]. For all datasets we use the standard training/validation/testing split as in [53], with 20 nodes per class for training, 500 validation nodes and 1,000 testing nodes and follow the training scheme of [19]. The statistics of the datasets are reported in Tab. 2. We compare our results using PDE- $\mathrm { G C N _ { D } }$ with recent and popular models like GCN [3], GAT [27], APPNP [30], JKNet [29] and DropEdge [28]. We note that our network does not over-smooth, as an increase in the number of layers does not cause performance degradation. For example, on CiteSeer, we obtain $7 5 . 6 \%$ accuracy with 32 layers, compared to $7 4 . 6 \%$ with two layers3. Overall, our results in Tab. 3 show that our PDE- $\mathrm { \cdot G C N _ { D } }$ achieves similar or better accuracy than the considered methods. + +# 4.4 Fully-supervised node classification + +We follow [54] and use 7 datasets: Cora, CiteSeer, PubMed, Chameleon, Cornell, Texas and Wisconsin. We also use the same train/validation/test splits of $6 0 \%$ , $2 0 \%$ , $2 0 \%$ , respectively. In addition, we report the average performance over 10 random splits from [54]. We fix the number of channels to 64 and perform grid search to determine the hyper-parameters parameters, which are reported in Appendix C. We compare our network with GCN, GAT, three variants of Geom-GCN [54], APPNP, JKNet, Incep and GCNII in Tab. 4. Our experiments read either similar or better than the state-of-the-art on Cora, CiteSeer and PubMed datasets. On Chameleon [55], Cornell, Texas and Wisconsin datasets, we improve state-of-the-art accuracy by a significant margin. For example, we obtain $9 3 . 2 4 \%$ accuracy on Texas with our PDE- $\mathrm { G C N _ { M } }$ , compared to $7 7 . 8 4 \%$ with $\mathrm { G C N I I ^ { \ast } }$ . Similar improvements hold for Cornell and Wisconsin datasets. The common factor for these datasets is their small size, as depicted from Tab. 2. We argue that the success of our network stems from its capability of apriori extracting features from graphs, due to its utilization of discretized differential operators and PDE guided construction. On Chameleon [55], using PDE- $\mathrm { G C N _ { M } }$ , we improve the current state-of-the-art accuracy of $\operatorname { G C N I I } ^ { * }$ from $6 2 . 4 8 \%$ to $6 6 . 0 1 \%$ . Also, we note that unlike in the semi-supervised case, where some of the labels are missing, here it is possible to obtain meaningful results with the hyperbolic equation based PDE- $\mathrm { G C N _ { H } }$ as we do not have unknown nodes in the fully-supervised case, which would be otherwise preserved using the hyperbolic equation dynamics. + +# 4.5 Inductive Learning + +We follow [19] and employ the PPI dataset [26] for the inductive learning task. We use a 8 layer PDE- $\mathrm { G C N _ { D } }$ network, without dropout or weight-decay, and a learning rate of 0.001. We compare our results with methods like GraphSAGE, GAT, JKNet, GeniePath, GCNII and others. As reported in Tab. 5, our PDE-GCN $\mathrm { D }$ achieves 99.07 Micro-averaged F1 score, superior to methods like GAT, JKNet and GeniePath, also close to state-of-the-art GCNII\* with a score of 99.58. + +Table 3: Semi-supervised node classification accuracy $( \% )$ ). – indicates not available results. + +
Layers
DatasetMethod248163264
CoraGCN [3]81.180.469.564.960.328.7
GCN (Drop) [28]82.882.075.875.762.549.5
JKNet [29]80.280.780.281.171.5
JKNet (Drop) [28]83.382.683.082.583.2
Incep [28]77.676.581.781.780.0
Incep (Drop)[28]82.982.583.183.183.5
GCNII [19]82.282.684.284.685.485.5
GCNII*[19]80.282.382.883.584.985.3
PDE-GCND (Ours)82.083.684.084.284.384.3
CiteseerGCN [3]70.867.630.218.325.020.0
GCN (Drop)[28]72.370.661.457.241.634.4
JKNet [29]168.767.769.868.263.4
JKNet (Drop) [28]72.671.872.670.872.2
Incep [28]69.368.470.268.067.5
Incep (Drop) [28]72.771.472.572.671.0
GCNII [19]68.268.870.672.973.473.4
GCNII* [19]66.166.770.672.073.273.1
PDE-GCND (Ours)74.675.075.275.575.675.5
PubmedGCN [3]79.076.561.240.922.435.3
GCN (Drop)[28]79.679.478.178.577.061.5
JKNet [29]78.078.172.672.474.5
JKNet (Drop) [28]78.778.779.779.278.9
Incep [28]77.777.974.911
Incep (Drop) [28]79.578.679.0
GCNII [19]78.278.879.380.279.879.7
GCNII* [19]77.778.278.880.379.880.1
PDE-GCND (Ours)79.380.680.180.480.280.3
+ +# 4.6 Dense shape correspondence + +Finding dense correspondences between shapes is a classical experiment for hyperbolic dynamics, as we are interested in modeling local motion dynamics. In essence, learning to find correspondences between shapes is similar to learning a transformation from one shape to the other. To this end, we use the FAUST dataset [51] containing 10 scanned human shapes in 10 different poses with 6,890 nodes each. We follow the train and test split from [4], where the first 80 subjects are used for training and the remaining 20 subjects for testing. Our metric is the correspondence percentage with zero geodesic error, i.e., the percentage of perfectly matched shapes from all our test cases. We follow the pre-processing and training scheme of [62] where we use Cartesian coordinates to describe distances between nodes, with initial features of a constant $\mathbf { 1 } \in \mathbb { R } ^ { n }$ where $n$ is the number of nodes. We use a 8 layer PDE- $\mathrm { G C N _ { D } }$ and PDE- $\mathrm { G C N _ { H } }$ variants, both with constant learning rate of 0.001 with no weight-decay or dropout, and compare to recent and popular methods like ACNN [60], MoNet [4], FMNet [61], and SplineCNN [62]. As expected from the discussion in Sec. 3.1, and reported in Tab. 6, the hyperbolic equation proves to be a better fit for this kind of problem. Also, our PDE- $\mathrm { G C N _ { H } }$ achieves a promising $9 9 . 9 \%$ correspondence rate with zero geodesic error, outperforming the rest of the considered methods. + +# 4.7 Ablation study + +Our method has two main dynamics – diffusion and hyperbolic. To verify our proposal in Sec. 3.1, we examine the performance of the hyperbolic PDE- $\mathrm { G C N _ { H } }$ on semi-supervised node-classification on Cora and CiteSeer datasets. Our results in Tab. 7 show that indeed for problems where we wish to obtain a piecewise-constant prediction, the diffusive formulation of our network, PDE- $\mathrm { \ G C N _ { D } }$ , is more suitable. Furthermore, we study the importance of the positive-semi definiteness of our learnt operator as described in Eq. (6)-(7), by removing the ${ \bf K } _ { l } ^ { T }$ term from the dynamics equations. This yields a non-symmetric operator that does not guarantee positive-semi-definiteness. We note that the enforcement of the latter is important to obtain higher accuracy which is improved by up to $3 \%$ with the introduction of a positive semi-definite operator. Also, we report on the use of $\sigma = R e L U$ , from the discussion in Sec. 3.3, where we favor tanh as an activation function, and the ability of our PDE- $\mathrm { \mathrm { . G C N _ { M } } }$ to reproduce the results of PDE- $\mathrm { G C N _ { D } }$ in the case of semi-supervised learning. + +Table 4: Fully-supervised node classification accuracy $( \% )$ . (L) indicates a $L$ layers network. + +
MethodCoraCite.Pubm.Cham.Corn.TexasWisc.
GCN [3]85.7773.6888.1328.1852.7052.1645.88
GAT[27]86.3774.3287.6242.9354.3258.3849.41
Geom-GCN-I [54]85.1977.9990.0560.3156.7657.5858.24
Geom-GCN-P [54]84.9375.1488.0960.9060.8167.5764.12
Geom-GCN-S [54]85.2774.7184.7559.9655.6859.7356.67
APPNP [30]87.8776.5389.4054.3073.5165.4169.02
JKNet [29]85.25 (16)75.85 (8)88.94 (64)60.07 (32)57.30 (4)56.49 (32)48.82 (8)
JKNet (Drop) [28]87.46 (16)75.96 (8)89.45 (64)62.08 (32)61.08 (4)57.30 (32)50.59 (8)
Incep (Drop) [28]86.86(8)76.83 (8)89.18 (4)61.71 (8)61.62 (16)57.84 (8)50.20 (8)
GCNII [19]88.49 (64)77.08 (64)89.57 (64)60.61 (8)74.86 (16)69.46 (32)74.12 (16)
GCNII*88.01 (64)77.13 (64)90.30 (64)62.48 (8)76.49 (16)77.84 (32)81.57 (16)
PDE-GCND (Ours)88.51 (16)78.36 (64)89.6 (64)64.12 (8)89.19 (2)90.81 (8)90.39 (8)
PDE-GCNH (Ours)87.71 (32)78.13 (16)89.16 (16)61.57 (64)89.45 (64)92.16 (64)91.37 (16)
PDE-GCNm (Ours)88.60 (16)78.48 (32)89.93 (16)66.01 (16)89.73 (64)93.24 (32)91.76 (16)
+ +Table 5: Protein-protein interaction (PPI). Results are reported in micro-averaged F1 score. + +
MethodMicro-averaged F1
GraphSAGE [26]61.20
VR-GCN [56]97.80
GaAN [57]98.71
GAT [27]97.30
JKNet [29]97.60
GeniePath [58]98.50
Cluster-GCN [59]99.36
GCNII [19]99.54
GCNII*[19]99.58
PDE-GCND (Ours)99.07
PDE-GCNm (Ours)99.18
+ +Table 6: Dense shape correspondence $( \% )$ with zero geodesic error + +
MethodFaust
ACNN [60]63.8
MoNet [4]89.1
FMNet [61] SplineCNN [62]98.2 99.2
PDE-GCND (Ours)
PDE-GCNH (Ours)64.2 99.9
+ +# 5 Summary + +In this paper we explored new architectures for graph neural networks. Our motivation stems from the similarities between graph networks and time dependent partial differential equations that are discretized on manifolds and graphs. By adopting an appropriate PDE, and embedding the finite graph in an infinite manifold, we are able to define networks that are either diffusive, conservative, or a combination of both. + +Not all natural phenomena are solved using the same PDE and we should not expect that all graph problems should be solved by the same network dynamics. To this end we allow the data to choose which type of network is appropriate for the solution of the problem (diffusive or hyperbolic). Indeed, numerical experiments show that the network gravitates towards a hyperbolic one for problems where conservation is required, and towards a diffusive one when anisotropic diffusion is favorable. + +Table 7: Ablation study of PDE-GCN accuracy $( \% )$ on semi-supervised node-classification. + +
Layers
MethodDataset248163264
PDE-GCNHCora CiteSeer79 70.979.378.0 72.178.077.877.5 72.4
PDE-GCND (non-symmetric)Cora83.571.7 83.383.672.3 83.172.5 82.781
PDE-GCND (σ = ReLU)CiteSeer Cora74.3 80.374.5 81.874.8 82.675.0 83.073.9 83.473.3 83.5
PDE-GCNmCiteSeer Cora CiteSeer73.1 82.073.2 83.472.8 83.973.3 84.273.6 84.374.0 84.5
+ +Finally, we showed that the proposed networks can be made deep without over-smoothing and, can deliver the state-of-the-art performance or improve it for virtually every problem we worked with. In particular, our network dramatically improved the state-of-the-art for problems that are data-poor. We believe that for such problems the structure imposed by our dynamics and operators regularizes the network and therefore yields implicit regularization. + +# Acknowledgments and Disclosure of Funding + +The research reported in this paper was supported by grant no. 2018209 from the United States - Israel Binational Science Foundation (BSF), Jerusalem, Israel. ME is supported by Kreitman High-tech scholarship. + +# References + +[1] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013. +[2] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. 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[Yes] +(b) Did you describe the limitations of your work? [Yes] Our theorems state that the activation function must be sign-preserving, bounded and monotone. +(c) Did you discuss any potential negative societal impacts of your work? [N/A] We did not find any societal impacts of the work. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] Proofs are provided in the supplementary material. + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code and the links to the (public) data sets are provided in the supplementary material. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] +(b) Did you mention the license of the assets? [N/A] +(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] + +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/wWtk6GxJB2x/wWtk6GxJB2x_content_list.json b/parse/train/wWtk6GxJB2x/wWtk6GxJB2x_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e6581769d72c6e68c7df692edf29f4e2534e4819 --- /dev/null +++ b/parse/train/wWtk6GxJB2x/wWtk6GxJB2x_content_list.json @@ -0,0 +1,1283 @@ +[ + { + "type": "text", + "text": "PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations ", + "text_level": 1, + "bbox": [ + 174, + 122, + 823, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Moshe Eliasof Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva, Israel eliasof@post.bgu.ac.il ", + "bbox": [ + 186, + 226, + 428, + 295 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Eldad Haber Department of Earth, Ocean and Atmospheric Sciences University of British Columbia Vancouver, Canada ehaber@eoas.ubc.ca ", + "bbox": [ + 447, + 226, + 810, + 295 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Eran Treister ", + "text_level": 1, + "bbox": [ + 450, + 318, + 545, + 330 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Computer Science Ben-Gurion University of the Negev Beer-Sheva, Israel erant@cs.bgu.ac.il ", + "bbox": [ + 379, + 332, + 619, + 386 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 421, + 535, + 438 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Graph neural networks are increasingly becoming the go-to approach in various fields such as computer vision, computational biology and chemistry, where data are naturally explained by graphs. However, unlike traditional convolutional neural networks, deep graph networks do not necessarily yield better performance than shallow graph networks. This behavior usually stems from the over-smoothing phenomenon. In this work, we propose a family of architectures to control this behavior by design. Our networks are motivated by numerical methods for solving Partial Differential Equations (PDEs) on manifolds, and as such, their behavior can be explained by similar analysis. Moreover, as we demonstrate using an extensive set of experiments, our PDE-motivated networks can generalize and be effective for various types of problems from different fields. Our architectures obtain better or on par with the current state-of-the-art results for problems that are typically approached using different architectures. ", + "bbox": [ + 233, + 453, + 764, + 632 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 656, + 310, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, Graph Convolutional Networks (GCNs) [1, 2, 3] have drawn the attention of researchers and practitioners in a variety of domains and applications, ranging from computer vision and graphics [4, 5, 6, 7, 8] to computational biology [9, 10, 11], recommendation systems [12] and social network analysis [13, 14]. However, GCNs still suffer from two main problems. First, they are usually shallow as opposed to the concept of deep convolutional neural networks (CNNs) [15, 16] due to the over-smoothing phenomenon [17, 18, 19], where the node feature vectors become almost identical, such that they are indistinguishable, which yields non-optimal performance. Furthermore, GCNs are typically customized to a specific domain and application. That is, as we demonstrate in Sec. 4.1, a successful point-cloud classification network [5] can perform poorly on a citation graph node-classification problem [19], and vice-versa. Furthermore, because many GCNs lack theoretical guarantees, it is difficult to reason about their success on one problem and lack on others. These observations motivate us to develop a profound understanding of graph networks and their dynamics. ", + "bbox": [ + 174, + 688, + 825, + 853 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To this end, we suggest a novel, universal approach to the design of GCN architectures. Our inspiration stems from the similarities and equivalence between Partial Differential Equations (PDEs) and deep networks explored in [20, 21, 22]. Furthermore, as GCNs can be thought of as a generalization of ", + "bbox": [ + 176, + 859, + 823, + 902 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "CNNs, and a standard convolution can be represented as a combination of differential operators on a structured grid [22], we adopt this interpretation to explore versions of GCNs as PDEs on graphs or manifolds. We therefore call our network architectures PDE-GCN, and demonstrate that our approach is general with respect to the given task. That is, our architectures behave similarly for different problems, and their performance is on par or better than other domain-specific GCNs. Furthermore, our family of architectures are backed by theoretical guarantees that allow us to explain the behavior of the GCNs in some of the results that we present. To be more specific, our contribution is as follows: ", + "bbox": [ + 173, + 92, + 825, + 189 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We introduce and implement general graph convolution operators, based on graph gradient and divergence. This abstraction of the spatial operation on the graph leads to a more general and flexible approach for architecture design. \n• We propose treating a variety of graph related problems as discretized PDEs, and formulate the dynamics that match different problems such as node-classification and dense shape-correspondence. This is in direct effort to propose a family of networks that can solve multiple problems, instead of GCNs which are tailored for a specific application. \n• Our method allows constructing a deep GCN without over-smoothing, with theoretical guarantees. \n• We validate our method by conducting numerous experiments on multiple datasets and applications, achieving significantly better or similar accuracy compared to state-of-the-art models. ", + "bbox": [ + 173, + 199, + 826, + 348 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related work ", + "text_level": 1, + "bbox": [ + 174, + 367, + 316, + 383 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Graph Convolutional Networks: GCNs are typically divided into spectral [1, 3, 2] and spatial [23, 24, 25, 4] categories. Most of those can be implemented using the Message-Passing Neural Network paradigm [25], where each node aggregates features (messages) from its neighbors, according to some scheme. The works [3, 2] use polynomials of the graph Laplacian to parameterize the convolution operator. DGCNN [5] constructs a $\\mathbf { k }$ -nearest-neighbors graph from point-clouds and dynamically updates it. MoNet [4] learns a Gaussian mixture model to weight the edges of meshes for shape analysis tasks. Works like GraphSAGE [26] and GAT [27] propose methods for inductive and transductive learning on non-geometrical graphs. ", + "bbox": [ + 174, + 398, + 825, + 510 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Several of the methods above suffer from over-smoothing [17, 19], leading to undesired node features similarity for deep networks. To overcome this problem, some approaches rely on imposing regularization and augmentation. For example, PairNorm [17] introduces a novel normalization layer, and DropEdge [28] randomly removes edges to decrease the degree of nodes. Other methods prevent over-smoothing by dedicated construction. For instance, JKNet [29] combines all intermediate representations at each stage of the network. APPNP [30] replaces learnt convolutional layers with a pre-defined kernel based on PageRank, yielding a shallow network which preserves locality, making it robust to over-smoothing. GCNII [19] proposes to add an identity residual where the initial features of the networks are added to the features of the $l$ -th layer, scaled by some coefficient. ", + "bbox": [ + 174, + 515, + 825, + 640 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Another approach is to construct a network that inherently does not over-smooth, as we suggest in this work. Our network is based on discretized PDEs, hence we are able to motivate our choices by well studied theory and numerical experiments [31]. On a similar note, the recent DiffGCN [8] also makes use of discretized operators to approximate the graph gradient and Laplacian. However, DiffGCN is specifically tailored for geometric tasks since it projects its components on the $x , y , z$ axes, and is applied using a ResNet [16] (diffusion) structure only. The recent GRAND [32] applies attention mechanism with diffusive dynamics, using several integration schemes. Here we propose a network for both geometric and non-geometric tasks, and also utilize both the diffusion or hyperbolic layer dynamics, including a mixture of the two. ", + "bbox": [ + 174, + 646, + 825, + 771 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "PDEs and CNNs: In a recent series of works, the connection between PDEs and CNNs was studied [20, 21, 22, 33, 34, 35, 36, 37]. It was shown that it is possible to treat a deep neural network as a dynamical system driven by some PDE, where each convolution layer is considered a time step of a discretized PDE. The connection between PDEs and CNNs was also used to reduce the computational burden [38]. Besides the interpretation of CNNs as a PDE solver, it was shown that it is also possible to learn a symbolic representation of PDEs in a task-and data driven approach in [39, 40]. In the context of GCNs and PDEs, it was recently shown [41, 42] that GCNs can be utilized to enhance the solution of PDEs for problems like fluid flow dynamics. However, in this work, we harness PDE concepts to design and construct GCNs for a variety of applications. ", + "bbox": [ + 174, + 786, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/84fba16b4d384652779751fd38ae51dc9abe79770ca77ec1b8facd8fdbdfb1ea.jpg", + "image_caption": [ + "Figure 1: Feature evolvement on an input mesh (left). Propagation in time is from left to right. Hyperbolic and diffusion equation dynamics are on the top and bottom row, respectively. While a diffusive graph network (similar to most common GCNs that rely on ResNets) smooths the information on the manifold, the hyperbolic network yields a non-uniform field. " + ], + "image_footnote": [], + "bbox": [ + 183, + 85, + 816, + 382 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Methods ", + "text_level": 1, + "bbox": [ + 174, + 476, + 279, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Partial Differential Equations on manifolds ", + "text_level": 1, + "bbox": [ + 174, + 506, + 514, + 522 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We show now that GCNs can be viewed as discretizations of PDEs on manifolds, similarly to CNNs that are viewed as discretized PDEs on regular grids [22]. Consider a general manifold $\\mathcal { M }$ where a vector function $f$ resides (also dubbed the features function), along with its continuous differential operators such as the gradient $\\nabla$ , divergence $\\nabla$ · and the Laplacian $\\Delta$ that reside on the manifold $\\mathcal { M }$ ", + "bbox": [ + 173, + 532, + 825, + 589 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given these differential operators, one can model different processes on $\\mathcal { M }$ . In particular, we consider two PDEs – the non-linear diffusion and the non-linear hyperbolic equations ", + "bbox": [ + 176, + 594, + 823, + 623 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/eb1518e442fa6d11189df90a0587dc7d4bfb2efa5380a76ee4f00f41b6a93e23.jpg", + "text": "$$\n\\begin{array} { r l } & { f _ { t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad t \\in [ 0 , T ] , } \\\\ & { f _ { t t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad f _ { t } ( t = 0 ) = 0 , \\quad t \\in [ 0 , T ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 628, + 736, + 670 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "respectively, equipped with appropriate boundary conditions. Here $K$ is a coefficient matrix that can change in time and represents the propagation over the manifold $\\mathcal { M }$ , $K ^ { * }$ is its conjugate transpose and $\\sigma ( \\cdot )$ is a non-linear activation function. Eq. (1)-(2) define a non-linear operator that takes initial feature vectors $f ^ { 0 }$ at time 0 and propagates them to time $T$ , yielding $f ^ { T }$ where they can be used for different tasks. We now provide two theorems that characterize the behavior of Eq. (1)-(2), based on ideas from [22]1. ", + "bbox": [ + 174, + 672, + 825, + 757 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 1. If the activation function $\\sigma ( \\cdot )$ is monotonically non-decreasing and sign-preserving, then the forward propagation through the diffusive PDE in (1) for $t \\in [ 0 , \\infty )$ yields a non-increasing feature norm, that is, ", + "bbox": [ + 173, + 761, + 825, + 803 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c906ab6527cec33da4e90f74c6bdb9dabd423377c5d1aafe833375b7d541b787.jpg", + "text": "$$\n\\frac { \\partial } { \\partial t } \\lvert | f \\rvert | ^ { 2 } \\leq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 454, + 801, + 544, + 832 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 2. Assume that the activation function $\\sigma ( \\cdot )$ is monotonically non-decreasing, signpreserving and satisfies $| \\sigma ( x ) | \\leq | x |$ , and define energy ", + "bbox": [ + 171, + 834, + 823, + 863 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/048f6c37a9c2c3b593390ce74289d0a715f20310955fe1076e1b5aa8b9895c6f.jpg", + "text": "$$\n\\mathcal { E } _ { n e t } = \\| f _ { t } \\| ^ { 2 } + \\left( K \\nabla f , \\sigma ( K \\nabla f ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 871, + 614, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "then the forward propagation through the hyperbolic PDE in (2) satisfies $\\mathcal { E } _ { n e t } \\leq c _ { K }$ , where $c _ { K }$ is $a$ constant that depends on $K$ but independent of time. ", + "bbox": [ + 173, + 90, + 825, + 119 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The outcome of those theorems is that the dynamics described in Eq. (1) is smoothing, while the one in Eq. (2) is bounded by a conserving mapping. An illustration of this behavior is presented in Fig. 1. ", + "bbox": [ + 173, + 130, + 823, + 159 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the physical world, diffusion and hyperbolic equations are used for different applications. Similarly, many computational models for image segmentation [43], denoising [44] and deblurring are based on anisotropic diffusion which are similar to the model in Eq. (1). On the other hand, applications that require conservation such as volume/distance preservation as in the dense shape correspondence task [45] and protein folding [11], are typically better treated using a hyperbolic equation as in Eq. (2). Those insights motivate us to construct two types of layers according to Eq. (1)-(2) using discretized differential operators on graphs. ", + "bbox": [ + 174, + 165, + 825, + 262 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Discretized differential operators on graphs ", + "text_level": 1, + "bbox": [ + 174, + 277, + 516, + 292 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The models (1) and (2) reside in a continuous manifold $\\mathcal { M }$ , on which a continuous function vector $f$ is defined. A graph can be thought of as a discretization of that manifold to a finite space. Assume we are given an undirected graph $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ where $\\nu \\in \\mathcal { M }$ is the set of $n$ vertices of the graph and $\\mathcal { E }$ is the set of $m$ edges of the graph. Let us denote by $\\mathbf { f } _ { i } \\in \\mathbb { R } ^ { c }$ the value of the discrete version of $f$ , on the $i$ -th node of $\\mathcal { G }$ . $c$ is the number of channels, which is the width of the neural network. We define $\\mathbf { G }$ , the discrete gradient operator on the graph, also known as the incidence matrix, as follows: ", + "bbox": [ + 173, + 303, + 825, + 387 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/59db653613e838769b356ae04120868f1ebcee0a0ec475e7660fb756b21ad375.jpg", + "text": "$$\n( \\mathbf { G f } ) _ { i j } = \\mathbf { W } _ { i j } ( \\mathbf { f } _ { i } - \\mathbf { f } _ { j } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 392, + 580, + 410 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where nodes $i$ and $j$ are connected via the $( i , j )$ -th edge, $\\mathbf { W } _ { i j }$ is an edge weight matrix which can be learnt, and $\\mathbf { f } _ { i }$ and $\\mathbf { f } _ { j }$ are the features on the $i$ -th and $j$ -th nodes, respectively. The gradient operator can be thought of as a weighted directional derivative of the function $f$ in the direction defined by the nodes $i$ and $j$ . Furthermore, if we choose the scaled identity matrix $\\mathbf { W } _ { i j } = d _ { i j } ^ { - 1 } \\mathbf { I }$ , where $d _ { i j }$ is the distance between the two nodes, then the discrete gradient is a second order approximation to the true gradient of the function $f$ on the edges of the graph. In this work, we use $\\mathbf { W } _ { i j } = \\gamma _ { i j } \\mathbf { I }$ , where the scale $\\gamma _ { i j }$ is the geometric mean of the degree of the nodes $i , j$ . Note, that the gradient operator is a mapping from the vertex space to the edge space. ", + "bbox": [ + 173, + 415, + 825, + 529 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given the gradient matrix, it is possible to define the divergence matrix [46], which is an approach that is used extensively in mimetic discretizations of PDEs. To this end, we define the inner product between an edge feature vector $\\mathbf { q }$ and the gradient of a node feature vector f as ", + "bbox": [ + 174, + 534, + 825, + 577 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4d2c94eaa2db622882c74f687aed6b3084bb2cb2baa97bea70a97c8af10bf501.jpg", + "text": "$$\n\\begin{array} { r } { ( \\mathbf { q } , \\mathbf { G } \\mathbf { f } ) = \\mathbf { q } ^ { \\top } \\mathbf { G } \\mathbf { f } = \\mathbf { f } ^ { \\top } \\mathbf { G } ^ { \\top } \\mathbf { q } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 582, + 583, + 602 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The divergence is naturally defined as the operator that maps edge operator $\\mathbf { q }$ to the node space, that is $\\nabla \\cdot \\approx - \\mathbf { G } ^ { \\top }$ . As usual, the graph Laplacian operator can be obtained by taking the divergence of the gradient. In graph theory it is defined as a positive matrix that is, $\\Delta \\widetilde { \\approx } \\mathbf { G } ^ { \\intercal } \\mathbf { G } ^ { \\intercal }$ . ", + "bbox": [ + 174, + 606, + 825, + 650 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We also define the weighted line integral over an edge. Similarly to Eq. (3)-(4), we define ", + "bbox": [ + 169, + 655, + 759, + 670 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4b4864d4e2c8be944528eabc784101d487b24ebc4e760c0f0234ccdbdd70d670.jpg", + "text": "$$\n( \\mathbf { A } \\mathbf { f } ) _ { i j } = { \\frac { 1 } { 2 } } \\mathbf { W } _ { i j } ( \\mathbf { f } _ { i } + \\mathbf { f } _ { j } ) , \\quad ( \\mathbf { q } , \\mathbf { A } \\mathbf { f } ) = \\mathbf { q } ^ { \\top } \\mathbf { A } \\mathbf { f } = \\mathbf { f } ^ { \\top } \\mathbf { A } ^ { \\top } \\mathbf { q } .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 675, + 694, + 705 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The operator $\\mathbf { A }$ approximates the mass operator on the edges. The right equation in Eq. (5) suggests that an appropriate averaging operator for edge features is the transpose of the nodal edge average. ", + "bbox": [ + 173, + 710, + 823, + 739 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The advantage of defining such operators is that we are able to design networks with architectures that mimic the continuous operators (1) and (2) on the discrete level, as we show in the next section. ", + "bbox": [ + 173, + 744, + 825, + 773 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 PDE-GCN: Graph Convolutional Networks by Partial Differential Equations ", + "text_level": 1, + "bbox": [ + 171, + 787, + 748, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to use the computational models in Eq. (1)-(2), we form their discrete versions: ", + "bbox": [ + 171, + 814, + 750, + 830 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7d2dd7c6a7bc8054bdffdeadf755a35d9cba3cd6d987550a2d55c5fc0a443a26.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\bf f } ^ { ( l + 1 ) } = { \\bf f } ^ { ( l ) } - h { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\boldsymbol { \\sigma } ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) , } \\\\ & { { \\bf f } ^ { ( l + 1 ) } = 2 { \\bf f } ^ { ( l ) } - { \\bf f } ^ { ( l - 1 ) } - h ^ { 2 } { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\boldsymbol { \\sigma } ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 833, + 663, + 877 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, in Eq. (6) we use the forward Euler to discretize Eq. (1), and in Eq. (7) we discretize the second order time derivative in Eq. (2), using the leapfrog method. In both cases, $\\mathbf { f } ^ { ( l ) }$ are the node features and $\\mathbf { K } _ { l }$ is a $1 \\times 1$ trainable convolution of the $l$ -th layer. The similarity between (6) and ResNet is well documented in the context of CNNs [22]. The hyper-parameter $h$ is the step-size, and it is chosen such that the stability of the discretization is kept. We use $\\sigma = \\operatorname { t a n h }$ for the activation function as it yields slightly better results in our experiments, although other functions such as ReLU can also be used, as reported in Sec. 4.7. Each of Eq. (6)-(7) defines a PDE-GCN block. We denote the former (diffusive equation) by PDE- $\\mathrm { G C N _ { D } }$ and the latter (hyperbolic equation) by PDE- $\\mathrm { G C N _ { H } }$ . ", + "bbox": [ + 174, + 90, + 825, + 204 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To complete the description of our network, a few more details are required, as follows: ", + "bbox": [ + 173, + 209, + 746, + 224 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The (opening) embedding layer. The input vertex features $\\mathbf { u } _ { \\nu }$ are fed through an embedding $( 1 \\times 1$ convolution) layer ${ \\bf K } _ { o }$ to obtain the initial features $\\mathbf { f } _ { 0 }$ of our PDE-GCN network: $\\mathbf { f } _ { 0 } = \\mathbf { K } _ { o } \\mathbf { u } _ { \\mathcal { V } }$ . In cases where input edge attributes (features) $\\mathbf { u } _ { \\mathcal { E } }$ are available (as in the experiment in Sec. 4.6), we transform them to the vertex space by taking their divergence and average, and concatenate them to the input of the embedding layer ${ \\bf K } _ { o }$ as follows: $\\mathbf { f } _ { 0 } = \\bar { \\mathbf { K } _ { o } } ( \\mathbf { u } _ { \\mathcal { V } } \\oplus \\mathbf { A } ^ { \\top } \\mathbf { u } _ { \\mathcal { E } } \\oplus \\mathbf { G } ^ { \\top } \\mathbf { u } _ { \\mathcal { E } } )$ . ", + "bbox": [ + 174, + 229, + 825, + 301 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The (closing) embedding layer. Given the final vertex features $\\mathbf { f } ^ { ( L ) }$ of our PDE-GCN with $L$ layers, we obtain the output of the network by performing: ${ \\bf u } _ { o u t } = { \\bf K } _ { c } { \\bf f } ^ { ( L ) }$ . Here $\\mathbf { K } _ { c }$ is a $1 \\times 1$ convolution layer mapping the hidden feature space to the output shape. ", + "bbox": [ + 174, + 308, + 825, + 352 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Initialization. The $1 \\times 1$ convolutions $\\mathbf { K } _ { l }$ in Eq. (6)-(7) are initialized as identity. This way, the network begins from a diffusion/hyperbolic equation, which serves as a prior and guides the network to initially behave like classical methods [43, 45] and to further improve during training. ", + "bbox": [ + 174, + 357, + 825, + 400 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The choice of dynamics. For some applications, anisotropic diffusion is appropriate, while for others conservation is more important. However, in some applications this may not be clear. To this end, it is possible to combine Eq. (1)-(2) to obtain the continuous process ", + "bbox": [ + 174, + 405, + 825, + 448 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b607c37ffd9c130475f3c24ab97ca4817b615915fec46c0f511f6110feeb0f08.jpg", + "text": "$$\n\\alpha f _ { t t } + ( 1 - \\alpha ) f _ { t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad f _ { t } ( t = 0 ) = 0 \\quad t \\in [ 0 , T ] ,\n$$", + "text_format": "latex", + "bbox": [ + 197, + 455, + 766, + 473 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\alpha = s i g m o i d ( \\beta )$ , meaning $0 \\leq \\alpha \\leq 1$ , and $\\beta$ is a single trainable parameter. The discretization of this PDE leads to the following network dynamics: ", + "bbox": [ + 171, + 481, + 823, + 510 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7eae4c85ed09f79b2fdb9c084e1247d56883808773b4f4f32acfc2e9e28e4d62.jpg", + "text": "$$\n\\alpha ( { \\bf f } ^ { ( l + 1 ) } - 2 { \\bf f } ^ { ( l ) } + { \\bf f } ^ { ( l - 1 ) } ) + h ( 1 - \\alpha ) ( { \\bf f } ^ { ( l + 1 ) } - { \\bf f } ^ { ( l ) } ) = - h ^ { 2 } { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\sigma ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 214, + 517, + 751, + 537 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbf { f } ^ { ( l + 1 ) }$ is updated by the known $\\mathbf { f } ^ { ( l ) }$ and $\\mathbf { f } ^ { ( l - 1 ) }$ . We denote a layer that is governed by Eq. (9) by PDE- $\\mathrm { \\bf G C N _ { M } }$ (where $\\mathbf { M }$ stands for mixture). Note, that it is also possible to learn a combination coefficient $\\alpha _ { i }$ per layer, although we did not read a benefit from such scheme. ", + "bbox": [ + 174, + 545, + 825, + 589 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As we show in our numerical experiments, learning $\\alpha$ yields results that are consistent with our understanding of the problem, that is, graph node-classification gravitates towards no second order derivatives while applications that require conservation gravitate towards the hyperbolic equation. ", + "bbox": [ + 174, + 594, + 825, + 637 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 657, + 312, + 675 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section we demonstrate our approach on various problems from different fields and applications ranging from 3D shape-classification [47] to protein-protein interaction [26] and node-classification [48]. The experiments also vary in their output type. Node classification is similar to segmentation problems that are typically solved by anisotropic diffusion while the dense shape correspondence problem is conservative and therefore can be thought of as a problem that does not require smoothing. ", + "bbox": [ + 174, + 689, + 825, + 760 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In all experiments, we use the suitable PDE-GCN (D, H or M) block as described in Sec. 3, with various depths (number of layers) and widths (number of channels), as well as the appropriate final convolution steps, depending on the task at hand. A detailed description of the architectures used in our experiments is given in Appendix B. We use the Adam [49] optimizer in all experiments, and perform grid search over the hyper-parameters of our network. The selected hyper-parameters are reported in Appendix C. Our objective function in all experiments is the cross-entropy loss, besides inductive learning on PPI where we use the binary cross-entropy loss. Our code is implemented using PyTorch [50], trained on an Nvidia Titan RTX GPU. ", + "bbox": [ + 174, + 765, + 825, + 877 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We show that for all the considered tasks and datasets, our method is either remarkably better or on par with state-of-the-art models. ", + "bbox": [ + 173, + 883, + 821, + 911 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/ebce0c204acde99b0a5558dac048d9aef67a145630997c94ea9debe6e8d33333.jpg", + "table_caption": [ + "Table 1: Generalization of GCNs to different domains and applications. $( L )$ denotes $L$ layers. " + ], + "table_footnote": [], + "table_body": "
MethodDatasetAccuracy (%)
DGCNN (4) [5]ModelNet-1092.8
DGCNN (2)/ (4)Cora34.9 /25.2
DGCNN + Diffusion (2) /(4)Cora71.0 / 66.1
GCNII (4) [19]ModelNet-1065.4
Cora82.6
PDE-GCND(4) (Ours)ModelNet-1092.2
Cora83.6
", + "bbox": [ + 282, + 117, + 715, + 257 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/046ce85cdd3e4f12bc605c2fa6f9a0bc85b25d19266eee7bfd91a71978336f2a.jpg", + "image_caption": [ + "Figure 2: Learnt mixture of the hyperbolic equation $\\alpha$ , to the dynamics of the network. The diffusion equation contribution is $1 - \\alpha$ . " + ], + "image_footnote": [], + "bbox": [ + 183, + 285, + 818, + 428 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Generalization of GCNs to different applications ", + "text_level": 1, + "bbox": [ + 174, + 496, + 550, + 511 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To gain deeper understanding about the effectiveness and generalization of various GCN methods to different tasks, we start by picking two datasets - ModelNet-10 [47] for 3D shape-classification, and Cora [48] for semi-supervised node classification. Those datasets are not only different in terms of application (global versus local classification), but also stem from different domains. While in ModelNet-10 the data has geometrical meaning, the data in Cora has no obvious geometrical interpretation. Therefore, we suggest that success in both applications should be obtained from a generalizable GCN. We compare our PDE- $\\mathrm { G C N _ { D } }$ with two recent and popular networks - DGCNN [5] and GCNII [19]. For ModelNet-10 shape-classification, we randomly sample 1,024 points from each shape to form a point cloud, and connect its points using k-nearest-neighbors (k-nn) algorithm with $k = 1 0$ to obtain a graph and follow the training scheme of [5]. On Cora, we follow the same procedure as in [19]. We evaluate all models with 4 layers, as well as 2 layers for DGCNN on Cora. ", + "bbox": [ + 174, + 522, + 825, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our results, reported in Tab. 1 suggest that while each of the considered methods obtains high accuracy on the dataset it originally was tested on (Cora for GCNII and ModelNet-10 for DGCNN), obtaining a similar measure of success on a different dataset was not possible when using the very same networks. Additionally, while our attempts to add a diffusion-equation dynamics to DGCNN (i.e., updating features as in Eq. (6)) showed an increase in performance – a large gap to state-of-theart model still exists. On top of that, we also see that DGCNN suffers from over-smoothing, as its accuracy significantly decreases when adding more layers. Last but not least, we observe that our PDE- $\\mathrm { \\cdot G C N _ { D } }$ obtains high accuracy on both datasets, similar or better than state-of-the-art models. ", + "bbox": [ + 174, + 681, + 825, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 Learning PDE network dynamics ", + "text_level": 1, + "bbox": [ + 176, + 814, + 444, + 829 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this experiment, we delve on the ability to learn the appropriate PDE that better models a given problem. To this end, we use the mixture model from Eq. (9) so that the resulting PDE is a combination of the diffusion and hyperbolic dynamics. We use a 8 layer mixed PDE-GCN, starting with $\\alpha = 0 . 5$ , such that it is balanced between a PDE- $\\mathrm { \\cdot G C N _ { D } }$ and PDE- $\\mathrm { \\cdot G C N _ { H } }$ . By learning the parameter $\\alpha$ in (9), we allow to choose a mixed PDE between a purely conservative network and a diffusive one. We consider two problems: semi-supervised node classification on Cora, and dense shape correspondence on FAUST [51]. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 823, + 119 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our results, reported in Fig. 2 suggest that just as in classical works [43, 52], problems like nodeclassification obtain better performance with an anisotropic diffusion like in Eq. (6), and for problems involving dense-correspondences like in [45, 11] that tend to conserve the energy of the underlying problem, a hyperbolic equation type of PDE as in Eq. (7) is more appropriate. ", + "bbox": [ + 174, + 126, + 825, + 181 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/79e6639a83f8b94c6d7d8e370729ee7431711112f12f61d80c15cdad4c80ff73.jpg", + "table_caption": [ + "Table 2: Statistics of datasets used in our semi-and fully supervised node-classification experiments. " + ], + "table_footnote": [], + "table_body": "
DatasetCoraCiteSeerPubMedChameleonCornellTexasWisconsinPPI
Classes7635555121
Nodes2,7083,32719,7172,27718318325156,944
Edges5,4294,73244,33836,101295309499818,716
Features1,4333,7035002,3251,7031,7031,70350
", + "bbox": [ + 176, + 224, + 825, + 310 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Semi-supervised node classification ", + "text_level": 1, + "bbox": [ + 176, + 343, + 457, + 358 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this set of experiments we use three datasets – Cora, Citeseer and PubMed [48]. For all datasets we use the standard training/validation/testing split as in [53], with 20 nodes per class for training, 500 validation nodes and 1,000 testing nodes and follow the training scheme of [19]. The statistics of the datasets are reported in Tab. 2. We compare our results using PDE- $\\mathrm { G C N _ { D } }$ with recent and popular models like GCN [3], GAT [27], APPNP [30], JKNet [29] and DropEdge [28]. We note that our network does not over-smooth, as an increase in the number of layers does not cause performance degradation. For example, on CiteSeer, we obtain $7 5 . 6 \\%$ accuracy with 32 layers, compared to $7 4 . 6 \\%$ with two layers3. Overall, our results in Tab. 3 show that our PDE- $\\mathrm { \\cdot G C N _ { D } }$ achieves similar or better accuracy than the considered methods. ", + "bbox": [ + 173, + 368, + 825, + 493 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4 Fully-supervised node classification ", + "text_level": 1, + "bbox": [ + 176, + 511, + 459, + 526 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We follow [54] and use 7 datasets: Cora, CiteSeer, PubMed, Chameleon, Cornell, Texas and Wisconsin. We also use the same train/validation/test splits of $6 0 \\%$ , $2 0 \\%$ , $2 0 \\%$ , respectively. In addition, we report the average performance over 10 random splits from [54]. We fix the number of channels to 64 and perform grid search to determine the hyper-parameters parameters, which are reported in Appendix C. We compare our network with GCN, GAT, three variants of Geom-GCN [54], APPNP, JKNet, Incep and GCNII in Tab. 4. Our experiments read either similar or better than the state-of-the-art on Cora, CiteSeer and PubMed datasets. On Chameleon [55], Cornell, Texas and Wisconsin datasets, we improve state-of-the-art accuracy by a significant margin. For example, we obtain $9 3 . 2 4 \\%$ accuracy on Texas with our PDE- $\\mathrm { G C N _ { M } }$ , compared to $7 7 . 8 4 \\%$ with $\\mathrm { G C N I I ^ { \\ast } }$ . Similar improvements hold for Cornell and Wisconsin datasets. The common factor for these datasets is their small size, as depicted from Tab. 2. We argue that the success of our network stems from its capability of apriori extracting features from graphs, due to its utilization of discretized differential operators and PDE guided construction. On Chameleon [55], using PDE- $\\mathrm { G C N _ { M } }$ , we improve the current state-of-the-art accuracy of $\\operatorname { G C N I I } ^ { * }$ from $6 2 . 4 8 \\%$ to $6 6 . 0 1 \\%$ . Also, we note that unlike in the semi-supervised case, where some of the labels are missing, here it is possible to obtain meaningful results with the hyperbolic equation based PDE- $\\mathrm { G C N _ { H } }$ as we do not have unknown nodes in the fully-supervised case, which would be otherwise preserved using the hyperbolic equation dynamics. ", + "bbox": [ + 173, + 537, + 825, + 772 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.5 Inductive Learning ", + "text_level": 1, + "bbox": [ + 176, + 790, + 346, + 805 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We follow [19] and employ the PPI dataset [26] for the inductive learning task. We use a 8 layer PDE- $\\mathrm { G C N _ { D } }$ network, without dropout or weight-decay, and a learning rate of 0.001. We compare our results with methods like GraphSAGE, GAT, JKNet, GeniePath, GCNII and others. As reported in Tab. 5, our PDE-GCN $\\mathrm { D }$ achieves 99.07 Micro-averaged F1 score, superior to methods like GAT, JKNet and GeniePath, also close to state-of-the-art GCNII\\* with a score of 99.58. ", + "bbox": [ + 174, + 815, + 825, + 885 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/66ac1492a044112ffee54cccc9874434b62b85f6bebd658ffdcc85b9e9f32484.jpg", + "table_caption": [ + "Table 3: Semi-supervised node classification accuracy $( \\% )$ ). – indicates not available results. " + ], + "table_footnote": [], + "table_body": "
Layers
DatasetMethod248163264
CoraGCN [3]81.180.469.564.960.328.7
GCN (Drop) [28]82.882.075.875.762.549.5
JKNet [29]80.280.780.281.171.5
JKNet (Drop) [28]83.382.683.082.583.2
Incep [28]77.676.581.781.780.0
Incep (Drop)[28]82.982.583.183.183.5
GCNII [19]82.282.684.284.685.485.5
GCNII*[19]80.282.382.883.584.985.3
PDE-GCND (Ours)82.083.684.084.284.384.3
CiteseerGCN [3]70.867.630.218.325.020.0
GCN (Drop)[28]72.370.661.457.241.634.4
JKNet [29]168.767.769.868.263.4
JKNet (Drop) [28]72.671.872.670.872.2
Incep [28]69.368.470.268.067.5
Incep (Drop) [28]72.771.472.572.671.0
GCNII [19]68.268.870.672.973.473.4
GCNII* [19]66.166.770.672.073.273.1
PDE-GCND (Ours)74.675.075.275.575.675.5
PubmedGCN [3]79.076.561.240.922.435.3
GCN (Drop)[28]79.679.478.178.577.061.5
JKNet [29]78.078.172.672.474.5
JKNet (Drop) [28]78.778.779.779.278.9
Incep [28]77.777.974.911
Incep (Drop) [28]79.578.679.0
GCNII [19]78.278.879.380.279.879.7
GCNII* [19]77.778.278.880.379.880.1
PDE-GCND (Ours)79.380.680.180.480.280.3
", + "bbox": [ + 243, + 117, + 754, + 546 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.6 Dense shape correspondence ", + "text_level": 1, + "bbox": [ + 176, + 579, + 411, + 594 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finding dense correspondences between shapes is a classical experiment for hyperbolic dynamics, as we are interested in modeling local motion dynamics. In essence, learning to find correspondences between shapes is similar to learning a transformation from one shape to the other. To this end, we use the FAUST dataset [51] containing 10 scanned human shapes in 10 different poses with 6,890 nodes each. We follow the train and test split from [4], where the first 80 subjects are used for training and the remaining 20 subjects for testing. Our metric is the correspondence percentage with zero geodesic error, i.e., the percentage of perfectly matched shapes from all our test cases. We follow the pre-processing and training scheme of [62] where we use Cartesian coordinates to describe distances between nodes, with initial features of a constant $\\mathbf { 1 } \\in \\mathbb { R } ^ { n }$ where $n$ is the number of nodes. We use a 8 layer PDE- $\\mathrm { G C N _ { D } }$ and PDE- $\\mathrm { G C N _ { H } }$ variants, both with constant learning rate of 0.001 with no weight-decay or dropout, and compare to recent and popular methods like ACNN [60], MoNet [4], FMNet [61], and SplineCNN [62]. As expected from the discussion in Sec. 3.1, and reported in Tab. 6, the hyperbolic equation proves to be a better fit for this kind of problem. Also, our PDE- $\\mathrm { G C N _ { H } }$ achieves a promising $9 9 . 9 \\%$ correspondence rate with zero geodesic error, outperforming the rest of the considered methods. ", + "bbox": [ + 173, + 606, + 825, + 813 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.7 Ablation study ", + "text_level": 1, + "bbox": [ + 174, + 830, + 313, + 844 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our method has two main dynamics – diffusion and hyperbolic. To verify our proposal in Sec. 3.1, we examine the performance of the hyperbolic PDE- $\\mathrm { G C N _ { H } }$ on semi-supervised node-classification on Cora and CiteSeer datasets. Our results in Tab. 7 show that indeed for problems where we wish to obtain a piecewise-constant prediction, the diffusive formulation of our network, PDE- $\\mathrm { \\ G C N _ { D } }$ , is more suitable. Furthermore, we study the importance of the positive-semi definiteness of our learnt operator as described in Eq. (6)-(7), by removing the ${ \\bf K } _ { l } ^ { T }$ term from the dynamics equations. This yields a non-symmetric operator that does not guarantee positive-semi-definiteness. We note that the enforcement of the latter is important to obtain higher accuracy which is improved by up to $3 \\%$ with the introduction of a positive semi-definite operator. Also, we report on the use of $\\sigma = R e L U$ , from the discussion in Sec. 3.3, where we favor tanh as an activation function, and the ability of our PDE- $\\mathrm { \\mathrm { . G C N _ { M } } }$ to reproduce the results of PDE- $\\mathrm { G C N _ { D } }$ in the case of semi-supervised learning. ", + "bbox": [ + 176, + 856, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e6c81074fe123b35c62b70204bb80ca7c8aeb530fa918c269d077e9944977f77.jpg", + "table_caption": [ + "Table 4: Fully-supervised node classification accuracy $( \\% )$ . (L) indicates a $L$ layers network. " + ], + "table_footnote": [], + "table_body": "
MethodCoraCite.Pubm.Cham.Corn.TexasWisc.
GCN [3]85.7773.6888.1328.1852.7052.1645.88
GAT[27]86.3774.3287.6242.9354.3258.3849.41
Geom-GCN-I [54]85.1977.9990.0560.3156.7657.5858.24
Geom-GCN-P [54]84.9375.1488.0960.9060.8167.5764.12
Geom-GCN-S [54]85.2774.7184.7559.9655.6859.7356.67
APPNP [30]87.8776.5389.4054.3073.5165.4169.02
JKNet [29]85.25 (16)75.85 (8)88.94 (64)60.07 (32)57.30 (4)56.49 (32)48.82 (8)
JKNet (Drop) [28]87.46 (16)75.96 (8)89.45 (64)62.08 (32)61.08 (4)57.30 (32)50.59 (8)
Incep (Drop) [28]86.86(8)76.83 (8)89.18 (4)61.71 (8)61.62 (16)57.84 (8)50.20 (8)
GCNII [19]88.49 (64)77.08 (64)89.57 (64)60.61 (8)74.86 (16)69.46 (32)74.12 (16)
GCNII*88.01 (64)77.13 (64)90.30 (64)62.48 (8)76.49 (16)77.84 (32)81.57 (16)
PDE-GCND (Ours)88.51 (16)78.36 (64)89.6 (64)64.12 (8)89.19 (2)90.81 (8)90.39 (8)
PDE-GCNH (Ours)87.71 (32)78.13 (16)89.16 (16)61.57 (64)89.45 (64)92.16 (64)91.37 (16)
PDE-GCNm (Ours)88.60 (16)78.48 (32)89.93 (16)66.01 (16)89.73 (64)93.24 (32)91.76 (16)
", + "bbox": [ + 173, + 117, + 826, + 315 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/a3056d2ba3c767d86f6d78b433befb46cbe6e9825a9df7d9201725bf522e9c13.jpg", + "table_caption": [ + "Table 5: Protein-protein interaction (PPI). Results are reported in micro-averaged F1 score. " + ], + "table_footnote": [], + "table_body": "
MethodMicro-averaged F1
GraphSAGE [26]61.20
VR-GCN [56]97.80
GaAN [57]98.71
GAT [27]97.30
JKNet [29]97.60
GeniePath [58]98.50
Cluster-GCN [59]99.36
GCNII [19]99.54
GCNII*[19]99.58
PDE-GCND (Ours)99.07
PDE-GCNm (Ours)99.18
", + "bbox": [ + 187, + 382, + 485, + 570 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/29cf4318615e7a62929b4cf2a4c1fe3368d6ffda096cd0c955de7e725651a95a.jpg", + "table_caption": [ + "Table 6: Dense shape correspondence $( \\% )$ with zero geodesic error " + ], + "table_footnote": [], + "table_body": "
MethodFaust
ACNN [60]63.8
MoNet [4]89.1
FMNet [61] SplineCNN [62]98.2 99.2
PDE-GCND (Ours)
PDE-GCNH (Ours)64.2 99.9
", + "bbox": [ + 589, + 416, + 795, + 536 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 608, + 825, + 705 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Summary ", + "text_level": 1, + "bbox": [ + 174, + 731, + 287, + 748 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper we explored new architectures for graph neural networks. Our motivation stems from the similarities between graph networks and time dependent partial differential equations that are discretized on manifolds and graphs. By adopting an appropriate PDE, and embedding the finite graph in an infinite manifold, we are able to define networks that are either diffusive, conservative, or a combination of both. ", + "bbox": [ + 174, + 766, + 825, + 835 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Not all natural phenomena are solved using the same PDE and we should not expect that all graph problems should be solved by the same network dynamics. To this end we allow the data to choose which type of network is appropriate for the solution of the problem (diffusive or hyperbolic). Indeed, numerical experiments show that the network gravitates towards a hyperbolic one for problems where conservation is required, and towards a diffusive one when anisotropic diffusion is favorable. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/6dfafb629feec34e7d664dce8724e9324765421b0b8af242a282644e68af51c1.jpg", + "table_caption": [ + "Table 7: Ablation study of PDE-GCN accuracy $( \\% )$ on semi-supervised node-classification. " + ], + "table_footnote": [], + "table_body": "
Layers
MethodDataset248163264
PDE-GCNHCora CiteSeer79 70.979.378.0 72.178.077.877.5 72.4
PDE-GCND (non-symmetric)Cora83.571.7 83.383.672.3 83.172.5 82.781
PDE-GCND (σ = ReLU)CiteSeer Cora74.3 80.374.5 81.874.8 82.675.0 83.073.9 83.473.3 83.5
PDE-GCNmCiteSeer Cora CiteSeer73.1 82.073.2 83.472.8 83.973.3 84.273.6 84.374.0 84.5
", + "bbox": [ + 207, + 117, + 790, + 291 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Finally, we showed that the proposed networks can be made deep without over-smoothing and, can deliver the state-of-the-art performance or improve it for virtually every problem we worked with. In particular, our network dramatically improved the state-of-the-art for problems that are data-poor. We believe that for such problems the structure imposed by our dynamics and operators regularizes the network and therefore yields implicit regularization. ", + "bbox": [ + 173, + 323, + 825, + 392 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 411, + 553, + 429 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The research reported in this paper was supported by grant no. 2018209 from the United States - Israel Binational Science Foundation (BSF), Jerusalem, Israel. ME is supported by Kreitman High-tech scholarship. ", + "bbox": [ + 174, + 443, + 825, + 484 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 510, + 266, + 526 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:1312.6203, 2013. \n[2] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in neural information processing systems, pages 3844– 3852, 2016. \n[3] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016. \n[4] Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. 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Furthermore, as GCNs can be thought of as a generalization of", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 681, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "CNNs, and a standard convolution can be represented as a combination of differential operators on a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "structured grid [22], we adopt this interpretation to explore versions of GCNs as PDEs on graphs or", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "manifolds. We therefore call our network architectures PDE-GCN, and demonstrate that our approach", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "is general with respect to the given task. That is, our architectures behave similarly for different", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "problems, and their performance is on par or better than other domain-specific GCNs. Furthermore,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "score": 1.0, + "content": "our family of architectures are backed by theoretical guarantees that allow us to explain the behavior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "of the GCNs in some of the results that we present. To be more specific, our contribution is as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 158, + 506, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 172 + ], + "score": 1.0, + "content": "• We introduce and implement general graph convolution operators, based on graph gradient and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 113, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 113, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "divergence. This abstraction of the spatial operation on the graph leads to a more general and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 114, + 180, + 279, + 194 + ], + "spans": [ + { + "bbox": [ + 114, + 180, + 279, + 194 + ], + "score": 1.0, + "content": "flexible approach for architecture design.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 208 + ], + "score": 1.0, + "content": "• We propose treating a variety of graph related problems as discretized PDEs, and formulate the dy-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 113, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 113, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "namics that match different problems such as node-classification and dense shape-correspondence.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 113, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 113, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "This is in direct effort to propose a family of networks that can solve multiple problems, instead of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 114, + 227, + 320, + 239 + ], + "spans": [ + { + "bbox": [ + 114, + 227, + 320, + 239 + ], + "score": 1.0, + "content": "GCNs which are tailored for a specific application.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 238, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 507, + 253 + ], + "score": 1.0, + "content": "• Our method allows constructing a deep GCN without over-smoothing, with theoretical guarantees.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 252, + 507, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 507, + 266 + ], + "score": 1.0, + "content": "• We validate our method by conducting numerous experiments on multiple datasets and applications,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 114, + 264, + 459, + 276 + ], + "spans": [ + { + "bbox": [ + 114, + 264, + 459, + 276 + ], + "score": 1.0, + "content": "achieving significantly better or similar accuracy compared to state-of-the-art models.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 291, + 194, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 196, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 196, + 306 + ], + "score": 1.0, + "content": "2 Related work", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "Graph Convolutional Networks: GCNs are typically divided into spectral [1, 3, 2] and spatial [23,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "24, 25, 4] categories. Most of those can be implemented using the Message-Passing Neural Network", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "paradigm [25], where each node aggregates features (messages) from its neighbors, according to some", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "scheme. The works [3, 2] use polynomials of the graph Laplacian to parameterize the convolution", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 249, + 372 + ], + "score": 1.0, + "content": "operator. DGCNN [5] constructs a", + "type": "text" + }, + { + "bbox": [ + 250, + 360, + 257, + 370 + ], + "score": 0.33, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "-nearest-neighbors graph from point-clouds and dynamically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "updates it. MoNet [4] learns a Gaussian mixture model to weight the edges of meshes for shape", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "analysis tasks. Works like GraphSAGE [26] and GAT [27] propose methods for inductive and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 394, + 303, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 303, + 405 + ], + "score": 1.0, + "content": "transductive learning on non-geometrical graphs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 504, + 420 + ], + "score": 1.0, + "content": "Several of the methods above suffer from over-smoothing [17, 19], leading to undesired node", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "score": 1.0, + "content": "features similarity for deep networks. To overcome this problem, some approaches rely on imposing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "regularization and augmentation. For example, PairNorm [17] introduces a novel normalization layer,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "and DropEdge [28] randomly removes edges to decrease the degree of nodes. Other methods prevent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "over-smoothing by dedicated construction. For instance, JKNet [29] combines all intermediate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "representations at each stage of the network. APPNP [30] replaces learnt convolutional layers with a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 473, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 488 + ], + "score": 1.0, + "content": "pre-defined kernel based on PageRank, yielding a shallow network which preserves locality, making", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "it robust to over-smoothing. GCNII [19] proposes to add an identity residual where the initial features", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 447, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 296, + 509 + ], + "score": 1.0, + "content": "of the networks are added to the features of the", + "type": "text" + }, + { + "bbox": [ + 296, + 497, + 300, + 506 + ], + "score": 0.55, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 495, + 447, + 509 + ], + "score": 1.0, + "content": "-th layer, scaled by some coefficient.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "Another approach is to construct a network that inherently does not over-smooth, as we suggest in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "this work. Our network is based on discretized PDEs, hence we are able to motivate our choices", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "by well studied theory and numerical experiments [31]. On a similar note, the recent DiffGCN [8]", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "also makes use of discretized operators to approximate the graph gradient and Laplacian. However,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 477, + 570 + ], + "score": 1.0, + "content": "DiffGCN is specifically tailored for geometric tasks since it projects its components on the", + "type": "text" + }, + { + "bbox": [ + 477, + 558, + 504, + 568 + ], + "score": 0.87, + "content": "x , y , z", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "axes, and is applied using a ResNet [16] (diffusion) structure only. The recent GRAND [32] applies", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "attention mechanism with diffusive dynamics, using several integration schemes. Here we propose a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "network for both geometric and non-geometric tasks, and also utilize both the diffusion or hyperbolic", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 600, + 298, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 298, + 613 + ], + "score": 1.0, + "content": "layer dynamics, including a mixture of the two.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "PDEs and CNNs: In a recent series of works, the connection between PDEs and CNNs was studied", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "[20, 21, 22, 33, 34, 35, 36, 37]. It was shown that it is possible to treat a deep neural network as a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "dynamical system driven by some PDE, where each convolution layer is considered a time step of a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "discretized PDE. The connection between PDEs and CNNs was also used to reduce the computational", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 680 + ], + "score": 1.0, + "content": "burden [38]. Besides the interpretation of CNNs as a PDE solver, it was shown that it is also possible", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "to learn a symbolic representation of PDEs in a task-and data driven approach in [39, 40]. In the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "context of GCNs and PDEs, it was recently shown [41, 42] that GCNs can be utilized to enhance", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "the solution of PDEs for problems like fluid flow dynamics. However, in this work, we harness PDE", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 381, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 381, + 724 + ], + "score": 1.0, + "content": "concepts to design and construct GCNs for a variety of applications.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "CNNs, and a standard convolution can be represented as a combination of differential operators on a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "structured grid [22], we adopt this interpretation to explore versions of GCNs as PDEs on graphs or", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "manifolds. We therefore call our network architectures PDE-GCN, and demonstrate that our approach", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "is general with respect to the given task. That is, our architectures behave similarly for different", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "problems, and their performance is on par or better than other domain-specific GCNs. 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To be more specific, our contribution is as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 72, + 506, + 150 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 158, + 506, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 172 + ], + "score": 1.0, + "content": "• We introduce and implement general graph convolution operators, based on graph gradient and", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 113, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "divergence. This abstraction of the spatial operation on the graph leads to a more general and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 114, + 180, + 279, + 194 + ], + "spans": [ + { + "bbox": [ + 114, + 180, + 279, + 194 + ], + "score": 1.0, + "content": "flexible approach for architecture design.", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 192, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 208 + ], + "score": 1.0, + "content": "• We propose treating a variety of graph related problems as discretized PDEs, and formulate the dy-", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 113, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "namics that match different problems such as node-classification and dense shape-correspondence.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 113, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 113, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "This is in direct effort to propose a family of networks that can solve multiple problems, instead of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 114, + 227, + 320, + 239 + ], + "spans": [ + { + "bbox": [ + 114, + 227, + 320, + 239 + ], + "score": 1.0, + "content": "GCNs which are tailored for a specific application.", + "type": "text" + } + ], + "index": 13, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 238, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 507, + 253 + ], + "score": 1.0, + "content": "• Our method allows constructing a deep GCN without over-smoothing, with theoretical guarantees.", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 252, + 507, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 507, + 266 + ], + "score": 1.0, + "content": "• We validate our method by conducting numerous experiments on multiple datasets and applications,", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 264, + 459, + 276 + ], + "spans": [ + { + "bbox": [ + 114, + 264, + 459, + 276 + ], + "score": 1.0, + "content": "achieving significantly better or similar accuracy compared to state-of-the-art models.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 157, + 507, + 276 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 291, + 194, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 196, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 196, + 306 + ], + "score": 1.0, + "content": "2 Related work", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "Graph Convolutional Networks: GCNs are typically divided into spectral [1, 3, 2] and spatial [23,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "24, 25, 4] categories. Most of those can be implemented using the Message-Passing Neural Network", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "paradigm [25], where each node aggregates features (messages) from its neighbors, according to some", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "scheme. The works [3, 2] use polynomials of the graph Laplacian to parameterize the convolution", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 249, + 372 + ], + "score": 1.0, + "content": "operator. DGCNN [5] constructs a", + "type": "text" + }, + { + "bbox": [ + 250, + 360, + 257, + 370 + ], + "score": 0.33, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "-nearest-neighbors graph from point-clouds and dynamically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "updates it. MoNet [4] learns a Gaussian mixture model to weight the edges of meshes for shape", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "analysis tasks. Works like GraphSAGE [26] and GAT [27] propose methods for inductive and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 394, + 303, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 303, + 405 + ], + "score": 1.0, + "content": "transductive learning on non-geometrical graphs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 315, + 506, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 504, + 420 + ], + "score": 1.0, + "content": "Several of the methods above suffer from over-smoothing [17, 19], leading to undesired node", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 434 + ], + "score": 1.0, + "content": "features similarity for deep networks. To overcome this problem, some approaches rely on imposing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "regularization and augmentation. For example, PairNorm [17] introduces a novel normalization layer,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "and DropEdge [28] randomly removes edges to decrease the degree of nodes. Other methods prevent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "over-smoothing by dedicated construction. For instance, JKNet [29] combines all intermediate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "representations at each stage of the network. APPNP [30] replaces learnt convolutional layers with a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 473, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 488 + ], + "score": 1.0, + "content": "pre-defined kernel based on PageRank, yielding a shallow network which preserves locality, making", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "it robust to over-smoothing. GCNII [19] proposes to add an identity residual where the initial features", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 447, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 296, + 509 + ], + "score": 1.0, + "content": "of the networks are added to the features of the", + "type": "text" + }, + { + "bbox": [ + 296, + 497, + 300, + 506 + ], + "score": 0.55, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 495, + 447, + 509 + ], + "score": 1.0, + "content": "-th layer, scaled by some coefficient.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 409, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "Another approach is to construct a network that inherently does not over-smooth, as we suggest in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "this work. Our network is based on discretized PDEs, hence we are able to motivate our choices", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "by well studied theory and numerical experiments [31]. On a similar note, the recent DiffGCN [8]", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "also makes use of discretized operators to approximate the graph gradient and Laplacian. However,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 477, + 570 + ], + "score": 1.0, + "content": "DiffGCN is specifically tailored for geometric tasks since it projects its components on the", + "type": "text" + }, + { + "bbox": [ + 477, + 558, + 504, + 568 + ], + "score": 0.87, + "content": "x , y , z", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "axes, and is applied using a ResNet [16] (diffusion) structure only. The recent GRAND [32] applies", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "attention mechanism with diffusive dynamics, using several integration schemes. Here we propose a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "network for both geometric and non-geometric tasks, and also utilize both the diffusion or hyperbolic", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 600, + 298, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 298, + 613 + ], + "score": 1.0, + "content": "layer dynamics, including a mixture of the two.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 511, + 506, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 635 + ], + "score": 1.0, + "content": "PDEs and CNNs: In a recent series of works, the connection between PDEs and CNNs was studied", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "[20, 21, 22, 33, 34, 35, 36, 37]. 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Besides the interpretation of CNNs as a PDE solver, it was shown that it is also possible", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "to learn a symbolic representation of PDEs in a task-and data driven approach in [39, 40]. In the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "context of GCNs and PDEs, it was recently shown [41, 42] that GCNs can be utilized to enhance", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "the solution of PDEs for problems like fluid flow dynamics. However, in this work, we harness PDE", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 381, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 381, + 724 + ], + "score": 1.0, + "content": "concepts to design and construct GCNs for a variety of applications.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 624, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 68, + 500, + 303 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 68, + 500, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 68, + 500, + 303 + ], + "spans": [ + { + "bbox": [ + 112, + 68, + 500, + 303 + ], + "score": 0.971, + "type": "image", + "image_path": "84fba16b4d384652779751fd38ae51dc9abe79770ca77ec1b8facd8fdbdfb1ea.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 68, + 500, + 146.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 146.33333333333331, + 500, + 224.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 224.66666666666663, + 500, + 302.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 313, + 506, + 358 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "Figure 1: Feature evolvement on an input mesh (left). Propagation in time is from left to right.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "Hyperbolic and diffusion equation dynamics are on the top and bottom row, respectively. 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Consider a general manifold", + "type": "text" + }, + { + "bbox": [ + 457, + 434, + 470, + 443 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "where a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 169, + 456 + ], + "score": 1.0, + "content": "vector function", + "type": "text" + }, + { + "bbox": [ + 170, + 445, + 177, + 456 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "resides (also dubbed the features function), along with its continuous differential", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 455, + 503, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 226, + 467 + ], + "score": 1.0, + "content": "operators such as the gradient", + "type": "text" + }, + { + "bbox": [ + 226, + 456, + 235, + 465 + ], + "score": 0.78, + "content": "\\nabla", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 455, + 284, + 467 + ], + "score": 1.0, + "content": ", divergence", + "type": "text" + }, + { + "bbox": [ + 285, + 455, + 294, + 465 + ], + "score": 0.78, + "content": "\\nabla", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 455, + 371, + 467 + ], + "score": 1.0, + "content": "· and the Laplacian", + "type": "text" + }, + { + "bbox": [ + 371, + 456, + 380, + 465 + ], + "score": 0.82, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 455, + 490, + 467 + ], + "score": 1.0, + "content": "that reside on the manifold", + "type": "text" + }, + { + "bbox": [ + 490, + 456, + 503, + 465 + ], + "score": 0.81, + "content": "\\mathcal { M }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 108, + 471, + 504, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 387, + 484 + ], + "score": 1.0, + "content": "Given these differential operators, one can model different processes on", + "type": "text" + }, + { + "bbox": [ + 387, + 471, + 400, + 482 + ], + "score": 0.84, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 471, + 505, + 484 + ], + "score": 1.0, + "content": ". In particular, we consider", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 482, + 414, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 414, + 495 + ], + "score": 1.0, + "content": "two PDEs – the non-linear diffusion and the non-linear hyperbolic equations", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 498, + 451, + 531 + ], + "lines": [ + { + "bbox": [ + 158, + 498, + 451, + 531 + ], + "spans": [ + { + "bbox": [ + 158, + 498, + 451, + 531 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { f _ { t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad t \\in [ 0 , T ] , } \\\\ & { f _ { t t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad f _ { t } ( t = 0 ) = 0 , \\quad t \\in [ 0 , T ] , } \\end{array}", + "type": "interline_equation", + "image_path": "eb1518e442fa6d11189df90a0587dc7d4bfb2efa5380a76ee4f00f41b6a93e23.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 158, + 498, + 451, + 509.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 158, + 509.0, + 451, + 520.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 158, + 520.0, + 451, + 531.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 372, + 547 + ], + "score": 1.0, + "content": "respectively, equipped with appropriate boundary conditions. Here", + "type": "text" + }, + { + "bbox": [ + 372, + 534, + 383, + 544 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "is a coefficient matrix that can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 368, + 559 + ], + "score": 1.0, + "content": "change in time and represents the propagation over the manifold", + "type": "text" + }, + { + "bbox": [ + 369, + 545, + 381, + 555 + ], + "score": 0.72, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 544, + 385, + 559 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 385, + 545, + 400, + 555 + ], + "score": 0.83, + "content": "K ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 544, + 506, + 559 + ], + "score": 1.0, + "content": "is its conjugate transpose", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 123, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 556, + 141, + 568 + ], + "score": 0.91, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 556, + 504, + 568 + ], + "score": 1.0, + "content": "is a non-linear activation function. Eq. (1)-(2) define a non-linear operator that takes initial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 168, + 579 + ], + "score": 1.0, + "content": "feature vectors", + "type": "text" + }, + { + "bbox": [ + 168, + 566, + 179, + 578 + ], + "score": 0.89, + "content": "f ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 566, + 333, + 579 + ], + "score": 1.0, + "content": "at time 0 and propagates them to time", + "type": "text" + }, + { + "bbox": [ + 334, + 567, + 342, + 577 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 566, + 381, + 579 + ], + "score": 1.0, + "content": ", yielding", + "type": "text" + }, + { + "bbox": [ + 381, + 566, + 394, + 578 + ], + "score": 0.9, + "content": "f ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "where they can be used for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "different tasks. We now provide two theorems that characterize the behavior of Eq. (1)-(2), based on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 588, + 177, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 177, + 600 + ], + "score": 1.0, + "content": "ideas from [22]1.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 264, + 617 + ], + "score": 1.0, + "content": "Theorem 1. If the activation function", + "type": "text" + }, + { + "bbox": [ + 264, + 603, + 282, + 615 + ], + "score": 0.9, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 602, + 506, + 617 + ], + "score": 1.0, + "content": "is monotonically non-decreasing and sign-preserving,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 366, + 628 + ], + "score": 1.0, + "content": "then the forward propagation through the diffusive PDE in (1) for", + "type": "text" + }, + { + "bbox": [ + 367, + 614, + 410, + 626 + ], + "score": 0.93, + "content": "t \\in [ 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 613, + 505, + 628 + ], + "score": 1.0, + "content": "yields a non-increasing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 624, + 193, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 193, + 638 + ], + "score": 1.0, + "content": "feature norm, that is,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 278, + 635, + 333, + 659 + ], + "lines": [ + { + "bbox": [ + 278, + 635, + 333, + 659 + ], + "spans": [ + { + "bbox": [ + 278, + 635, + 333, + 659 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial t } \\lvert | f \\rvert | ^ { 2 } \\leq 0 .", + "type": "interline_equation", + "image_path": "c906ab6527cec33da4e90f74c6bdb9dabd423377c5d1aafe833375b7d541b787.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 278, + 635, + 333, + 659 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 661, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 318, + 675 + ], + "score": 1.0, + "content": "Theorem 2. Assume that the activation function", + "type": "text" + }, + { + "bbox": [ + 318, + 661, + 336, + 673 + ], + "score": 0.9, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 659, + 506, + 675 + ], + "score": 1.0, + "content": "is monotonically non-decreasing, sign-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 671, + 330, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 203, + 686 + ], + "score": 1.0, + "content": "preserving and satisfies", + "type": "text" + }, + { + "bbox": [ + 203, + 672, + 253, + 684 + ], + "score": 0.93, + "content": "| \\sigma ( x ) | \\leq | x |", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 671, + 330, + 686 + ], + "score": 1.0, + "content": ", and define energy", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 690, + 376, + 704 + ], + "lines": [ + { + "bbox": [ + 233, + 690, + 376, + 704 + ], + "spans": [ + { + "bbox": [ + 233, + 690, + 376, + 704 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { n e t } = \\| f _ { t } \\| ^ { 2 } + \\left( K \\nabla f , \\sigma ( K \\nabla f ) \\right) ,", + "type": "interline_equation", + "image_path": "048f6c37a9c2c3b593390ce74289d0a715f20310955fe1076e1b5aa8b9895c6f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 233, + 690, + 376, + 704 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 711, + 219, + 722 + ], + "lines": [ + { + "bbox": [ + 120, + 709, + 221, + 724 + ], + "spans": [ + { + "bbox": [ + 120, + 709, + 221, + 724 + ], + "score": 1.0, + "content": "1See proofs in Appendix A.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 68, + 500, + 303 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 68, + 500, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 68, + 500, + 303 + ], + "spans": [ + { + "bbox": [ + 112, + 68, + 500, + 303 + ], + "score": 0.971, + "type": "image", + "image_path": "84fba16b4d384652779751fd38ae51dc9abe79770ca77ec1b8facd8fdbdfb1ea.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 68, + 500, + 146.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 146.33333333333331, + 500, + 224.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 224.66666666666663, + 500, + 302.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 313, + 506, + 358 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "Figure 1: Feature evolvement on an input mesh (left). Propagation in time is from left to right.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "Hyperbolic and diffusion equation dynamics are on the top and bottom row, respectively. While", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "a diffusive graph network (similar to most common GCNs that rely on ResNets) smooths the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 346, + 427, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 427, + 358 + ], + "score": 1.0, + "content": "information on the manifold, the hyperbolic network yields a non-uniform field.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 107, + 377, + 171, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 375, + 172, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 172, + 393 + ], + "score": 1.0, + "content": "3 Methods", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 401, + 315, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 316, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 316, + 416 + ], + "score": 1.0, + "content": "3.1 Partial Differential Equations on manifolds", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "We show now that GCNs can be viewed as discretizations of PDEs on manifolds, similarly to CNNs", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 457, + 445 + ], + "score": 1.0, + "content": "that are viewed as discretized PDEs on regular grids [22]. Consider a general manifold", + "type": "text" + }, + { + "bbox": [ + 457, + 434, + 470, + 443 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "where a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 169, + 456 + ], + "score": 1.0, + "content": "vector function", + "type": "text" + }, + { + "bbox": [ + 170, + 445, + 177, + 456 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "resides (also dubbed the features function), along with its continuous differential", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 455, + 503, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 226, + 467 + ], + "score": 1.0, + "content": "operators such as the gradient", + "type": "text" + }, + { + "bbox": [ + 226, + 456, + 235, + 465 + ], + "score": 0.78, + "content": "\\nabla", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 455, + 284, + 467 + ], + "score": 1.0, + "content": ", divergence", + "type": "text" + }, + { + "bbox": [ + 285, + 455, + 294, + 465 + ], + "score": 0.78, + "content": "\\nabla", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 455, + 371, + 467 + ], + "score": 1.0, + "content": "· and the Laplacian", + "type": "text" + }, + { + "bbox": [ + 371, + 456, + 380, + 465 + ], + "score": 0.82, + "content": "\\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 455, + 490, + 467 + ], + "score": 1.0, + "content": "that reside on the manifold", + "type": "text" + }, + { + "bbox": [ + 490, + 456, + 503, + 465 + ], + "score": 0.81, + "content": "\\mathcal { M }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 421, + 506, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 471, + 504, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 387, + 484 + ], + "score": 1.0, + "content": "Given these differential operators, one can model different processes on", + "type": "text" + }, + { + "bbox": [ + 387, + 471, + 400, + 482 + ], + "score": 0.84, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 471, + 505, + 484 + ], + "score": 1.0, + "content": ". In particular, we consider", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 482, + 414, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 414, + 495 + ], + "score": 1.0, + "content": "two PDEs – the non-linear diffusion and the non-linear hyperbolic equations", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 471, + 505, + 495 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 498, + 451, + 531 + ], + "lines": [ + { + "bbox": [ + 158, + 498, + 451, + 531 + ], + "spans": [ + { + "bbox": [ + 158, + 498, + 451, + 531 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { f _ { t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad t \\in [ 0 , T ] , } \\\\ & { f _ { t t } = \\nabla \\cdot K ^ { * } \\sigma ( K \\nabla f ) , \\quad f ( t = 0 ) = f ^ { 0 } , \\quad f _ { t } ( t = 0 ) = 0 , \\quad t \\in [ 0 , T ] , } \\end{array}", + "type": "interline_equation", + "image_path": "eb1518e442fa6d11189df90a0587dc7d4bfb2efa5380a76ee4f00f41b6a93e23.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 158, + 498, + 451, + 509.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 158, + 509.0, + 451, + 520.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 158, + 520.0, + 451, + 531.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 372, + 547 + ], + "score": 1.0, + "content": "respectively, equipped with appropriate boundary conditions. Here", + "type": "text" + }, + { + "bbox": [ + 372, + 534, + 383, + 544 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "is a coefficient matrix that can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 368, + 559 + ], + "score": 1.0, + "content": "change in time and represents the propagation over the manifold", + "type": "text" + }, + { + "bbox": [ + 369, + 545, + 381, + 555 + ], + "score": 0.72, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 544, + 385, + 559 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 385, + 545, + 400, + 555 + ], + "score": 0.83, + "content": "K ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 544, + 506, + 559 + ], + "score": 1.0, + "content": "is its conjugate transpose", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 556, + 504, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 123, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 556, + 141, + 568 + ], + "score": 0.91, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 556, + 504, + 568 + ], + "score": 1.0, + "content": "is a non-linear activation function. Eq. (1)-(2) define a non-linear operator that takes initial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 168, + 579 + ], + "score": 1.0, + "content": "feature vectors", + "type": "text" + }, + { + "bbox": [ + 168, + 566, + 179, + 578 + ], + "score": 0.89, + "content": "f ^ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 566, + 333, + 579 + ], + "score": 1.0, + "content": "at time 0 and propagates them to time", + "type": "text" + }, + { + "bbox": [ + 334, + 567, + 342, + 577 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 566, + 381, + 579 + ], + "score": 1.0, + "content": ", yielding", + "type": "text" + }, + { + "bbox": [ + 381, + 566, + 394, + 578 + ], + "score": 0.9, + "content": "f ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "where they can be used for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "different tasks. We now provide two theorems that characterize the behavior of Eq. (1)-(2), based on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 588, + 177, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 177, + 600 + ], + "score": 1.0, + "content": "ideas from [22]1.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 534, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 264, + 617 + ], + "score": 1.0, + "content": "Theorem 1. If the activation function", + "type": "text" + }, + { + "bbox": [ + 264, + 603, + 282, + 615 + ], + "score": 0.9, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 602, + 506, + 617 + ], + "score": 1.0, + "content": "is monotonically non-decreasing and sign-preserving,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 613, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 366, + 628 + ], + "score": 1.0, + "content": "then the forward propagation through the diffusive PDE in (1) for", + "type": "text" + }, + { + "bbox": [ + 367, + 614, + 410, + 626 + ], + "score": 0.93, + "content": "t \\in [ 0 , \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 613, + 505, + 628 + ], + "score": 1.0, + "content": "yields a non-increasing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 624, + 193, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 193, + 638 + ], + "score": 1.0, + "content": "feature norm, that is,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 602, + 506, + 638 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 278, + 635, + 333, + 659 + ], + "lines": [ + { + "bbox": [ + 278, + 635, + 333, + 659 + ], + "spans": [ + { + "bbox": [ + 278, + 635, + 333, + 659 + ], + "score": 0.92, + "content": "\\frac { \\partial } { \\partial t } \\lvert | f \\rvert | ^ { 2 } \\leq 0 .", + "type": "interline_equation", + "image_path": "c906ab6527cec33da4e90f74c6bdb9dabd423377c5d1aafe833375b7d541b787.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 278, + 635, + 333, + 659 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 661, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 318, + 675 + ], + "score": 1.0, + "content": "Theorem 2. Assume that the activation function", + "type": "text" + }, + { + "bbox": [ + 318, + 661, + 336, + 673 + ], + "score": 0.9, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 659, + 506, + 675 + ], + "score": 1.0, + "content": "is monotonically non-decreasing, sign-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 671, + 330, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 203, + 686 + ], + "score": 1.0, + "content": "preserving and satisfies", + "type": "text" + }, + { + "bbox": [ + 203, + 672, + 253, + 684 + ], + "score": 0.93, + "content": "| \\sigma ( x ) | \\leq | x |", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 671, + 330, + 686 + ], + "score": 1.0, + "content": ", and define energy", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 659, + 506, + 686 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 690, + 376, + 704 + ], + "lines": [ + { + "bbox": [ + 233, + 690, + 376, + 704 + ], + "spans": [ + { + "bbox": [ + 233, + 690, + 376, + 704 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { n e t } = \\| f _ { t } \\| ^ { 2 } + \\left( K \\nabla f , \\sigma ( K \\nabla f ) \\right) ,", + "type": "interline_equation", + "image_path": "048f6c37a9c2c3b593390ce74289d0a715f20310955fe1076e1b5aa8b9895c6f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 233, + 690, + 376, + 704 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 504, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 399, + 86 + ], + "score": 1.0, + "content": "then the forward propagation through the hyperbolic PDE in (2) satisfies", + "type": "text" + }, + { + "bbox": [ + 400, + 73, + 443, + 84 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { n e t } \\leq c _ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 71, + 474, + 86 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 74, + 487, + 84 + ], + "score": 0.85, + "content": "c _ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 71, + 497, + 86 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 497, + 74, + 504, + 83 + ], + "score": 0.31, + "content": "a", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 318, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 208, + 97 + ], + "score": 1.0, + "content": "constant that depends on", + "type": "text" + }, + { + "bbox": [ + 209, + 84, + 219, + 93 + ], + "score": 0.39, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 83, + 318, + 97 + ], + "score": 1.0, + "content": "but independent of time.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "The outcome of those theorems is that the dynamics described in Eq. (1) is smoothing, while the one", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 114, + 507, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 507, + 127 + ], + "score": 1.0, + "content": "in Eq. (2) is bounded by a conserving mapping. An illustration of this behavior is presented in Fig. 1.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 507, + 145 + ], + "score": 1.0, + "content": "In the physical world, diffusion and hyperbolic equations are used for different applications. Similarly,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 506, + 155 + ], + "score": 1.0, + "content": "many computational models for image segmentation [43], denoising [44] and deblurring are based on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 506, + 165 + ], + "score": 1.0, + "content": "anisotropic diffusion which are similar to the model in Eq. (1). On the other hand, applications that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "require conservation such as volume/distance preservation as in the dense shape correspondence task", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "[45] and protein folding [11], are typically better treated using a hyperbolic equation as in Eq. (2).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "score": 1.0, + "content": "Those insights motivate us to construct two types of layers according to Eq. (1)-(2) using discretized", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 236, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 236, + 209 + ], + "score": 1.0, + "content": "differential operators on graphs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 220, + 316, + 232 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 318, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 318, + 236 + ], + "score": 1.0, + "content": "3.2 Discretized differential operators on graphs", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 326, + 253 + ], + "score": 1.0, + "content": "The models (1) and (2) reside in a continuous manifold", + "type": "text" + }, + { + "bbox": [ + 326, + 241, + 339, + 251 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 240, + 496, + 253 + ], + "score": 1.0, + "content": ", on which a continuous function vector", + "type": "text" + }, + { + "bbox": [ + 497, + 241, + 504, + 253 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "is defined. A graph can be thought of as a discretization of that manifold to a finite space. Assume we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 225, + 275 + ], + "score": 1.0, + "content": "are given an undirected graph", + "type": "text" + }, + { + "bbox": [ + 225, + 263, + 271, + 275 + ], + "score": 0.93, + "content": "\\mathcal { G } = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 262, + 299, + 275 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 299, + 263, + 331, + 273 + ], + "score": 0.91, + "content": "\\nu \\in \\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 262, + 380, + 275 + ], + "score": 1.0, + "content": "is the set of", + "type": "text" + }, + { + "bbox": [ + 380, + 264, + 387, + 272 + ], + "score": 0.8, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 262, + 487, + 275 + ], + "score": 1.0, + "content": "vertices of the graph and", + "type": "text" + }, + { + "bbox": [ + 487, + 263, + 495, + 272 + ], + "score": 0.82, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 146, + 286 + ], + "score": 1.0, + "content": "the set of", + "type": "text" + }, + { + "bbox": [ + 146, + 275, + 156, + 284 + ], + "score": 0.75, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 273, + 306, + 286 + ], + "score": 1.0, + "content": "edges of the graph. Let us denote by", + "type": "text" + }, + { + "bbox": [ + 306, + 274, + 339, + 285 + ], + "score": 0.91, + "content": "\\mathbf { f } _ { i } \\in \\mathbb { R } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 273, + 481, + 286 + ], + "score": 1.0, + "content": "the value of the discrete version of", + "type": "text" + }, + { + "bbox": [ + 482, + 274, + 489, + 285 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 273, + 506, + 286 + ], + "score": 1.0, + "content": ", on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 285, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 121, + 296 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 285, + 126, + 294 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 285, + 171, + 296 + ], + "score": 1.0, + "content": "-th node of", + "type": "text" + }, + { + "bbox": [ + 171, + 285, + 178, + 295 + ], + "score": 0.75, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 285, + 182, + 296 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 183, + 287, + 189, + 294 + ], + "score": 0.6, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 285, + 505, + 296 + ], + "score": 1.0, + "content": "is the number of channels, which is the width of the neural network. We define", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 295, + 484, + 308 + ], + "spans": [ + { + "bbox": [ + 107, + 295, + 117, + 306 + ], + "score": 0.65, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 295, + 484, + 308 + ], + "score": 1.0, + "content": ", the discrete gradient operator on the graph, also known as the incidence matrix, as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 311, + 355, + 325 + ], + "lines": [ + { + "bbox": [ + 255, + 311, + 355, + 325 + ], + "spans": [ + { + "bbox": [ + 255, + 311, + 355, + 325 + ], + "score": 0.93, + "content": "( \\mathbf { G f } ) _ { i j } = \\mathbf { W } _ { i j } ( \\mathbf { f } _ { i } - \\mathbf { f } _ { j } ) ,", + "type": "interline_equation", + "image_path": "59db653613e838769b356ae04120868f1ebcee0a0ec475e7660fb756b21ad375.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 255, + 311, + 355, + 325 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 158, + 343 + ], + "score": 1.0, + "content": "where nodes", + "type": "text" + }, + { + "bbox": [ + 158, + 331, + 163, + 340 + ], + "score": 0.81, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 329, + 180, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 331, + 186, + 341 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 329, + 273, + 343 + ], + "score": 1.0, + "content": "are connected via the", + "type": "text" + }, + { + "bbox": [ + 274, + 330, + 294, + 342 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 329, + 330, + 343 + ], + "score": 1.0, + "content": "-th edge,", + "type": "text" + }, + { + "bbox": [ + 330, + 330, + 350, + 342 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "is an edge weight matrix which can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 151, + 353 + ], + "score": 1.0, + "content": "learnt, and", + "type": "text" + }, + { + "bbox": [ + 151, + 341, + 160, + 352 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 341, + 178, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 342, + 187, + 353 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 341, + 280, + 353 + ], + "score": 1.0, + "content": "are the features on the", + "type": "text" + }, + { + "bbox": [ + 280, + 342, + 285, + 351 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 341, + 314, + 353 + ], + "score": 1.0, + "content": "-th and", + "type": "text" + }, + { + "bbox": [ + 315, + 342, + 320, + 352 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "-th nodes, respectively. The gradient operator", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 377, + 364 + ], + "score": 1.0, + "content": "can be thought of as a weighted directional derivative of the function", + "type": "text" + }, + { + "bbox": [ + 378, + 352, + 384, + 363 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "in the direction defined by the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 362, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 132, + 379 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 133, + 365, + 138, + 374 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 362, + 155, + 379 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 364, + 162, + 375 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 362, + 381, + 379 + ], + "score": 1.0, + "content": ". Furthermore, if we choose the scaled identity matrix", + "type": "text" + }, + { + "bbox": [ + 381, + 362, + 435, + 377 + ], + "score": 0.94, + "content": "\\mathbf { W } _ { i j } = d _ { i j } ^ { - 1 } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 362, + 466, + 379 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 466, + 364, + 479, + 376 + ], + "score": 0.9, + "content": "d _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 362, + 506, + 379 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "distance between the two nodes, then the discrete gradient is a second order approximation to the true", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 205, + 399 + ], + "score": 1.0, + "content": "gradient of the function", + "type": "text" + }, + { + "bbox": [ + 205, + 386, + 213, + 397 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 385, + 408, + 399 + ], + "score": 1.0, + "content": "on the edges of the graph. In this work, we use", + "type": "text" + }, + { + "bbox": [ + 408, + 385, + 459, + 398 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { i j } = \\gamma _ { i j } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 385, + 505, + 399 + ], + "score": 1.0, + "content": ", where the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 128, + 410 + ], + "score": 1.0, + "content": "scale", + "type": "text" + }, + { + "bbox": [ + 129, + 397, + 142, + 408 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 396, + 342, + 410 + ], + "score": 1.0, + "content": "is the geometric mean of the degree of the nodes", + "type": "text" + }, + { + "bbox": [ + 342, + 397, + 356, + 408 + ], + "score": 0.87, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 396, + 506, + 410 + ], + "score": 1.0, + "content": ". Note, that the gradient operator is a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 408, + 305, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 305, + 420 + ], + "score": 1.0, + "content": "mapping from the vertex space to the edge space.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "Given the gradient matrix, it is possible to define the divergence matrix [46], which is an approach", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "that is used extensively in mimetic discretizations of PDEs. To this end, we define the inner product", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 423, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 232, + 458 + ], + "score": 1.0, + "content": "between an edge feature vector", + "type": "text" + }, + { + "bbox": [ + 233, + 447, + 241, + 457 + ], + "score": 0.39, + "content": "\\mathbf { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 446, + 423, + 458 + ], + "score": 1.0, + "content": "and the gradient of a node feature vector f as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 461, + 357, + 477 + ], + "lines": [ + { + "bbox": [ + 234, + 461, + 357, + 477 + ], + "spans": [ + { + "bbox": [ + 234, + 461, + 357, + 477 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( \\mathbf { q } , \\mathbf { G } \\mathbf { f } ) = \\mathbf { q } ^ { \\top } \\mathbf { G } \\mathbf { f } = \\mathbf { f } ^ { \\top } \\mathbf { G } ^ { \\top } \\mathbf { q } . } \\end{array}", + "type": "interline_equation", + "image_path": "4d2c94eaa2db622882c74f687aed6b3084bb2cb2baa97bea70a97c8af10bf501.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 234, + 461, + 357, + 477 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 405, + 494 + ], + "score": 1.0, + "content": "The divergence is naturally defined as the operator that maps edge operator", + "type": "text" + }, + { + "bbox": [ + 406, + 483, + 414, + 492 + ], + "score": 0.5, + "content": "\\mathbf { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "to the node space, that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 116, + 505 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 491, + 164, + 502 + ], + "score": 0.9, + "content": "\\nabla \\cdot \\approx - \\mathbf { G } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 490, + 506, + 505 + ], + "score": 1.0, + "content": ". As usual, the graph Laplacian operator can be obtained by taking the divergence of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 502, + 439, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 381, + 515 + ], + "score": 1.0, + "content": "the gradient. In graph theory it is defined as a positive matrix that is,", + "type": "text" + }, + { + "bbox": [ + 381, + 502, + 435, + 514 + ], + "score": 0.91, + "content": "\\Delta \\widetilde { \\approx } \\mathbf { G } ^ { \\intercal } \\mathbf { G } ^ { \\intercal }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 502, + 439, + 515 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 104, + 519, + 465, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 466, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 466, + 532 + ], + "score": 1.0, + "content": "We also define the weighted line integral over an edge. Similarly to Eq. (3)-(4), we define", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 535, + 425, + 559 + ], + "lines": [ + { + "bbox": [ + 186, + 535, + 425, + 559 + ], + "spans": [ + { + "bbox": [ + 186, + 535, + 425, + 559 + ], + "score": 0.93, + "content": "( \\mathbf { A } \\mathbf { f } ) _ { i j } = { \\frac { 1 } { 2 } } \\mathbf { W } _ { i j } ( \\mathbf { f } _ { i } + \\mathbf { f } _ { j } ) , \\quad ( \\mathbf { q } , \\mathbf { A } \\mathbf { f } ) = \\mathbf { q } ^ { \\top } \\mathbf { A } \\mathbf { f } = \\mathbf { f } ^ { \\top } \\mathbf { A } ^ { \\top } \\mathbf { q } .", + "type": "interline_equation", + "image_path": "4b4864d4e2c8be944528eabc784101d487b24ebc4e760c0f0234ccdbdd70d670.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 186, + 535, + 425, + 559 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 563, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 159, + 575 + ], + "score": 1.0, + "content": "The operator", + "type": "text" + }, + { + "bbox": [ + 160, + 563, + 169, + 573 + ], + "score": 0.45, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "approximates the mass operator on the edges. The right equation in Eq. (5) suggests", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 573, + 501, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 501, + 587 + ], + "score": 1.0, + "content": "that an appropriate averaging operator for edge features is the transpose of the nodal edge average.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "The advantage of defining such operators is that we are able to design networks with architectures", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "that mimic the continuous operators (1) and (2) on the discrete level, as we show in the next section.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 105, + 624, + 458, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 459, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 459, + 639 + ], + "score": 1.0, + "content": "3.3 PDE-GCN: Graph Convolutional Networks by Partial Differential Equations", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 105, + 645, + 459, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 460, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 460, + 658 + ], + "score": 1.0, + "content": "In order to use the computational models in Eq. (1)-(2), we form their discrete versions:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 660, + 406, + 695 + ], + "lines": [ + { + "bbox": [ + 204, + 660, + 406, + 695 + ], + "spans": [ + { + "bbox": [ + 204, + 660, + 406, + 695 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { { \\bf f } ^ { ( l + 1 ) } = { \\bf f } ^ { ( l ) } - h { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\boldsymbol { \\sigma } ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) , } \\\\ & { { \\bf f } ^ { ( l + 1 ) } = 2 { \\bf f } ^ { ( l ) } - { \\bf f } ^ { ( l - 1 ) } - h ^ { 2 } { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\boldsymbol { \\sigma } ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "7d2dd7c6a7bc8054bdffdeadf755a35d9cba3cd6d987550a2d55c5fc0a443a26.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 660, + 406, + 677.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 204, + 677.5, + 406, + 695.0 + ], + "spans": [], + "index": 43 + } + ] + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 700, + 507, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 342, + 713 + ], + "score": 1.0, + "content": "2In the field of numerical PDEs, the Laplacian is defined as", + "type": "text" + }, + { + "bbox": [ + 342, + 700, + 372, + 711 + ], + "score": 0.9, + "content": "- \\mathbf { G } ^ { \\intercal } \\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ", but the combinatorial Laplacian is", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 173, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 145, + 722 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + }, + { + "bbox": [ + 145, + 711, + 169, + 721 + ], + "score": 0.9, + "content": "\\mathbf { G } ^ { \\top } \\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 711, + 173, + 722 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 504, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 399, + 86 + ], + "score": 1.0, + "content": "then the forward propagation through the hyperbolic PDE in (2) satisfies", + "type": "text" + }, + { + "bbox": [ + 400, + 73, + 443, + 84 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { n e t } \\leq c _ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 71, + 474, + 86 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 74, + 487, + 84 + ], + "score": 0.85, + "content": "c _ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 71, + 497, + 86 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 497, + 74, + 504, + 83 + ], + "score": 0.31, + "content": "a", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 318, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 208, + 97 + ], + "score": 1.0, + "content": "constant that depends on", + "type": "text" + }, + { + "bbox": [ + 209, + 84, + 219, + 93 + ], + "score": 0.39, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 83, + 318, + 97 + ], + "score": 1.0, + "content": "but independent of time.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 71, + 504, + 97 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "The outcome of those theorems is that the dynamics described in Eq. (1) is smoothing, while the one", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 114, + 507, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 507, + 127 + ], + "score": 1.0, + "content": "in Eq. (2) is bounded by a conserving mapping. An illustration of this behavior is presented in Fig. 1.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 103, + 507, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 507, + 145 + ], + "score": 1.0, + "content": "In the physical world, diffusion and hyperbolic equations are used for different applications. 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On the other hand, applications that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "require conservation such as volume/distance preservation as in the dense shape correspondence task", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "[45] and protein folding [11], are typically better treated using a hyperbolic equation as in Eq. (2).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "score": 1.0, + "content": "Those insights motivate us to construct two types of layers according to Eq. (1)-(2) using discretized", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 236, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 236, + 209 + ], + "score": 1.0, + "content": "differential operators on graphs.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 130, + 507, + 209 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 220, + 316, + 232 + ], + "lines": [ + { + "bbox": [ + 104, + 218, + 318, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 318, + 236 + ], + "score": 1.0, + "content": "3.2 Discretized differential operators on graphs", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 326, + 253 + ], + "score": 1.0, + "content": "The models (1) and (2) reside in a continuous manifold", + "type": "text" + }, + { + "bbox": [ + 326, + 241, + 339, + 251 + ], + "score": 0.83, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 240, + 496, + 253 + ], + "score": 1.0, + "content": ", on which a continuous function vector", + "type": "text" + }, + { + "bbox": [ + 497, + 241, + 504, + 253 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "is defined. A graph can be thought of as a discretization of that manifold to a finite space. 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Let us denote by", + "type": "text" + }, + { + "bbox": [ + 306, + 274, + 339, + 285 + ], + "score": 0.91, + "content": "\\mathbf { f } _ { i } \\in \\mathbb { R } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 273, + 481, + 286 + ], + "score": 1.0, + "content": "the value of the discrete version of", + "type": "text" + }, + { + "bbox": [ + 482, + 274, + 489, + 285 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 273, + 506, + 286 + ], + "score": 1.0, + "content": ", on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 285, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 121, + 296 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 285, + 126, + 294 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 285, + 171, + 296 + ], + "score": 1.0, + "content": "-th node of", + "type": "text" + }, + { + "bbox": [ + 171, + 285, + 178, + 295 + ], + "score": 0.75, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 285, + 182, + 296 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 183, + 287, + 189, + 294 + ], + "score": 0.6, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 285, + 505, + 296 + ], + "score": 1.0, + "content": "is the number of channels, which is the width of the neural network. We define", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 295, + 484, + 308 + ], + "spans": [ + { + "bbox": [ + 107, + 295, + 117, + 306 + ], + "score": 0.65, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 295, + 484, + 308 + ], + "score": 1.0, + "content": ", the discrete gradient operator on the graph, also known as the incidence matrix, as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 240, + 506, + 308 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 311, + 355, + 325 + ], + "lines": [ + { + "bbox": [ + 255, + 311, + 355, + 325 + ], + "spans": [ + { + "bbox": [ + 255, + 311, + 355, + 325 + ], + "score": 0.93, + "content": "( \\mathbf { G f } ) _ { i j } = \\mathbf { W } _ { i j } ( \\mathbf { f } _ { i } - \\mathbf { f } _ { j } ) ,", + "type": "interline_equation", + "image_path": "59db653613e838769b356ae04120868f1ebcee0a0ec475e7660fb756b21ad375.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 255, + 311, + 355, + 325 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 158, + 343 + ], + "score": 1.0, + "content": "where nodes", + "type": "text" + }, + { + "bbox": [ + 158, + 331, + 163, + 340 + ], + "score": 0.81, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 329, + 180, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 331, + 186, + 341 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 329, + 273, + 343 + ], + "score": 1.0, + "content": "are connected via the", + "type": "text" + }, + { + "bbox": [ + 274, + 330, + 294, + 342 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 329, + 330, + 343 + ], + "score": 1.0, + "content": "-th edge,", + "type": "text" + }, + { + "bbox": [ + 330, + 330, + 350, + 342 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "is an edge weight matrix which can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 151, + 353 + ], + "score": 1.0, + "content": "learnt, and", + "type": "text" + }, + { + "bbox": [ + 151, + 341, + 160, + 352 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 341, + 178, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 342, + 187, + 353 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 341, + 280, + 353 + ], + "score": 1.0, + "content": "are the features on the", + "type": "text" + }, + { + "bbox": [ + 280, + 342, + 285, + 351 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 341, + 314, + 353 + ], + "score": 1.0, + "content": "-th and", + "type": "text" + }, + { + "bbox": [ + 315, + 342, + 320, + 352 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "-th nodes, respectively. The gradient operator", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 377, + 364 + ], + "score": 1.0, + "content": "can be thought of as a weighted directional derivative of the function", + "type": "text" + }, + { + "bbox": [ + 378, + 352, + 384, + 363 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "in the direction defined by the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 362, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 132, + 379 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 133, + 365, + 138, + 374 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 362, + 155, + 379 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 156, + 364, + 162, + 375 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 362, + 381, + 379 + ], + "score": 1.0, + "content": ". 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In this work, we use", + "type": "text" + }, + { + "bbox": [ + 408, + 385, + 459, + 398 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { i j } = \\gamma _ { i j } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 385, + 505, + 399 + ], + "score": 1.0, + "content": ", where the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 128, + 410 + ], + "score": 1.0, + "content": "scale", + "type": "text" + }, + { + "bbox": [ + 129, + 397, + 142, + 408 + ], + "score": 0.88, + "content": "\\gamma _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 396, + 342, + 410 + ], + "score": 1.0, + "content": "is the geometric mean of the degree of the nodes", + "type": "text" + }, + { + "bbox": [ + 342, + 397, + 356, + 408 + ], + "score": 0.87, + "content": "i , j", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 396, + 506, + 410 + ], + "score": 1.0, + "content": ". Note, that the gradient operator is a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 408, + 305, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 305, + 420 + ], + "score": 1.0, + "content": "mapping from the vertex space to the edge space.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 329, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "Given the gradient matrix, it is possible to define the divergence matrix [46], which is an approach", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "that is used extensively in mimetic discretizations of PDEs. To this end, we define the inner product", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 423, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 232, + 458 + ], + "score": 1.0, + "content": "between an edge feature vector", + "type": "text" + }, + { + "bbox": [ + 233, + 447, + 241, + 457 + ], + "score": 0.39, + "content": "\\mathbf { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 446, + 423, + 458 + ], + "score": 1.0, + "content": "and the gradient of a node feature vector f as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 423, + 505, + 458 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 461, + 357, + 477 + ], + "lines": [ + { + "bbox": [ + 234, + 461, + 357, + 477 + ], + "spans": [ + { + "bbox": [ + 234, + 461, + 357, + 477 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( \\mathbf { q } , \\mathbf { G } \\mathbf { f } ) = \\mathbf { q } ^ { \\top } \\mathbf { G } \\mathbf { f } = \\mathbf { f } ^ { \\top } \\mathbf { G } ^ { \\top } \\mathbf { q } . } \\end{array}", + "type": "interline_equation", + "image_path": "4d2c94eaa2db622882c74f687aed6b3084bb2cb2baa97bea70a97c8af10bf501.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 234, + 461, + 357, + 477 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 405, + 494 + ], + "score": 1.0, + "content": "The divergence is naturally defined as the operator that maps edge operator", + "type": "text" + }, + { + "bbox": [ + 406, + 483, + 414, + 492 + ], + "score": 0.5, + "content": "\\mathbf { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "to the node space, that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 490, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 116, + 505 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 117, + 491, + 164, + 502 + ], + "score": 0.9, + "content": "\\nabla \\cdot \\approx - \\mathbf { G } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 490, + 506, + 505 + ], + "score": 1.0, + "content": ". 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In graph theory it is defined as a positive matrix that is,", + "type": "text" + }, + { + "bbox": [ + 381, + 502, + 435, + 514 + ], + "score": 0.91, + "content": "\\Delta \\widetilde { \\approx } \\mathbf { G } ^ { \\intercal } \\mathbf { G } ^ { \\intercal }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 502, + 439, + 515 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 480, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 519, + 465, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 466, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 466, + 532 + ], + "score": 1.0, + "content": "We also define the weighted line integral over an edge. Similarly to Eq. (3)-(4), we define", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 518, + 466, + 532 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 535, + 425, + 559 + ], + "lines": [ + { + "bbox": [ + 186, + 535, + 425, + 559 + ], + "spans": [ + { + "bbox": [ + 186, + 535, + 425, + 559 + ], + "score": 0.93, + "content": "( \\mathbf { A } \\mathbf { f } ) _ { i j } = { \\frac { 1 } { 2 } } \\mathbf { W } _ { i j } ( \\mathbf { f } _ { i } + \\mathbf { f } _ { j } ) , \\quad ( \\mathbf { q } , \\mathbf { A } \\mathbf { f } ) = \\mathbf { q } ^ { \\top } \\mathbf { A } \\mathbf { f } = \\mathbf { f } ^ { \\top } \\mathbf { A } ^ { \\top } \\mathbf { q } .", + "type": "interline_equation", + "image_path": "4b4864d4e2c8be944528eabc784101d487b24ebc4e760c0f0234ccdbdd70d670.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 186, + 535, + 425, + 559 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 563, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 159, + 575 + ], + "score": 1.0, + "content": "The operator", + "type": "text" + }, + { + "bbox": [ + 160, + 563, + 169, + 573 + ], + "score": 0.45, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "approximates the mass operator on the edges. The right equation in Eq. (5) suggests", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 573, + 501, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 501, + 587 + ], + "score": 1.0, + "content": "that an appropriate averaging operator for edge features is the transpose of the nodal edge average.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 562, + 505, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "The advantage of defining such operators is that we are able to design networks with architectures", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "that mimic the continuous operators (1) and (2) on the discrete level, as we show in the next section.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 589, + 506, + 614 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 624, + 458, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 459, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 459, + 639 + ], + "score": 1.0, + "content": "3.3 PDE-GCN: Graph Convolutional Networks by Partial Differential Equations", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 105, + 645, + 459, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 460, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 460, + 658 + ], + "score": 1.0, + "content": "In order to use the computational models in Eq. (1)-(2), we form their discrete versions:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 645, + 460, + 658 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 660, + 406, + 695 + ], + "lines": [ + { + "bbox": [ + 204, + 660, + 406, + 695 + ], + "spans": [ + { + "bbox": [ + 204, + 660, + 406, + 695 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { { \\bf f } ^ { ( l + 1 ) } = { \\bf f } ^ { ( l ) } - h { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\boldsymbol { \\sigma } ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) , } \\\\ & { { \\bf f } ^ { ( l + 1 ) } = 2 { \\bf f } ^ { ( l ) } - { \\bf f } ^ { ( l - 1 ) } - h ^ { 2 } { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\boldsymbol { \\sigma } ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "7d2dd7c6a7bc8054bdffdeadf755a35d9cba3cd6d987550a2d55c5fc0a443a26.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 660, + 406, + 677.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 204, + 677.5, + 406, + 695.0 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "Here, in Eq. (6) we use the forward Euler to discretize Eq. (1), and in Eq. (7) we discretize the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 436, + 97 + ], + "score": 1.0, + "content": "second order time derivative in Eq. (2), using the leapfrog method. 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The similarity between (6) and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 417, + 119 + ], + "score": 1.0, + "content": "ResNet is well documented in the context of CNNs [22]. The hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 417, + 107, + 424, + 117 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "is the step-size, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 391, + 129 + ], + "score": 1.0, + "content": "it is chosen such that the stability of the discretization is kept. We use", + "type": "text" + }, + { + "bbox": [ + 392, + 118, + 433, + 128 + ], + "score": 0.45, + "content": "\\sigma = \\operatorname { t a n h }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "for the activation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "function as it yields slightly better results in our experiments, although other functions such as ReLU", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "can also be used, as reported in Sec. 4.7. Each of Eq. (6)-(7) defines a PDE-GCN block. We denote", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 266, + 164 + ], + "score": 1.0, + "content": "the former (diffusive equation) by PDE-", + "type": "text" + }, + { + "bbox": [ + 266, + 150, + 294, + 162 + ], + "score": 0.57, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 149, + 473, + 164 + ], + "score": 1.0, + "content": "and the latter (hyperbolic equation) by PDE-", + "type": "text" + }, + { + "bbox": [ + 473, + 150, + 500, + 162 + ], + "score": 0.77, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 149, + 505, + 164 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 457, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 459, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 459, + 180 + ], + "score": 1.0, + "content": "To complete the description of our network, a few more details are required, as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 343, + 195 + ], + "score": 1.0, + "content": "The (opening) embedding layer. The input vertex features", + "type": "text" + }, + { + "bbox": [ + 344, + 185, + 357, + 194 + ], + "score": 0.85, + "content": "\\mathbf { u } _ { \\nu }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 182, + 481, + 195 + ], + "score": 1.0, + "content": "are fed through an embedding", + "type": "text" + }, + { + "bbox": [ + 481, + 183, + 505, + 194 + ], + "score": 0.87, + "content": "( 1 \\times 1", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 193, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 183, + 207 + ], + "score": 1.0, + "content": "convolution) layer", + "type": "text" + }, + { + "bbox": [ + 183, + 194, + 198, + 205 + ], + "score": 0.87, + "content": "{ \\bf K } _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 193, + 315, + 207 + ], + "score": 1.0, + "content": "to obtain the initial features", + "type": "text" + }, + { + "bbox": [ + 315, + 194, + 324, + 205 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 193, + 439, + 207 + ], + "score": 1.0, + "content": "of our PDE-GCN network:", + "type": "text" + }, + { + "bbox": [ + 439, + 194, + 489, + 205 + ], + "score": 0.91, + "content": "\\mathbf { f } _ { 0 } = \\mathbf { K } _ { o } \\mathbf { u } _ { \\mathcal { V } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 193, + 505, + 207 + ], + "score": 1.0, + "content": ". 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The similarity between (6) and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 417, + 119 + ], + "score": 1.0, + "content": "ResNet is well documented in the context of CNNs [22]. The hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 417, + 107, + 424, + 117 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "is the step-size, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 391, + 129 + ], + "score": 1.0, + "content": "it is chosen such that the stability of the discretization is kept. We use", + "type": "text" + }, + { + "bbox": [ + 392, + 118, + 433, + 128 + ], + "score": 0.45, + "content": "\\sigma = \\operatorname { t a n h }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "for the activation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "function as it yields slightly better results in our experiments, although other functions such as ReLU", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 152 + ], + "score": 1.0, + "content": "can also be used, as reported in Sec. 4.7. Each of Eq. (6)-(7) defines a PDE-GCN block. We denote", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 266, + 164 + ], + "score": 1.0, + "content": "the former (diffusive equation) by PDE-", + "type": "text" + }, + { + "bbox": [ + 266, + 150, + 294, + 162 + ], + "score": 0.57, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 149, + 473, + 164 + ], + "score": 1.0, + "content": "and the latter (hyperbolic equation) by PDE-", + "type": "text" + }, + { + "bbox": [ + 473, + 150, + 500, + 162 + ], + "score": 0.77, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 149, + 505, + 164 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 73, + 506, + 164 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 457, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 459, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 459, + 180 + ], + "score": 1.0, + "content": "To complete the description of our network, a few more details are required, as follows:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 164, + 459, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 343, + 195 + ], + "score": 1.0, + "content": "The (opening) embedding layer. The input vertex features", + "type": "text" + }, + { + "bbox": [ + 344, + 185, + 357, + 194 + ], + "score": 0.85, + "content": "\\mathbf { u } _ { \\nu }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 182, + 481, + 195 + ], + "score": 1.0, + "content": "are fed through an embedding", + "type": "text" + }, + { + "bbox": [ + 481, + 183, + 505, + 194 + ], + "score": 0.87, + "content": "( 1 \\times 1", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 193, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 183, + 207 + ], + "score": 1.0, + "content": "convolution) layer", + "type": "text" + }, + { + "bbox": [ + 183, + 194, + 198, + 205 + ], + "score": 0.87, + "content": "{ \\bf K } _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 193, + 315, + 207 + ], + "score": 1.0, + "content": "to obtain the initial features", + "type": "text" + }, + { + "bbox": [ + 315, + 194, + 324, + 205 + ], + "score": 0.86, + "content": "\\mathbf { f } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 193, + 439, + 207 + ], + "score": 1.0, + "content": "of our PDE-GCN network:", + "type": "text" + }, + { + "bbox": [ + 439, + 194, + 489, + 205 + ], + "score": 0.91, + "content": "\\mathbf { f } _ { 0 } = \\mathbf { K } _ { o } \\mathbf { u } _ { \\mathcal { V } }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 193, + 505, + 207 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 284, + 217 + ], + "score": 1.0, + "content": "cases where input edge attributes (features)", + "type": "text" + }, + { + "bbox": [ + 284, + 206, + 297, + 216 + ], + "score": 0.88, + "content": "\\mathbf { u } _ { \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "are available (as in the experiment in Sec. 4.6), we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 215, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 506, + 229 + ], + "score": 1.0, + "content": "transform them to the vertex space by taking their divergence and average, and concatenate them to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 440, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 239, + 239 + ], + "score": 1.0, + "content": "the input of the embedding layer", + "type": "text" + }, + { + "bbox": [ + 239, + 227, + 253, + 237 + ], + "score": 0.89, + "content": "{ \\bf K } _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 226, + 301, + 239 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + }, + { + "bbox": [ + 301, + 226, + 434, + 239 + ], + "score": 0.92, + "content": "\\mathbf { f } _ { 0 } = \\bar { \\mathbf { K } _ { o } } ( \\mathbf { u } _ { \\mathcal { V } } \\oplus \\mathbf { A } ^ { \\top } \\mathbf { u } _ { \\mathcal { E } } \\oplus \\mathbf { G } ^ { \\top } \\mathbf { u } _ { \\mathcal { E } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 226, + 440, + 239 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 182, + 506, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 507, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 358, + 258 + ], + "score": 1.0, + "content": "The (closing) embedding layer. 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This way, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 307 + ], + "score": 1.0, + "content": "network begins from a diffusion/hyperbolic equation, which serves as a prior and guides the network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 304, + 461, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 461, + 319 + ], + "score": 1.0, + "content": "to initially behave like classical methods [43, 45] and to further improve during training.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 282, + 506, + 319 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 321, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "The choice of dynamics. For some applications, anisotropic diffusion is appropriate, while for others", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "conservation is more important. However, in some applications this may not be clear. To this end, it", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 344, + 375, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 375, + 356 + ], + "score": 1.0, + "content": "is possible to combine Eq. 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The discretization", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 322, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 322, + 405 + ], + "score": 1.0, + "content": "of this PDE leads to the following network dynamics:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 380, + 505, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 410, + 460, + 426 + ], + "lines": [ + { + "bbox": [ + 131, + 410, + 460, + 426 + ], + "spans": [ + { + "bbox": [ + 131, + 410, + 460, + 426 + ], + "score": 0.91, + "content": "\\alpha ( { \\bf f } ^ { ( l + 1 ) } - 2 { \\bf f } ^ { ( l ) } + { \\bf f } ^ { ( l - 1 ) } ) + h ( 1 - \\alpha ) ( { \\bf f } ^ { ( l + 1 ) } - { \\bf f } ^ { ( l ) } ) = - h ^ { 2 } { \\bf G } ^ { \\top } { \\bf K } _ { l } ^ { \\top } \\sigma ( { \\bf K } _ { l } { \\bf G } { \\bf f } ^ { ( l ) } ) ,", + "type": "interline_equation", + "image_path": "7eae4c85ed09f79b2fdb9c084e1247d56883808773b4f4f32acfc2e9e28e4d62.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 131, + 410, + 460, + 426 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 133, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 432, + 158, + 443 + ], + "score": 0.85, + "content": "\\mathbf { f } ^ { ( l + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 430, + 259, + 447 + ], + "score": 1.0, + "content": "is updated by the known", + "type": "text" + }, + { + "bbox": [ + 259, + 432, + 274, + 443 + ], + "score": 0.87, + "content": "\\mathbf { f } ^ { ( l ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 430, + 292, + 447 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 432, + 317, + 443 + ], + "score": 0.89, + "content": "\\mathbf { f } ^ { ( l - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 430, + 506, + 447 + ], + "score": 1.0, + "content": ". We denote a layer that is governed by Eq. (9)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 141, + 457 + ], + "score": 1.0, + "content": "by PDE-", + "type": "text" + }, + { + "bbox": [ + 142, + 444, + 171, + 456 + ], + "score": 0.83, + "content": "\\mathrm { \\bf G C N _ { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 443, + 203, + 457 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 203, + 445, + 214, + 455 + ], + "score": 0.45, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "stands for mixture). Note, that it is also possible to learn a combination", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 455, + 419, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 151, + 468 + ], + "score": 1.0, + "content": "coefficient", + "type": "text" + }, + { + "bbox": [ + 151, + 457, + 162, + 466 + ], + "score": 0.85, + "content": "\\alpha _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 455, + 419, + 468 + ], + "score": 1.0, + "content": "per layer, although we did not read a benefit from such scheme.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 430, + 506, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 323, + 484 + ], + "score": 1.0, + "content": "As we show in our numerical experiments, learning", + "type": "text" + }, + { + "bbox": [ + 324, + 474, + 331, + 482 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "yields results that are consistent with our", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "understanding of the problem, that is, graph node-classification gravitates towards no second order", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 493, + 498, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 498, + 506 + ], + "score": 1.0, + "content": "derivatives while applications that require conservation gravitate towards the hyperbolic equation.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 472, + 505, + 506 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 521, + 191, + 535 + ], + "lines": [ + { + "bbox": [ + 104, + 519, + 193, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 193, + 538 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "In this section we demonstrate our approach on various problems from different fields and applications", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "ranging from 3D shape-classification [47] to protein-protein interaction [26] and node-classification", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "[48]. The experiments also vary in their output type. Node classification is similar to segmentation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "problems that are typically solved by anisotropic diffusion while the dense shape correspondence", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 591, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 507, + 604 + ], + "score": 1.0, + "content": "problem is conservative and therefore can be thought of as a problem that does not require smoothing.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 546, + 507, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "In all experiments, we use the suitable PDE-GCN (D, H or M) block as described in Sec. 3, with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "various depths (number of layers) and widths (number of channels), as well as the appropriate final", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "score": 1.0, + "content": "convolution steps, depending on the task at hand. A detailed description of the architectures used in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "our experiments is given in Appendix B. We use the Adam [49] optimizer in all experiments, and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "score": 1.0, + "content": "perform grid search over the hyper-parameters of our network. The selected hyper-parameters are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "reported in Appendix C. Our objective function in all experiments is the cross-entropy loss, besides", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 506, + 686 + ], + "score": 1.0, + "content": "inductive learning on PPI where we use the binary cross-entropy loss. Our code is implemented using", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 318, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 318, + 695 + ], + "score": 1.0, + "content": "PyTorch [50], trained on an Nvidia Titan RTX GPU.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 607, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 713 + ], + "score": 1.0, + "content": "We show that for all the considered tasks and datasets, our method is either remarkably better or on", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 711, + 237, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 237, + 722 + ], + "score": 1.0, + "content": "par with state-of-the-art models.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 698, + 505, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 173, + 93, + 438, + 204 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 114, + 77, + 492, + 89 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 75, + 493, + 91 + ], + "spans": [ + { + "bbox": [ + 117, + 75, + 406, + 91 + ], + "score": 1.0, + "content": "Table 1: Generalization of GCNs to different domains and applications.", + "type": "text" + }, + { + "bbox": [ + 406, + 77, + 421, + 90 + ], + "score": 0.64, + "content": "( L )", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 75, + 455, + 91 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + }, + { + "bbox": [ + 456, + 78, + 464, + 87 + ], + "score": 0.59, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 75, + 493, + 91 + ], + "score": 1.0, + "content": "layers.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 173, + 93, + 438, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 93, + 438, + 204 + ], + "spans": [ + { + "bbox": [ + 173, + 93, + 438, + 204 + ], + "score": 0.972, + "html": "
MethodDatasetAccuracy (%)
DGCNN (4) [5]ModelNet-1092.8
DGCNN (2)/ (4)Cora34.9 /25.2
DGCNN + Diffusion (2) /(4)Cora71.0 / 66.1
GCNII (4) [19]ModelNet-1065.4
Cora82.6
PDE-GCND(4) (Ours)ModelNet-1092.2
Cora83.6
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The diffusion", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 356, + 232, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 203, + 369 + ], + "score": 1.0, + "content": "equation contribution is", + "type": "text" + }, + { + "bbox": [ + 204, + 357, + 228, + 367 + ], + "score": 0.89, + "content": "1 - \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 356, + 232, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + } + ], + "index": 6.25 + }, + { + "type": "title", + "bbox": [ + 107, + 393, + 337, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 338, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 338, + 407 + ], + "score": 1.0, + "content": "4.1 Generalization of GCNs to different applications", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "To gain deeper understanding about the effectiveness and generalization of various GCN methods", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "to different tasks, we start by picking two datasets - ModelNet-10 [47] for 3D shape-classification,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "and Cora [48] for semi-supervised node classification. Those datasets are not only different in terms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "of application (global versus local classification), but also stem from different domains. While", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "in ModelNet-10 the data has geometrical meaning, the data in Cora has no obvious geometrical", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "interpretation. Therefore, we suggest that success in both applications should be obtained from a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 279, + 493 + ], + "score": 1.0, + "content": "generalizable GCN. We compare our PDE-", + "type": "text" + }, + { + "bbox": [ + 279, + 480, + 307, + 492 + ], + "score": 0.6, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "with two recent and popular networks - DGCNN", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "[5] and GCNII [19]. For ModelNet-10 shape-classification, we randomly sample 1,024 points from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "each shape to form a point cloud, and connect its points using k-nearest-neighbors (k-nn) algorithm", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 512, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 126, + 527 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 513, + 157, + 523 + ], + "score": 0.91, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 512, + 506, + 527 + ], + "score": 1.0, + "content": "to obtain a graph and follow the training scheme of [5]. On Cora, we follow the same", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 524, + 507, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 507, + 537 + ], + "score": 1.0, + "content": "procedure as in [19]. We evaluate all models with 4 layers, as well as 2 layers for DGCNN on Cora.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "Our results, reported in Tab. 1 suggest that while each of the considered methods obtains high", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "accuracy on the dataset it originally was tested on (Cora for GCNII and ModelNet-10 for DGCNN),", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "score": 1.0, + "content": "obtaining a similar measure of success on a different dataset was not possible when using the very", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "same networks. Additionally, while our attempts to add a diffusion-equation dynamics to DGCNN", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 585, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 596 + ], + "score": 1.0, + "content": "(i.e., updating features as in Eq. (6)) showed an increase in performance – a large gap to state-of-the-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "art model still exists. On top of that, we also see that DGCNN suffers from over-smoothing, as its", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "accuracy significantly decreases when adding more layers. Last but not least, we observe that our", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 616, + 498, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 128, + 630 + ], + "score": 1.0, + "content": "PDE-", + "type": "text" + }, + { + "bbox": [ + 129, + 617, + 156, + 628 + ], + "score": 0.66, + "content": "\\mathrm { \\cdot G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 616, + 498, + 630 + ], + "score": 1.0, + "content": "obtains high accuracy on both datasets, similar or better than state-of-the-art models.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 645, + 272, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 273, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 273, + 658 + ], + "score": 1.0, + "content": "4.2 Learning PDE network dynamics", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "In this experiment, we delve on the ability to learn the appropriate PDE that better models a given", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "problem. To this end, we use the mixture model from Eq. (9) so that the resulting PDE is a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "score": 1.0, + "content": "combination of the diffusion and hyperbolic dynamics. 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MethodDatasetAccuracy (%)
DGCNN (4) [5]ModelNet-1092.8
DGCNN (2)/ (4)Cora34.9 /25.2
DGCNN + Diffusion (2) /(4)Cora71.0 / 66.1
GCNII (4) [19]ModelNet-1065.4
Cora82.6
PDE-GCND(4) (Ours)ModelNet-1092.2
Cora83.6
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The diffusion", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 356, + 232, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 203, + 369 + ], + "score": 1.0, + "content": "equation contribution is", + "type": "text" + }, + { + "bbox": [ + 204, + 357, + 228, + 367 + ], + "score": 0.89, + "content": "1 - \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 356, + 232, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + } + ], + "index": 6.25 + }, + { + "type": "title", + "bbox": [ + 107, + 393, + 337, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 338, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 338, + 407 + ], + "score": 1.0, + "content": "4.1 Generalization of GCNs to different applications", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "To gain deeper understanding about the effectiveness and generalization of various GCN methods", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "to different tasks, we start by picking two datasets - ModelNet-10 [47] for 3D shape-classification,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "and Cora [48] for semi-supervised node classification. Those datasets are not only different in terms", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "of application (global versus local classification), but also stem from different domains. While", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "in ModelNet-10 the data has geometrical meaning, the data in Cora has no obvious geometrical", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "interpretation. Therefore, we suggest that success in both applications should be obtained from a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 279, + 493 + ], + "score": 1.0, + "content": "generalizable GCN. We compare our PDE-", + "type": "text" + }, + { + "bbox": [ + 279, + 480, + 307, + 492 + ], + "score": 0.6, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "with two recent and popular networks - DGCNN", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "[5] and GCNII [19]. For ModelNet-10 shape-classification, we randomly sample 1,024 points from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "each shape to form a point cloud, and connect its points using k-nearest-neighbors (k-nn) algorithm", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 512, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 126, + 527 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 513, + 157, + 523 + ], + "score": 0.91, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 512, + 506, + 527 + ], + "score": 1.0, + "content": "to obtain a graph and follow the training scheme of [5]. On Cora, we follow the same", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 524, + 507, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 507, + 537 + ], + "score": 1.0, + "content": "procedure as in [19]. We evaluate all models with 4 layers, as well as 2 layers for DGCNN on Cora.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 415, + 507, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "Our results, reported in Tab. 1 suggest that while each of the considered methods obtains high", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "accuracy on the dataset it originally was tested on (Cora for GCNII and ModelNet-10 for DGCNN),", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "score": 1.0, + "content": "obtaining a similar measure of success on a different dataset was not possible when using the very", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "same networks. Additionally, while our attempts to add a diffusion-equation dynamics to DGCNN", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 585, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 596 + ], + "score": 1.0, + "content": "(i.e., updating features as in Eq. (6)) showed an increase in performance – a large gap to state-of-the-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "art model still exists. On top of that, we also see that DGCNN suffers from over-smoothing, as its", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "accuracy significantly decreases when adding more layers. 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By learning the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 149, + 723 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 150, + 713, + 157, + 721 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "in (9), we allow to choose a mixed PDE between a purely conservative network and a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "diffusive one. We consider two problems: semi-supervised node classification on Cora, and dense", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 263, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 263, + 96 + ], + "score": 1.0, + "content": "shape correspondence on FAUST [51].", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 667, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "diffusive one. We consider two problems: semi-supervised node classification on Cora, and dense", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 263, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 263, + 96 + ], + "score": 1.0, + "content": "shape correspondence on FAUST [51].", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 112 + ], + "score": 1.0, + "content": "Our results, reported in Fig. 2 suggest that just as in classical works [43, 52], problems like node-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "classification obtain better performance with an anisotropic diffusion like in Eq. (6), and for problems", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 120, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 505, + 136 + ], + "score": 1.0, + "content": "involving dense-correspondences like in [45, 11] that tend to conserve the energy of the underlying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 420, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 420, + 146 + ], + "score": 1.0, + "content": "problem, a hyperbolic equation type of PDE as in Eq. (7) is more appropriate.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "table", + "bbox": [ + 108, + 178, + 505, + 246 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 162, + 504, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 176 + ], + "score": 1.0, + "content": "Table 2: Statistics of datasets used in our semi-and fully supervised node-classification experiments.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 108, + 178, + 505, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 178, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 108, + 178, + 505, + 246 + ], + "score": 0.978, + "html": "
DatasetCoraCiteSeerPubMedChameleonCornellTexasWisconsinPPI
Classes7635555121
Nodes2,7083,32719,7172,27718318325156,944
Edges5,4294,73244,33836,101295309499818,716
Features1,4333,7035002,3251,7031,7031,70350
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For all datasets", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 302, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 507, + 318 + ], + "score": 1.0, + "content": "we use the standard training/validation/testing split as in [53], with 20 nodes per class for training,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "500 validation nodes and 1,000 testing nodes and follow the training scheme of [19]. The statistics", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 410, + 338 + ], + "score": 1.0, + "content": "of the datasets are reported in Tab. 2. We compare our results using PDE-", + "type": "text" + }, + { + "bbox": [ + 410, + 325, + 438, + 336 + ], + "score": 0.81, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "with recent and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "popular models like GCN [3], GAT [27], APPNP [30], JKNet [29] and DropEdge [28]. We note that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "our network does not over-smooth, as an increase in the number of layers does not cause performance", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 301, + 371 + ], + "score": 1.0, + "content": "degradation. 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Overall, our results in Tab. 3 show that our PDE-", + "type": "text" + }, + { + "bbox": [ + 373, + 369, + 401, + 380 + ], + "score": 0.72, + "content": "\\mathrm { \\cdot G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 368, + 506, + 382 + ], + "score": 1.0, + "content": "achieves similar or better", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 380, + 263, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 263, + 392 + ], + "score": 1.0, + "content": "accuracy than the considered methods.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 405, + 281, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 282, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 282, + 418 + ], + "score": 1.0, + "content": "4.4 Fully-supervised node classification", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "We follow [54] and use 7 datasets: Cora, CiteSeer, PubMed, Chameleon, Cornell, Texas and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 368, + 450 + ], + "score": 1.0, + "content": "Wisconsin. We also use the same train/validation/test splits of", + "type": "text" + }, + { + "bbox": [ + 368, + 437, + 388, + 448 + ], + "score": 0.86, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 437, + 390, + 450 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 391, + 437, + 411, + 448 + ], + "score": 0.84, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 437, + 414, + 450 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 414, + 437, + 433, + 448 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ", respectively. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "addition, we report the average performance over 10 random splits from [54]. We fix the number", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "of channels to 64 and perform grid search to determine the hyper-parameters parameters, which are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "reported in Appendix C. We compare our network with GCN, GAT, three variants of Geom-GCN", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 480, + 504, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 504, + 492 + ], + "score": 1.0, + "content": "[54], APPNP, JKNet, Incep and GCNII in Tab. 4. Our experiments read either similar or better than", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "the state-of-the-art on Cora, CiteSeer and PubMed datasets. On Chameleon [55], Cornell, Texas and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "Wisconsin datasets, we improve state-of-the-art accuracy by a significant margin. For example, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 133, + 527 + ], + "score": 1.0, + "content": "obtain", + "type": "text" + }, + { + "bbox": [ + 133, + 513, + 166, + 524 + ], + "score": 0.9, + "content": "9 3 . 2 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 513, + 299, + 527 + ], + "score": 1.0, + "content": "accuracy on Texas with our PDE-", + "type": "text" + }, + { + "bbox": [ + 299, + 514, + 327, + 524 + ], + "score": 0.85, + "content": "\\mathrm { G C N _ { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 513, + 382, + 527 + ], + "score": 1.0, + "content": ", compared to", + "type": "text" + }, + { + "bbox": [ + 383, + 513, + 415, + 524 + ], + "score": 0.9, + "content": "7 7 . 8 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 513, + 436, + 527 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 436, + 514, + 469, + 524 + ], + "score": 0.46, + "content": "\\mathrm { G C N I I ^ { \\ast } }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 513, + 506, + 527 + ], + "score": 1.0, + "content": ". Similar", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "improvements hold for Cornell and Wisconsin datasets. The common factor for these datasets is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "their small size, as depicted from Tab. 2. We argue that the success of our network stems from its", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "capability of apriori extracting features from graphs, due to its utilization of discretized differential", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 406, + 570 + ], + "score": 1.0, + "content": "operators and PDE guided construction. On Chameleon [55], using PDE-", + "type": "text" + }, + { + "bbox": [ + 406, + 557, + 435, + 568 + ], + "score": 0.87, + "content": "\\mathrm { G C N _ { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 556, + 506, + 570 + ], + "score": 1.0, + "content": ", we improve the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 245, + 580 + ], + "score": 1.0, + "content": "current state-of-the-art accuracy of", + "type": "text" + }, + { + "bbox": [ + 245, + 568, + 278, + 578 + ], + "score": 0.41, + "content": "\\operatorname { G C N I I } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 568, + 300, + 580 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 300, + 568, + 333, + 578 + ], + "score": 0.88, + "content": "6 2 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 568, + 344, + 580 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 344, + 568, + 376, + 578 + ], + "score": 0.9, + "content": "6 6 . 0 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 568, + 505, + 580 + ], + "score": 1.0, + "content": ". 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As reported", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 679, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 207, + 692 + ], + "score": 1.0, + "content": "in Tab. 5, our PDE-GCN", + "type": "text" + }, + { + "bbox": [ + 208, + 681, + 215, + 690 + ], + "score": 0.26, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 679, + 507, + 692 + ], + "score": 1.0, + "content": "achieves 99.07 Micro-averaged F1 score, superior to methods like GAT,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 690, + 435, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 435, + 702 + ], + "score": 1.0, + "content": "JKNet and GeniePath, also close to state-of-the-art GCNII* with a score of 99.58.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 712, + 334, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 335, + 724 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 335, + 724 + ], + "score": 1.0, + "content": "3Note that this result is also a new state-of-the-art accuracy.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 72, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 112 + ], + "score": 1.0, + "content": "Our results, reported in Fig. 2 suggest that just as in classical works [43, 52], problems like node-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "classification obtain better performance with an anisotropic diffusion like in Eq. (6), and for problems", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 120, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 505, + 136 + ], + "score": 1.0, + "content": "involving dense-correspondences like in [45, 11] that tend to conserve the energy of the underlying", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 420, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 420, + 146 + ], + "score": 1.0, + "content": "problem, a hyperbolic equation type of PDE as in Eq. (7) is more appropriate.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 100, + 506, + 146 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 178, + 505, + 246 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 162, + 504, + 174 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 176 + ], + "score": 1.0, + "content": "Table 2: Statistics of datasets used in our semi-and fully supervised node-classification experiments.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 108, + 178, + 505, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 178, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 108, + 178, + 505, + 246 + ], + "score": 0.978, + "html": "
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For all datasets", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 302, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 507, + 318 + ], + "score": 1.0, + "content": "we use the standard training/validation/testing split as in [53], with 20 nodes per class for training,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "500 validation nodes and 1,000 testing nodes and follow the training scheme of [19]. The statistics", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 410, + 338 + ], + "score": 1.0, + "content": "of the datasets are reported in Tab. 2. We compare our results using PDE-", + "type": "text" + }, + { + "bbox": [ + 410, + 325, + 438, + 336 + ], + "score": 0.81, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "with recent and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "popular models like GCN [3], GAT [27], APPNP [30], JKNet [29] and DropEdge [28]. We note that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "our network does not over-smooth, as an increase in the number of layers does not cause performance", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 301, + 371 + ], + "score": 1.0, + "content": "degradation. For example, on CiteSeer, we obtain", + "type": "text" + }, + { + "bbox": [ + 302, + 358, + 329, + 369 + ], + "score": 0.87, + "content": "7 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 357, + 477, + 371 + ], + "score": 1.0, + "content": "accuracy with 32 layers, compared to", + "type": "text" + }, + { + "bbox": [ + 477, + 358, + 505, + 369 + ], + "score": 0.87, + "content": "7 4 . 6 \\%", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 368, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 373, + 382 + ], + "score": 1.0, + "content": "with two layers3. Overall, our results in Tab. 3 show that our PDE-", + "type": "text" + }, + { + "bbox": [ + 373, + 369, + 401, + 380 + ], + "score": 0.72, + "content": "\\mathrm { \\cdot G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 368, + 506, + 382 + ], + "score": 1.0, + "content": "achieves similar or better", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 380, + 263, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 263, + 392 + ], + "score": 1.0, + "content": "accuracy than the considered methods.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 293, + 507, + 392 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 405, + 281, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 282, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 282, + 418 + ], + "score": 1.0, + "content": "4.4 Fully-supervised node classification", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 426, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "We follow [54] and use 7 datasets: Cora, CiteSeer, PubMed, Chameleon, Cornell, Texas and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 368, + 450 + ], + "score": 1.0, + "content": "Wisconsin. We also use the same train/validation/test splits of", + "type": "text" + }, + { + "bbox": [ + 368, + 437, + 388, + 448 + ], + "score": 0.86, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 437, + 390, + 450 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 391, + 437, + 411, + 448 + ], + "score": 0.84, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 437, + 414, + 450 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 414, + 437, + 433, + 448 + ], + "score": 0.87, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ", respectively. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "addition, we report the average performance over 10 random splits from [54]. We fix the number", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "of channels to 64 and perform grid search to determine the hyper-parameters parameters, which are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "reported in Appendix C. We compare our network with GCN, GAT, three variants of Geom-GCN", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 480, + 504, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 504, + 492 + ], + "score": 1.0, + "content": "[54], APPNP, JKNet, Incep and GCNII in Tab. 4. Our experiments read either similar or better than", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "the state-of-the-art on Cora, CiteSeer and PubMed datasets. On Chameleon [55], Cornell, Texas and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "Wisconsin datasets, we improve state-of-the-art accuracy by a significant margin. For example, we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 133, + 527 + ], + "score": 1.0, + "content": "obtain", + "type": "text" + }, + { + "bbox": [ + 133, + 513, + 166, + 524 + ], + "score": 0.9, + "content": "9 3 . 2 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 513, + 299, + 527 + ], + "score": 1.0, + "content": "accuracy on Texas with our PDE-", + "type": "text" + }, + { + "bbox": [ + 299, + 514, + 327, + 524 + ], + "score": 0.85, + "content": "\\mathrm { G C N _ { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 513, + 382, + 527 + ], + "score": 1.0, + "content": ", compared to", + "type": "text" + }, + { + "bbox": [ + 383, + 513, + 415, + 524 + ], + "score": 0.9, + "content": "7 7 . 8 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 513, + 436, + 527 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 436, + 514, + 469, + 524 + ], + "score": 0.46, + "content": "\\mathrm { G C N I I ^ { \\ast } }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 513, + 506, + 527 + ], + "score": 1.0, + "content": ". Similar", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "improvements hold for Cornell and Wisconsin datasets. The common factor for these datasets is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "their small size, as depicted from Tab. 2. We argue that the success of our network stems from its", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "capability of apriori extracting features from graphs, due to its utilization of discretized differential", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 406, + 570 + ], + "score": 1.0, + "content": "operators and PDE guided construction. On Chameleon [55], using PDE-", + "type": "text" + }, + { + "bbox": [ + 406, + 557, + 435, + 568 + ], + "score": 0.87, + "content": "\\mathrm { G C N _ { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 556, + 506, + 570 + ], + "score": 1.0, + "content": ", we improve the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 245, + 580 + ], + "score": 1.0, + "content": "current state-of-the-art accuracy of", + "type": "text" + }, + { + "bbox": [ + 245, + 568, + 278, + 578 + ], + "score": 0.41, + "content": "\\operatorname { G C N I I } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 568, + 300, + 580 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 300, + 568, + 333, + 578 + ], + "score": 0.88, + "content": "6 2 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 568, + 344, + 580 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 344, + 568, + 376, + 578 + ], + "score": 0.9, + "content": "6 6 . 0 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 568, + 505, + 580 + ], + "score": 1.0, + "content": ". Also, we note that unlike in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "semi-supervised case, where some of the labels are missing, here it is possible to obtain meaningful", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 305, + 603 + ], + "score": 1.0, + "content": "results with the hyperbolic equation based PDE-", + "type": "text" + }, + { + "bbox": [ + 306, + 590, + 334, + 601 + ], + "score": 0.88, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "as we do not have unknown nodes in the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 507, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 507, + 614 + ], + "score": 1.0, + "content": "fully-supervised case, which would be otherwise preserved using the hyperbolic equation dynamics.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 425, + 507, + 614 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 626, + 212, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 214, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 214, + 640 + ], + "score": 1.0, + "content": "4.5 Inductive Learning", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "We follow [19] and employ the PPI dataset [26] for the inductive learning task. We use a 8 layer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 656, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 129, + 671 + ], + "score": 1.0, + "content": "PDE-", + "type": "text" + }, + { + "bbox": [ + 129, + 658, + 157, + 669 + ], + "score": 0.87, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 656, + 505, + 671 + ], + "score": 1.0, + "content": "network, without dropout or weight-decay, and a learning rate of 0.001. We compare", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "our results with methods like GraphSAGE, GAT, JKNet, GeniePath, GCNII and others. As reported", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 679, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 207, + 692 + ], + "score": 1.0, + "content": "in Tab. 5, our PDE-GCN", + "type": "text" + }, + { + "bbox": [ + 208, + 681, + 215, + 690 + ], + "score": 0.26, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 679, + 507, + 692 + ], + "score": 1.0, + "content": "achieves 99.07 Micro-averaged F1 score, superior to methods like GAT,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 690, + 435, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 435, + 702 + ], + "score": 1.0, + "content": "JKNet and GeniePath, also close to state-of-the-art GCNII* with a score of 99.58.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 647, + 507, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 149, + 93, + 462, + 433 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 120, + 77, + 488, + 89 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 77, + 490, + 90 + ], + "spans": [ + { + "bbox": [ + 120, + 77, + 339, + 90 + ], + "score": 1.0, + "content": "Table 3: Semi-supervised node classification accuracy", + "type": "text" + }, + { + "bbox": [ + 340, + 77, + 354, + 88 + ], + "score": 0.78, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 77, + 490, + 90 + ], + "score": 1.0, + "content": "). – indicates not available results.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 149, + 93, + 462, + 433 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 93, + 462, + 433 + ], + "spans": [ + { + "bbox": [ + 149, + 93, + 462, + 433 + ], + "score": 0.984, + "html": "
Layers
DatasetMethod248163264
CoraGCN [3]81.180.469.564.960.328.7
GCN (Drop) [28]82.882.075.875.762.549.5
JKNet [29]80.280.780.281.171.5
JKNet (Drop) [28]83.382.683.082.583.2
Incep [28]77.676.581.781.780.0
Incep (Drop)[28]82.982.583.183.183.5
GCNII [19]82.282.684.284.685.485.5
GCNII*[19]80.282.382.883.584.985.3
PDE-GCND (Ours)82.083.684.084.284.384.3
CiteseerGCN [3]70.867.630.218.325.020.0
GCN (Drop)[28]72.370.661.457.241.634.4
JKNet [29]168.767.769.868.263.4
JKNet (Drop) [28]72.671.872.670.872.2
Incep [28]69.368.470.268.067.5
Incep (Drop) [28]72.771.472.572.671.0
GCNII [19]68.268.870.672.973.473.4
GCNII* [19]66.166.770.672.073.273.1
PDE-GCND (Ours)74.675.075.275.575.675.5
PubmedGCN [3]79.076.561.240.922.435.3
GCN (Drop)[28]79.679.478.178.577.061.5
JKNet [29]78.078.172.672.474.5
JKNet (Drop) [28]78.778.779.779.278.9
Incep [28]77.777.974.911
Incep (Drop) [28]79.578.679.0
GCNII [19]78.278.879.380.279.879.7
GCNII* [19]77.778.278.880.379.880.1
PDE-GCND (Ours)79.380.680.180.480.280.3
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In essence, learning to find correspondences", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "between shapes is similar to learning a transformation from one shape to the other. To this end, we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "use the FAUST dataset [51] containing 10 scanned human shapes in 10 different poses with 6,890", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "score": 1.0, + "content": "nodes each. We follow the train and test split from [4], where the first 80 subjects are used for training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "score": 1.0, + "content": "and the remaining 20 subjects for testing. Our metric is the correspondence percentage with zero", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 546, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 557 + ], + "score": 1.0, + "content": "geodesic error, i.e., the percentage of perfectly matched shapes from all our test cases. We follow the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "pre-processing and training scheme of [62] where we use Cartesian coordinates to describe distances", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 307, + 579 + ], + "score": 1.0, + "content": "between nodes, with initial features of a constant", + "type": "text" + }, + { + "bbox": [ + 307, + 567, + 338, + 577 + ], + "score": 0.91, + "content": "\\mathbf { 1 } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 566, + 367, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 367, + 569, + 375, + 577 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "is the number of nodes. We use", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 167, + 590 + ], + "score": 1.0, + "content": "a 8 layer PDE-", + "type": "text" + }, + { + "bbox": [ + 167, + 578, + 195, + 589 + ], + "score": 0.74, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 577, + 236, + 590 + ], + "score": 1.0, + "content": "and PDE-", + "type": "text" + }, + { + "bbox": [ + 237, + 578, + 265, + 589 + ], + "score": 0.79, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "variants, both with constant learning rate of 0.001 with no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "weight-decay or dropout, and compare to recent and popular methods like ACNN [60], MoNet [4],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "FMNet [61], and SplineCNN [62]. As expected from the discussion in Sec. 3.1, and reported in Tab.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 609, + 504, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 475, + 624 + ], + "score": 1.0, + "content": "6, the hyperbolic equation proves to be a better fit for this kind of problem. Also, our PDE-", + "type": "text" + }, + { + "bbox": [ + 476, + 611, + 504, + 622 + ], + "score": 0.85, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 192, + 635 + ], + "score": 1.0, + "content": "achieves a promising", + "type": "text" + }, + { + "bbox": [ + 192, + 621, + 219, + 632 + ], + "score": 0.89, + "content": "9 9 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "correspondence rate with zero geodesic error, outperforming the rest of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 633, + 205, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 205, + 644 + ], + "score": 1.0, + "content": "the considered methods.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 658, + 192, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 194, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 194, + 673 + ], + "score": 1.0, + "content": "4.7 Ablation study", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 507, + 691 + ], + "score": 1.0, + "content": "Our method has two main dynamics – diffusion and hyperbolic. To verify our proposal in Sec. 3.1,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 318, + 702 + ], + "score": 1.0, + "content": "we examine the performance of the hyperbolic PDE-", + "type": "text" + }, + { + "bbox": [ + 318, + 689, + 346, + 700 + ], + "score": 0.8, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "on semi-supervised node-classification", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "on Cora and CiteSeer datasets. Our results in Tab. 7 show that indeed for problems where we wish", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 464, + 723 + ], + "score": 1.0, + "content": "to obtain a piecewise-constant prediction, the diffusive formulation of our network, PDE-", + "type": "text" + }, + { + "bbox": [ + 464, + 711, + 492, + 722 + ], + "score": 0.68, + "content": "\\mathrm { \\ G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 711, + 506, + 723 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 149, + 93, + 462, + 433 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 120, + 77, + 488, + 89 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 77, + 490, + 90 + ], + "spans": [ + { + "bbox": [ + 120, + 77, + 339, + 90 + ], + "score": 1.0, + "content": "Table 3: Semi-supervised node classification accuracy", + "type": "text" + }, + { + "bbox": [ + 340, + 77, + 354, + 88 + ], + "score": 0.78, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 77, + 490, + 90 + ], + "score": 1.0, + "content": "). – indicates not available results.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 149, + 93, + 462, + 433 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 93, + 462, + 433 + ], + "spans": [ + { + "bbox": [ + 149, + 93, + 462, + 433 + ], + "score": 0.984, + "html": "
Layers
DatasetMethod248163264
CoraGCN [3]81.180.469.564.960.328.7
GCN (Drop) [28]82.882.075.875.762.549.5
JKNet [29]80.280.780.281.171.5
JKNet (Drop) [28]83.382.683.082.583.2
Incep [28]77.676.581.781.780.0
Incep (Drop)[28]82.982.583.183.183.5
GCNII [19]82.282.684.284.685.485.5
GCNII*[19]80.282.382.883.584.985.3
PDE-GCND (Ours)82.083.684.084.284.384.3
CiteseerGCN [3]70.867.630.218.325.020.0
GCN (Drop)[28]72.370.661.457.241.634.4
JKNet [29]168.767.769.868.263.4
JKNet (Drop) [28]72.671.872.670.872.2
Incep [28]69.368.470.268.067.5
Incep (Drop) [28]72.771.472.572.671.0
GCNII [19]68.268.870.672.973.473.4
GCNII* [19]66.166.770.672.073.273.1
PDE-GCND (Ours)74.675.075.275.575.675.5
PubmedGCN [3]79.076.561.240.922.435.3
GCN (Drop)[28]79.679.478.178.577.061.5
JKNet [29]78.078.172.672.474.5
JKNet (Drop) [28]78.778.779.779.278.9
Incep [28]77.777.974.911
Incep (Drop) [28]79.578.679.0
GCNII [19]78.278.879.380.279.879.7
GCNII* [19]77.778.278.880.379.880.1
PDE-GCND (Ours)79.380.680.180.480.280.3
", + "type": "table", + "image_path": "66ac1492a044112ffee54cccc9874434b62b85f6bebd658ffdcc85b9e9f32484.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 149, + 93, + 462, + 206.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 149, + 206.33333333333331, + 462, + 319.66666666666663 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 149, + 319.66666666666663, + 462, + 432.99999999999994 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "title", + "bbox": [ + 108, + 459, + 252, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 253, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 253, + 474 + ], + "score": 1.0, + "content": "4.6 Dense shape correspondence", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "Finding dense correspondences between shapes is a classical experiment for hyperbolic dynamics, as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 491, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 504, + 502 + ], + "score": 1.0, + "content": "we are interested in modeling local motion dynamics. In essence, learning to find correspondences", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "between shapes is similar to learning a transformation from one shape to the other. To this end, we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "use the FAUST dataset [51] containing 10 scanned human shapes in 10 different poses with 6,890", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 538 + ], + "score": 1.0, + "content": "nodes each. We follow the train and test split from [4], where the first 80 subjects are used for training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "score": 1.0, + "content": "and the remaining 20 subjects for testing. Our metric is the correspondence percentage with zero", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 546, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 557 + ], + "score": 1.0, + "content": "geodesic error, i.e., the percentage of perfectly matched shapes from all our test cases. We follow the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "pre-processing and training scheme of [62] where we use Cartesian coordinates to describe distances", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 307, + 579 + ], + "score": 1.0, + "content": "between nodes, with initial features of a constant", + "type": "text" + }, + { + "bbox": [ + 307, + 567, + 338, + 577 + ], + "score": 0.91, + "content": "\\mathbf { 1 } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 566, + 367, + 579 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 367, + 569, + 375, + 577 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "is the number of nodes. We use", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 167, + 590 + ], + "score": 1.0, + "content": "a 8 layer PDE-", + "type": "text" + }, + { + "bbox": [ + 167, + 578, + 195, + 589 + ], + "score": 0.74, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 577, + 236, + 590 + ], + "score": 1.0, + "content": "and PDE-", + "type": "text" + }, + { + "bbox": [ + 237, + 578, + 265, + 589 + ], + "score": 0.79, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "variants, both with constant learning rate of 0.001 with no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "weight-decay or dropout, and compare to recent and popular methods like ACNN [60], MoNet [4],", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "FMNet [61], and SplineCNN [62]. As expected from the discussion in Sec. 3.1, and reported in Tab.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 609, + 504, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 475, + 624 + ], + "score": 1.0, + "content": "6, the hyperbolic equation proves to be a better fit for this kind of problem. Also, our PDE-", + "type": "text" + }, + { + "bbox": [ + 476, + 611, + 504, + 622 + ], + "score": 0.85, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 192, + 635 + ], + "score": 1.0, + "content": "achieves a promising", + "type": "text" + }, + { + "bbox": [ + 192, + 621, + 219, + 632 + ], + "score": 0.89, + "content": "9 9 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "correspondence rate with zero geodesic error, outperforming the rest of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 633, + 205, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 205, + 644 + ], + "score": 1.0, + "content": "the considered methods.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 480, + 506, + 644 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 658, + 192, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 194, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 194, + 673 + ], + "score": 1.0, + "content": "4.7 Ablation study", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 507, + 691 + ], + "score": 1.0, + "content": "Our method has two main dynamics – diffusion and hyperbolic. To verify our proposal in Sec. 3.1,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 318, + 702 + ], + "score": 1.0, + "content": "we examine the performance of the hyperbolic PDE-", + "type": "text" + }, + { + "bbox": [ + 318, + 689, + 346, + 700 + ], + "score": 0.8, + "content": "\\mathrm { G C N _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "on semi-supervised node-classification", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "on Cora and CiteSeer datasets. Our results in Tab. 7 show that indeed for problems where we wish", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 464, + 723 + ], + "score": 1.0, + "content": "to obtain a piecewise-constant prediction, the diffusive formulation of our network, PDE-", + "type": "text" + }, + { + "bbox": [ + 464, + 711, + 492, + 722 + ], + "score": 0.68, + "content": "\\mathrm { \\ G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 711, + 506, + 723 + ], + "score": 1.0, + "content": ", is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "more suitable. Furthermore, we study the importance of the positive-semi definiteness of our learnt", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 322, + 506 + ], + "score": 1.0, + "content": "operator as described in Eq. (6)-(7), by removing the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 322, + 492, + 339, + 505 + ], + "score": 0.91, + "content": "{ \\bf K } _ { l } ^ { T }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 339, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "term from the dynamics equations. This", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "yields a non-symmetric operator that does not guarantee positive-semi-definiteness. We note that", + "type": "text", + "cross_page": true + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 514, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 489, + 527 + ], + "score": 1.0, + "content": "the enforcement of the latter is important to obtain higher accuracy which is improved by up to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 489, + 515, + 504, + 525 + ], + "score": 0.87, + "content": "3 \\%", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 455, + 538 + ], + "score": 1.0, + "content": "with the introduction of a positive semi-definite operator. Also, we report on the use of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 455, + 526, + 502, + 536 + ], + "score": 0.9, + "content": "\\sigma = R e L U", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 502, + 526, + 506, + 538 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "from the discussion in Sec. 3.3, where we favor tanh as an activation function, and the ability of our", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 546, + 477, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 128, + 561 + ], + "score": 1.0, + "content": "PDE-", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 129, + 548, + 158, + 559 + ], + "score": 0.71, + "content": "\\mathrm { \\mathrm { . G C N _ { M } } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 158, + 546, + 288, + 561 + ], + "score": 1.0, + "content": "to reproduce the results of PDE-", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 288, + 548, + 315, + 559 + ], + "score": 0.48, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 316, + 546, + 477, + 561 + ], + "score": 1.0, + "content": "in the case of semi-supervised learning.", + "type": "text", + "cross_page": true + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 678, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 93, + 506, + 250 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 117, + 77, + 491, + 90 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 77, + 491, + 91 + ], + "spans": [ + { + "bbox": [ + 119, + 77, + 338, + 91 + ], + "score": 1.0, + "content": "Table 4: Fully-supervised node classification accuracy", + "type": "text" + }, + { + "bbox": [ + 338, + 77, + 354, + 89 + ], + "score": 0.8, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 77, + 418, + 91 + ], + "score": 1.0, + "content": ". (L) indicates a", + "type": "text" + }, + { + "bbox": [ + 419, + 78, + 427, + 87 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 77, + 491, + 91 + ], + "score": 1.0, + "content": "layers network.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 93, + 506, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 93, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 250 + ], + "score": 0.984, + "html": "
MethodCoraCite.Pubm.Cham.Corn.TexasWisc.
GCN [3]85.7773.6888.1328.1852.7052.1645.88
GAT[27]86.3774.3287.6242.9354.3258.3849.41
Geom-GCN-I [54]85.1977.9990.0560.3156.7657.5858.24
Geom-GCN-P [54]84.9375.1488.0960.9060.8167.5764.12
Geom-GCN-S [54]85.2774.7184.7559.9655.6859.7356.67
APPNP [30]87.8776.5389.4054.3073.5165.4169.02
JKNet [29]85.25 (16)75.85 (8)88.94 (64)60.07 (32)57.30 (4)56.49 (32)48.82 (8)
JKNet (Drop) [28]87.46 (16)75.96 (8)89.45 (64)62.08 (32)61.08 (4)57.30 (32)50.59 (8)
Incep (Drop) [28]86.86(8)76.83 (8)89.18 (4)61.71 (8)61.62 (16)57.84 (8)50.20 (8)
GCNII [19]88.49 (64)77.08 (64)89.57 (64)60.61 (8)74.86 (16)69.46 (32)74.12 (16)
GCNII*88.01 (64)77.13 (64)90.30 (64)62.48 (8)76.49 (16)77.84 (32)81.57 (16)
PDE-GCND (Ours)88.51 (16)78.36 (64)89.6 (64)64.12 (8)89.19 (2)90.81 (8)90.39 (8)
PDE-GCNH (Ours)87.71 (32)78.13 (16)89.16 (16)61.57 (64)89.45 (64)92.16 (64)91.37 (16)
PDE-GCNm (Ours)88.60 (16)78.48 (32)89.93 (16)66.01 (16)89.73 (64)93.24 (32)91.76 (16)
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MethodMicro-averaged F1
GraphSAGE [26]61.20
VR-GCN [56]97.80
GaAN [57]98.71
GAT [27]97.30
JKNet [29]97.60
GeniePath [58]98.50
Cluster-GCN [59]99.36
GCNII [19]99.54
GCNII*[19]99.58
PDE-GCND (Ours)99.07
PDE-GCNm (Ours)99.18
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MethodFaust
ACNN [60]63.8
MoNet [4]89.1
FMNet [61] SplineCNN [62]98.2 99.2
PDE-GCND (Ours)
PDE-GCNH (Ours)64.2 99.9
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(6)-(7), by removing the", + "type": "text" + }, + { + "bbox": [ + 322, + 492, + 339, + 505 + ], + "score": 0.91, + "content": "{ \\bf K } _ { l } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "term from the dynamics equations. This", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "yields a non-symmetric operator that does not guarantee positive-semi-definiteness. We note that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 514, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 489, + 527 + ], + "score": 1.0, + "content": "the enforcement of the latter is important to obtain higher accuracy which is improved by up to", + "type": "text" + }, + { + "bbox": [ + 489, + 515, + 504, + 525 + ], + "score": 0.87, + "content": "3 \\%", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 455, + 538 + ], + "score": 1.0, + "content": "with the introduction of a positive semi-definite operator. Also, we report on the use of", + "type": "text" + }, + { + "bbox": [ + 455, + 526, + 502, + 536 + ], + "score": 0.9, + "content": "\\sigma = R e L U", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 526, + 506, + 538 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "from the discussion in Sec. 3.3, where we favor tanh as an activation function, and the ability of our", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 546, + 477, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 128, + 561 + ], + "score": 1.0, + "content": "PDE-", + "type": "text" + }, + { + "bbox": [ + 129, + 548, + 158, + 559 + ], + "score": 0.71, + "content": "\\mathrm { \\mathrm { . G C N _ { M } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 546, + 288, + 561 + ], + "score": 1.0, + "content": "to reproduce the results of PDE-", + "type": "text" + }, + { + "bbox": [ + 288, + 548, + 315, + 559 + ], + "score": 0.48, + "content": "\\mathrm { G C N _ { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 546, + 477, + 561 + ], + "score": 1.0, + "content": "in the case of semi-supervised learning.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 579, + 176, + 593 + ], + "lines": [ + { + "bbox": [ + 104, + 578, + 178, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 178, + 597 + ], + "score": 1.0, + "content": "5 Summary", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "In this paper we explored new architectures for graph neural networks. Our motivation stems from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "the similarities between graph networks and time dependent partial differential equations that are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "discretized on manifolds and graphs. By adopting an appropriate PDE, and embedding the finite", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 639, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 104, + 639, + 507, + 654 + ], + "score": 1.0, + "content": "graph in an infinite manifold, we are able to define networks that are either diffusive, conservative, or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 651, + 199, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 199, + 662 + ], + "score": 1.0, + "content": "a combination of both.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "Not all natural phenomena are solved using the same PDE and we should not expect that all graph", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "problems should be solved by the same network dynamics. To this end we allow the data to choose", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "which type of network is appropriate for the solution of the problem (diffusive or hyperbolic). Indeed,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "numerical experiments show that the network gravitates towards a hyperbolic one for problems where", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 711, + 478, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 478, + 723 + ], + "score": 1.0, + "content": "conservation is required, and towards a diffusive one when anisotropic diffusion is favorable.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 93, + 506, + 250 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 117, + 77, + 491, + 90 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 77, + 491, + 91 + ], + "spans": [ + { + "bbox": [ + 119, + 77, + 338, + 91 + ], + "score": 1.0, + "content": "Table 4: Fully-supervised node classification accuracy", + "type": "text" + }, + { + "bbox": [ + 338, + 77, + 354, + 89 + ], + "score": 0.8, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 77, + 418, + 91 + ], + "score": 1.0, + "content": ". (L) indicates a", + "type": "text" + }, + { + "bbox": [ + 419, + 78, + 427, + 87 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 77, + 491, + 91 + ], + "score": 1.0, + "content": "layers network.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 93, + 506, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 93, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 250 + ], + "score": 0.984, + "html": "
MethodCoraCite.Pubm.Cham.Corn.TexasWisc.
GCN [3]85.7773.6888.1328.1852.7052.1645.88
GAT[27]86.3774.3287.6242.9354.3258.3849.41
Geom-GCN-I [54]85.1977.9990.0560.3156.7657.5858.24
Geom-GCN-P [54]84.9375.1488.0960.9060.8167.5764.12
Geom-GCN-S [54]85.2774.7184.7559.9655.6859.7356.67
APPNP [30]87.8776.5389.4054.3073.5165.4169.02
JKNet [29]85.25 (16)75.85 (8)88.94 (64)60.07 (32)57.30 (4)56.49 (32)48.82 (8)
JKNet (Drop) [28]87.46 (16)75.96 (8)89.45 (64)62.08 (32)61.08 (4)57.30 (32)50.59 (8)
Incep (Drop) [28]86.86(8)76.83 (8)89.18 (4)61.71 (8)61.62 (16)57.84 (8)50.20 (8)
GCNII [19]88.49 (64)77.08 (64)89.57 (64)60.61 (8)74.86 (16)69.46 (32)74.12 (16)
GCNII*88.01 (64)77.13 (64)90.30 (64)62.48 (8)76.49 (16)77.84 (32)81.57 (16)
PDE-GCND (Ours)88.51 (16)78.36 (64)89.6 (64)64.12 (8)89.19 (2)90.81 (8)90.39 (8)
PDE-GCNH (Ours)87.71 (32)78.13 (16)89.16 (16)61.57 (64)89.45 (64)92.16 (64)91.37 (16)
PDE-GCNm (Ours)88.60 (16)78.48 (32)89.93 (16)66.01 (16)89.73 (64)93.24 (32)91.76 (16)
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MethodMicro-averaged F1
GraphSAGE [26]61.20
VR-GCN [56]97.80
GaAN [57]98.71
GAT [27]97.30
JKNet [29]97.60
GeniePath [58]98.50
Cluster-GCN [59]99.36
GCNII [19]99.54
GCNII*[19]99.58
PDE-GCND (Ours)99.07
PDE-GCNm (Ours)99.18
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MethodFaust
ACNN [60]63.8
MoNet [4]89.1
FMNet [61] SplineCNN [62]98.2 99.2
PDE-GCND (Ours)
PDE-GCNH (Ours)64.2 99.9
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Layers
MethodDataset248163264
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PDE-GCNmCiteSeer Cora CiteSeer73.1 82.073.2 83.472.8 83.973.3 84.273.6 84.374.0 84.5
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ME is supported by Kreitman High-tech", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 370, + 157, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 157, + 387 + ], + "score": 1.0, + "content": "scholarship.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 404, + 163, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 165, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 165, + 419 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 109, + 424, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 109, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 109, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "[1] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. 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Layers
MethodDataset248163264
PDE-GCNHCora CiteSeer79 70.979.378.0 72.178.077.877.5 72.4
PDE-GCND (non-symmetric)Cora83.571.7 83.383.672.3 83.172.5 82.781
PDE-GCND (σ = ReLU)CiteSeer Cora74.3 80.374.5 81.874.8 82.675.0 83.073.9 83.473.3 83.5
PDE-GCNmCiteSeer Cora CiteSeer73.1 82.073.2 83.472.8 83.973.3 84.273.6 84.374.0 84.5
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For all authors...", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 146, + 362, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 146, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 146, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 162, + 373, + 288, + 386 + ], + "spans": [ + { + "bbox": [ + 162, + 373, + 288, + 386 + ], + "score": 1.0, + "content": "contributions and scope? [Yes]", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 145, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 145, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? 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[Yes] Proofs are provided in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 162, + 498, + 275, + 510 + ], + "spans": [ + { + "bbox": [ + 162, + 498, + 275, + 510 + ], + "score": 1.0, + "content": "the supplementary material.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 130, + 513, + 241, + 524 + ], + "lines": [ + { + "bbox": [ + 128, + 511, + 243, + 527 + ], + "spans": [ + { + "bbox": [ + 128, + 511, + 243, + 527 + ], + "score": 1.0, + "content": "3. 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[Yes] The code and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 161, + 549, + 470, + 562 + ], + "spans": [ + { + "bbox": [ + 161, + 549, + 470, + 562 + ], + "score": 1.0, + "content": "the links to the (public) data sets are provided in the supplementary material.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 145, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 145, + 561, + 506, + 576 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 162, + 573, + 246, + 585 + ], + "spans": [ + { + "bbox": [ + 162, + 573, + 246, + 585 + ], + "score": 1.0, + "content": "were chosen)? [Yes]", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 146, + 586, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 146, + 586, + 507, + 599 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 162, + 597, + 278, + 609 + ], + "spans": [ + { + "bbox": [ + 162, + 597, + 278, + 609 + ], + "score": 1.0, + "content": "ments multiple times)? [No]", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 146, + 608, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 146, + 608, + 505, + 624 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 162, + 620, + 370, + 634 + ], + "spans": [ + { + "bbox": [ + 162, + 620, + 370, + 634 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? [Yes]", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 134, + 635, + 505, + 647 + ], + "lines": [ + { + "bbox": [ + 130, + 634, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 130, + 634, + 507, + 650 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 146, + 650, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 145, + 649, + 424, + 663 + ], + "spans": [ + { + "bbox": [ + 145, + 649, + 424, + 663 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? [Yes]", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 145, + 663, + 360, + 675 + ], + "spans": [ + { + "bbox": [ + 145, + 663, + 360, + 675 + ], + "score": 1.0, + "content": "(b) Did you mention the license of the assets? [N/A]", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 145, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 145, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "(c) Did you include any new assets either in the supplemental material or as a URL? 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MethodDatasetAccuracy (%)
DGCNN (4) [5]ModelNet-1092.8
DGCNN (2)/ (4)Cora34.9 /25.2
DGCNN + Diffusion (2) /(4)Cora71.0 / 66.1
GCNII (4) [19]ModelNet-1065.4
Cora82.6
PDE-GCND(4) (Ours)ModelNet-1092.2
Cora83.6
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Layers
DatasetMethod248163264
CoraGCN [3]81.180.469.564.960.328.7
GCN (Drop) [28]82.882.075.875.762.549.5
JKNet [29]80.280.780.281.171.5
JKNet (Drop) [28]83.382.683.082.583.2
Incep [28]77.676.581.781.780.0
Incep (Drop)[28]82.982.583.183.183.5
GCNII [19]82.282.684.284.685.485.5
GCNII*[19]80.282.382.883.584.985.3
PDE-GCND (Ours)82.083.684.084.284.384.3
CiteseerGCN [3]70.867.630.218.325.020.0
GCN (Drop)[28]72.370.661.457.241.634.4
JKNet [29]168.767.769.868.263.4
JKNet (Drop) [28]72.671.872.670.872.2
Incep [28]69.368.470.268.067.5
Incep (Drop) [28]72.771.472.572.671.0
GCNII [19]68.268.870.672.973.473.4
GCNII* [19]66.166.770.672.073.273.1
PDE-GCND (Ours)74.675.075.275.575.675.5
PubmedGCN [3]79.076.561.240.922.435.3
GCN (Drop)[28]79.679.478.178.577.061.5
JKNet [29]78.078.172.672.474.5
JKNet (Drop) [28]78.778.779.779.278.9
Incep [28]77.777.974.911
Incep (Drop) [28]79.578.679.0
GCNII [19]78.278.879.380.279.879.7
GCNII* [19]77.778.278.880.379.880.1
PDE-GCND (Ours)79.380.680.180.480.280.3
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MethodCoraCite.Pubm.Cham.Corn.TexasWisc.
GCN [3]85.7773.6888.1328.1852.7052.1645.88
GAT[27]86.3774.3287.6242.9354.3258.3849.41
Geom-GCN-I [54]85.1977.9990.0560.3156.7657.5858.24
Geom-GCN-P [54]84.9375.1488.0960.9060.8167.5764.12
Geom-GCN-S [54]85.2774.7184.7559.9655.6859.7356.67
APPNP [30]87.8776.5389.4054.3073.5165.4169.02
JKNet [29]85.25 (16)75.85 (8)88.94 (64)60.07 (32)57.30 (4)56.49 (32)48.82 (8)
JKNet (Drop) [28]87.46 (16)75.96 (8)89.45 (64)62.08 (32)61.08 (4)57.30 (32)50.59 (8)
Incep (Drop) [28]86.86(8)76.83 (8)89.18 (4)61.71 (8)61.62 (16)57.84 (8)50.20 (8)
GCNII [19]88.49 (64)77.08 (64)89.57 (64)60.61 (8)74.86 (16)69.46 (32)74.12 (16)
GCNII*88.01 (64)77.13 (64)90.30 (64)62.48 (8)76.49 (16)77.84 (32)81.57 (16)
PDE-GCND (Ours)88.51 (16)78.36 (64)89.6 (64)64.12 (8)89.19 (2)90.81 (8)90.39 (8)
PDE-GCNH (Ours)87.71 (32)78.13 (16)89.16 (16)61.57 (64)89.45 (64)92.16 (64)91.37 (16)
PDE-GCNm (Ours)88.60 (16)78.48 (32)89.93 (16)66.01 (16)89.73 (64)93.24 (32)91.76 (16)
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MethodMicro-averaged F1
GraphSAGE [26]61.20
VR-GCN [56]97.80
GaAN [57]98.71
GAT [27]97.30
JKNet [29]97.60
GeniePath [58]98.50
Cluster-GCN [59]99.36
GCNII [19]99.54
GCNII*[19]99.58
PDE-GCND (Ours)99.07
PDE-GCNm (Ours)99.18
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MethodFaust
ACNN [60]63.8
MoNet [4]89.1
FMNet [61] SplineCNN [62]98.2 99.2
PDE-GCND (Ours)
PDE-GCNH (Ours)64.2 99.9
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Layers
MethodDataset248163264
PDE-GCNHCora CiteSeer79 70.979.378.0 72.178.077.877.5 72.4
PDE-GCND (non-symmetric)Cora83.571.7 83.383.672.3 83.172.5 82.781
PDE-GCND (σ = ReLU)CiteSeer Cora74.3 80.374.5 81.874.8 82.675.0 83.073.9 83.473.3 83.5
PDE-GCNmCiteSeer Cora CiteSeer73.1 82.073.2 83.472.8 83.973.3 84.273.6 84.374.0 84.5
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